22691:
21428:
21421:
22686:{\displaystyle {\begin{aligned}E_{x'}&=F^{0'1'}={\Lambda ^{0}}_{\mu }{\Lambda ^{1}}_{\nu }F^{\mu \nu }={\Lambda ^{0}}_{1}{\Lambda ^{1}}_{0}F^{10}+{\Lambda ^{0}}_{0}{\Lambda ^{1}}_{1}F^{01}\\&=(-\gamma \beta )(-\gamma \beta )(-E_{x})+\gamma \gamma E_{x}=-\gamma ^{2}\beta ^{2}(E_{x})+\gamma ^{2}E_{x}=E_{x}(1-\beta ^{2})\gamma ^{2}\\&=E_{x},\\E_{y'}&=F^{0'2'}={\Lambda ^{0}}_{\mu }{\Lambda ^{2}}_{\nu }F^{\mu \nu }={\Lambda ^{0}}_{\mu }{\Lambda ^{2}}_{2}F^{\mu 2}={\Lambda ^{0}}_{0}{\Lambda ^{2}}_{2}F^{02}+{\Lambda ^{0}}_{1}{\Lambda ^{2}}_{2}F^{12}\\&=\gamma \times 1\times E_{y}+(-\beta \gamma )\times 1\times B_{z}=\gamma E_{y}-\beta \gamma B_{z}\\&=\gamma \left(\mathbf {E} +{\boldsymbol {\beta }}\times \mathbf {B} \right)_{y}\\E_{z'}&=F^{0'3'}={\Lambda ^{0}}_{\mu }{\Lambda ^{3}}_{\nu }F^{\mu \nu }={\Lambda ^{0}}_{\mu }{\Lambda ^{3}}_{3}F^{\mu 3}={\Lambda ^{0}}_{0}{\Lambda ^{3}}_{3}F^{03}+{\Lambda ^{0}}_{1}{\Lambda ^{3}}_{3}F^{13}\\&=\gamma \times 1\times E_{z}-\beta \gamma \times 1\times (-B_{y})=\gamma E_{z}+\beta \gamma B_{y}\\&=\gamma \left(\mathbf {E} +{\boldsymbol {\beta }}\times \mathbf {B} \right)_{z}.\end{aligned}}}
20372:
15221:
21416:{\displaystyle {\begin{aligned}B_{x'}&=F^{2'3'}={\Lambda ^{2}}_{\mu }{\Lambda ^{3}}_{\nu }F^{\mu \nu }={\Lambda ^{2}}_{2}{\Lambda ^{3}}_{3}F^{23}=1\times 1\times B_{x}\\&=B_{x},\\B_{y'}&=F^{3'1'}={\Lambda ^{3}}_{\mu }{\Lambda ^{1}}_{\nu }F^{\mu \nu }={\Lambda ^{3}}_{3}{\Lambda ^{1}}_{\nu }F^{3\nu }={\Lambda ^{3}}_{3}{\Lambda ^{1}}_{0}F^{30}+{\Lambda ^{3}}_{3}{\Lambda ^{1}}_{1}F^{31}\\&=1\times (-\beta \gamma )(-E_{z})+1\times \gamma B_{y}=\gamma B_{y}+\beta \gamma E_{z}\\&=\gamma \left(\mathbf {B} -{\boldsymbol {\beta }}\times \mathbf {E} \right)_{y}\\B_{z'}&=F^{1'2'}={\Lambda ^{1}}_{\mu }{\Lambda ^{2}}_{\nu }F^{\mu \nu }={\Lambda ^{1}}_{\mu }{\Lambda ^{2}}_{2}F^{\mu 2}={\Lambda ^{1}}_{0}{\Lambda ^{2}}_{2}F^{02}+{\Lambda ^{1}}_{1}{\Lambda ^{2}}_{2}F^{12}\\&=(-\gamma \beta )\times 1\times E_{y}+\gamma \times 1\times B_{z}=\gamma B_{z}-\beta \gamma E_{y}\\&=\gamma \left(\mathbf {B} -{\boldsymbol {\beta }}\times \mathbf {E} \right)_{z}\end{aligned}}}
4974:
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4243:
15216:{\displaystyle {\begin{alignedat}{3}K_{x}&={\begin{bmatrix}0&1&0&0\\1&0&0&0\\0&0&0&0\\0&0&0&0\\\end{bmatrix}}\,,\quad &K_{y}&={\begin{bmatrix}0&0&1&0\\0&0&0&0\\1&0&0&0\\0&0&0&0\end{bmatrix}}\,,\quad &K_{z}&={\begin{bmatrix}0&0&0&1\\0&0&0&0\\0&0&0&0\\1&0&0&0\end{bmatrix}}\\J_{x}&={\begin{bmatrix}0&0&0&0\\0&0&0&0\\0&0&0&-1\\0&0&1&0\\\end{bmatrix}}\,,\quad &J_{y}&={\begin{bmatrix}0&0&0&0\\0&0&0&1\\0&0&0&0\\0&-1&0&0\end{bmatrix}}\,,\quad &J_{z}&={\begin{bmatrix}0&0&0&0\\0&0&-1&0\\0&1&0&0\\0&0&0&0\end{bmatrix}}\end{alignedat}}}
11922:
4969:{\displaystyle {\begin{bmatrix}ct'\\\\\\\\x'\\\\\\\\y'\\\\\\z'\end{bmatrix}}={\begin{bmatrix}\gamma &-\gamma \beta _{x}&-\gamma \beta _{y}&-\gamma \beta _{z}\\-\gamma \beta _{x}&1+{\frac {\gamma ^{2}}{1+\gamma }}\beta _{x}^{2}&{\frac {\gamma ^{2}}{1+\gamma }}\beta _{x}\beta _{y}&{\frac {\gamma ^{2}}{1+\gamma }}\beta _{x}\beta _{z}\\-\gamma \beta _{y}&{\frac {\gamma ^{2}}{1+\gamma }}\beta _{x}\beta _{y}&1+{\frac {\gamma ^{2}}{1+\gamma }}\beta _{y}^{2}&{\frac {\gamma ^{2}}{1+\gamma }}\beta _{y}\beta _{z}\\-\gamma \beta _{z}&{\frac {\gamma ^{2}}{1+\gamma }}\beta _{x}\beta _{z}&{\frac {\gamma ^{2}}{1+\gamma }}\beta _{y}\beta _{z}&1+{\frac {\gamma ^{2}}{1+\gamma }}\beta _{z}^{2}\\\end{bmatrix}}{\begin{bmatrix}ct\\\\\\x\\\\\\\\y\\\\\\\\z\end{bmatrix}},}
12765:{\displaystyle B(\mathbf {v} )={\begin{bmatrix}\gamma &-\gamma v_{x}/c&-\gamma v_{y}/c&-\gamma v_{z}/c\\-\gamma v_{x}/c&1+(\gamma -1){\dfrac {v_{x}^{2}}{v^{2}}}&(\gamma -1){\dfrac {v_{x}v_{y}}{v^{2}}}&(\gamma -1){\dfrac {v_{x}v_{z}}{v^{2}}}\\-\gamma v_{y}/c&(\gamma -1){\dfrac {v_{y}v_{x}}{v^{2}}}&1+(\gamma -1){\dfrac {v_{y}^{2}}{v^{2}}}&(\gamma -1){\dfrac {v_{y}v_{z}}{v^{2}}}\\-\gamma v_{z}/c&(\gamma -1){\dfrac {v_{z}v_{x}}{v^{2}}}&(\gamma -1){\dfrac {v_{z}v_{y}}{v^{2}}}&1+(\gamma -1){\dfrac {v_{z}^{2}}{v^{2}}}\end{bmatrix}}={\begin{bmatrix}\gamma &-\gamma {\vec {\beta }}^{T}\\-\gamma {\vec {\beta }}&I+(\gamma -1){\dfrac {{\vec {\beta }}{\vec {\beta }}^{T}}{\beta ^{2}}}\end{bmatrix}},}
213:
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23048:{\displaystyle {\begin{aligned}\mathbf {E} _{\parallel '}&=\mathbf {E} _{\parallel }\\\mathbf {B} _{\parallel '}&=\mathbf {B} _{\parallel }\\\mathbf {E} _{\bot '}&=\gamma \left(\mathbf {E} _{\bot }+{\boldsymbol {\beta }}\times \mathbf {B} _{\bot }\right)=\gamma \left(\mathbf {E} +{\boldsymbol {\beta }}\times \mathbf {B} \right)_{\bot },\\\mathbf {B} _{\bot '}&=\gamma \left(\mathbf {B} _{\bot }-{\boldsymbol {\beta }}\times \mathbf {E} _{\bot }\right)=\gamma \left(\mathbf {B} -{\boldsymbol {\beta }}\times \mathbf {E} \right)_{\bot },\end{aligned}}}
24118:
24157:
19353:
17406:{\displaystyle {\begin{bmatrix}{x'}^{0}\\{x'}^{1}\\{x'}^{2}\\{x'}^{3}\end{bmatrix}}={\begin{bmatrix}{\Lambda ^{0}}_{0}&{\Lambda ^{0}}_{1}&{\Lambda ^{0}}_{2}&{\Lambda ^{0}}_{3}\\{\Lambda ^{1}}_{0}&{\Lambda ^{1}}_{1}&{\Lambda ^{1}}_{2}&{\Lambda ^{1}}_{3}\\{\Lambda ^{2}}_{0}&{\Lambda ^{2}}_{1}&{\Lambda ^{2}}_{2}&{\Lambda ^{2}}_{3}\\{\Lambda ^{3}}_{0}&{\Lambda ^{3}}_{1}&{\Lambda ^{3}}_{2}&{\Lambda ^{3}}_{3}\\\end{bmatrix}}{\begin{bmatrix}x^{0}\\x^{1}\\x^{2}\\x^{3}\end{bmatrix}}}
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24776:{\displaystyle {\begin{aligned}&U(\Lambda ,a)\Psi _{p_{1}\sigma _{1}n_{1};p_{2}\sigma _{2}n_{2};\cdots }\\={}&e^{-ia_{\mu }\left}{\sqrt {\frac {(\Lambda p_{1})^{0}(\Lambda p_{2})^{0}\cdots }{p_{1}^{0}p_{2}^{0}\cdots }}}\left(\sum _{\sigma _{1}'\sigma _{2}'\cdots }D_{\sigma _{1}'\sigma _{1}}^{(j_{1})}\leftD_{\sigma _{2}'\sigma _{2}}^{(j_{2})}\left\cdots \right)\Psi _{\Lambda p_{1}\sigma _{1}'n_{1};\Lambda p_{2}\sigma _{2}'n_{2};\cdots },\end{aligned}}}
28148:
226:
36:
2428:
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1921:
20244:{\displaystyle {\Lambda ^{\mu }}_{\nu }={\begin{bmatrix}\gamma &-\gamma \beta &0&0\\-\gamma \beta &\gamma &0&0\\0&0&1&0\\0&0&0&1\\\end{bmatrix}},\qquad F^{\mu \nu }={\begin{bmatrix}0&E_{x}&E_{y}&E_{z}\\-E_{x}&0&B_{z}&-B_{y}\\-E_{y}&-B_{z}&0&B_{x}\\-E_{z}&B_{y}&-B_{x}&0\end{bmatrix}}{\text{(Gaussian units, signature }}(-,+,+,+){\text{)}},}
2870:, and the relative velocity between the frames is the parameter of the transformation. The other basic type of Lorentz transformation is rotation in the spatial coordinates only, these like boosts are inertial transformations since there is no relative motion, the frames are simply tilted (and not continuously rotating), and in this case quantities defining the rotation are the parameters of the transformation (e.g.,
24113:{\displaystyle {\begin{aligned}u\otimes v\rightarrow \Pi (\Lambda )u\otimes \Pi (\Lambda )v&={\Pi (\Lambda )^{\alpha }}_{\beta }u^{\beta }\otimes {\Pi (\Lambda )^{\rho }}_{\sigma }v^{\sigma }\\&={\Pi (\Lambda )^{\alpha }}_{\beta }{\Pi (\Lambda )^{\rho }}_{\sigma }u^{\beta }\otimes v^{\sigma }\\&\equiv {\Pi (\Lambda )^{\alpha }}_{\beta }{\Pi (\Lambda )^{\rho }}_{\sigma }w^{\beta \sigma }\end{aligned}}}
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19476:
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15796:{\displaystyle {\begin{aligned}\Lambda &=(I-{\boldsymbol {\zeta }}\cdot \mathbf {K} +\cdots )(I+{\boldsymbol {\theta }}\cdot \mathbf {J} +\cdots )\\&=(I+{\boldsymbol {\theta }}\cdot \mathbf {J} +\cdots )(I-{\boldsymbol {\zeta }}\cdot \mathbf {K} +\cdots )\\&=I-{\boldsymbol {\zeta }}\cdot \mathbf {K} +{\boldsymbol {\theta }}\cdot \mathbf {J} +\cdots \end{aligned}}}
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19348:{\displaystyle T_{\theta '\iota '\cdots \kappa '}^{\alpha '\beta '\cdots \zeta '}={\Lambda ^{\alpha '}}_{\mu }{\Lambda ^{\beta '}}_{\nu }\cdots {\Lambda ^{\zeta '}}_{\rho }{\Lambda _{\theta '}}^{\sigma }{\Lambda _{\iota '}}^{\upsilon }\cdots {\Lambda _{\kappa '}}^{\zeta }T_{\sigma \upsilon \cdots \zeta }^{\mu \nu \cdots \rho },}
19801:{\displaystyle F^{\mu \nu }={\begin{bmatrix}0&-{\frac {1}{c}}E_{x}&-{\frac {1}{c}}E_{y}&-{\frac {1}{c}}E_{z}\\{\frac {1}{c}}E_{x}&0&-B_{z}&B_{y}\\{\frac {1}{c}}E_{y}&B_{z}&0&-B_{x}\\{\frac {1}{c}}E_{z}&-B_{y}&B_{x}&0\end{bmatrix}}{\text{(SI units, signature }}(+,-,-,-){\text{)}}.}
13817:. An explicit form of the general Lorentz transformation is cumbersome to write down and will not be given here. Nevertheless, closed form expressions for the transformation matrices will be given below using group theoretical arguments. It will be easier to use the rapidity parametrization for boosts, in which case one writes
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Linking terminology used in mathematics and physics: A group generator is any element of the Lie algebra. A group parameter is a component of a coordinate vector representing an arbitrary element of the Lie algebra with respect to some basis. A basis, then, is a set of generators being a basis of the
19038:{\displaystyle u\otimes v\rightarrow \Lambda u\otimes \Lambda v={\Lambda ^{\mu }}_{\nu }u^{\nu }\otimes {\Lambda ^{\rho }}_{\sigma }v^{\sigma }={\Lambda ^{\mu }}_{\nu }{\Lambda ^{\rho }}_{\sigma }u^{\nu }\otimes v^{\sigma }\equiv {\Lambda ^{\mu }}_{\nu }{\Lambda ^{\rho }}_{\sigma }w^{\nu \sigma }.}
9513:, etc., its properties can be fixed in the rest frame of that object. Then the Lorentz transformations give the corresponding properties in a frame moving relative to the object with constant velocity. This breaks some notions taken for granted in non-relativistic physics. For example, the energy
2904:
Boosts should not be conflated with mere displacements in spacetime; in this case, the coordinate systems are simply shifted and there is no relative motion. However, these also count as symmetries forced by special relativity since they leave the spacetime interval invariant. A combination of a
13629:
Spatial rotations alone are also
Lorentz transformations since they leave the spacetime interval invariant. Like boosts, successive rotations about different axes do not commute. Unlike boosts, the composition of any two rotations is equivalent to a single rotation. Some other similarities and
976:). The Galilean transformation is a good approximation only at relative speeds much less than the speed of light. Lorentz transformations have a number of unintuitive features that do not appear in Galilean transformations. For example, they reflect the fact that observers moving at different
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1132:, the relative velocity of the two reference frames normalized to the speed of light) as the consequence of clock synchronization, under the assumption that the speed of light is constant in moving frames. Larmor is credited to have been the first to understand the crucial
24925:
One can imagine that in each inertial frame there are observers positioned throughout space, each with a synchronized clock and at rest in the particular inertial frame. These observers then report to a central office, where all reports are collected. When one speaks of a
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1097:. FitzGerald then conjectured that Heaviside's distortion result might be applied to a theory of intermolecular forces. Some months later, FitzGerald published the conjecture that bodies in motion are being contracted, in order to explain the baffling outcome of the
13291:
a single boost, each composition is still a
Lorentz transformation as it preserves the spacetime interval. It turns out the composition of any two Lorentz boosts is equivalent to a boost followed or preceded by a rotation on the spatial coordinates, in the form of
2788:
13934:
For simplicity, look at the infinitesimal
Lorentz boost in the x direction (examining a boost in any other direction, or rotation about any axis, follows an identical procedure). The infinitesimal boost is a small boost away from the identity, obtained by the
7513:
23671:{\displaystyle {\begin{aligned}\mathbf {j} '&=\mathbf {j} -\gamma \rho v\mathbf {n} +\left(\gamma -1\right)(\mathbf {j} \cdot \mathbf {n} )\mathbf {n} \\\rho '&=\gamma \left(\rho -\mathbf {j} \cdot {\frac {v\mathbf {n} }{c^{2}}}\right),\end{aligned}}}
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1916:{\displaystyle {\begin{aligned}&c^{2}(t_{2}-t_{1})^{2}-(x_{2}-x_{1})^{2}-(y_{2}-y_{1})^{2}-(z_{2}-z_{1})^{2}\\={}&c^{2}(t_{2}'-t_{1}')^{2}-(x_{2}'-x_{1}')^{2}-(y_{2}'-y_{1}')^{2}-(z_{2}'-z_{1}')^{2}\quad {\text{(all events 1, 2)}}.\end{aligned}}}
7410:{\displaystyle {\begin{aligned}t'&=\gamma \left(t-{\frac {\mathbf {r} _{\parallel }\cdot \mathbf {v} }{c^{2}}}\right)\\\mathbf {r} _{\|}'&=\gamma (\mathbf {r} _{\|}-\mathbf {v} t)\\\mathbf {r} _{\perp }'&=\mathbf {r} _{\perp }\end{aligned}}}
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3989:
remains unchanged. This "trick" of simply reversing the direction of relative velocity while preserving its magnitude, and exchanging primed and unprimed variables, always applies to finding the inverse transformation of every boost in any direction.
9139:{\displaystyle {\begin{aligned}A'&=\gamma \left(A-{\frac {v\mathbf {n} \cdot \mathbf {Z} }{c}}\right)\,,\\\mathbf {Z} '&=\mathbf {Z} +(\gamma -1)(\mathbf {Z} \cdot \mathbf {n} )\mathbf {n} -{\frac {\gamma Av\mathbf {n} }{c}}\,.\end{aligned}}}
6957:
than the end points of an identical rod at rest in his own frame. Length contraction affects any geometric quantity related to lengths, so from the perspective of a moving observer, areas and volumes will also appear to shrink along the direction of
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2858:
are the inverse transformation. Depending on how the frames move relative to each other, and how they are oriented in space relative to each other, other parameters that describe direction, speed, and orientation enter the transformation equations.
23495:, and time dilation has an effect on the rate of flow of charge (current), so charge and current distributions must transform in a related way under a boost. It turns out they transform exactly like the space-time and energy-momentum four-vectors,
10019:{\displaystyle X'={\begin{bmatrix}c\,t'\\x'\\y'\\z'\end{bmatrix}}\,,\quad \eta ={\begin{bmatrix}-1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{bmatrix}}\,,\quad X={\begin{bmatrix}c\,t\\x\\y\\z\end{bmatrix}}}
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8093:{\displaystyle {\begin{aligned}t&=\gamma \left(t'+{\frac {\mathbf {r} '\cdot v\mathbf {n} }{c^{2}}}\right)\,,\\\mathbf {r} &=\mathbf {r} '+(\gamma -1)(\mathbf {r} '\cdot \mathbf {n} )\mathbf {n} +\gamma t'v\mathbf {n} \,,\end{aligned}}}
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11233:
10995:
7821:{\displaystyle {\begin{aligned}t'&=\gamma \left(t-{\frac {v\mathbf {n} \cdot \mathbf {r} }{c^{2}}}\right)\,,\\\mathbf {r} '&=\mathbf {r} +(\gamma -1)(\mathbf {r} \cdot \mathbf {n} )\mathbf {n} -\gamma tv\mathbf {n} \,.\end{aligned}}}
2423:{\displaystyle {\begin{aligned}&c^{2}t^{2}-x^{2}-y^{2}-z^{2}=c^{2}t'^{2}-x'^{2}-y'^{2}-z'^{2}\\{\text{or}}\quad &c^{2}t_{1}t_{2}-x_{1}x_{2}-y_{1}y_{2}-z_{1}z_{2}=c^{2}t'_{1}t'_{2}-x'_{1}x'_{2}-y'_{1}y'_{2}-z'_{1}z'_{2}\end{aligned}}}
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of an object is a scalar in non-relativistic mechanics, but not in relativistic mechanics because energy changes under
Lorentz transformations; its value is different for various inertial frames. In the rest frame of an object, it has a
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provides a one-to-one correspondence between small enough neighborhoods of the origin of the Lie algebra and neighborhoods of the identity element of the Lie group. In the case of the
Lorentz group, the exponential map is just the
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is the
Lorentz factor. This formula represents a passive transformation, as it describes how the coordinates of the measured quantity changes from the unprimed frame to the primed frame. The active transformation is given by
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An observer measures a charge at rest in frame F. The observer will detect a static electric field. As the charge is stationary in this frame, there is no electric current, so the observer does not observe any magnetic
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Two other spacetime symmetries have not been accounted for. In order for the spacetime interval to be invariant, it can be shown that it is necessary and sufficient for the coordinate transformation to be of the form
1038:
between any two events. This property is the defining property of a
Lorentz transformation. They describe only the transformations in which the spacetime event at the origin is left fixed. They can be considered as a
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of the
Lorentz transformations that two values of space and time coordinates can be chosen, the Lorentz transformations can be applied to each, then subtracted to get the Lorentz transformations of the differences;
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11487:. This is not always the case: the composition of two antichronous Lorentz transformations is orthochronous, and the composition of two improper Lorentz transformations is proper. In other words, while the sets
8490:{\displaystyle \mathbf {u} ={\frac {d\mathbf {r} }{dt}}\,,\quad \mathbf {u} '={\frac {d\mathbf {r} '}{dt'}}\,,\quad \gamma _{\mathbf {v} }={\frac {1}{\sqrt {1-{\dfrac {\mathbf {v} \cdot \mathbf {v} }{c^{2}}}}}}}
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in calculations and experiments, it is lengths between two points or time intervals that are measured or of interest (e.g., the length of a moving vehicle, or time duration it takes to travel from one place to
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23468:{\displaystyle F^{\mu '\nu '}\left(x'\right)={\Lambda ^{\mu '}}_{\mu }{\Lambda ^{\nu '}}_{\nu }F^{\mu \nu }\left(\Lambda ^{-1}x'\right)={\Lambda ^{\mu '}}_{\mu }{\Lambda ^{\nu '}}_{\nu }F^{\mu \nu }(x).}
10506:{\displaystyle \eta ={\begin{bmatrix}-1&0\\0&\mathbf {I} \end{bmatrix}}\,,\quad \Lambda ={\begin{bmatrix}\Gamma &-\mathbf {a} ^{\mathrm {T} }\\-\mathbf {b} &\mathbf {M} \end{bmatrix}}\,,}
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2034:
1535:
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16027:{\displaystyle e^{-{\boldsymbol {\zeta }}\cdot \mathbf {K} +{\boldsymbol {\theta }}\cdot \mathbf {J} }\neq e^{-{\boldsymbol {\zeta }}\cdot \mathbf {K} }e^{{\boldsymbol {\theta }}\cdot \mathbf {J} },}
9524:
and zero momentum. In a boosted frame its energy is different and it appears to have a momentum. Similarly, in non-relativistic quantum mechanics the spin of a particle is a constant vector, but in
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The separate requirements of the three equations lead to three different groups. The second equation is satisfied for spacetime translations in addition to
Lorentz transformations leading to the
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the transformations of velocity can be readily derived by making the difference infinitesimally small and dividing the equations, and the process repeated for the transformation of acceleration,
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indices simply take the values 1, 2, 3, for spatial components (the opposite for Landau and
Lifshitz). Note that the first index (reading left to right) corresponds in the matrix notation to a
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between the squares of the time and spatial coordinates in the spacetime interval, rather than a sum. The geometric significance of the hyperbolic functions can be visualized by taking
17888:. (The linked article also provides more information about what the operation of raising and lowering indices really is mathematically.) The inverse of this transformation is given by
14163:
is defined by this result (its significance will be explained shortly). In the limit of an infinite number of infinitely small steps, the finite boost transformation in the form of a
11864:
6556:
A critical requirement of the Lorentz transformations is the invariance of the speed of light, a fact used in their derivation, and contained in the transformations themselves. If in
6374:{\displaystyle {\begin{aligned}\Delta t&=\gamma \left(\Delta t'+{\frac {v\,\Delta x'}{c^{2}}}\right)\,,\\\Delta x&=\gamma \left(\Delta x'+v\,\Delta t'\right)\,.\end{aligned}}}
6211:{\displaystyle {\begin{aligned}\Delta t'&=\gamma \left(\Delta t-{\frac {v\,\Delta x}{c^{2}}}\right)\,,\\\Delta x'&=\gamma \left(\Delta x-v\,\Delta t\right)\,,\end{aligned}}}
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7611:{\displaystyle \mathbf {r} _{\parallel }=(\mathbf {r} \cdot \mathbf {n} )\mathbf {n} \,,\quad \mathbf {r} _{\perp }=\mathbf {r} -(\mathbf {r} \cdot \mathbf {n} )\mathbf {n} }
1183:
is something that happens at a certain point in spacetime, or more generally, the point in spacetime itself. In any inertial frame an event is specified by a time coordinate
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1518:
events, not necessarily separated by light signals, is in fact invariant, i.e., independent of the state of relative motion of observers in different inertial frames, as is
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15234:
above, although the minus signs in the boost generators are conventional. Physically, the generators of the Lorentz group correspond to important symmetries in spacetime:
14388:{\displaystyle B({\boldsymbol {\zeta }})=e^{-{\boldsymbol {\zeta }}\cdot \mathbf {K} }\,,\quad R({\boldsymbol {\theta }})=e^{{\boldsymbol {\theta }}\cdot \mathbf {J} }\,.}
11917:
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18650:
11436:{\displaystyle {\mathcal {L}}={\mathcal {L}}_{+}^{\uparrow }\cup {\mathcal {L}}_{-}^{\uparrow }\cup {\mathcal {L}}_{+}^{\downarrow }\cup {\mathcal {L}}_{-}^{\downarrow }}
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9534:
depends on relative motion. In the rest frame of the particle, the spin pseudovector can be fixed to be its ordinary non-relativistic spin with a zero timelike quantity
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10183:
8506:
15298:, and this reflects an infinitesimal transformation away from the identity. The smooth curve can always be taken as an exponential as the exponential will always map
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Only time and the coordinates parallel to the direction of relative motion change, while those coordinates perpendicular do not. With this in mind, split the spatial
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There are also vector quantities with covariant indices. They are generally obtained from their corresponding objects with contravariant indices by the operation of
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298:
relative to the former. The respective inverse transformation is then parameterized by the negative of this velocity. The transformations are named after the Dutch
7043:
The use of vectors allows positions and velocities to be expressed in arbitrary directions compactly. A single boost in any direction depends on the full relative
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The spacetime coordinates of an event, as measured by each observer in their inertial reference frame (in standard configuration) are shown in the speech bubbles.
15920:{\displaystyle \Lambda ({\boldsymbol {\zeta }},{\boldsymbol {\theta }})=e^{-{\boldsymbol {\zeta }}\cdot \mathbf {K} +{\boldsymbol {\theta }}\cdot \mathbf {J} }.}
329:
19073:
can be written as a sum of a coefficient (component!) times tensor products of basis vectors and basis covectors, one arrives at the transformation law for any
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3486:
of the transformation, for a given boost it is a constant number, but can take a continuous range of values. In the setup used here, positive relative velocity
13481:, which rotates any 3d vector in one sense (active transformation), or equivalently the coordinate frame in the opposite sense (passive transformation). It is
5322:). Given the strong resemblance to rotations of spatial coordinates in 3d space in the Cartesian xy, yz, and zx planes, a Lorentz boost can be thought of as a
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The electric and magnetic fields transform differently from space and time, but exactly the same way as relativistic angular momentum and the boost vector.
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6510:, then they can use that event as the origin, and the spacetime coordinate differences are the differences between their coordinates and this origin, e.g.,
16034:
because the generators do not commute. For a description of how to find the factors of a general Lorentz transformation in terms of a boost and a rotation
10531:
3404:
9712:. A method of deriving the EM field transformations in an efficient way which also illustrates the unit of the electromagnetic field uses tensor algebra,
5456:
3973:{\displaystyle {\begin{aligned}t&=\gamma \left(t'+{\frac {vx'}{c^{2}}}\right)\\x&=\gamma \left(x'+vt'\right)\\y&=y'\\z&=z',\end{aligned}}}
19054:
The second step uses the bilinearity of the tensor product and the last step defines a 2-tensor on component form, or rather, it just renames the tensor
14061:
is the matrix of derivatives (of the entries, with respect to the same variable), and it is understood the derivatives are found first then evaluated at
13249:
are not collinear but in different directions, the situation is considerably more complicated. Lorentz boosts along different directions do not commute:
24972:
are isomorphic. It is widely believed that the choice between the two metric signatures has no physical relevance, even though some objects related to
14463:
12994:
3369:{\displaystyle {\begin{aligned}t'&=\gamma \left(t-{\frac {vx}{c^{2}}}\right)\\x'&=\gamma \left(x-vt\right)\\y'&=y\\z'&=z\end{aligned}}}
497:{\displaystyle {\begin{aligned}t'&=\gamma \left(t-{\frac {vx}{c^{2}}}\right)\\x'&=\gamma \left(x-vt\right)\\y'&=y\\z'&=z\end{aligned}}}
19406:
17535:
18513:
17607:
14409:
are altogether six continuous variables which make up the group parameters (in this particular representation), and the generators of the group are
3611:, each of which make the transformations unphysical. The space and time coordinates are measurable quantities and numerically must be real numbers.
909:{\displaystyle {\begin{aligned}ct'&=\gamma \left(ct-\beta x\right)\\x'&=\gamma \left(x-\beta ct\right)\\y'&=y\\z'&=z.\end{aligned}}}
23202:
smooth coordinate transformation) are geometric objects. In the geometric view, the electromagnetic field is a six-dimensional geometric object in
18299:
17428:
23683:
10715:
10293:
and this matrix equation contains the general conditions on the Lorentz transformation to ensure invariance of the spacetime interval. Taking the
7024:
and orthogonal. The position vector as measured in each frame is split into components parallel and perpendicular to the relative velocity vector
15569:{\displaystyle R({\boldsymbol {\theta }})=I+\sin \theta (\mathbf {e} \cdot \mathbf {J} )+(1-\cos \theta )(\mathbf {e} \cdot \mathbf {J} )^{2}\,.}
13596:
6027:{\displaystyle {\begin{aligned}ct&=ct'\cosh \zeta +x'\sinh \zeta \\x&=x'\cosh \zeta +ct'\sinh \zeta \\y&=y'\\z&=z'\end{aligned}}}
16584:(onto). Hence any group element in the connected component of the identity can be expressed as an exponential of an element of the Lie algebra.
10255:
11660:
all form subgroups, the sets containing improper and/or antichronous transformations without enough proper orthochronous transformations (e.g.
10784:
9602:. Applying this definition using the transformations of coordinates and momentum leads to the transformation of angular momentum. It turns out
5290:{\displaystyle {\begin{aligned}ct'&=ct\cosh \zeta -x\sinh \zeta \\x'&=x\cosh \zeta -ct\sinh \zeta \\y'&=y\\z'&=z\end{aligned}}}
16224:
1407:{\displaystyle c^{2}(t_{2}-t_{1})^{2}-(x_{2}-x_{1})^{2}-(y_{2}-y_{1})^{2}-(z_{2}-z_{1})^{2}=0\quad {\text{(lightlike separated events 1, 2)}}}
580:
19437:
and observers. The fact that the electromagnetic field shows relativistic effects becomes clear by carrying out a simple thought experiment.
15843:
count as higher order terms and are negligible). Taking the limit as before leads to the finite transformation in the form of an exponential
15448:{\displaystyle B({\boldsymbol {\zeta }})=I-\sinh \zeta (\mathbf {n} \cdot \mathbf {K} )+(\cosh \zeta -1)(\mathbf {n} \cdot \mathbf {K} )^{2}}
14287:
5827:
The inverse transformations are obtained by exchanging primed and unprimed quantities to switch the coordinate frames, and negating rapidity
15929:
The converse is also true, but the decomposition of a finite general Lorentz transformation into such factors is nontrivial. In particular,
13090:
5374:
1108:
Lorentz (1892–1904) and Larmor (1897–1900), who believed the luminiferous aether hypothesis, also looked for the transformation under which
11090:
10852:
5704:{\displaystyle {\begin{aligned}\beta &=\tanh \zeta \,,\\\gamma &=\cosh \zeta \,,\\\beta \gamma &=\sinh \zeta \,.\end{aligned}}}
5588:
Comparing the Lorentz transformations in terms of the relative velocity and rapidity, or using the above formulae, the connections between
27068:
27898:
27237:
5365:
in the transformations. Squaring and subtracting the results, one can derive hyperbolic curves of constant coordinate values but varying
3145:. In other words, the times and positions are coincident at this event. If all these hold, then the coordinate systems are said to be in
2862:
Transformations describing relative motion with constant (uniform) velocity and without rotation of the space coordinate axes are called
17891:
17825:
8788:
3700:
is the "moving" frame. According to the principle of relativity, there is no privileged frame of reference, so the transformations from
28133:
24870:
17767:
16856:
12855:
7219:{\displaystyle \mathbf {r} =\mathbf {r} _{\perp }+\mathbf {r} _{\|}\,,\quad \mathbf {r} '=\mathbf {r} _{\perp }'+\mathbf {r} _{\|}'\,,}
2918:
1168:
164:
27231:. This webpage poses a problem, the solution of which is the Lorentz transformation, which is presented graphically in its next page.
10528:
to ensure relativistic invariance. Not much information can be directly extracted from all the conditions, however one of the results
4982:
27922:
27668:
7431:
retains its definition for a boost in any direction, since it depends only on the magnitude of the relative velocity. The definition
4136:{\displaystyle {\begin{aligned}ct'&=\gamma \left(ct-\beta x\right)\,,\\x'&=\gamma \left(x-\beta ct\right)\,,\\\end{aligned}}}
6738:
17497:
16059:
139:
26035:
13459:{\displaystyle \quad R({\boldsymbol {\rho }})={\begin{bmatrix}1&0\\0&\mathbf {R} ({\boldsymbol {\rho }})\end{bmatrix}}\,,}
6436:
if the coordinate systems are never coincident (i.e., not in standard configuration), and if both observers can agree on an event
13920:{\displaystyle \{B({\boldsymbol {\zeta }}),R({\boldsymbol {\theta }}),\Lambda ({\boldsymbol {\zeta }},{\boldsymbol {\theta }})\}}
9743:
5716:
23754:
Charge density transforms as the time component of a four-vector. It is a rotational scalar. The current density is a 3-vector.
16070:
Lorentz generators can be added together, or multiplied by real numbers, to obtain more Lorentz generators. In other words, the
4203:
3687:) can be found by algebraically solving the original set of equations. A more efficient way is to use physical principles. Here
27704:
27658:
25246:, a different convention is used for these matrices; the right hand sides are all multiplied by a factor of the imaginary unit
16481:. Here the operation is the commutator which satisfies all of these axioms, the vector space is the set of Lorentz generators
25894:
27842:
27191:
27099:
27078:
26987:
26961:
26927:
26897:
26833:
26799:
26774:
26753:
26732:
26710:
26689:
26666:
26644:
26625:
26601:
26571:
26545:
26526:
26507:
26488:
26467:
26446:
26416:
26194:
26025:
25601:
25429:
27228:
27225:. This web page contains a more detailed derivation of the Lorentz transformation with special emphasis on group properties.
19067:
These observations generalize in an obvious way to more factors, and using the fact that a general tensor on a vector space
18192:{\displaystyle \eta _{\rho \nu }{\Lambda ^{\rho }}_{\sigma }\eta ^{\mu \sigma }={\left(\Lambda ^{-1}\right)^{\mu }}_{\nu },}
13927:
with matrix multiplication as the operation of composition forms a group, called the "restricted Lorentz group", and is the
27326:
12777:
27148:
23198:
view can be obtained and understood. Only objects that have well defined Lorentz transformation properties (in fact under
19816:
is often preferred over SI units, even in texts whose main choice of units is SI units, because in it the electric field
13928:
11824:
258:
26102:
25308:
of the generators is also a generator. They just live in a different space to the position vectors in ordinary 3d space.
18580:
of the standard representation of the Lorentz group. This notion generalizes to general representations, simply replace
8791:
can be similarly obtained by taking differentials in the velocity vectors, and dividing these by the time differential.
27579:
27057:
27036:
27013:
24895:
8960:
implies the quantities transform under Lorentz transformations similar to the transformation of spacetime coordinates;
1113:
1056:
104:
11699:
11663:
8743:
relative to F. The inverse transformations can be obtained in a similar way, or as with position coordinates exchange
8500:
taking the differentials in the coordinates and time of the vector transformations, then dividing equations, leads to
27950:
27584:
27129:
27089:
26867:
26791:
9167:
is exactly the same as for the position vector, as is the process of obtaining the inverse transformations (exchange
189:
19383:) operating on column vectors. This latter form is sometimes preferred; e.g., for the electromagnetic field tensor.
16784:
11735:
11490:
11453:
under the same operation of the group (here matrix multiplication). In other words, for two Lorentz transformations
10233:
of signature (3,1) on spacetime, and the group of transformations which leaves this quadratic form invariant is the
16726:
which negates the time coordinate only, because these transformations leave the spacetime interval invariant. Here
16137:
5340:
of rotation, analogous to the ordinary angle for circular rotations. This transformation can be illustrated with a
1208:
997:
27904:
10123:
1101:. In 1892, Lorentz independently presented the same idea in a more detailed manner, which was subsequently called
212:
27617:
25930:
25866:
23226:(alone) do not have well defined Lorentz transformation properties. The mathematical underpinnings are equations
13539:. These articles give the explicit formulae for the composite transformation matrices, including expressions for
8288:
5100:
The Lorentz transformations can also be derived in a way that resembles circular rotations in 3d space using the
4238:, then the transformation from an unprimed spacetime coordinate system to a primed coordinate system is given by
1098:
24984:
corresponding to the different signatures of the bilinear form associated to the two groups, are non-isomorphic.
11557:
6873:. Either way, each observer measures the time interval between ticks of a moving clock to be longer by a factor
1003:
Historically, the transformations were the result of attempts by Lorentz and others to explain how the speed of
27720:
26809:
25239:
18476:{\displaystyle {A'}_{\nu }={\Lambda _{\nu }}^{\mu }A_{\mu }={\left(\Lambda ^{-1}\right)^{\mu }}_{\nu }A_{\mu }}
15579:
It has been stated that the general proper Lorentz transformation is a product of a boost and rotation. At the
9525:
9449:
11796:, and the same measurement made in another inertial frame (with the same orientation and origin) gives result
9748:
Throughout, italic non-bold capital letters are 4×4 matrices, while non-italic bold letters are 3×3 matrices.
5522:
axis in spacetime. A consequence these two hyperbolic formulae is an identity that matches the Lorentz factor
28175:
28151:
27893:
27798:
27728:
26855:
14283:
9634:
6058:. If there are two events, there is a spatial separation and time interval between them. It follows from the
2871:
27861:
26184:
25970:
25284:, etc. The term "vector" applies much more broadly than Euclidean vectors, row or column vectors, etc., see
25219:{\displaystyle {\boldsymbol {\theta }}\cdot \mathbf {J} =\theta _{x}J_{x}+\theta _{y}J_{y}+\theta _{z}J_{z}}
20251:
where the field tensor is displayed side by side for easiest possible reference in the manipulations below.
5078:
1043:
of Minkowski space. The more general set of transformations that also includes translations is known as the
27715:
16493:
16411:
3618:, an observer in F′ notices the coordinates of the event to be "boosted" in the negative directions of the
923:
10241:. In other words, the Lorentz group is O(3,1). As presented in this article, any Lie groups mentioned are
6950:. So each observer measures the distance between the end points of a moving rod to be shorter by a factor
6695:. It is sometimes said that nonrelativistic physics is a physics of "instantaneous action at a distance".
28185:
28170:
27756:
27490:
27475:
26654:
25899:
24951:. The first equation (or the second restricted to lightlike separation) leads to a yet larger group, the
10377:
Writing the Minkowski metric as a block matrix, and the Lorentz transformation in the most general form,
10234:
8781:
919:
27049:
Relativity, Groups Particles. Special Relativity and Relativistic Symmetry in Field and Particle Physics
25121:{\displaystyle {\boldsymbol {\zeta }}\cdot \mathbf {K} =\zeta _{x}K_{x}+\zeta _{y}K_{y}+\zeta _{z}K_{z}}
16125:{\displaystyle V=\{{\boldsymbol {\zeta }}\cdot \mathbf {K} +{\boldsymbol {\theta }}\cdot \mathbf {J} \}}
11526:
11321:
where "+" and "−" indicate the determinant sign, while "↑" for ≥ and "↓" for ≤ denote the inequalities.
8115:
to be reinstated when convenient, and the rapidity parametrization is immediately obtained by replacing
6944:
the two measurements are no longer simultaneous, but this does not matter because the rod is at rest in
3746:(i.e., the relative velocity has the same magnitude but is oppositely directed). Thus if an observer in
2783:{\displaystyle (a,a)=(\Lambda a,\Lambda a)=(a',a'),\quad \Lambda \in \mathrm {O} (1,3),\quad a,a'\in M,}
718:
28043:
27485:
27438:
27259:
27047:
27005:
26408:
24811:
17413:
allows the transformation of other physical quantities that cannot be expressed as four-vectors; e.g.,
14031:{\displaystyle B_{x}=I+\zeta \left.{\frac {\partial B_{x}}{\partial \zeta }}\right|_{\zeta =0}+\cdots }
8279:
6702:
4181:
2591:
1117:
989:
25928:
Macfarlane, A. J. (1962). "On the Restricted Lorentz Group and Groups Homomorphically Related to It".
16835:
14275:{\displaystyle B_{x}=\lim _{N\to \infty }\left(I-{\frac {\zeta }{N}}K_{x}\right)^{N}=e^{-\zeta K_{x}}}
12918:
11653:{\displaystyle {\mathcal {L}}_{0}={\mathcal {L}}_{+}^{\uparrow }\cup {\mathcal {L}}_{-}^{\downarrow }}
28073:
27699:
27380:
27121:
26859:
26204:
Ungar, A. A. (1988). "Thomas rotation and the parameterization of the Lorentz transformation group".
24885:
24880:
18576:
This means exactly that covariant vectors (thought of as column matrices) transform according to the
18085:{\displaystyle {A'}_{\nu }=\eta _{\rho \nu }{\Lambda ^{\rho }}_{\sigma }\eta ^{\mu \sigma }A_{\mu }.}
13565:
13367:
13349:
11869:
10695:
10664:
9475:
8293:
8283:
6977:
1066:
283:
26052:
18757:{\displaystyle (A\otimes B)(u\otimes v)=Au\otimes Bv,\qquad u\in U,v\in V,u\otimes v\in U\otimes V.}
11228:{\displaystyle {\mathcal {L}}_{-}^{\downarrow }={\mathcal {L}}_{-}\cap {\mathcal {L}}^{\downarrow }}
10990:{\displaystyle {\mathcal {L}}_{+}^{\downarrow }={\mathcal {L}}_{+}\cap {\mathcal {L}}^{\downarrow }}
10188:
1034:—the mathematical model of spacetime in special relativity—the Lorentz transformations preserve the
28190:
28098:
27648:
27408:
27319:
26305:
16737:
16058:
in terms of the generators, and one wants to find the product in terms of the generators, then the
6922:) measurements at opposite ends. Under these conditions, the inverse Lorentz transform shows that
6692:
2930:
2890:
1156:
114:
26359:
Mocanu, C. I. (1992). "On the relativistic velocity composition paradox and the Thomas rotation".
11900:
5043:
27827:
27749:
27369:
27360:
26979:
26146:"Electromagnetic phenomena in a system moving with any velocity smaller than that of light"
24930:
observer, one refers to someone having, at least in principle, a copy of this report. See, e.g.,
24890:
24875:
24855:
13678:
8693:{\displaystyle \mathbf {u} '={\frac {1}{1-{\frac {\mathbf {v} \cdot \mathbf {u} }{c^{2}}}}}\left}
8299:, the ordering of vectors is chosen to reflect the ordering of the addition of velocities; first
6615:
6614:
For relative speeds much less than the speed of light, the Lorentz transformations reduce to the
1152:
969:
961:
949:
is something that happens at a point in space at an instant of time, or more formally a point in
251:
26825:
26813:
26330:
Mocanu, C. I. (1986). "Some difficulties within the framework of relativistic electrodynamics".
26166:
26084:
13931:
SO(3,1). (The plus sign indicates that it preserves the orientation of the temporal dimension).
11307:{\displaystyle {\mathcal {L}}_{-}^{\uparrow }={\mathcal {L}}_{-}\cap {\mathcal {L}}^{\uparrow }}
11069:{\displaystyle {\mathcal {L}}_{+}^{\uparrow }={\mathcal {L}}_{+}\cap {\mathcal {L}}^{\uparrow }}
9191:
to switch observed quantities, and reverse the direction of relative motion by the substitution
3643:
axes, while the event does not change and is simply represented in another coordinate system, a
28053:
27916:
27822:
27780:
27418:
26945:
26581:
26555:
26361:
26300:
26291:
26252:
26047:
23194:
and make a strong point of the ease with which results that are difficult to achieve using the
19829:
19400:
16580:. Globally, the exponential map is not one-to-one, but in the case of the Lorentz group, it is
16405:
These commutation relations, and the vector space of generators, fulfill the definition of the
16174:
11897:
represents the rotation-free Lorentz transformation between the unprimed and primed frames and
5037:
3644:
3108:
axes are parallel), remain mutually perpendicular, and relative motion is along the coincident
1109:
1082:
27297:
Online Flash animations of Galilean and Lorentz frames, various paradoxes, EM wave phenomena,
27234:
26250:
Ungar, A. A. (1989). "The relativistic velocity composition paradox and the Thomas rotation".
25740:
25608:
25436:
23187:{\displaystyle F^{\mu '\nu '}={\Lambda ^{\mu '}}_{\mu }{\Lambda ^{\nu '}}_{\nu }F^{\mu \nu },}
20360:{\displaystyle F^{\mu '\nu '}={\Lambda ^{\mu '}}_{\mu }{\Lambda ^{\nu '}}_{\nu }F^{\mu \nu }.}
15258:
which correspond to the motion of the system in spacetime. The derivative of any smooth curve
10168:
8953:{\displaystyle A^{2}-\mathbf {Z} \cdot \mathbf {Z} ={A'}^{2}-\mathbf {Z} '\cdot \mathbf {Z} '}
1077:
himself—had been discussing the physics implied by these equations since 1887. Early in 1889,
28058:
27663:
27602:
27222:
25462:
24850:
19430:
17530:
17422:
11450:
10650:{\displaystyle \Gamma ^{2}\geq 1\quad \Rightarrow \quad \Gamma \leq -1\,,\quad \Gamma \geq 1}
10246:
9543:, however a boosted observer will perceive a nonzero timelike component and an altered spin.
8103:
The unit vector has the advantage of simplifying equations for a single boost, allows either
6059:
4143:
which shows much more clearly the symmetry in the transformation. From the allowed ranges of
3615:
2447:
2446:(a solution satisfying the first formula automatically satisfies the second one as well; see
1139:
In 1905, Poincaré was the first to recognize that the transformation has the properties of a
1023:
942:
683:
280:
174:
74:
15576:
which compactly reproduce the boost and rotation matrices as given in the previous section.
11919:
is the velocity of the primed frame as seen from the unprimed frame. The matrix is given by
2450:). Finding the solution to the simpler problem is just a matter of look-up in the theory of
27963:
27889:
27733:
27335:
27163:
26821:
26370:
26261:
26213:
26117:
26005:
25939:
25908:
25668:
25243:
24900:
24146:
17993:, first raise its index, then transform it according to the same rule as for contravariant
17508:
indices that take the value 0 for time components, and 1, 2, 3 for space components, while
17501:
16740:), and each have determinant −1. This latter property makes them improper transformations.
16218:
14145:{\displaystyle \left.{\frac {\partial B_{x}}{\partial \zeta }}\right|_{\zeta =0}=-K_{x}\,.}
13167:
9731:
2823:
1188:
27291:
26702:
Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity
18289:{\displaystyle {\Lambda _{\nu }}^{\mu }\equiv {\left(\Lambda ^{-1}\right)^{\mu }}_{\nu },}
8716:
are the velocity of some massive object. They can also be for a third inertial frame (say
8241:{\displaystyle {\boldsymbol {\zeta }}=\zeta \mathbf {n} =\mathbf {n} \tanh ^{-1}\beta \,,}
2909:, an element of the Poincaré group, which is also called the inhomogeneous Lorentz group.
996:
is the same in all inertial reference frames. The invariance of light speed is one of the
8:
28038:
28033:
28023:
27855:
27804:
27622:
27607:
27433:
27364:
27312:
27272:
explaining and visualizing the Lorentz transformation with a mechanical Minkowski diagram
25293:
24860:
18577:
17788:
be thought of as spacetime indices (sometimes called Lorentz indices), and they run from
16581:
16458:
10210:
10117:
5323:
5101:
1140:
1094:
1040:
1026:. It may include a rotation of space; a rotation-free Lorentz transformation is called a
973:
28028:
27167:
26374:
26265:
26217:
26121:
26009:
25943:
25912:
25672:
13777:
includes a boost and rotation together, and is a nonsymmetric matrix. As special cases,
11799:
10694:
only one inequality. There are four sets which include every possible pair given by the
2889:
also contains special transformations that are neither rotations nor boosts, but rather
311:
28180:
28103:
28003:
27927:
27785:
27766:
27760:
27711:
27653:
27562:
27480:
27398:
27375:
27343:
27025:
26911:
26881:
26386:
26347:
26318:
26277:
26237:
26065:
25305:
17677:
in which the primed indices denote the indices of A in the primed frame. For a general
16577:
16399:
14164:
11779:
11325:
9454:
8773:
6882:
1179:
1148:
1102:
1090:
1035:
1016:
1008:
981:
954:
946:
938:
287:
244:
230:
194:
51:
46:
13166:(parallel or antiparallel along the same line of relative motion), the boost matrices
6037:
The inverse transformations can be similarly visualized by considering the cases when
2021:
solution preserving the origin of the simpler problem solves the general problem too:
28113:
27945:
27937:
27538:
27495:
27279:
27187:
27125:
27095:
27074:
27053:
27032:
27009:
26983:
26957:
26937:
26923:
26893:
26863:
26829:
26795:
26783:
26770:
26749:
26728:
26706:
26685:
26662:
26640:
26621:
26597:
26589:
26567:
26541:
26522:
26503:
26484:
26463:
26442:
26412:
26402:
26390:
26351:
26322:
26241:
26229:
26190:
26021:
25684:
25597:
25425:
25235:
23758:
19395:
Lorentz boost of an electric charge, the charge is at rest in one frame or the other.
16211:
16071:
13744:
13536:
10162:
9649:
fields, the transformations cannot be obtained as directly using vector algebra. The
9351:
8777:
8177:{\displaystyle {\boldsymbol {\beta }}=\beta \mathbf {n} =\mathbf {n} \tanh \zeta \,,}
7507:
6972:
5341:
3588:
965:
169:
27983:
26281:
26079:
24944:
2835:
2830:
and mixes thereof. If the spacetime translations are included, then one obtains the
1121:
1112:
are invariant when transformed from the aether to a moving frame. They extended the
1044:
28018:
28008:
27955:
27932:
27510:
27250:. A computer program demonstrating the Lorentz transformations on everyday objects.
26971:
26378:
26339:
26310:
26269:
26221:
26136:
26125:
26057:
26013:
25960:
25955:
25947:
25916:
25676:
25656:
25269:
24981:
24905:
19463:
16593:
16462:
15245:
13936:
13724:
10242:
9757:
9547:
6968:
6684:{\displaystyle {\begin{aligned}t'&\approx t\\x'&\approx x-vt\end{aligned}}}
6425:
rather than spatial points or instants of time are useful for a number of reasons:
5337:
2855:
2637:
1078:
1012:
930:, etc.). The term "Lorentz transformations" only refers to transformations between
927:
184:
28078:
26289:
Ungar, A. A. (2000). "The relativistic composite-velocity reciprocity principle".
28093:
28068:
27993:
27988:
27871:
27832:
27794:
27738:
27612:
27548:
27241:
27113:
26907:
26877:
26847:
26741:
26720:
26700:
26681:
26615:
26611:
26559:
26478:
26457:
26436:
26398:
26180:
26162:
25591:
25419:
24952:
23487:
19434:
16733:
16478:
16474:
16133:
16051:
14058:
13590:
13532:
13478:
9546:
Not all quantities are invariant in the form as shown above, for example orbital
9436:
9294:
8248:
each of which serves as a useful abbreviation in some contexts. The magnitude of
7086:
7044:
5327:
2851:
2620:
2451:
1144:
1074:
1031:
302:
84:
27876:
25876:
19391:
16873:
Writing the general matrix transformation of coordinates as the matrix equation
1204:
to specify position in space in that frame. Subscripts label individual events.
557:=0, where the primed frame is seen from the unprimed frame as moving with speed
28118:
27790:
27774:
27770:
27673:
27643:
27500:
27403:
27285:
27275:
27200:
27171:
25318:
25301:
25285:
24845:
23478:
19813:
19421:
19412:
18623:
17755:{\displaystyle {X'}^{\alpha }={\Pi (\Lambda )^{\alpha }}_{\beta }X^{\beta }\,,}
17509:
17505:
16862:
16469:
in this context) on the elements of the vector space, satisfying the axioms of
16038:(this usually does not yield an intelligible expression in terms of generators
13748:
13634:
10230:
9739:
9735:
9565:
9556:
9510:
9502:
9419:
9401:
6388:
3608:
3468:
3398:
1212:
1116:
hypothesis and found out that the time coordinate has to be modified as well ("
1086:
1062:
993:
945:
in this context) to measure lengths, and a clock to measure time intervals. An
653:
574:
217:
27265:
27253:
26314:
24121:
19455:
observer sees a different electric field because the charge moves at velocity
19356:
19046:
18765:
18491:. Thus, in terms of matrices, this transformation should be thought of as the
11776:
If a Lorentz covariant 4-vector is measured in one inertial frame with result
308:
The most common form of the transformation, parametrized by the real constant
28164:
28108:
28088:
28083:
27998:
27866:
27694:
27638:
27470:
27423:
26953:
26915:
26885:
26762:
26233:
26130:
25688:
24865:
19373:
is defined above. This form can generally be reduced to the form for general
17885:
10215:
9725:
9650:
9307:
6801:
6698:
Three counterintuitive, but correct, predictions of the transformations are:
2633:
2455:
1172:
1133:
1070:
985:
544:
are the coordinates of an event in two frames with the origins coinciding at
26083:
26061:
26017:
24129:
The above equation could, for instance, be the transformation of a state in
16719:{\displaystyle T={\begin{bmatrix}-1&0\\0&\mathbf {I} \end{bmatrix}}}
16652:{\displaystyle P={\begin{bmatrix}1&0\\0&-\mathbf {I} \end{bmatrix}}}
10765:{\displaystyle {\mathcal {L}}^{\downarrow }=\{\Lambda \,:\,\Gamma \leq -1\}}
10513:
carrying out the block matrix multiplications obtains general conditions on
10109:{\displaystyle X\cdot X=X^{\mathrm {T} }\eta X={X'}^{\mathrm {T} }\eta {X'}}
2846:
The relations between the primed and unprimed spacetime coordinates are the
28128:
28048:
28013:
27543:
27505:
27027:
Relativistic Mechanics - Special Relativity and Classical Particle Dynamics
25292:
for details. The generators of a Lie group also form a vector space over a
25289:
23775:
hold unmodified for any representation of the Lorentz group, including the
16470:
16451:
16141:
13375:
13371:
11329:
9506:
9414:
5326:
of spacetime coordinates in the xt, yt, and zt Cartesian-time planes of 4d
179:
27205:
Nachrichten von der Königlicher Gesellschaft den Wissenschaft zu Göttingen
26540:. Manchester Physics Series. John Wiley & Sons Ltd. pp. 124–126.
25833:
6812:. If a time interval is measured at the same point in that frame, so that
6798:, so the events are no longer simultaneous according to a moving observer.
27912:
27881:
27428:
26997:
25297:
17520:
16466:
16406:
13706:
13575:
10831:{\displaystyle {\mathcal {L}}^{\uparrow }=\{\Lambda \,:\,\Gamma \geq 1\}}
10294:
9521:
9372:
9346:
9272:
9218:
7457:
7422:
5033:
2875:
645:{\textstyle \gamma =\left({\sqrt {1-{\frac {v^{2}}{c^{2}}}}}\right)^{-1}}
159:
26678:
Lie Groups, Lie Algebras, and Representations An Elementary Introduction
25990:
25421:
The Rotation and Lorentz Groups and Their Representations for Physicists
16402:
of x, y, z components (i.e. change x to y, y to z, and z to x, repeat).
2897:
in which the spatial coordinates of all events are reversed in sign and
2563:{\displaystyle (a,a)=(a',a')\quad {\text{or}}\quad a\cdot a=a'\cdot a',}
28123:
27689:
27533:
27528:
27109:
26941:
26843:
26585:
26382:
26343:
26273:
26225:
26145:
24130:
16660:
16394:
10368:{\displaystyle \left^{2}=1\quad \Rightarrow \quad \det(\Lambda )=\pm 1}
5837:
since this is equivalent to negating the relative velocity. Therefore,
5579:{\displaystyle \cosh \zeta ={\frac {1}{\sqrt {1-\tanh ^{2}\zeta }}}\,.}
2898:
2894:
2609:). The alternative notation defined on the right is referred to as the
1522:. The transformation sought after thus must possess the property that:
926:(accelerating, moving in curved paths, rotational motion with constant
25951:
25920:
25878:
Michelson, FitzGerald and Lorentz: the Origins of Relativity Revisited
25680:
23055:
and are independent of the metric signature. For SI units, substitute
16568:{\displaystyle \exp \,:\,{\mathfrak {so}}(3,1)\to \mathrm {SO} (3,1),}
6849:. If an interval is measured at the same point in that frame, so that
5075:. The inverse of the transformation is given by reversing the sign of
1959:
are the spacetime coordinates used to define events in one frame, and
1151:, by deriving the Lorentz transformation under the assumptions of the
27520:
15803:
is commutative because only linear terms are required (products like
13734:
13163:
11149:{\displaystyle {\mathcal {L}}_{-}=\{\Lambda \,:\,\det(\Lambda )=-1\}}
10911:{\displaystyle {\mathcal {L}}_{+}=\{\Lambda \,:\,\det(\Lambda )=+1\}}
10576:{\displaystyle \Gamma ^{2}=1+\mathbf {b} ^{\mathrm {T} }\mathbf {b} }
10238:
10031:
9385:
3482:
3456:{\displaystyle \gamma ={\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}}
1162:
950:
299:
291:
69:
35:
27:
27288:
on Desmos showing Lorentz transformations with points and hyperbolas
27091:
Symmetry in quantum mechanics:From angular momentum to supersymmetry
27070:
Special Relativity in General Frames: From Particles to Astrophysics
23244:. One should note that the primed and unprimed tensors refer to the
18367:
Now for a subtlety. The implied summation on the right hand side of
5505:{\displaystyle \tanh \zeta ={\frac {\sinh \zeta }{\cosh \zeta }}\,,}
2901:
in which the time coordinate for each event gets its sign reversed.
1089:
surrounding a spherical distribution of charge should cease to have
28063:
27413:
26090:
Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences
25363:
is expressed as a linear combination of the Cartesian unit vectors
23776:
23206:
as opposed to two interdependent, but separate, 3-vector fields in
19809:
17803:
16825:
is a constant column containing translations in time and space. If
11446:
9333:
5312:
977:
953:. The transformations connect the space and time coordinates of an
295:
17421:
of any order in 4d spacetime, to be defined. In the corresponding
16847:. Poincaré transformations are not dealt further in this article.
13074:
and the composition of the two boosts connects the coordinates in
12907:{\textstyle \gamma ={\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}}
11463:
from a particular subgroup, the composite Lorentz transformations
27269:
27247:
26151:
Proceedings of the Royal Netherlands Academy of Arts and Sciences
13067:{\displaystyle X''=B(\mathbf {v} )X'\,,\quad X'=B(\mathbf {u} )X}
8130:
The vectorial relation between relative velocity and rapidity is
7851:
to switch observed coordinates, and negate the relative velocity
272:
134:
27282:
showing Lorentz transformations with a virtual Minkowski diagram
19462:
in their rest frame. The motion of the charge corresponds to an
19411:
Lorentz transformations can also be used to illustrate that the
9713:
9489:
For a given object (e.g., particle, fluid, field, material), if
5512:
provides the link between a constant value of rapidity, and the
5025:{\displaystyle \gamma =1/{\sqrt {1-{\boldsymbol {\beta }}^{2}}}}
2893:
in a plane through the origin. Two of these can be singled out;
1124:
gave a physical interpretation to local time (to first order in
27743:
19466:, and thus the observer in frame F′ also sees a magnetic field.
19074:
18217:
Lorentz transformation. One defines (as a matter of notation),
17595:{\displaystyle {A'}^{\nu }={\Lambda ^{\nu }}_{\mu }A^{\mu }\,.}
17418:
17414:
16485:
as given previously, and the field is the set of real numbers.
9312:
7113:, each into components perpendicular (⊥) and parallel ( ‖ ) to
2905:
rotation with a boost, followed by a shift in spacetime, is an
1979:
are the coordinates in another frame. First one observes that (
27149:"First proposal of the universal speed of light by Voigt 1887"
19386:
18567:{\displaystyle A'=\left(\Lambda ^{-1}\right)^{\mathrm {T} }A.}
17670:{\displaystyle A^{\nu '}={\Lambda ^{\nu '}}_{\mu }A^{\mu }\,.}
14038:
where the higher order terms not shown are negligible because
13589:(positive anticlockwise, negative clockwise, according to the
7481:
in the direction of relative motion, the relative velocity is
6791:{\displaystyle \Delta t'=\gamma {\frac {-v\,\Delta x}{c^{2}}}}
3632:
in the transformations. This has the equivalent effect of the
2017:
and are not dealt with further here. Then one observes that a
27304:
27120:. Course of Theoretical Physics. Vol. 2 (4th ed.).
26617:
Spacetime and Geometry: An Introduction to General Relativity
18358:{\displaystyle {A'}_{\nu }={\Lambda _{\nu }}^{\mu }A_{\mu }.}
17487:{\displaystyle {x'}^{\nu }={\Lambda ^{\nu }}_{\mu }x^{\mu },}
16736:. These are both symmetric, they are their own inverses (see
16181:, and the components of the axis-angle and rapidity vectors,
13630:
differences between the boost and rotation matrices include:
13205:. This composite transformation happens to be another boost,
10698:("n"-shaped symbol meaning "and") of these classifying sets.
10674:
The determinant and inequality provide four ways to classify
6914:, so its length must be measured by taking two simultaneous (
5513:
5447:
axes can be constructed for varying coordinates but constant
3464:
2628:. The Lorentz transformation is thus an element of the group
1004:
26639:. Manchester Physics (2nd ed.). John Wiley & Sons.
26438:
Electrodynamics and Classical Theory of Fields and Particles
25424:(illustrated ed.). John Wiley & Sons. p. 213.
23745:{\displaystyle j^{\mu '}={\Lambda ^{\mu '}}_{\mu }j^{\mu }.}
15292:, serves as a definition of a corresponding group generator
3713:
must take exactly the same form as the transformations from
668:, the Lorentz factor is negligibly different from 1, but as
27724:
14077:
13977:
13619:{\displaystyle {\boldsymbol {\theta }}=\theta \mathbf {e} }
9277:
5307:
4008:
1159:, and by abandoning the mechanistic aether as unnecessary.
27186:(5th ed.), Belmont, : Brooks/Cole, pp. 546–579,
23248:. Thus the complete equation with spacetime dependence is
15337:
Expanding the exponentials in their Taylor series obtains
10286:{\displaystyle \eta =\Lambda ^{\mathrm {T} }\eta \Lambda }
10245:. In this context the operation of composition amounts to
6707:
Suppose two events occur along the x axis simultaneously (
5371:, which parametrizes the curves according to the identity
4231:{\displaystyle {\boldsymbol {\beta }}={\boldsymbol {v}}/c}
4200:
is in an arbitrary vector direction with the boost vector
26480:
An Introduction to the Standard Model of Particle Physics
25268:
Until now the term "vector" has exclusively referred to "
10663:
multiplies the time coordinate and this has an effect on
10297:
of the equation using the product rule gives immediately
10252:
From the invariance of the spacetime interval it follows
9587:, and in a boosted frame the altered angular momentum is
9501:
correspond to properties specific to the object like its
8331:
Defining the coordinate velocities and Lorentz factor by
8292:
The transformation of velocities provides the definition
8254:
is the absolute value of the rapidity scalar confined to
7620:
Accumulating the results gives the full transformations,
5713:
Taking the inverse hyperbolic tangent gives the rapidity
2912:
941:, an observer can use a local coordinate system (usually
16374:{\displaystyle =J_{z}\,,\quad =-J_{z}\,,\quad =K_{z}\,,}
9555:
does not have a timelike quantity, and neither does the
8724:. Denote either entity by X. Then X moves with velocity
6054:
So far the Lorentz transformations have been applied to
5817:
is relative motion along the negative directions of the
3514:
is relative motion along the negative directions of the
3507:
is no relative motion, while negative relative velocity
1143:, and he named it after Lorentz. Later in the same year
26661:(3rd ed.). Pearson Education, Dorling Kindersley.
16747:
is a proper orthochronous Lorentz transformation, then
10209:. This set together with matrix multiplication forms a
3125:, the origins of both coordinate systems are the same,
20004:
19872:
19501:
19473:
The electromagnetic field strength tensor is given by
17340:
16984:
16885:
16680:
16613:
15111:
14984:
14857:
14736:
14612:
14488:
13574:. The rotation is about an axis in the direction of a
13408:
13287:
are not equal. Although each of these compositions is
13143:{\displaystyle X''=B(\mathbf {v} )B(\mathbf {u} )X\,.}
12858:
12845:{\textstyle v={\sqrt {v_{x}^{2}+v_{y}^{2}+v_{z}^{2}}}}
12780:
12624:
11948:
10447:
10395:
9977:
9863:
9788:
6879:
than the time interval between ticks of his own clock.
6857:, then the transformations give this interval in F by
5426:{\displaystyle \cosh ^{2}\zeta -\sinh ^{2}\zeta =1\,.}
4901:
4342:
4252:
2822:) is satisfied. The elements of the Lorentz group are
2806:
which is precisely preservation of the bilinear form (
934:
frames, usually in the context of special relativity.
721:
583:
25864:
25484:
25133:
25035:
24160:
23802:
23686:
23501:
23254:
23093:
22714:
21431:
20375:
20266:
19844:
19479:
19429:
are simply different aspects of the same force — the
19379:-component objects given above with a single matrix (
19092:
18822:
18653:
18516:
18373:
18302:
18223:
18100:
18003:
17894:
17828:
17689:
17610:
17538:
17431:
16879:
16787:
16668:
16601:
16502:
16414:
16227:
16080:
15935:
15849:
15589:
15460:
15343:
14461:
14298:
14173:
14074:
13952:
13854:
13599:
13384:
13093:
12997:
12921:
12703:
12578:
12517:
12462:
12379:
12331:
12270:
12187:
12132:
12084:
11925:
11903:
11872:
11827:
11802:
11782:
11738:
11702:
11666:
11591:
11560:
11529:
11493:
11337:
11250:
11171:
11093:
11012:
10933:
10855:
10787:
10718:
10599:
10534:
10383:
10303:
10258:
10191:
10171:
10126:
10040:
9771:
8966:
8877:
8794:
8509:
8454:
8340:
8190:
8136:
7907:
7650:
7516:
7232:
7125:
6741:
6623:
6224:
6071:
5864:
5719:
5612:
5528:
5459:
5377:
5137:
5081:
5046:
4985:
4246:
4206:
4184:
4023:
3804:
3407:
3213:
2655:
2475:
2032:
1533:
1228:
1015:. The transformations later became a cornerstone for
757:
686:
341:
314:
26906:
26876:
26620:(illustrated ed.). Addison Wesley. p. 22.
17516:. The second index corresponds to the column index.
16659:
which negates all the spatial coordinates only, and
13675:(rotation in the opposite sense about the same axis)
7022:
The coordinate axes of each frame are still parallel
3069:
The coordinate axes in each frame are parallel (the
2878:, etc.). A combination of a rotation and boost is a
918:
Frames of reference can be divided into two groups:
26936:
26580:
26476:
26144:
25809:
25458:
24136:
23076:
17523:, not just 4-dimensional spacetime coordinates. If
16214:of a Lorentz generator with respect to this basis.
15285:with respect to that group parameter, evaluated at
14453:, each vectors of matrices with the explicit forms
10690:). Any particular LT has only one determinant sign
10659:The negative inequality may be unexpected, because
10158:is a square matrix which can depend on parameters.
10026:the spacetime interval takes the form (superscript
2854:of all the coordinates in the other frame, and the
1099:
1887 aether-wind experiment of Michelson and Morley
27024:
25657:"Lorentz Transformation and the Thomas Precession"
25392:which form a basis, and the Cartesian coordinates
25218:
25120:
24775:
24112:
23744:
23670:
23467:
23186:
23047:
22685:
21415:
20359:
20243:
19800:
19347:
19037:
18756:
18566:
18475:
18357:
18288:
18191:
18084:
17937:{\displaystyle x^{\mu }=\eta ^{\mu \nu }x_{\nu },}
17936:
17871:{\displaystyle x_{\nu }=\eta _{\mu \nu }x^{\mu },}
17870:
17754:
17669:
17594:
17486:
17405:
16813:
16718:
16651:
16567:
16442:
16373:
16124:
16026:
15919:
15795:
15568:
15447:
15215:
14387:
14274:
14144:
14030:
13919:
13721:. This property makes them proper transformations.
13618:
13458:
13142:
13066:
12941:
12906:
12844:
12764:
11911:
11889:
11858:
11813:
11788:
11760:
11724:
11688:
11652:
11577:
11546:
11515:
11435:
11306:
11227:
11148:
11068:
10989:
10910:
10830:
10764:
10649:
10575:
10505:
10367:
10285:
10201:
10177:
10146:
10108:
10018:
9756:Writing the coordinates in column vectors and the
9138:
8952:
8692:
8489:
8240:
8176:
8092:
7820:
7610:
7409:
7218:
6842:. Conversely, suppose there is a clock at rest in
6790:
6683:
6373:
6210:
6026:
5751:
5703:
5578:
5504:
5425:
5289:
5089:
5067:
5024:
4968:
4230:
4192:
4135:
3972:
3455:
3368:
2882:, which transforms the origin back to the origin.
2782:
2562:
2422:
1915:
1406:
1163:Derivation of the group of Lorentz transformations
1105:. Their explanation was widely known before 1905.
1011:, and to understand the symmetries of the laws of
908:
743:
692:
644:
496:
323:
27087:
26189:. New York: Three Rivers Press (published 1995).
25642:
25596:(illustrated ed.). OUP Oxford. p. 124.
19828:have the same units making the appearance of the
19445:The other observer in frame F′ moves at velocity
19407:classical electromagnetism and special relativity
16398:, and the other relations can be found by taking
13656:(relative motion in the opposite direction), and
6820:, then the transformations give this interval in
28162:
26455:
25865:O'Connor, John J.; Robertson, Edmund F. (1996),
25736:
25589:
14188:
11725:{\displaystyle {\mathcal {L}}_{-}^{\downarrow }}
11689:{\displaystyle {\mathcal {L}}_{+}^{\downarrow }}
11122:
10884:
10344:
10310:
9161:) into components perpendicular and parallel to
3571:is outside these limits. At the speed of light (
27182:Thornton, Stephen T.; Marion, Jerry B. (2004),
17519:The transformation matrix is universal for all
17504:is applied. It is a standard convention to use
15279:in the group depending on some group parameter
13765:The most general proper Lorentz transformation
13150:Successive transformations act on the left. If
8737:relative to F′, in turn F′ moves with velocity
8273:
5810:is no relative motion, while negative rapidity
5796:is motion along the positive directions of the
3493:is motion along the positive directions of the
3385:is the relative velocity between frames in the
1430:in all inertial frames for events connected by
1155:and the constancy of the speed of light in any
27181:
27108:
26842:
26199:– via Albert Einstein Reference Archive.
19832:more natural. Consider a Lorentz boost in the
19433:, as a consequence of relative motion between
16775:
11761:{\displaystyle {\mathcal {L}}_{-}^{\uparrow }}
11516:{\displaystyle {\mathcal {L}}_{+}^{\uparrow }}
6391:) indicates a difference of quantities; e.g.,
27320:
26634:
26554:
26497:
25775:
25567:
23761:are invariant under Lorentz transformations.
16489:Lie algebra in the usual vector space sense.
15225:These are all defined in an analogous way to
14288:characterizations of the exponential function
8730:relative to F, or equivalently with velocity
8312:(the velocity of X relative to F′) to obtain
7831:The projection and rejection also applies to
3636:F′ boosted in the positive directions of the
1520:shown using homogeneity and isotropy of space
1093:once the charge is in motion relative to the
922:(relative motion with constant velocity) and
252:
27045:
26976:Relativity Special, General and Cosmological
26635:Grant, I. S.; Phillips, W. R. (2008). "14".
26535:
26521:. Schaum Series. Mc Graw Hill. p. 688.
26483:(2nd ed.). Cambridge University Press.
26477:Cottingham, W. N.; Greenwood, D. A. (2007).
26456:Dennery, Philippe; Krzywicki, André (2012).
25578:
25542:The reference is within the following paper:
25447:
22708:is used. These results can be summarized by
16450:. In summary, a Lie algebra is defined as a
16119:
16087:
13914:
13855:
11143:
11111:
10905:
10873:
10825:
10805:
10759:
10736:
9751:
8772:The transformation of velocity is useful in
8184:and the "rapidity vector" can be defined as
8127:. It is not convenient for multiple boosts.
7838:. For the inverse transformations, exchange
7346:
7318:
7204:
7156:
6562:the equation for a pulse of light along the
2923:
2619:endowed with this bilinear form is known as
751:an equivalent form of the transformation is
27184:Classical dynamics of particles and systems
27066:
26748:, vol. 1, Cambridge University Press,
25398:are coordinates with respect to this basis.
24145:multi-particle state (Fock space state) in
19387:Transformation of the electromagnetic field
16587:
9719:
5752:{\displaystyle \zeta =\tanh ^{-1}\beta \,.}
5036:of the transformation matrix is +1 and its
2603:exposed by the right hand side formula in (
27327:
27313:
26186:Relativity: The Special and General Theory
26168:Relativity: The Special and General Theory
25927:
24871:Representation theory of the Lorentz group
18773:From this it is immediately clear that if
16857:Representation theory of the Lorentz group
16065:
11578:{\displaystyle {\mathcal {L}}^{\uparrow }}
9653:is the definition of these fields, and in
6721:, but separated by a nonzero displacement
2919:Derivations of the Lorentz transformations
2644:(also called the Lorentz group). One has:
1169:Derivations of the Lorentz transformations
957:as measured by an observer in each frame.
294:to another frame that moves at a constant
259:
245:
27223:Derivation of the Lorentz transformations
27203:(1887), "Über das Doppler'sche princip",
27088:Chaichian, Masud; Hagedorn, Rolf (1997).
26782:
26653:
26502:(12th ed.). Pearson Addison-Wesley.
26304:
26129:
26051:
26033:
25991:"The Genesis of the theory of relativity"
25959:
25786:
25530:
17748:
17663:
17588:
16510:
16506:
16367:
16320:
16270:
16132:together with the operations of ordinary
15562:
15082:
14955:
14707:
14583:
14381:
14339:
14138:
13939:of the boost matrix to first order about
13843:
13452:
13136:
13031:
11771:
11328:("u"-shaped symbol meaning "or") of four
11121:
11117:
10883:
10879:
10815:
10811:
10746:
10742:
10633:
10499:
10431:
9983:
9961:
9847:
9794:
9128:
9030:
8421:
8371:
8234:
8170:
8082:
7983:
7810:
7721:
7556:
7212:
7161:
6962:
6768:
6363:
6346:
6300:
6270:
6200:
6188:
6142:
6117:
5745:
5693:
5663:
5636:
5572:
5498:
5419:
4125:
4074:
3521:axes. The magnitude of relative velocity
3040:, and an observer in this "moving" frame
2458:of various signature. First equation in (
1434:. The quantity on the left is called the
1103:FitzGerald–Lorentz contraction hypothesis
27146:
26740:
26719:
26698:
26500:University Physics – With Modern Physics
26397:
26161:
26100:
26078:
25988:
25820:
25712:
25701:
25619:
25555:
25543:
25518:
19390:
17997:-vectors, then finally lower the index;
16868:
9608:transforms with another vector quantity
9573:. The definition of angular momentum is
8305:(the velocity of F′ relative to F) then
8287:
6976:
6551:
5347:The hyperbolic functions arise from the
5316:(many other symbols are used, including
3548:are allowed. The corresponding range of
2929:
331:representing a velocity confined to the
27262:visualizing the Lorentz transformation.
26970:
26808:
26761:
26610:
26516:
26142:
26085:"On the Dynamics of the Electron"
25968:
25892:
25797:
25763:
25724:
25506:
25473:
25135:
25037:
23785:one simply replaces all occurrences of
23015:
22970:
22892:
22847:
22653:
22239:
21386:
20972:
16496:from the Lie algebra to the Lie group,
16107:
16091:
16007:
15987:
15961:
15945:
15900:
15884:
15865:
15857:
15771:
15755:
15717:
15686:
15645:
15614:
15468:
15351:
14367:
14351:
14325:
14306:
13907:
13899:
13882:
13865:
13601:
13437:
13393:
11324:The full Lorentz group splits into the
10667:. If the positive equality holds, then
9744:rotation formalisms in three dimensions
9234:the "spacelike component". Examples of
8833:and their Lorentz-boosted counterparts
8789:Lorentz transformations of acceleration
8192:
8138:
5083:
5010:
4216:
4208:
4186:
4175:is standard throughout the literature.
3993:Sometimes it is more convenient to use
3565:The transformations are not defined if
28163:
27046:Sexl, R. U.; Urbantke, H. K. (2001) .
26498:Young, H. D.; Freedman, R. A. (2008).
26358:
26329:
26036:"A Note on Relativity Before Einstein"
18604:are linear operators on vector spaces
17498:covariant and contravariant components
16144:over the real numbers. The generators
16138:multiplication of a matrix by a number
6893:aligned along the x axis, with length
5090:{\displaystyle {\boldsymbol {\beta }}}
2913:Physical formulation of Lorentz boosts
1007:was observed to be independent of the
712:for the transformation to make sense.
27308:
27199:
26996:
26746:The quantum theory of fields (3 vol.)
26536:Forshaw, J. R.; Smith, A. G. (2009).
26434:
26288:
26249:
26203:
25874:
25654:
25630:
25495:
23087:represented by the tensor expression
16850:
16592:Lorentz transformations also include
16443:{\displaystyle {\mathfrak {so}}(3,1)}
13626:will serve as a useful abbreviation.
12852:is the magnitude of the velocity and
11821:, the two results will be related by
3047:defines events using the coordinates
2613:. Spacetime mathematically viewed as
140:Newton's law of universal gravitation
27022:
26675:
26103:"Zur Elektrodynamik bewegter Körper"
25751:
24931:
24151:
23477:Length contraction has an effect on
18510:. That is, in pure matrix notation,
17813:
17496:where lower and upper indices label
16831:inhomogeneous Lorentz transformation
6806:Suppose there is a clock at rest in
2907:inhomogeneous Lorentz transformation
2850:, each coordinate in one frame is a
2646:
2636:or, for those that prefer the other
2466:
2464:) can be written more compactly as:
2023:
1524:
1219:
1136:property inherent in his equations.
27899:Tolman–Oppenheimer–Volkoff equation
27852:Friedmann–Lemaître–Robertson–Walker
27244:– a chapter from an online textbook
26404:The Quantum Theory of Fields, vol I
25417:
23680:or, in the simpler geometric view,
23077:Misner, Thorne & Wheeler (1973)
20369:For the magnetic field one obtains
17784:. In this case, the indices should
17768:representation of the Lorentz group
16516:
16513:
16420:
16417:
14284:limit definition of the exponential
13929:special indefinite orthogonal group
13513:) to the original boost parameters
11859:{\displaystyle X'=B(\mathbf {v} )X}
2816:and bilinearity of the form) that (
744:{\textstyle \beta ={\frac {v}{c}},}
13:
27140:
24722:
24683:
24679:
24646:
24563:
24398:
24372:
24329:
24300:
24185:
24172:
24073:
24067:
24045:
24039:
23984:
23978:
23956:
23950:
23908:
23902:
23867:
23861:
23843:
23837:
23825:
23819:
23708:
23419:
23395:
23364:
23324:
23300:
23147:
23123:
23033:
22984:
22961:
22930:
22910:
22861:
22838:
22807:
22495:
22476:
22444:
22425:
22390:
22371:
22336:
22317:
22081:
22062:
22030:
22011:
21976:
21957:
21922:
21903:
21611:
21592:
21560:
21541:
21506:
21487:
21228:
21209:
21177:
21158:
21123:
21104:
21069:
21050:
20811:
20792:
20760:
20741:
20706:
20687:
20652:
20633:
20504:
20485:
20450:
20431:
20320:
20296:
19848:
19288:
19261:
19237:
19213:
19186:
19162:
19003:
18984:
18939:
18920:
18888:
18856:
18844:
18835:
18552:
18534:
18434:
18397:
18326:
18254:
18227:
18157:
18117:
18040:
17982:. To transform a covariant vector
17718:
17712:
17632:
17562:
17455:
17311:
17290:
17269:
17248:
17225:
17204:
17183:
17162:
17139:
17118:
17097:
17076:
17053:
17032:
17011:
16990:
16845:homogeneous Lorentz transformation
16799:
16543:
16540:
15850:
15594:
14198:
14097:
14082:
14053:is simply the boost matrix in the
13997:
13982:
13892:
13525:. In a composition of boosts, the
11742:
11706:
11670:
11634:
11612:
11595:
11564:
11547:{\displaystyle {\mathcal {L}}_{+}}
11533:
11497:
11417:
11395:
11373:
11351:
11340:
11293:
11276:
11254:
11214:
11197:
11175:
11128:
11114:
11097:
11055:
11038:
11016:
10976:
10959:
10937:
10890:
10876:
10859:
10816:
10808:
10791:
10747:
10739:
10722:
10638:
10621:
10601:
10562:
10536:
10466:
10450:
10436:
10350:
10316:
10280:
10271:
10266:
10194:
10172:
10138:
10087:
10059:
8799:In general, given four quantities
8795:Transformation of other quantities
8327:(the velocity of X relative to F).
6887:Suppose there is a rod at rest in
6769:
6742:
6347:
6329:
6308:
6271:
6250:
6229:
6189:
6176:
6150:
6118:
6102:
6076:
2734:
2727:
2686:
2677:
2013:. Such transformations are called
1057:History of Lorentz transformations
700:grows without bound. The value of
105:Introduction to general relativity
14:
28202:
27669:Hamilton–Jacobi–Einstein equation
27248:Warp Special Relativity Simulator
27229:The Paradox of Special Relativity
27216:
25810:Misner, Thorne & Wheeler 1973
24149:transforms according to the rule
17425:, the above matrix expression is
15304:smoothly back into the group via
13681:for no relative motion/rotation:
9637:for details. For the case of the
9228:is the "timelike component", and
6585:the Lorentz transformations give
4193:{\displaystyle {\boldsymbol {v}}}
2997:A "stationary" observer in frame
2812:) which implies (by linearity of
1401:(lightlike separated events 1, 2)
190:Mathematics of general relativity
110:Mathematics of general relativity
28147:
28146:
26143:Lorentz, Hendrik Antoon (1904).
25895:"Vector Lorentz transformations"
25143:
25045:
24137:Transformation of general fields
23639:
23625:
23587:
23579:
23571:
23541:
23524:
23508:
23023:
23007:
22979:
22956:
22924:
22900:
22884:
22856:
22833:
22801:
22785:
22761:
22745:
22721:
22661:
22645:
22247:
22231:
21394:
21378:
20980:
20964:
20203:(Gaussian units, signature
17944:where, when viewed as matrices,
17683:-component object one may write
16704:
16637:
16115:
16099:
16060:Baker–Campbell–Hausdorff formula
16015:
15995:
15969:
15953:
15908:
15892:
15779:
15763:
15725:
15694:
15653:
15622:
15548:
15540:
15505:
15497:
15431:
15423:
15388:
15380:
14375:
14333:
13612:
13429:
13126:
13112:
13054:
13016:
12942:{\displaystyle B(-\mathbf {v} )}
12932:
11933:
11905:
11880:
11846:
11477:must be in the same subgroup as
10569:
10556:
10487:
10480:
10460:
10419:
9118:
9098:
9090:
9082:
9056:
9040:
9015:
9007:
8942:
8929:
8900:
8892:
8720:′′), in which case they must be
8681:
8671:
8663:
8642:
8629:
8597:
8585:
8575:
8547:
8539:
8512:
8466:
8458:
8432:
8397:
8378:
8356:
8342:
8211:
8203:
8157:
8149:
8078:
8056:
8048:
8036:
8005:
7992:
7961:
7946:
7806:
7789:
7781:
7773:
7747:
7731:
7699:
7691:
7604:
7596:
7588:
7577:
7563:
7552:
7544:
7536:
7519:
7510:and rejection give respectively
7393:
7371:
7355:
7341:
7313:
7285:
7271:
7199:
7181:
7168:
7151:
7136:
7127:
3694:is the "stationary" frame while
3003:defines events with coordinates
1022:The Lorentz transformation is a
998:postulates of special relativity
225:
224:
211:
34:
16:Family of linear transformations
26922:. Vol. 2. Addison Wesley.
26920:The Feynman Lectures on Physics
26892:. Vol. 1. Addison Wesley.
26890:The Feynman Lectures on Physics
26659:Introduction to Electrodynamics
26519:3000 Solved Problems in Physics
25931:Journal of Mathematical Physics
25868:A History of Special Relativity
25826:
25814:
25803:
25791:
25780:
25769:
25757:
25745:
25730:
25718:
25706:
25695:
25648:
25636:
25624:
25613:
25593:Relativity Made Relatively Easy
25583:
25572:
25561:
25549:
25536:
25524:
25459:Cottingham & Greenwood 2007
25311:
25262:
25228:
25023:
24987:
24958:
24937:
24133:describing two free electrons.
23789:by the bispinor representation
23079:refer to this last form as the
21425:For the electric field results
20254:The general transformation law
19982:
18705:
18296:and may in this notation write
16761:is improper orthochronous, and
16324:
16274:
15086:
14959:
14711:
14587:
14343:
13385:
13035:
11890:{\displaystyle B(\mathbf {v} )}
10637:
10620:
10616:
10435:
10343:
10339:
10213:, in this context known as the
10165:of all Lorentz transformations
10120:under a Lorentz transformation
9965:
9851:
8425:
8375:
7560:
7165:
2841:
2756:
2726:
2590:refers to the bilinear form of
2525:
2519:
2188:
1995:of numbers are added to events
1985:) is satisfied if an arbitrary
1900:
1398:
27476:Mass–energy equivalence (E=mc)
27334:
27147:Ernst, A.; Hsu, J.-P. (2001),
27118:The Classical Theory of Fields
26858:. Vol. 2 (4th ed.).
26852:The Classical Theory of Fields
26769:, Cambridge University Press,
26206:Foundations of Physics Letters
26034:Macrossan, Michael N. (1986),
25512:
25500:
25489:
25478:
25467:
25452:
25441:
25411:
25240:relativistic quantum mechanics
24919:
24896:Prandtl–Glauert transformation
24662:
24643:
24630:
24617:
24579:
24560:
24547:
24534:
24412:
24395:
24386:
24369:
24343:
24326:
24314:
24297:
24181:
24169:
24123:
24077:
24070:
24049:
24042:
23988:
23981:
23960:
23953:
23912:
23905:
23871:
23864:
23846:
23840:
23828:
23822:
23816:
23781:
23771:
23583:
23567:
23459:
23453:
23240:
23234:
23228:
22587:
22571:
22154:
22142:
21812:
21793:
21754:
21741:
21693:
21677:
21674:
21662:
21659:
21647:
21276:
21264:
20884:
20868:
20865:
20853:
20256:
20230:
20206:
19787:
19763:
19451:relative to F and the charge.
19358:
19048:
18832:
18767:
18681:
18669:
18666:
18654:
17806:, then the indices are called
17722:
17715:
16814:{\displaystyle X'=\Lambda X+C}
16559:
16547:
16536:
16533:
16521:
16437:
16425:
16351:
16325:
16301:
16275:
16254:
16228:
16221:of the Lorentz generators are
15869:
15853:
15735:
15707:
15704:
15676:
15663:
15635:
15632:
15604:
15553:
15536:
15533:
15515:
15509:
15493:
15472:
15464:
15436:
15419:
15416:
15398:
15392:
15376:
15355:
15347:
14355:
14347:
14310:
14302:
14195:
13911:
13895:
13886:
13878:
13869:
13861:
13441:
13433:
13397:
13389:
13366:are rotation parameters (e.g.
13130:
13122:
13116:
13108:
13058:
13050:
13020:
13012:
12936:
12925:
12725:
12712:
12699:
12687:
12673:
12645:
12574:
12562:
12513:
12501:
12458:
12446:
12375:
12363:
12327:
12315:
12266:
12254:
12183:
12171:
12128:
12116:
12080:
12068:
11937:
11929:
11884:
11876:
11850:
11842:
11753:
11717:
11681:
11645:
11623:
11570:
11508:
11428:
11406:
11384:
11362:
11299:
11265:
11220:
11186:
11131:
11125:
11061:
11027:
10982:
10948:
10893:
10887:
10797:
10728:
10617:
10353:
10347:
10340:
10319:
10313:
10202:{\displaystyle {\mathcal {L}}}
9526:relativistic quantum mechanics
9450:Electromagnetic four-potential
9094:
9078:
9075:
9063:
8294:relativistic velocity addition
8262:, which agrees with the range
8052:
8031:
8028:
8016:
7877:is always positive) to obtain
7785:
7769:
7766:
7754:
7600:
7584:
7548:
7532:
7453:is also used by some authors.
7362:
7336:
6907:, the rod moves with velocity
5062:
5050:
2750:
2738:
2720:
2698:
2692:
2674:
2668:
2656:
2516:
2494:
2488:
2476:
1891:
1858:
1846:
1813:
1801:
1768:
1756:
1723:
1693:
1666:
1654:
1627:
1615:
1588:
1576:
1549:
1383:
1356:
1344:
1317:
1305:
1278:
1266:
1239:
1209:second postulate of relativity
1114:FitzGerald–Lorentz contraction
279:are a six-parameter family of
1:
26856:Course of Theoretical Physics
25853:
25643:Chaichian & Hagedorn 1997
25485:O'Connor & Robertson 1996
25418:Rao, K. N. Srinivasa (1988).
15327:again when differentiated at
10219:. Also, the above expression
9635:relativistic angular momentum
7226:then the transformations are
3500:axes, zero relative velocity
1147:published what is now called
27031:. New York: W. A. Benjamin.
26790:(2nd ed.). Reading MA:
26767:Relativistic Quantum Physics
25737:Dennery & Krzywicki 2012
24912:
19830:electromagnetic field tensor
18499:acting on the column vector
17529:is any four-vector, then in
11912:{\displaystyle \mathbf {v} }
10147:{\displaystyle X'=\Lambda X}
8274:Transformation of velocities
7065:that cannot equal or exceed
5777:. From the relation between
5068:{\displaystyle 2(1+\gamma )}
4979:where the Lorentz factor is
3533:, so only subluminal speeds
2818:
2808:
2796:
2605:
2576:
2460:
2436:
1981:
1929:
1420:
335:-direction, is expressed as
7:
27491:Relativistic Doppler effect
27004:(2nd ed.). Cambridge:
25971:"Lost in Einstein's Shadow"
25900:American Journal of Physics
25858:
25661:American Journal of Physics
25655:Furry, W. H. (1955-11-01).
24949:inhomogeneous Lorentz group
24838:
24789:
19838:-direction. It is given by
19822:and the magnetic induction
18487:of the matrix representing
16861:For the notation used, see
16776:Inhomogeneous Lorentz group
13848:The set of transformations
13593:). The "axis-angle vector"
10710:(or non-orthochronous) LTs
10235:indefinite orthogonal group
10185:in this article is denoted
8782:relativistic Doppler effect
7861:(or simply the unit vector
5110:direction, the results are
3095:axes are parallel, and the
2832:inhomogeneous Lorentz group
992:, but always such that the
135:Introduction to gravitation
10:
28207:
27962:In computational physics:
27486:Relativity of simultaneity
27156:Chinese Journal of Physics
27067:Gourgoulhon, Eric (2013).
27006:Cambridge University Press
26459:Mathematics for Physicists
26435:Barut, Asim Orhan (1964).
26409:Cambridge University Press
25989:Darrigol, Olivier (2005),
23764:
19760:(SI units, signature
19404:
19398:
18591:
17604:Alternatively, one writes
16860:
16854:
16754:is improper antichronous,
16074:of all Lorentz generators
16056:the decomposition is given
12991:, the separate boosts are
10593:always so it follows that
9729:
9723:
8280:differential of a function
8277:
6966:
6703:Relativity of simultaneity
6596:, and vice versa, for any
2916:
2880:homogeneous transformation
1166:
1061:Many physicists—including
1054:
1050:
28144:
27976:
27841:
27813:
27799:Lense–Thirring precession
27682:
27631:
27593:
27572:
27561:
27519:
27463:
27447:
27389:
27381:Doubly special relativity
27353:
27342:
27073:. Springer. p. 213.
26818:Classical Electrodynamics
26699:Weinberg, Steven (1972).
26332:Archiv für Elektrotechnik
26101:Einstein, Albert (1905),
25885:
25875:Brown, Harvey R. (2003),
25776:Grant & Phillips 2008
25590:Andrew M. Steane (2012).
25568:Young & Freedman 2008
25437:Equation 6-3.24, page 210
25272:", examples are position
24886:Algebra of physical space
24881:Velocity-addition formula
18612:, then a linear operator
17978:. This is referred to as
16054:. If, on the other hand,
13566:axis-angle representation
13374:, etc.). The rotation in
12978:is boosted with velocity
12959:is boosted with velocity
11768:) do not form subgroups.
9752:Homogeneous Lorentz group
9476:Magnetic vector potential
8871:, a relation of the form
8284:velocity addition formula
6421:These transformations on
3726:. The only difference is
2924:Coordinate transformation
2872:axis–angle representation
165:Derivations of relativity
27659:Post-Newtonian formalism
27649:Einstein field equations
27585:Mathematical formulation
27409:Hyperbolic orthogonality
26978:(2nd ed.). Dallas:
26427:
26131:10.1002/andp.19053221004
25579:Forshaw & Smith 2009
25448:Forshaw & Smith 2009
25405:
24993:For two square matrices
24980:respectively, e.g., the
19814:Gaussian system of units
16772:is proper antichronous.
16738:involution (mathematics)
16588:Improper transformations
15321:; this curve will yield
14286:has been used (see also
14059:derivative of the matrix
13535:, and gives rise to the
10178:{\displaystyle \Lambda }
9720:Mathematical formulation
7035:Standard configuration.
6693:correspondence principle
6418:coordinates, and so on.
5850:direction with rapidity
5310:) is a parameter called
5123:direction with rapidity
4178:When the boost velocity
2611:relativistic dot product
1157:inertial reference frame
715:Expressing the speed as
115:Einstein field equations
27370:Galilean transformation
27361:Principle of relativity
27292:Lorentz Frames Animated
26980:Oxford University Press
26946:Wheeler, John Archibald
26676:Hall, Brian C. (2003).
26538:Dynamics and Relativity
26462:. Courier Corporation.
26315:10.1023/A:1003653302643
26018:10.1007/3-7643-7436-5_1
25961:2027/mdp.39015095220474
25893:Cushing, J. T. (1967).
24891:Relativistic aberration
24876:Principle of relativity
24856:Galilean transformation
23246:same event in spacetime
23238:that immediately yield
23083:view as opposed to the
16836:Poincaré transformation
16461:of numbers, and with a
16066:The Lie algebra so(3,1)
13679:identity transformation
11866:where the boost matrix
10671:is the Lorentz factor.
9633:related to boosts, see
9216:collectively make up a
6691:in accordance with the
6616:Galilean transformation
6218:with inverse relations
5104:. For the boost in the
3782:event with coordinates
3650:The inverse relations (
3527:cannot equal or exceed
3191:event with coordinates
3082:axes are parallel, the
2885:The full Lorentz group
2848:Lorentz transformations
1514:. The interval between
1153:principle of relativity
970:absolute space and time
962:Galilean transformation
693:{\displaystyle \gamma }
277:Lorentz transformations
80:Lorentz transformations
27455:Lorentz transformation
26974:(2006) . "Chapter 9".
26362:Foundations of Physics
26292:Foundations of Physics
26253:Foundations of Physics
25969:Rothman, Tony (2006),
25321:, the position vector
25220:
25122:
24777:
24114:
23746:
23672:
23469:
23188:
23049:
22687:
21417:
20361:
20245:
19802:
19401:Electromagnetic tensor
19396:
19349:
19039:
18781:are a four-vectors in
18758:
18622:may be defined on the
18568:
18477:
18359:
18290:
18193:
18086:
17938:
17872:
17756:
17671:
17596:
17500:respectively, and the
17488:
17407:
16815:
16720:
16653:
16569:
16444:
16375:
16126:
16028:
15921:
15797:
15570:
15449:
15217:
14397:The axis-angle vector
14389:
14276:
14146:
14032:
13921:
13620:
13460:
13144:
13068:
12943:
12908:
12846:
12766:
11913:
11891:
11860:
11815:
11790:
11772:Proper transformations
11762:
11726:
11690:
11654:
11579:
11548:
11517:
11437:
11308:
11240:Improper orthochronous
11229:
11150:
11070:
10991:
10912:
10832:
10766:
10651:
10577:
10507:
10369:
10287:
10203:
10179:
10148:
10110:
10020:
9140:
8954:
8694:
8491:
8328:
8242:
8178:
8094:
7822:
7612:
7411:
7220:
7040:
7039:Inverse configuration.
7013:to move with velocity
6994:to move with velocity
6963:Vector transformations
6792:
6685:
6375:
6212:
6028:
5753:
5705:
5580:
5506:
5427:
5291:
5091:
5069:
5026:
4970:
4232:
4194:
4149:and the definition of
4137:
3974:
3772:, then an observer in
3645:passive transformation
3457:
3370:
3180:, then an observer in
3147:standard configuration
2994:
2784:
2564:
2424:
2015:spacetime translations
1917:
1408:
980:may measure different
910:
745:
694:
646:
498:
325:
27923:Weyl−Lewis−Papapetrou
27664:Raychaudhuri equation
27603:Equivalence principle
27122:Butterworth–Heinemann
26860:Butterworth–Heinemann
26822:John Wiley & Sons
26062:10.1093/bjps/37.2.232
25221:
25123:
24851:Electromagnetic field
24812:Wigner's little group
24778:
24115:
23747:
23673:
23470:
23189:
23050:
22688:
21418:
20362:
20246:
19812:. In relativity, the
19803:
19431:electromagnetic force
19405:Further information:
19394:
19350:
19040:
18759:
18569:
18478:
18360:
18291:
18194:
18087:
17939:
17873:
17757:
17672:
17597:
17531:tensor index notation
17489:
17423:tensor index notation
17408:
16869:Contravariant vectors
16816:
16721:
16654:
16570:
16445:
16376:
16219:commutation relations
16127:
16029:
15922:
15798:
15571:
15450:
15218:
14390:
14277:
14147:
14033:
13922:
13844:The Lie group SO(3,1)
13733:is symmetric (equals
13621:
13461:
13145:
13069:
12944:
12909:
12847:
12767:
11914:
11892:
11861:
11816:
11791:
11763:
11727:
11691:
11655:
11580:
11549:
11518:
11438:
11309:
11230:
11161:Improper antichronous
11151:
11071:
10992:
10913:
10833:
10767:
10652:
10578:
10508:
10370:
10288:
10247:matrix multiplication
10204:
10180:
10149:
10111:
10021:
9730:Further information:
9148:The decomposition of
9141:
8955:
8695:
8492:
8291:
8278:Further information:
8243:
8179:
8095:
7881:Inverse Lorentz boost
7823:
7613:
7425:. The Lorentz factor
7412:
7221:
6981:An observer in frame
6980:
6967:Further information:
6793:
6686:
6552:Physical implications
6376:
6213:
6029:
5841:Inverse Lorentz boost
5754:
5706:
5581:
5507:
5428:
5292:
5092:
5070:
5027:
4971:
4233:
4195:
4138:
3975:
3786:Inverse Lorentz boost
3625:axes, because of the
3616:active transformation
3458:
3371:
2933:
2917:Further information:
2785:
2565:
2448:polarization identity
2425:
1918:
1409:
1189:Cartesian coordinates
1024:linear transformation
988:, and even different
943:Cartesian coordinates
911:
746:
706:must be smaller than
695:
662:is much smaller than
647:
499:
326:
175:Differential geometry
75:Equivalence principle
28176:Mathematical physics
27964:Numerical relativity
27805:pulsar timing arrays
27295:from John de Pillis.
27094:. IoP. p. 239.
27023:Sard, R. D. (1970).
27002:Quantum Field Theory
26517:Halpern, A. (1988).
25546:, pp. 1504–1508
25244:quantum field theory
25131:
25033:
24901:Split-complex number
24158:
24147:quantum field theory
23800:
23684:
23499:
23252:
23091:
22712:
21429:
20373:
20264:
19842:
19477:
19090:
18820:
18651:
18514:
18371:
18300:
18221:
18098:
18001:
17892:
17826:
17687:
17608:
17536:
17502:summation convention
17429:
16877:
16785:
16666:
16599:
16500:
16412:
16225:
16078:
15933:
15847:
15587:
15458:
15341:
15244:which correspond to
14459:
14403:and rapidity vector
14296:
14171:
14072:
13950:
13852:
13743:is nonsymmetric but
13597:
13564:In this article the
13531:matrix is named the
13382:
13350:composite velocities
13091:
12995:
12971:, and another frame
12919:
12856:
12778:
11923:
11901:
11870:
11825:
11800:
11780:
11736:
11700:
11664:
11589:
11558:
11527:
11491:
11335:
11248:
11169:
11091:
11010:
11002:Proper orthochronous
10931:
10853:
10785:
10716:
10597:
10532:
10381:
10301:
10256:
10189:
10169:
10124:
10038:
9769:
9732:Matrix (mathematics)
8964:
8875:
8507:
8338:
8188:
8134:
7905:
7871:since the magnitude
7648:
7514:
7230:
7123:
6739:
6621:
6222:
6069:
5862:
5803:axes, zero rapidity
5789:, positive rapidity
5717:
5610:
5526:
5457:
5375:
5135:
5102:hyperbolic functions
5079:
5044:
4983:
4244:
4204:
4182:
4021:
3802:
3732:moves with velocity
3405:
3211:
3028:moves with velocity
2653:
2473:
2030:
1531:
1226:
755:
719:
684:
581:
339:
312:
153:Relevant mathematics
27856:Friedmann equations
27750:Hulse–Taylor binary
27712:Gravitational waves
27608:Riemannian geometry
27434:Proper acceleration
27419:Maxwell's equations
27365:Galilean relativity
27266:MinutePhysics video
27168:2001ChJPh..39..211E
26788:Classical Mechanics
26375:1992FoPhL...5..443M
26266:1989FoPh...19.1385U
26218:1988FoPhL...1...57U
26137:English translation
26122:1905AnP...322..891E
26040:Br. J. Philos. Sci.
26010:2006eins.book....1D
25944:1962JMP.....3.1116M
25913:1967AmJPh..35..858C
25673:1955AmJPh..23..517F
25609:Extract of page 124
24861:Hyperbolic rotation
24747:
24708:
24634:
24604:
24551:
24521:
24498:
24485:
24455:
24440:
23779:representation. In
19341:
19155:
18578:dual representation
18201:That is, it is the
17766:is the appropriate
16400:cyclic permutations
15242:rotation generators
12839:
12821:
12803:
12594:
12347:
12100:
11757:
11721:
11685:
11649:
11627:
11512:
11449:of a group must be
11432:
11410:
11388:
11366:
11269:
11190:
11031:
10952:
10923:Proper antichronous
9765:as a square matrix
9246:are the following:
7383:
7325:
7211:
7193:
5324:hyperbolic rotation
4887:
4678:
4469:
2973:moves at velocity −
2415:
2402:
2386:
2373:
2357:
2344:
2328:
2315:
1889:
1873:
1844:
1828:
1799:
1783:
1754:
1738:
1217:) it follows that:
1110:Maxwell's equations
1095:luminiferous aether
1083:Maxwell's equations
1041:hyperbolic rotation
990:orderings of events
974:Galilean relativity
968:, which assumes an
960:They supersede the
22:Part of a series on
28186:Coordinate systems
28171:Special relativity
27905:Reissner–Nordström
27823:Brans–Dicke theory
27654:Linearized gravity
27481:Length contraction
27399:Frame of reference
27376:Special relativity
27240:2011-08-29 at the
26938:Misner, Charles W.
26383:10.1007/bf00690425
26344:10.1007/bf01574845
26274:10.1007/BF00732759
26226:10.1007/BF00661317
26110:Annalen der Physik
25998:Séminaire Poincaré
25978:American Scientist
25558:, pp. 891–921
25306:linear combination
25216:
25118:
24831:representation of
24773:
24771:
24735:
24696:
24592:
24587:
24509:
24504:
24503:
24486:
24473:
24441:
24426:
24110:
24108:
23742:
23668:
23666:
23465:
23184:
23045:
23043:
22683:
22681:
21413:
21411:
20357:
20241:
20195:
19973:
19798:
19752:
19397:
19345:
19309:
19093:
19035:
18754:
18564:
18473:
18355:
18286:
18213:-component of the
18189:
18082:
17950:is the inverse of
17934:
17868:
17752:
17667:
17592:
17484:
17403:
17397:
17329:
16970:
16851:Tensor formulation
16811:
16716:
16710:
16649:
16643:
16578:matrix exponential
16565:
16440:
16381:where the bracket
16371:
16122:
16024:
15917:
15793:
15791:
15583:level the product
15566:
15445:
15213:
15211:
15203:
15076:
14949:
14825:
14701:
14577:
14385:
14290:). More generally
14272:
14202:
14165:matrix exponential
14142:
14028:
13917:
13747:(transpose equals
13616:
13485:simple to connect
13456:
13446:
13222:is collinear with
13140:
13064:
12965:relative to frame
12939:
12904:
12842:
12825:
12807:
12789:
12762:
12753:
12749:
12610:
12606:
12580:
12552:
12497:
12414:
12359:
12333:
12305:
12222:
12167:
12112:
12086:
11909:
11887:
11856:
11814:{\displaystyle X'}
11811:
11786:
11758:
11739:
11722:
11703:
11686:
11667:
11650:
11631:
11609:
11575:
11544:
11513:
11494:
11433:
11414:
11392:
11370:
11348:
11304:
11251:
11225:
11172:
11146:
11066:
11013:
10987:
10934:
10908:
10828:
10762:
10647:
10573:
10503:
10493:
10425:
10365:
10283:
10199:
10175:
10144:
10106:
10016:
10010:
9955:
9841:
9455:Electric potential
9136:
9134:
8950:
8774:stellar aberration
8690:
8487:
8482:
8329:
8238:
8174:
8090:
8088:
7818:
7816:
7608:
7407:
7405:
7369:
7311:
7216:
7197:
7179:
7041:
6883:Length contraction
6788:
6681:
6679:
6412:for two values of
6371:
6369:
6208:
6206:
6024:
6022:
5749:
5701:
5699:
5576:
5502:
5423:
5287:
5285:
5087:
5065:
5022:
4966:
4957:
4890:
4873:
4664:
4455:
4328:
4228:
4190:
4133:
4131:
3970:
3968:
3453:
3366:
3364:
3156:If an observer in
2995:
2946:moves at velocity
2899:temporal inversion
2780:
2560:
2420:
2418:
2403:
2390:
2374:
2361:
2345:
2332:
2316:
2303:
1913:
1911:
1877:
1861:
1832:
1816:
1787:
1771:
1742:
1726:
1436:spacetime interval
1404:
1149:special relativity
1141:mathematical group
1091:spherical symmetry
1036:spacetime interval
1017:special relativity
906:
904:
741:
690:
642:
494:
492:
324:{\displaystyle v,}
321:
218:Physics portal
195:Spacetime topology
170:Spacetime diagrams
98:General relativity
70:Spacetime manifold
63:Spacetime concepts
52:General relativity
47:Special relativity
28158:
28157:
27972:
27971:
27951:Ozsváth–Schücking
27557:
27556:
27539:Minkowski diagram
27496:Thomas precession
27439:Relativistic mass
27286:Interactive graph
27280:Desmos (graphing)
27276:Interactive graph
27193:978-0-534-40896-1
27101:978-0-7503-0408-5
27080:978-3-642-37276-6
26989:978-0-19-856732-5
26963:978-0-7167-0344-0
26952:. San Francisco:
26929:978-0-201-02117-2
26899:978-0-201-02117-2
26862:. pp. 9–12.
26835:978-0-471-43132-9
26801:978-0-201-02918-5
26776:978-0-521-76726-2
26755:978-0-521-67053-1
26734:978-0-19-852682-7
26712:978-0-471-92567-5
26691:978-0-387-40122-5
26668:978-81-7758-293-2
26646:978-0-471-92712-9
26627:978-0-8053-8732-2
26603:978-0-7167-0344-0
26573:978-0-7167-0336-5
26564:Spacetime Physics
26547:978-0-470-01460-8
26528:978-0-07-025734-4
26509:978-0-321-50130-1
26490:978-1-139-46221-1
26469:978-0-486-15712-2
26448:978-0-486-64038-9
26418:978-0-521-55001-7
26260:(11): 1385–1396.
26196:978-0-517-88441-6
26027:978-3-7643-7435-8
25952:10.1063/1.1703854
25921:10.1119/1.1974267
25681:10.1119/1.1934085
25603:978-0-19-966286-9
25533:, pp. 232–34
25431:978-0-470-21044-4
25296:of numbers (e.g.
25236:quantum mechanics
24982:Clifford algebras
24797:
24796:
24468:
24461:
24460:
23759:Maxwell equations
23654:
20236:
20204:
19793:
19761:
19706:
19650:
19594:
19570:
19545:
19520:
19083:. It is given by
18493:inverse transpose
17961:. As it happens,
17820:lowering an index
17814:Covariant vectors
17780:matrix for every
14223:
14187:
14104:
14004:
13537:Thomas precession
12902:
12901:
12899:
12840:
12748:
12728:
12715:
12676:
12648:
12605:
12551:
12496:
12413:
12358:
12304:
12221:
12166:
12111:
11789:{\displaystyle X}
11319:
11318:
10243:matrix Lie groups
9487:
9486:
9352:angular frequency
9126:
9023:
8778:Fizeau experiment
8655:
8619:
8591:
8565:
8562:
8485:
8484:
8481:
8419:
8369:
7976:
7714:
7508:vector projection
7300:
7052:with a magnitude
6973:vector projection
6786:
6293:
6135:
5570:
5569:
5496:
5453:. The definition
5342:Minkowski diagram
5020:
4871:
4818:
4771:
4704:
4662:
4609:
4542:
4495:
4453:
3983:and the value of
3863:
3634:coordinate system
3589:faster than light
3587:is infinite, and
3451:
3450:
3448:
3267:
3162:records an event
2895:spatial inversion
2856:inverse functions
2804:
2803:
2584:
2583:
2523:
2444:
2443:
2186:
1937:
1936:
1904:
1903:(all events 1, 2)
1428:
1427:
1402:
1067:George FitzGerald
966:Newtonian physics
736:
627:
625:
395:
269:
268:
128:Classical gravity
28198:
28150:
28149:
27933:van Stockum dust
27705:Two-body problem
27623:Mach's principle
27570:
27569:
27511:Terrell rotation
27351:
27350:
27329:
27322:
27315:
27306:
27305:
27256:
27212:
27196:
27178:
27176:
27170:, archived from
27153:
27135:
27105:
27084:
27063:
27042:
27030:
27019:
26993:
26967:
26933:
26903:
26873:
26839:
26820:(2nd ed.).
26805:
26779:
26758:
26737:
26716:
26695:
26672:
26655:Griffiths, D. J.
26650:
26637:Electromagnetism
26631:
26607:
26577:
26551:
26532:
26513:
26494:
26473:
26452:
26422:
26394:
26355:
26326:
26308:
26285:
26245:
26200:
26178:
26176:
26175:
26158:
26148:
26134:
26133:
26107:
26097:
26087:
26075:
26074:
26073:
26064:, archived from
26055:
26030:
25995:
25985:
25975:
25965:
25963:
25938:(6): 1116–1129.
25924:
25881:
25871:
25848:
25847:
25845:
25844:
25830:
25824:
25818:
25812:
25807:
25801:
25795:
25789:
25784:
25778:
25773:
25767:
25761:
25755:
25749:
25743:
25734:
25728:
25722:
25716:
25715:, pp. 55–58
25710:
25704:
25699:
25693:
25692:
25652:
25646:
25640:
25634:
25628:
25622:
25617:
25611:
25607:
25587:
25581:
25576:
25570:
25565:
25559:
25553:
25547:
25540:
25534:
25528:
25522:
25516:
25510:
25509:, pp. 112f.
25504:
25498:
25493:
25487:
25482:
25476:
25471:
25465:
25456:
25450:
25445:
25439:
25435:
25415:
25399:
25397:
25391:
25362:
25315:
25309:
25283:
25277:
25270:Euclidean vector
25266:
25260:
25258:
25257:
25256:
25232:
25226:
25225:
25223:
25222:
25217:
25215:
25214:
25205:
25204:
25192:
25191:
25182:
25181:
25169:
25168:
25159:
25158:
25146:
25138:
25127:
25125:
25124:
25119:
25117:
25116:
25107:
25106:
25094:
25093:
25084:
25083:
25071:
25070:
25061:
25060:
25048:
25040:
25027:
25021:
25020:
25004:
24998:
24991:
24985:
24979:
24975:
24971:
24967:
24962:
24956:
24941:
24935:
24923:
24906:Gyrovector space
24834:
24830:
24828:
24819:
24809:
24791:
24782:
24780:
24779:
24774:
24772:
24765:
24764:
24757:
24756:
24743:
24734:
24733:
24718:
24717:
24704:
24695:
24694:
24677:
24673:
24669:
24665:
24661:
24660:
24633:
24629:
24628:
24615:
24614:
24613:
24600:
24586:
24582:
24578:
24577:
24550:
24546:
24545:
24532:
24531:
24530:
24517:
24502:
24494:
24481:
24462:
24459:
24454:
24449:
24439:
24434:
24424:
24420:
24419:
24410:
24409:
24394:
24393:
24384:
24383:
24367:
24366:
24364:
24363:
24362:
24358:
24351:
24350:
24341:
24340:
24322:
24321:
24312:
24311:
24291:
24290:
24268:
24259:
24258:
24251:
24250:
24241:
24240:
24231:
24230:
24218:
24217:
24208:
24207:
24198:
24197:
24164:
24152:
24125:
24122:
24119:
24117:
24116:
24111:
24109:
24105:
24104:
24092:
24091:
24086:
24085:
24084:
24064:
24063:
24058:
24057:
24056:
24030:
24026:
24025:
24013:
24012:
24003:
24002:
23997:
23996:
23995:
23975:
23974:
23969:
23968:
23967:
23941:
23937:
23936:
23927:
23926:
23921:
23920:
23919:
23896:
23895:
23886:
23885:
23880:
23879:
23878:
23792:
23788:
23751:
23749:
23748:
23743:
23738:
23737:
23728:
23727:
23722:
23721:
23720:
23719:
23701:
23700:
23699:
23677:
23675:
23674:
23669:
23667:
23660:
23656:
23655:
23653:
23652:
23643:
23642:
23633:
23628:
23602:
23590:
23582:
23574:
23566:
23562:
23544:
23527:
23515:
23511:
23494:
23485:
23474:
23472:
23471:
23466:
23452:
23451:
23439:
23438:
23433:
23432:
23431:
23430:
23415:
23414:
23409:
23408:
23407:
23406:
23388:
23384:
23383:
23375:
23374:
23357:
23356:
23344:
23343:
23338:
23337:
23336:
23335:
23320:
23319:
23314:
23313:
23312:
23311:
23293:
23289:
23277:
23276:
23275:
23267:
23225:
23219:
23197:
23193:
23191:
23190:
23185:
23180:
23179:
23167:
23166:
23161:
23160:
23159:
23158:
23143:
23142:
23137:
23136:
23135:
23134:
23116:
23115:
23114:
23106:
23082:
23074:
23073:
23072:
23066:
23054:
23052:
23051:
23046:
23044:
23037:
23036:
23031:
23027:
23026:
23018:
23010:
22993:
22989:
22988:
22987:
22982:
22973:
22965:
22964:
22959:
22938:
22937:
22936:
22927:
22914:
22913:
22908:
22904:
22903:
22895:
22887:
22870:
22866:
22865:
22864:
22859:
22850:
22842:
22841:
22836:
22815:
22814:
22813:
22804:
22794:
22793:
22788:
22775:
22774:
22773:
22764:
22754:
22753:
22748:
22735:
22734:
22733:
22724:
22707:
22692:
22690:
22689:
22684:
22682:
22675:
22674:
22669:
22665:
22664:
22656:
22648:
22628:
22624:
22623:
22605:
22604:
22586:
22585:
22552:
22551:
22524:
22520:
22519:
22510:
22509:
22504:
22503:
22502:
22491:
22490:
22485:
22484:
22483:
22469:
22468:
22459:
22458:
22453:
22452:
22451:
22440:
22439:
22434:
22433:
22432:
22418:
22417:
22405:
22404:
22399:
22398:
22397:
22386:
22385:
22380:
22379:
22378:
22364:
22363:
22351:
22350:
22345:
22344:
22343:
22332:
22331:
22326:
22325:
22324:
22310:
22309:
22308:
22300:
22280:
22279:
22278:
22261:
22260:
22255:
22251:
22250:
22242:
22234:
22214:
22210:
22209:
22191:
22190:
22175:
22174:
22138:
22137:
22110:
22106:
22105:
22096:
22095:
22090:
22089:
22088:
22077:
22076:
22071:
22070:
22069:
22055:
22054:
22045:
22044:
22039:
22038:
22037:
22026:
22025:
22020:
22019:
22018:
22004:
22003:
21991:
21990:
21985:
21984:
21983:
21972:
21971:
21966:
21965:
21964:
21950:
21949:
21937:
21936:
21931:
21930:
21929:
21918:
21917:
21912:
21911:
21910:
21896:
21895:
21894:
21886:
21866:
21865:
21864:
21844:
21843:
21828:
21824:
21823:
21811:
21810:
21792:
21791:
21779:
21778:
21769:
21768:
21753:
21752:
21740:
21739:
21730:
21729:
21714:
21713:
21692:
21691:
21640:
21636:
21635:
21626:
21625:
21620:
21619:
21618:
21607:
21606:
21601:
21600:
21599:
21585:
21584:
21575:
21574:
21569:
21568:
21567:
21556:
21555:
21550:
21549:
21548:
21534:
21533:
21521:
21520:
21515:
21514:
21513:
21502:
21501:
21496:
21495:
21494:
21480:
21479:
21478:
21470:
21450:
21449:
21448:
21422:
21420:
21419:
21414:
21412:
21408:
21407:
21402:
21398:
21397:
21389:
21381:
21361:
21357:
21356:
21338:
21337:
21322:
21321:
21297:
21296:
21257:
21253:
21252:
21243:
21242:
21237:
21236:
21235:
21224:
21223:
21218:
21217:
21216:
21202:
21201:
21192:
21191:
21186:
21185:
21184:
21173:
21172:
21167:
21166:
21165:
21151:
21150:
21138:
21137:
21132:
21131:
21130:
21119:
21118:
21113:
21112:
21111:
21097:
21096:
21084:
21083:
21078:
21077:
21076:
21065:
21064:
21059:
21058:
21057:
21043:
21042:
21041:
21033:
21013:
21012:
21011:
20994:
20993:
20988:
20984:
20983:
20975:
20967:
20947:
20943:
20942:
20924:
20923:
20908:
20907:
20883:
20882:
20840:
20836:
20835:
20826:
20825:
20820:
20819:
20818:
20807:
20806:
20801:
20800:
20799:
20785:
20784:
20775:
20774:
20769:
20768:
20767:
20756:
20755:
20750:
20749:
20748:
20734:
20733:
20721:
20720:
20715:
20714:
20713:
20702:
20701:
20696:
20695:
20694:
20680:
20679:
20667:
20666:
20661:
20660:
20659:
20648:
20647:
20642:
20641:
20640:
20626:
20625:
20624:
20616:
20596:
20595:
20594:
20574:
20573:
20558:
20554:
20553:
20529:
20528:
20519:
20518:
20513:
20512:
20511:
20500:
20499:
20494:
20493:
20492:
20478:
20477:
20465:
20464:
20459:
20458:
20457:
20446:
20445:
20440:
20439:
20438:
20424:
20423:
20422:
20414:
20394:
20393:
20392:
20366:
20364:
20363:
20358:
20353:
20352:
20340:
20339:
20334:
20333:
20332:
20331:
20316:
20315:
20310:
20309:
20308:
20307:
20289:
20288:
20287:
20279:
20250:
20248:
20247:
20242:
20237:
20234:
20205:
20202:
20200:
20199:
20187:
20186:
20172:
20171:
20160:
20159:
20143:
20142:
20126:
20125:
20111:
20110:
20094:
20093:
20079:
20078:
20062:
20061:
20045:
20044:
20033:
20032:
20021:
20020:
19995:
19994:
19978:
19977:
19863:
19862:
19857:
19856:
19855:
19837:
19827:
19821:
19807:
19805:
19804:
19799:
19794:
19791:
19762:
19759:
19757:
19756:
19744:
19743:
19732:
19731:
19717:
19716:
19707:
19699:
19693:
19692:
19673:
19672:
19661:
19660:
19651:
19643:
19637:
19636:
19625:
19624:
19605:
19604:
19595:
19587:
19581:
19580:
19571:
19563:
19556:
19555:
19546:
19538:
19531:
19530:
19521:
19513:
19492:
19491:
19464:electric current
19461:
19450:
19435:electric charges
19428:
19419:
19382:
19378:
19372:
19360:
19357:
19354:
19352:
19351:
19346:
19340:
19326:
19308:
19307:
19302:
19301:
19300:
19299:
19281:
19280:
19275:
19274:
19273:
19272:
19257:
19256:
19251:
19250:
19249:
19248:
19233:
19232:
19227:
19226:
19225:
19224:
19206:
19205:
19200:
19199:
19198:
19197:
19182:
19181:
19176:
19175:
19174:
19173:
19154:
19153:
19142:
19134:
19125:
19124:
19113:
19105:
19082:
19072:
19063:
19050:
19047:
19044:
19042:
19041:
19036:
19031:
19030:
19018:
19017:
19012:
19011:
19010:
18999:
18998:
18993:
18992:
18991:
18977:
18976:
18964:
18963:
18954:
18953:
18948:
18947:
18946:
18935:
18934:
18929:
18928:
18927:
18913:
18912:
18903:
18902:
18897:
18896:
18895:
18881:
18880:
18871:
18870:
18865:
18864:
18863:
18812:
18784:
18780:
18776:
18769:
18766:
18763:
18761:
18760:
18755:
18643:
18633:
18629:
18621:
18611:
18607:
18603:
18599:
18587:
18583:
18573:
18571:
18570:
18565:
18557:
18556:
18555:
18549:
18545:
18544:
18524:
18509:
18498:
18490:
18483:is running over
18482:
18480:
18479:
18474:
18472:
18471:
18462:
18461:
18456:
18455:
18454:
18449:
18445:
18444:
18422:
18421:
18412:
18411:
18406:
18405:
18404:
18390:
18389:
18384:
18383:
18364:
18362:
18361:
18356:
18351:
18350:
18341:
18340:
18335:
18334:
18333:
18319:
18318:
18313:
18312:
18295:
18293:
18292:
18287:
18282:
18281:
18276:
18275:
18274:
18269:
18265:
18264:
18242:
18241:
18236:
18235:
18234:
18212:
18198:
18196:
18195:
18190:
18185:
18184:
18179:
18178:
18177:
18172:
18168:
18167:
18145:
18144:
18132:
18131:
18126:
18125:
18124:
18113:
18112:
18091:
18089:
18088:
18083:
18078:
18077:
18068:
18067:
18055:
18054:
18049:
18048:
18047:
18036:
18035:
18020:
18019:
18014:
18013:
17996:
17992:
17980:raising an index
17977:
17976:
17960:
17949:
17943:
17941:
17940:
17935:
17930:
17929:
17920:
17919:
17904:
17903:
17883:
17877:
17875:
17874:
17869:
17864:
17863:
17854:
17853:
17838:
17837:
17801:
17797:
17791:
17783:
17779:
17765:
17761:
17759:
17758:
17753:
17747:
17746:
17737:
17736:
17731:
17730:
17729:
17706:
17705:
17700:
17699:
17682:
17676:
17674:
17673:
17668:
17662:
17661:
17652:
17651:
17646:
17645:
17644:
17643:
17625:
17624:
17623:
17601:
17599:
17598:
17593:
17587:
17586:
17577:
17576:
17571:
17570:
17569:
17555:
17554:
17549:
17548:
17528:
17493:
17491:
17490:
17485:
17480:
17479:
17470:
17469:
17464:
17463:
17462:
17448:
17447:
17442:
17441:
17412:
17410:
17409:
17404:
17402:
17401:
17394:
17393:
17380:
17379:
17366:
17365:
17352:
17351:
17334:
17333:
17326:
17325:
17320:
17319:
17318:
17305:
17304:
17299:
17298:
17297:
17284:
17283:
17278:
17277:
17276:
17263:
17262:
17257:
17256:
17255:
17240:
17239:
17234:
17233:
17232:
17219:
17218:
17213:
17212:
17211:
17198:
17197:
17192:
17191:
17190:
17177:
17176:
17171:
17170:
17169:
17154:
17153:
17148:
17147:
17146:
17133:
17132:
17127:
17126:
17125:
17112:
17111:
17106:
17105:
17104:
17091:
17090:
17085:
17084:
17083:
17068:
17067:
17062:
17061:
17060:
17047:
17046:
17041:
17040:
17039:
17026:
17025:
17020:
17019:
17018:
17005:
17004:
16999:
16998:
16997:
16975:
16974:
16967:
16966:
16961:
16960:
16946:
16945:
16940:
16939:
16925:
16924:
16919:
16918:
16904:
16903:
16898:
16897:
16829:≠ 0, this is an
16820:
16818:
16817:
16812:
16795:
16771:
16760:
16753:
16746:
16731:
16725:
16723:
16722:
16717:
16715:
16714:
16707:
16658:
16656:
16655:
16650:
16648:
16647:
16640:
16594:parity inversion
16574:
16572:
16571:
16566:
16546:
16520:
16519:
16463:binary operation
16449:
16447:
16446:
16441:
16424:
16423:
16392:is known as the
16391:
16380:
16378:
16377:
16372:
16366:
16365:
16350:
16349:
16337:
16336:
16319:
16318:
16300:
16299:
16287:
16286:
16269:
16268:
16253:
16252:
16240:
16239:
16209:
16172:
16131:
16129:
16128:
16123:
16118:
16110:
16102:
16094:
16049:
16043:
16033:
16031:
16030:
16025:
16020:
16019:
16018:
16010:
16000:
15999:
15998:
15990:
15974:
15973:
15972:
15964:
15956:
15948:
15926:
15924:
15923:
15918:
15913:
15912:
15911:
15903:
15895:
15887:
15868:
15860:
15842:
15822:
15802:
15800:
15799:
15794:
15792:
15782:
15774:
15766:
15758:
15741:
15728:
15720:
15697:
15689:
15669:
15656:
15648:
15625:
15617:
15575:
15573:
15572:
15567:
15561:
15560:
15551:
15543:
15508:
15500:
15471:
15454:
15452:
15451:
15446:
15444:
15443:
15434:
15426:
15391:
15383:
15354:
15333:
15326:
15320:
15314:
15303:
15297:
15291:
15284:
15278:
15268:
15256:boost generators
15253:
15246:angular momentum
15239:
15233:
15222:
15220:
15219:
15214:
15212:
15208:
15207:
15098:
15097:
15081:
15080:
14971:
14970:
14954:
14953:
14844:
14843:
14830:
14829:
14723:
14722:
14706:
14705:
14599:
14598:
14582:
14581:
14475:
14474:
14452:
14430:
14408:
14402:
14394:
14392:
14391:
14386:
14380:
14379:
14378:
14370:
14354:
14338:
14337:
14336:
14328:
14309:
14281:
14279:
14278:
14273:
14271:
14270:
14269:
14268:
14245:
14244:
14239:
14235:
14234:
14233:
14224:
14216:
14201:
14183:
14182:
14162:
14151:
14149:
14148:
14143:
14137:
14136:
14121:
14120:
14109:
14105:
14103:
14095:
14094:
14093:
14080:
14067:
14052:
14043:
14037:
14035:
14034:
14029:
14021:
14020:
14009:
14005:
14003:
13995:
13994:
13993:
13980:
13962:
13961:
13945:
13937:Taylor expansion
13926:
13924:
13923:
13918:
13910:
13902:
13885:
13868:
13839:
13828:
13816:
13796:
13776:
13760:
13742:
13732:
13720:
13702:
13674:
13655:
13625:
13623:
13622:
13617:
13615:
13604:
13588:
13583:, through angle
13582:
13573:
13560:
13559:
13553:
13530:
13524:
13518:
13512:
13511:
13504:
13503:
13496:
13490:
13476:
13465:
13463:
13462:
13457:
13451:
13450:
13440:
13432:
13396:
13365:
13364:
13357:
13347:
13346:
13339:
13333:
13331:
13321:
13310:
13286:
13267:
13248:
13242:
13233:
13227:
13221:
13215:
13204:
13161:
13155:
13149:
13147:
13146:
13141:
13129:
13115:
13101:
13086:
13080:
13073:
13071:
13070:
13065:
13057:
13043:
13030:
13019:
13005:
12990:
12983:
12977:
12970:
12964:
12958:
12948:
12946:
12945:
12940:
12935:
12913:
12911:
12910:
12905:
12903:
12900:
12898:
12897:
12888:
12887:
12878:
12870:
12866:
12851:
12849:
12848:
12843:
12841:
12838:
12833:
12820:
12815:
12802:
12797:
12788:
12771:
12769:
12768:
12763:
12758:
12757:
12750:
12747:
12746:
12737:
12736:
12735:
12730:
12729:
12721:
12717:
12716:
12708:
12704:
12678:
12677:
12669:
12656:
12655:
12650:
12649:
12641:
12615:
12614:
12607:
12604:
12603:
12593:
12588:
12579:
12553:
12550:
12549:
12540:
12539:
12538:
12529:
12528:
12518:
12498:
12495:
12494:
12485:
12484:
12483:
12474:
12473:
12463:
12440:
12435:
12434:
12415:
12412:
12411:
12402:
12401:
12400:
12391:
12390:
12380:
12360:
12357:
12356:
12346:
12341:
12332:
12306:
12303:
12302:
12293:
12292:
12291:
12282:
12281:
12271:
12248:
12243:
12242:
12223:
12220:
12219:
12210:
12209:
12208:
12199:
12198:
12188:
12168:
12165:
12164:
12155:
12154:
12153:
12144:
12143:
12133:
12113:
12110:
12109:
12099:
12094:
12085:
12056:
12051:
12050:
12028:
12023:
12022:
12002:
11997:
11996:
11976:
11971:
11970:
11936:
11918:
11916:
11915:
11910:
11908:
11896:
11894:
11893:
11888:
11883:
11865:
11863:
11862:
11857:
11849:
11835:
11820:
11818:
11817:
11812:
11810:
11795:
11793:
11792:
11787:
11767:
11765:
11764:
11759:
11756:
11751:
11746:
11745:
11731:
11729:
11728:
11723:
11720:
11715:
11710:
11709:
11695:
11693:
11692:
11687:
11684:
11679:
11674:
11673:
11659:
11657:
11656:
11651:
11648:
11643:
11638:
11637:
11626:
11621:
11616:
11615:
11605:
11604:
11599:
11598:
11584:
11582:
11581:
11576:
11574:
11573:
11568:
11567:
11553:
11551:
11550:
11545:
11543:
11542:
11537:
11536:
11522:
11520:
11519:
11514:
11511:
11506:
11501:
11500:
11486:
11480:
11476:
11469:
11462:
11456:
11442:
11440:
11439:
11434:
11431:
11426:
11421:
11420:
11409:
11404:
11399:
11398:
11387:
11382:
11377:
11376:
11365:
11360:
11355:
11354:
11344:
11343:
11313:
11311:
11310:
11305:
11303:
11302:
11297:
11296:
11286:
11285:
11280:
11279:
11268:
11263:
11258:
11257:
11234:
11232:
11231:
11226:
11224:
11223:
11218:
11217:
11207:
11206:
11201:
11200:
11189:
11184:
11179:
11178:
11155:
11153:
11152:
11147:
11107:
11106:
11101:
11100:
11075:
11073:
11072:
11067:
11065:
11064:
11059:
11058:
11048:
11047:
11042:
11041:
11030:
11025:
11020:
11019:
10996:
10994:
10993:
10988:
10986:
10985:
10980:
10979:
10969:
10968:
10963:
10962:
10951:
10946:
10941:
10940:
10917:
10915:
10914:
10909:
10869:
10868:
10863:
10862:
10837:
10835:
10834:
10829:
10801:
10800:
10795:
10794:
10771:
10769:
10768:
10763:
10732:
10731:
10726:
10725:
10704:Intersection, ∩
10701:
10700:
10682:ransformations (
10670:
10662:
10656:
10654:
10653:
10648:
10609:
10608:
10592:
10582:
10580:
10579:
10574:
10572:
10567:
10566:
10565:
10559:
10544:
10543:
10527:
10512:
10510:
10509:
10504:
10498:
10497:
10490:
10483:
10471:
10470:
10469:
10463:
10430:
10429:
10422:
10374:
10372:
10371:
10366:
10332:
10331:
10326:
10322:
10292:
10290:
10289:
10284:
10276:
10275:
10274:
10228:
10208:
10206:
10205:
10200:
10198:
10197:
10184:
10182:
10181:
10176:
10157:
10153:
10151:
10150:
10145:
10134:
10115:
10113:
10112:
10107:
10105:
10104:
10092:
10091:
10090:
10084:
10083:
10064:
10063:
10062:
10029:
10025:
10023:
10022:
10017:
10015:
10014:
9960:
9959:
9846:
9845:
9838:
9826:
9814:
9802:
9779:
9764:
9758:Minkowski metric
9711:
9688:
9681:
9658:
9648:
9642:
9632:
9607:
9601:
9586:
9572:
9563:
9554:
9548:angular momentum
9542:
9533:
9518:
9500:
9494:
9483:
9472:
9462:
9444:
9433:
9427:
9409:
9398:
9380:
9369:
9359:
9347:Four-wave vector
9341:
9330:
9320:
9302:
9291:
9285:
9266:
9259:
9249:
9248:
9245:
9239:
9233:
9227:
9215:
9200:
9190:
9178:
9166:
9160:
9153:
9145:
9143:
9142:
9137:
9135:
9127:
9122:
9121:
9106:
9101:
9093:
9085:
9059:
9047:
9043:
9029:
9025:
9024:
9019:
9018:
9010:
9001:
8978:
8959:
8957:
8956:
8951:
8949:
8945:
8936:
8932:
8923:
8922:
8917:
8916:
8903:
8895:
8887:
8886:
8870:
8839:
8832:
8804:
8768:
8761:
8755:
8748:
8742:
8736:
8729:
8715:
8708:
8699:
8697:
8696:
8691:
8689:
8685:
8684:
8679:
8675:
8674:
8666:
8656:
8654:
8647:
8646:
8645:
8634:
8633:
8632:
8622:
8620:
8618:
8617:
8605:
8600:
8592:
8590:
8589:
8588:
8578:
8573:
8566:
8564:
8563:
8561:
8560:
8551:
8550:
8542:
8536:
8524:
8519:
8515:
8496:
8494:
8493:
8488:
8486:
8483:
8480:
8479:
8470:
8469:
8461:
8455:
8446:
8442:
8437:
8436:
8435:
8420:
8418:
8417:
8405:
8404:
8400:
8390:
8385:
8381:
8370:
8368:
8360:
8359:
8350:
8345:
8326:
8311:
8304:
8298:
8269:
8261:
8253:
8247:
8245:
8244:
8239:
8227:
8226:
8214:
8206:
8195:
8183:
8181:
8180:
8175:
8160:
8152:
8141:
8126:
8120:
8114:
8108:
8099:
8097:
8096:
8091:
8089:
8081:
8073:
8059:
8051:
8043:
8039:
8012:
8008:
7995:
7982:
7978:
7977:
7975:
7974:
7965:
7964:
7953:
7949:
7942:
7937:
7899:
7891:
7876:
7870:
7860:
7850:
7843:
7837:
7827:
7825:
7824:
7819:
7817:
7809:
7792:
7784:
7776:
7750:
7738:
7734:
7720:
7716:
7715:
7713:
7712:
7703:
7702:
7694:
7685:
7662:
7642:
7634:
7617:
7615:
7614:
7609:
7607:
7599:
7591:
7580:
7572:
7571:
7566:
7555:
7547:
7539:
7528:
7527:
7522:
7505:
7499:
7493:
7480:
7452:
7444:
7430:
7420:
7416:
7414:
7413:
7408:
7406:
7402:
7401:
7396:
7379:
7374:
7358:
7350:
7349:
7344:
7321:
7316:
7306:
7302:
7301:
7299:
7298:
7289:
7288:
7280:
7279:
7274:
7267:
7244:
7225:
7223:
7222:
7217:
7207:
7202:
7189:
7184:
7175:
7171:
7160:
7159:
7154:
7145:
7144:
7139:
7130:
7118:
7112:
7106:
7099:
7093:
7081:
7070:
7064:
7059:
7051:
7029:
7023:
7019:
7012:
7006:
6999:
6993:
6986:
6969:Euclidean vector
6956:
6949:
6943:
6937:
6921:
6913:
6906:
6899:
6892:
6878:
6872:
6856:
6848:
6841:
6826:
6819:
6811:
6797:
6795:
6794:
6789:
6787:
6785:
6784:
6775:
6760:
6752:
6734:
6727:
6720:
6714:
6690:
6688:
6687:
6682:
6680:
6657:
6635:
6610:
6595:
6584:
6577:
6567:
6561:
6546:
6527:
6509:
6502:
6471:
6465:
6417:
6411:
6386:
6380:
6378:
6377:
6372:
6370:
6362:
6358:
6357:
6339:
6299:
6295:
6294:
6292:
6291:
6282:
6281:
6265:
6260:
6217:
6215:
6214:
6209:
6207:
6199:
6195:
6160:
6141:
6137:
6136:
6134:
6133:
6124:
6112:
6086:
6050:
6043:
6033:
6031:
6030:
6025:
6023:
6019:
5997:
5966:
5943:
5912:
5892:
5856:
5848:
5836:
5823:
5816:
5809:
5802:
5795:
5788:
5782:
5776:
5768:
5758:
5756:
5755:
5750:
5738:
5737:
5710:
5708:
5707:
5702:
5700:
5605:
5599:
5593:
5585:
5583:
5582:
5577:
5571:
5562:
5561:
5546:
5542:
5521:
5511:
5509:
5508:
5503:
5497:
5495:
5484:
5473:
5452:
5446:
5440:
5432:
5430:
5429:
5424:
5406:
5405:
5387:
5386:
5370:
5364:
5357:
5338:hyperbolic angle
5335:
5330:. The parameter
5321:
5319:θ, ϕ, φ, η, ψ, ξ
5305:
5296:
5294:
5293:
5288:
5286:
5272:
5250:
5201:
5152:
5129:
5121:
5109:
5096:
5094:
5093:
5088:
5086:
5074:
5072:
5071:
5066:
5031:
5029:
5028:
5023:
5021:
5019:
5018:
5013:
5001:
4999:
4975:
4973:
4972:
4967:
4962:
4961:
4948:
4945:
4942:
4932:
4929:
4926:
4916:
4913:
4895:
4894:
4886:
4881:
4872:
4870:
4859:
4858:
4849:
4839:
4838:
4829:
4828:
4819:
4817:
4806:
4805:
4796:
4792:
4791:
4782:
4781:
4772:
4770:
4759:
4758:
4749:
4745:
4744:
4725:
4724:
4715:
4714:
4705:
4703:
4692:
4691:
4682:
4677:
4672:
4663:
4661:
4650:
4649:
4640:
4630:
4629:
4620:
4619:
4610:
4608:
4597:
4596:
4587:
4583:
4582:
4563:
4562:
4553:
4552:
4543:
4541:
4530:
4529:
4520:
4516:
4515:
4506:
4505:
4496:
4494:
4483:
4482:
4473:
4468:
4463:
4454:
4452:
4441:
4440:
4431:
4421:
4420:
4401:
4400:
4383:
4382:
4365:
4364:
4333:
4332:
4325:
4314:
4311:
4307:
4296:
4293:
4290:
4286:
4275:
4272:
4269:
4265:
4237:
4235:
4234:
4229:
4224:
4219:
4211:
4199:
4197:
4196:
4191:
4189:
4174:
4168:
4162:
4154:
4148:
4142:
4140:
4139:
4134:
4132:
4124:
4120:
4089:
4073:
4069:
4038:
4016:
4006:
3988:
3979:
3977:
3976:
3971:
3969:
3962:
3940:
3918:
3914:
3913:
3899:
3869:
3865:
3864:
3862:
3861:
3852:
3851:
3839:
3834:
3793:
3777:
3771:
3752:
3745:
3738:
3731:
3725:
3718:
3712:
3706:
3699:
3693:
3686:
3667:
3642:
3631:
3624:
3606:
3600:
3586:
3580:
3570:
3561:
3553:
3547:
3532:
3526:
3520:
3513:
3506:
3499:
3492:
3479:
3462:
3460:
3459:
3454:
3452:
3449:
3447:
3446:
3437:
3436:
3427:
3419:
3415:
3396:
3390:
3384:
3375:
3373:
3372:
3367:
3365:
3351:
3329:
3317:
3313:
3285:
3273:
3269:
3268:
3266:
3265:
3256:
3248:
3225:
3202:
3186:
3179:
3161:
3144:
3124:
3113:
3107:
3100:
3094:
3087:
3081:
3074:
3065:
3046:
3039:
3033:
3027:
3021:. Another frame
3020:
3002:
2992:
2985:
2978:
2972:
2961:
2955:
2945:
2888:
2815:
2798:
2789:
2787:
2786:
2781:
2770:
2737:
2719:
2708:
2647:
2643:
2638:metric signature
2631:
2627:
2618:
2602:
2596:
2589:
2578:
2569:
2567:
2566:
2561:
2556:
2545:
2524:
2521:
2515:
2504:
2467:
2452:classical groups
2438:
2429:
2427:
2426:
2421:
2419:
2411:
2398:
2382:
2369:
2353:
2340:
2324:
2311:
2302:
2301:
2289:
2288:
2279:
2278:
2266:
2265:
2256:
2255:
2243:
2242:
2233:
2232:
2220:
2219:
2210:
2209:
2200:
2199:
2187:
2184:
2178:
2177:
2176:
2160:
2159:
2158:
2142:
2141:
2140:
2124:
2123:
2122:
2109:
2108:
2096:
2095:
2083:
2082:
2070:
2069:
2057:
2056:
2047:
2046:
2036:
2024:
2012:
2003:
1994:
1988:
1978:
1958:
1931:
1922:
1920:
1919:
1914:
1912:
1905:
1902:
1899:
1898:
1885:
1869:
1854:
1853:
1840:
1824:
1809:
1808:
1795:
1779:
1764:
1763:
1750:
1734:
1722:
1721:
1710:
1701:
1700:
1691:
1690:
1678:
1677:
1662:
1661:
1652:
1651:
1639:
1638:
1623:
1622:
1613:
1612:
1600:
1599:
1584:
1583:
1574:
1573:
1561:
1560:
1548:
1547:
1537:
1525:
1513:
1475:
1422:
1413:
1411:
1410:
1405:
1403:
1400:
1391:
1390:
1381:
1380:
1368:
1367:
1352:
1351:
1342:
1341:
1329:
1328:
1313:
1312:
1303:
1302:
1290:
1289:
1274:
1273:
1264:
1263:
1251:
1250:
1238:
1237:
1220:
1207:From Einstein's
1203:
1079:Oliver Heaviside
1013:electromagnetism
928:angular velocity
915:
913:
912:
907:
905:
888:
866:
854:
850:
819:
807:
803:
772:
750:
748:
747:
742:
737:
729:
711:
705:
699:
697:
696:
691:
679:
673:
667:
661:
651:
649:
648:
643:
641:
640:
632:
628:
626:
624:
623:
614:
613:
604:
596:
572:
566:
562:
556:
549:
543:
523:
503:
501:
500:
495:
493:
479:
457:
445:
441:
413:
401:
397:
396:
394:
393:
384:
376:
353:
334:
330:
328:
327:
322:
288:coordinate frame
261:
254:
247:
233:
228:
227:
220:
216:
215:
185:Curved spacetime
38:
19:
18:
28206:
28205:
28201:
28200:
28199:
28197:
28196:
28195:
28191:Hendrik Lorentz
28161:
28160:
28159:
28154:
28140:
27968:
27872:BKL singularity
27862:Lemaître–Tolman
27837:
27833:Quantum gravity
27815:
27809:
27795:geodetic effect
27769:(together with
27739:LISA Pathfinder
27678:
27627:
27613:Penrose diagram
27595:
27589:
27564:
27553:
27549:Minkowski space
27515:
27459:
27443:
27391:
27385:
27345:
27338:
27333:
27254:
27242:Wayback Machine
27219:
27201:Voigt, Woldemar
27194:
27174:
27151:
27143:
27141:Further reading
27138:
27132:
27102:
27081:
27060:
27039:
27016:
26990:
26964:
26930:
26918:(1977) . "13".
26912:Leighton, R. B.
26900:
26888:(1977) . "15".
26882:Leighton, R. B.
26870:
26848:Lifshitz, E. M.
26836:
26802:
26777:
26756:
26735:
26713:
26692:
26669:
26647:
26628:
26604:
26574:
26548:
26529:
26510:
26491:
26470:
26449:
26430:
26425:
26419:
26197:
26179:
26173:
26171:
26116:(10): 891–921,
26105:
26080:Poincaré, Henri
26071:
26069:
26053:10.1.1.679.5898
26028:
25993:
25973:
25888:
25861:
25856:
25851:
25842:
25840:
25832:
25831:
25827:
25819:
25815:
25808:
25804:
25796:
25792:
25785:
25781:
25774:
25770:
25762:
25758:
25750:
25746:
25735:
25731:
25723:
25719:
25711:
25707:
25700:
25696:
25653:
25649:
25641:
25637:
25633:, p. 18–19
25629:
25625:
25618:
25614:
25604:
25588:
25584:
25577:
25573:
25566:
25562:
25554:
25550:
25541:
25537:
25529:
25525:
25521:, pp. 1–22
25517:
25513:
25505:
25501:
25494:
25490:
25483:
25479:
25472:
25468:
25457:
25453:
25446:
25442:
25432:
25416:
25412:
25408:
25403:
25402:
25393:
25390:
25381:
25372:
25364:
25361:
25349:
25337:
25322:
25317:In ordinary 3d
25316:
25312:
25302:complex numbers
25279:
25273:
25267:
25263:
25254:
25252:
25247:
25233:
25229:
25210:
25206:
25200:
25196:
25187:
25183:
25177:
25173:
25164:
25160:
25154:
25150:
25142:
25134:
25132:
25129:
25128:
25112:
25108:
25102:
25098:
25089:
25085:
25079:
25075:
25066:
25062:
25056:
25052:
25044:
25036:
25034:
25031:
25030:
25028:
25024:
25006:
25000:
24994:
24992:
24988:
24977:
24973:
24969:
24965:
24963:
24959:
24953:conformal group
24942:
24938:
24924:
24920:
24915:
24910:
24841:
24832:
24822:
24821:
24815:
24800:
24770:
24769:
24752:
24748:
24739:
24729:
24725:
24713:
24709:
24700:
24690:
24686:
24682:
24678:
24656:
24652:
24639:
24635:
24624:
24620:
24616:
24609:
24605:
24596:
24591:
24573:
24569:
24556:
24552:
24541:
24537:
24533:
24526:
24522:
24513:
24508:
24490:
24477:
24472:
24467:
24463:
24450:
24445:
24435:
24430:
24425:
24415:
24411:
24405:
24401:
24389:
24385:
24379:
24375:
24368:
24365:
24346:
24342:
24336:
24332:
24317:
24313:
24307:
24303:
24296:
24292:
24286:
24282:
24275:
24271:
24269:
24267:
24261:
24260:
24246:
24242:
24236:
24232:
24226:
24222:
24213:
24209:
24203:
24199:
24193:
24189:
24188:
24184:
24161:
24159:
24156:
24155:
24139:
24127:
24120:
24107:
24106:
24097:
24093:
24087:
24080:
24076:
24066:
24065:
24059:
24052:
24048:
24038:
24037:
24028:
24027:
24021:
24017:
24008:
24004:
23998:
23991:
23987:
23977:
23976:
23970:
23963:
23959:
23949:
23948:
23939:
23938:
23932:
23928:
23922:
23915:
23911:
23901:
23900:
23891:
23887:
23881:
23874:
23870:
23860:
23859:
23852:
23803:
23801:
23798:
23797:
23790:
23786:
23767:
23733:
23729:
23723:
23712:
23711:
23707:
23706:
23705:
23692:
23691:
23687:
23685:
23682:
23681:
23665:
23664:
23648:
23644:
23638:
23634:
23632:
23624:
23617:
23613:
23603:
23595:
23592:
23591:
23586:
23578:
23570:
23552:
23548:
23540:
23523:
23516:
23507:
23506:
23502:
23500:
23497:
23496:
23490:
23488:current density
23481:
23444:
23440:
23434:
23423:
23422:
23418:
23417:
23416:
23410:
23399:
23398:
23394:
23393:
23392:
23376:
23367:
23363:
23362:
23358:
23349:
23345:
23339:
23328:
23327:
23323:
23322:
23321:
23315:
23304:
23303:
23299:
23298:
23297:
23282:
23278:
23268:
23260:
23259:
23255:
23253:
23250:
23249:
23221:
23215:
23195:
23172:
23168:
23162:
23151:
23150:
23146:
23145:
23144:
23138:
23127:
23126:
23122:
23121:
23120:
23107:
23099:
23098:
23094:
23092:
23089:
23088:
23080:
23068:
23062:
23061:
23056:
23042:
23041:
23032:
23022:
23014:
23006:
23005:
23001:
23000:
22983:
22978:
22977:
22969:
22960:
22955:
22954:
22953:
22949:
22939:
22929:
22928:
22923:
22922:
22919:
22918:
22909:
22899:
22891:
22883:
22882:
22878:
22877:
22860:
22855:
22854:
22846:
22837:
22832:
22831:
22830:
22826:
22816:
22806:
22805:
22800:
22799:
22796:
22795:
22789:
22784:
22783:
22776:
22766:
22765:
22760:
22759:
22756:
22755:
22749:
22744:
22743:
22736:
22726:
22725:
22720:
22719:
22715:
22713:
22710:
22709:
22696:
22680:
22679:
22670:
22660:
22652:
22644:
22643:
22639:
22638:
22626:
22625:
22619:
22615:
22600:
22596:
22581:
22577:
22547:
22543:
22522:
22521:
22515:
22511:
22505:
22498:
22494:
22493:
22492:
22486:
22479:
22475:
22474:
22473:
22464:
22460:
22454:
22447:
22443:
22442:
22441:
22435:
22428:
22424:
22423:
22422:
22410:
22406:
22400:
22393:
22389:
22388:
22387:
22381:
22374:
22370:
22369:
22368:
22356:
22352:
22346:
22339:
22335:
22334:
22333:
22327:
22320:
22316:
22315:
22314:
22301:
22293:
22292:
22288:
22281:
22271:
22270:
22266:
22263:
22262:
22256:
22246:
22238:
22230:
22229:
22225:
22224:
22212:
22211:
22205:
22201:
22186:
22182:
22170:
22166:
22133:
22129:
22108:
22107:
22101:
22097:
22091:
22084:
22080:
22079:
22078:
22072:
22065:
22061:
22060:
22059:
22050:
22046:
22040:
22033:
22029:
22028:
22027:
22021:
22014:
22010:
22009:
22008:
21996:
21992:
21986:
21979:
21975:
21974:
21973:
21967:
21960:
21956:
21955:
21954:
21942:
21938:
21932:
21925:
21921:
21920:
21919:
21913:
21906:
21902:
21901:
21900:
21887:
21879:
21878:
21874:
21867:
21857:
21856:
21852:
21849:
21848:
21839:
21835:
21826:
21825:
21819:
21815:
21806:
21802:
21787:
21783:
21774:
21770:
21764:
21760:
21748:
21744:
21735:
21731:
21725:
21721:
21709:
21705:
21687:
21683:
21638:
21637:
21631:
21627:
21621:
21614:
21610:
21609:
21608:
21602:
21595:
21591:
21590:
21589:
21580:
21576:
21570:
21563:
21559:
21558:
21557:
21551:
21544:
21540:
21539:
21538:
21526:
21522:
21516:
21509:
21505:
21504:
21503:
21497:
21490:
21486:
21485:
21484:
21471:
21463:
21462:
21458:
21451:
21441:
21440:
21436:
21432:
21430:
21427:
21426:
21410:
21409:
21403:
21393:
21385:
21377:
21376:
21372:
21371:
21359:
21358:
21352:
21348:
21333:
21329:
21317:
21313:
21292:
21288:
21255:
21254:
21248:
21244:
21238:
21231:
21227:
21226:
21225:
21219:
21212:
21208:
21207:
21206:
21197:
21193:
21187:
21180:
21176:
21175:
21174:
21168:
21161:
21157:
21156:
21155:
21143:
21139:
21133:
21126:
21122:
21121:
21120:
21114:
21107:
21103:
21102:
21101:
21089:
21085:
21079:
21072:
21068:
21067:
21066:
21060:
21053:
21049:
21048:
21047:
21034:
21026:
21025:
21021:
21014:
21004:
21003:
20999:
20996:
20995:
20989:
20979:
20971:
20963:
20962:
20958:
20957:
20945:
20944:
20938:
20934:
20919:
20915:
20903:
20899:
20878:
20874:
20838:
20837:
20831:
20827:
20821:
20814:
20810:
20809:
20808:
20802:
20795:
20791:
20790:
20789:
20780:
20776:
20770:
20763:
20759:
20758:
20757:
20751:
20744:
20740:
20739:
20738:
20726:
20722:
20716:
20709:
20705:
20704:
20703:
20697:
20690:
20686:
20685:
20684:
20672:
20668:
20662:
20655:
20651:
20650:
20649:
20643:
20636:
20632:
20631:
20630:
20617:
20609:
20608:
20604:
20597:
20587:
20586:
20582:
20579:
20578:
20569:
20565:
20556:
20555:
20549:
20545:
20524:
20520:
20514:
20507:
20503:
20502:
20501:
20495:
20488:
20484:
20483:
20482:
20470:
20466:
20460:
20453:
20449:
20448:
20447:
20441:
20434:
20430:
20429:
20428:
20415:
20407:
20406:
20402:
20395:
20385:
20384:
20380:
20376:
20374:
20371:
20370:
20345:
20341:
20335:
20324:
20323:
20319:
20318:
20317:
20311:
20300:
20299:
20295:
20294:
20293:
20280:
20272:
20271:
20267:
20265:
20262:
20261:
20233:
20201:
20194:
20193:
20188:
20182:
20178:
20173:
20167:
20163:
20161:
20155:
20151:
20145:
20144:
20138:
20134:
20132:
20127:
20121:
20117:
20112:
20106:
20102:
20096:
20095:
20089:
20085:
20080:
20074:
20070:
20068:
20063:
20057:
20053:
20047:
20046:
20040:
20036:
20034:
20028:
20024:
20022:
20016:
20012:
20010:
20000:
19999:
19987:
19983:
19972:
19971:
19966:
19961:
19956:
19950:
19949:
19944:
19939:
19934:
19928:
19927:
19922:
19917:
19912:
19900:
19899:
19894:
19889:
19878:
19868:
19867:
19858:
19851:
19847:
19846:
19845:
19843:
19840:
19839:
19833:
19823:
19817:
19790:
19758:
19751:
19750:
19745:
19739:
19735:
19733:
19727:
19723:
19718:
19712:
19708:
19698:
19695:
19694:
19688:
19684:
19679:
19674:
19668:
19664:
19662:
19656:
19652:
19642:
19639:
19638:
19632:
19628:
19626:
19620:
19616:
19611:
19606:
19600:
19596:
19586:
19583:
19582:
19576:
19572:
19562:
19557:
19551:
19547:
19537:
19532:
19526:
19522:
19512:
19507:
19497:
19496:
19484:
19480:
19478:
19475:
19474:
19456:
19446:
19424:
19415:
19409:
19403:
19389:
19380:
19374:
19370:
19365:
19362:
19355:
19327:
19313:
19303:
19292:
19291:
19287:
19286:
19285:
19276:
19265:
19264:
19260:
19259:
19258:
19252:
19241:
19240:
19236:
19235:
19234:
19228:
19217:
19216:
19212:
19211:
19210:
19201:
19190:
19189:
19185:
19184:
19183:
19177:
19166:
19165:
19161:
19160:
19159:
19146:
19135:
19127:
19126:
19117:
19106:
19098:
19097:
19091:
19088:
19087:
19078:
19068:
19055:
19052:
19045:
19023:
19019:
19013:
19006:
19002:
19001:
19000:
18994:
18987:
18983:
18982:
18981:
18972:
18968:
18959:
18955:
18949:
18942:
18938:
18937:
18936:
18930:
18923:
18919:
18918:
18917:
18908:
18904:
18898:
18891:
18887:
18886:
18885:
18876:
18872:
18866:
18859:
18855:
18854:
18853:
18821:
18818:
18817:
18800:
18786:
18782:
18778:
18774:
18771:
18764:
18652:
18649:
18648:
18635:
18631:
18627:
18613:
18609:
18605:
18601:
18597:
18594:
18585:
18581:
18551:
18550:
18537:
18533:
18529:
18528:
18517:
18515:
18512:
18511:
18508:
18500:
18496:
18488:
18467:
18463:
18457:
18450:
18437:
18433:
18429:
18428:
18427:
18426:
18417:
18413:
18407:
18400:
18396:
18395:
18394:
18385:
18376:
18375:
18374:
18372:
18369:
18368:
18346:
18342:
18336:
18329:
18325:
18324:
18323:
18314:
18305:
18304:
18303:
18301:
18298:
18297:
18277:
18270:
18257:
18253:
18249:
18248:
18247:
18246:
18237:
18230:
18226:
18225:
18224:
18222:
18219:
18218:
18202:
18180:
18173:
18160:
18156:
18152:
18151:
18150:
18149:
18137:
18133:
18127:
18120:
18116:
18115:
18114:
18105:
18101:
18099:
18096:
18095:
18073:
18069:
18060:
18056:
18050:
18043:
18039:
18038:
18037:
18028:
18024:
18015:
18006:
18005:
18004:
18002:
17999:
17998:
17994:
17991:
17983:
17975:
17967:
17962:
17959:
17951:
17945:
17925:
17921:
17912:
17908:
17899:
17895:
17893:
17890:
17889:
17879:
17859:
17855:
17846:
17842:
17833:
17829:
17827:
17824:
17823:
17816:
17799:
17793:
17789:
17781:
17771:
17763:
17742:
17738:
17732:
17725:
17721:
17711:
17710:
17701:
17692:
17691:
17690:
17688:
17685:
17684:
17678:
17657:
17653:
17647:
17636:
17635:
17631:
17630:
17629:
17616:
17615:
17611:
17609:
17606:
17605:
17582:
17578:
17572:
17565:
17561:
17560:
17559:
17550:
17541:
17540:
17539:
17537:
17534:
17533:
17524:
17475:
17471:
17465:
17458:
17454:
17453:
17452:
17443:
17434:
17433:
17432:
17430:
17427:
17426:
17396:
17395:
17389:
17385:
17382:
17381:
17375:
17371:
17368:
17367:
17361:
17357:
17354:
17353:
17347:
17343:
17336:
17335:
17328:
17327:
17321:
17314:
17310:
17309:
17308:
17306:
17300:
17293:
17289:
17288:
17287:
17285:
17279:
17272:
17268:
17267:
17266:
17264:
17258:
17251:
17247:
17246:
17245:
17242:
17241:
17235:
17228:
17224:
17223:
17222:
17220:
17214:
17207:
17203:
17202:
17201:
17199:
17193:
17186:
17182:
17181:
17180:
17178:
17172:
17165:
17161:
17160:
17159:
17156:
17155:
17149:
17142:
17138:
17137:
17136:
17134:
17128:
17121:
17117:
17116:
17115:
17113:
17107:
17100:
17096:
17095:
17094:
17092:
17086:
17079:
17075:
17074:
17073:
17070:
17069:
17063:
17056:
17052:
17051:
17050:
17048:
17042:
17035:
17031:
17030:
17029:
17027:
17021:
17014:
17010:
17009:
17008:
17006:
17000:
16993:
16989:
16988:
16987:
16980:
16979:
16969:
16968:
16962:
16953:
16952:
16951:
16948:
16947:
16941:
16932:
16931:
16930:
16927:
16926:
16920:
16911:
16910:
16909:
16906:
16905:
16899:
16890:
16889:
16888:
16881:
16880:
16878:
16875:
16874:
16871:
16866:
16859:
16853:
16843:= 0, this is a
16788:
16786:
16783:
16782:
16778:
16762:
16755:
16748:
16744:
16734:identity matrix
16727:
16709:
16708:
16703:
16701:
16695:
16694:
16689:
16676:
16675:
16667:
16664:
16663:
16642:
16641:
16636:
16631:
16625:
16624:
16619:
16609:
16608:
16600:
16597:
16596:
16590:
16539:
16512:
16511:
16501:
16498:
16497:
16494:exponential map
16479:Jacobi identity
16475:alternatization
16416:
16415:
16413:
16410:
16409:
16382:
16361:
16357:
16345:
16341:
16332:
16328:
16314:
16310:
16295:
16291:
16282:
16278:
16264:
16260:
16248:
16244:
16235:
16231:
16226:
16223:
16222:
16207:
16203:
16199:
16195:
16191:
16187:
16182:
16170:
16166:
16162:
16158:
16154:
16150:
16145:
16134:matrix addition
16114:
16106:
16098:
16090:
16079:
16076:
16075:
16068:
16052:Wigner rotation
16045:
16039:
16014:
16006:
16005:
16001:
15994:
15986:
15982:
15978:
15968:
15960:
15952:
15944:
15940:
15936:
15934:
15931:
15930:
15907:
15899:
15891:
15883:
15879:
15875:
15864:
15856:
15848:
15845:
15844:
15824:
15804:
15790:
15789:
15778:
15770:
15762:
15754:
15739:
15738:
15724:
15716:
15693:
15685:
15667:
15666:
15652:
15644:
15621:
15613:
15597:
15590:
15588:
15585:
15584:
15556:
15552:
15547:
15539:
15504:
15496:
15467:
15459:
15456:
15455:
15439:
15435:
15430:
15422:
15387:
15379:
15350:
15342:
15339:
15338:
15328:
15322:
15316:
15305:
15299:
15293:
15286:
15280:
15270:
15259:
15249:
15235:
15231:
15226:
15210:
15209:
15202:
15201:
15196:
15191:
15186:
15180:
15179:
15174:
15169:
15164:
15158:
15157:
15152:
15144:
15139:
15133:
15132:
15127:
15122:
15117:
15107:
15106:
15099:
15093:
15089:
15087:
15075:
15074:
15069:
15064:
15056:
15050:
15049:
15044:
15039:
15034:
15028:
15027:
15022:
15017:
15012:
15006:
15005:
15000:
14995:
14990:
14980:
14979:
14972:
14966:
14962:
14960:
14948:
14947:
14942:
14937:
14932:
14926:
14925:
14917:
14912:
14907:
14901:
14900:
14895:
14890:
14885:
14879:
14878:
14873:
14868:
14863:
14853:
14852:
14845:
14839:
14835:
14832:
14831:
14824:
14823:
14818:
14813:
14808:
14802:
14801:
14796:
14791:
14786:
14780:
14779:
14774:
14769:
14764:
14758:
14757:
14752:
14747:
14742:
14732:
14731:
14724:
14718:
14714:
14712:
14700:
14699:
14694:
14689:
14684:
14678:
14677:
14672:
14667:
14662:
14656:
14655:
14650:
14645:
14640:
14634:
14633:
14628:
14623:
14618:
14608:
14607:
14600:
14594:
14590:
14588:
14576:
14575:
14570:
14565:
14560:
14554:
14553:
14548:
14543:
14538:
14532:
14531:
14526:
14521:
14516:
14510:
14509:
14504:
14499:
14494:
14484:
14483:
14476:
14470:
14466:
14462:
14460:
14457:
14456:
14449:
14445:
14441:
14432:
14427:
14423:
14419:
14410:
14404:
14398:
14374:
14366:
14365:
14361:
14350:
14332:
14324:
14320:
14316:
14305:
14297:
14294:
14293:
14264:
14260:
14253:
14249:
14240:
14229:
14225:
14215:
14208:
14204:
14203:
14191:
14178:
14174:
14172:
14169:
14168:
14160:
14155:
14132:
14128:
14110:
14096:
14089:
14085:
14081:
14079:
14076:
14075:
14073:
14070:
14069:
14062:
14057:direction. The
14050:
14045:
14039:
14010:
13996:
13989:
13985:
13981:
13979:
13976:
13975:
13957:
13953:
13951:
13948:
13947:
13940:
13906:
13898:
13881:
13864:
13853:
13850:
13849:
13846:
13830:
13818:
13798:
13778:
13766:
13752:
13738:
13728:
13725:matrix symmetry
13710:
13682:
13657:
13638:
13611:
13600:
13598:
13595:
13594:
13591:right-hand rule
13584:
13578:
13569:
13555:
13549:
13540:
13533:Wigner rotation
13526:
13520:
13514:
13507:
13506:
13499:
13498:
13492:
13486:
13479:rotation matrix
13467:
13445:
13444:
13436:
13428:
13426:
13420:
13419:
13414:
13404:
13403:
13392:
13383:
13380:
13379:
13378:form is simply
13360:
13359:
13353:
13342:
13341:
13335:
13327:
13317:
13312:
13293:
13269:
13250:
13244:
13238:
13229:
13223:
13217:
13206:
13171:
13157:
13151:
13125:
13111:
13094:
13092:
13089:
13088:
13082:
13075:
13053:
13036:
13023:
13015:
12998:
12996:
12993:
12992:
12985:
12979:
12972:
12966:
12960:
12953:
12931:
12920:
12917:
12916:
12893:
12889:
12883:
12879:
12877:
12865:
12857:
12854:
12853:
12834:
12829:
12816:
12811:
12798:
12793:
12787:
12779:
12776:
12775:
12752:
12751:
12742:
12738:
12731:
12720:
12719:
12718:
12707:
12706:
12705:
12702:
12679:
12668:
12667:
12658:
12657:
12651:
12640:
12639:
12638:
12630:
12620:
12619:
12609:
12608:
12599:
12595:
12589:
12584:
12577:
12554:
12545:
12541:
12534:
12530:
12524:
12520:
12519:
12516:
12499:
12490:
12486:
12479:
12475:
12469:
12465:
12464:
12461:
12444:
12436:
12430:
12426:
12417:
12416:
12407:
12403:
12396:
12392:
12386:
12382:
12381:
12378:
12361:
12352:
12348:
12342:
12337:
12330:
12307:
12298:
12294:
12287:
12283:
12277:
12273:
12272:
12269:
12252:
12244:
12238:
12234:
12225:
12224:
12215:
12211:
12204:
12200:
12194:
12190:
12189:
12186:
12169:
12160:
12156:
12149:
12145:
12139:
12135:
12134:
12131:
12114:
12105:
12101:
12095:
12090:
12083:
12060:
12052:
12046:
12042:
12033:
12032:
12024:
12018:
12014:
12006:
11998:
11992:
11988:
11980:
11972:
11966:
11962:
11954:
11944:
11943:
11932:
11924:
11921:
11920:
11904:
11902:
11899:
11898:
11879:
11871:
11868:
11867:
11845:
11828:
11826:
11823:
11822:
11803:
11801:
11798:
11797:
11781:
11778:
11777:
11774:
11752:
11747:
11741:
11740:
11737:
11734:
11733:
11716:
11711:
11705:
11704:
11701:
11698:
11697:
11680:
11675:
11669:
11668:
11665:
11662:
11661:
11644:
11639:
11633:
11632:
11622:
11617:
11611:
11610:
11600:
11594:
11593:
11592:
11590:
11587:
11586:
11569:
11563:
11562:
11561:
11559:
11556:
11555:
11538:
11532:
11531:
11530:
11528:
11525:
11524:
11507:
11502:
11496:
11495:
11492:
11489:
11488:
11482:
11478:
11471:
11464:
11458:
11454:
11427:
11422:
11416:
11415:
11405:
11400:
11394:
11393:
11383:
11378:
11372:
11371:
11361:
11356:
11350:
11349:
11339:
11338:
11336:
11333:
11332:
11298:
11292:
11291:
11290:
11281:
11275:
11274:
11273:
11264:
11259:
11253:
11252:
11249:
11246:
11245:
11219:
11213:
11212:
11211:
11202:
11196:
11195:
11194:
11185:
11180:
11174:
11173:
11170:
11167:
11166:
11102:
11096:
11095:
11094:
11092:
11089:
11088:
11060:
11054:
11053:
11052:
11043:
11037:
11036:
11035:
11026:
11021:
11015:
11014:
11011:
11008:
11007:
10981:
10975:
10974:
10973:
10964:
10958:
10957:
10956:
10947:
10942:
10936:
10935:
10932:
10929:
10928:
10864:
10858:
10857:
10856:
10854:
10851:
10850:
10796:
10790:
10789:
10788:
10786:
10783:
10782:
10727:
10721:
10720:
10719:
10717:
10714:
10713:
10668:
10660:
10604:
10600:
10598:
10595:
10594:
10584:
10568:
10561:
10560:
10555:
10554:
10539:
10535:
10533:
10530:
10529:
10514:
10492:
10491:
10486:
10484:
10479:
10473:
10472:
10465:
10464:
10459:
10458:
10453:
10443:
10442:
10424:
10423:
10418:
10416:
10410:
10409:
10404:
10391:
10390:
10382:
10379:
10378:
10327:
10309:
10305:
10304:
10302:
10299:
10298:
10270:
10269:
10265:
10257:
10254:
10253:
10220:
10193:
10192:
10190:
10187:
10186:
10170:
10167:
10166:
10155:
10127:
10125:
10122:
10121:
10097:
10096:
10086:
10085:
10076:
10075:
10074:
10058:
10057:
10053:
10039:
10036:
10035:
10027:
10009:
10008:
10002:
10001:
9995:
9994:
9988:
9987:
9973:
9972:
9954:
9953:
9948:
9943:
9938:
9932:
9931:
9926:
9921:
9916:
9910:
9909:
9904:
9899:
9894:
9888:
9887:
9882:
9877:
9872:
9859:
9858:
9840:
9839:
9831:
9828:
9827:
9819:
9816:
9815:
9807:
9804:
9803:
9795:
9784:
9783:
9772:
9770:
9767:
9766:
9760:
9754:
9746:
9728:
9722:
9690:
9683:
9660:
9654:
9644:
9638:
9609:
9603:
9588:
9574:
9568:
9559:
9550:
9540:
9535:
9529:
9514:
9496:
9490:
9479:
9464:
9458:
9440:
9437:Current density
9429:
9423:
9422:(multiplied by
9405:
9397:
9391:
9376:
9361:
9355:
9337:
9322:
9316:
9298:
9295:Position vector
9287:
9281:
9280:(multiplied by
9262:
9255:
9241:
9235:
9229:
9223:
9205:
9204:The quantities
9192:
9180:
9168:
9162:
9155:
9149:
9133:
9132:
9117:
9107:
9105:
9097:
9089:
9081:
9055:
9048:
9039:
9038:
9035:
9034:
9014:
9006:
9002:
9000:
8993:
8989:
8979:
8971:
8967:
8965:
8962:
8961:
8941:
8940:
8928:
8927:
8918:
8909:
8908:
8907:
8899:
8891:
8882:
8878:
8876:
8873:
8872:
8868:
8860:
8852:
8841:
8834:
8830:
8823:
8816:
8806:
8800:
8797:
8763:
8757:
8750:
8744:
8738:
8731:
8725:
8710:
8704:
8703:The velocities
8680:
8670:
8662:
8661:
8657:
8641:
8640:
8636:
8635:
8628:
8627:
8623:
8621:
8613:
8609:
8604:
8596:
8584:
8583:
8579:
8574:
8572:
8571:
8567:
8556:
8552:
8546:
8538:
8537:
8535:
8528:
8523:
8511:
8510:
8508:
8505:
8504:
8475:
8471:
8465:
8457:
8456:
8453:
8441:
8431:
8430:
8426:
8410:
8406:
8396:
8395:
8391:
8389:
8377:
8376:
8361:
8355:
8351:
8349:
8341:
8339:
8336:
8335:
8313:
8306:
8300:
8296:
8286:
8276:
8263:
8255:
8249:
8219:
8215:
8210:
8202:
8191:
8189:
8186:
8185:
8156:
8148:
8137:
8135:
8132:
8131:
8122:
8116:
8110:
8104:
8101:
8087:
8086:
8077:
8066:
8055:
8047:
8035:
8034:
8004:
8003:
7996:
7991:
7988:
7987:
7970:
7966:
7960:
7945:
7944:
7943:
7941:
7930:
7929:
7925:
7915:
7908:
7906:
7903:
7902:
7895:
7893:with magnitude
7887:
7872:
7862:
7852:
7845:
7839:
7832:
7829:
7815:
7814:
7805:
7788:
7780:
7772:
7746:
7739:
7730:
7729:
7726:
7725:
7708:
7704:
7698:
7690:
7686:
7684:
7677:
7673:
7663:
7655:
7651:
7649:
7646:
7645:
7638:
7636:with magnitude
7630:
7603:
7595:
7587:
7576:
7567:
7562:
7561:
7551:
7543:
7535:
7523:
7518:
7517:
7515:
7512:
7511:
7501:
7495:
7494:with magnitude
7482:
7460:
7446:
7445:with magnitude
7432:
7426:
7418:
7404:
7403:
7397:
7392:
7391:
7384:
7375:
7370:
7366:
7365:
7354:
7345:
7340:
7339:
7326:
7317:
7312:
7308:
7307:
7294:
7290:
7284:
7275:
7270:
7269:
7268:
7266:
7259:
7255:
7245:
7237:
7233:
7231:
7228:
7227:
7203:
7198:
7185:
7180:
7167:
7166:
7155:
7150:
7149:
7140:
7135:
7134:
7126:
7124:
7121:
7120:
7114:
7108:
7107:as measured in
7101:
7095:
7094:as measured in
7089:
7087:position vector
7072:
7066:
7055:
7053:
7047:
7045:velocity vector
7031:
7025:
7021:
7014:
7008:
7001:
6995:
6988:
6982:
6975:
6965:
6951:
6945:
6939:
6923:
6915:
6908:
6901:
6894:
6888:
6874:
6858:
6850:
6843:
6828:
6821:
6813:
6807:
6780:
6776:
6761:
6759:
6745:
6740:
6737:
6736:
6735:, we find that
6729:
6722:
6716:
6708:
6678:
6677:
6658:
6650:
6647:
6646:
6636:
6628:
6624:
6622:
6619:
6618:
6597:
6586:
6579:
6569:
6563:
6557:
6554:
6544:
6529:
6526:
6511:
6504:
6500:
6493:
6486:
6479:
6473:
6467:
6464:
6457:
6450:
6443:
6437:
6413:
6410:
6403:
6392:
6384:
6368:
6367:
6350:
6332:
6328:
6324:
6314:
6305:
6304:
6287:
6283:
6274:
6266:
6264:
6253:
6249:
6245:
6235:
6225:
6223:
6220:
6219:
6205:
6204:
6175:
6171:
6161:
6153:
6147:
6146:
6129:
6125:
6113:
6111:
6101:
6097:
6087:
6079:
6072:
6070:
6067:
6066:
6045:
6038:
6035:
6021:
6020:
6012:
6005:
5999:
5998:
5990:
5983:
5977:
5976:
5959:
5936:
5929:
5923:
5922:
5905:
5885:
5875:
5865:
5863:
5860:
5859:
5852:
5844:
5828:
5818:
5811:
5804:
5797:
5790:
5784:
5778:
5770:
5762:
5730:
5726:
5718:
5715:
5714:
5698:
5697:
5677:
5668:
5667:
5647:
5641:
5640:
5620:
5613:
5611:
5608:
5607:
5601:
5595:
5589:
5557:
5553:
5541:
5527:
5524:
5523:
5517:
5485:
5474:
5472:
5458:
5455:
5454:
5448:
5442:
5436:
5435:Conversely the
5401:
5397:
5382:
5378:
5376:
5373:
5372:
5366:
5359:
5352:
5331:
5328:Minkowski space
5317:
5301:
5298:
5284:
5283:
5273:
5265:
5262:
5261:
5251:
5243:
5240:
5239:
5202:
5194:
5191:
5190:
5153:
5145:
5138:
5136:
5133:
5132:
5125:
5117:
5105:
5082:
5080:
5077:
5076:
5045:
5042:
5041:
5014:
5009:
5008:
5000:
4995:
4984:
4981:
4980:
4978:
4956:
4955:
4949:
4946:
4943:
4940:
4939:
4933:
4930:
4927:
4924:
4923:
4917:
4914:
4911:
4910:
4897:
4896:
4889:
4888:
4882:
4877:
4860:
4854:
4850:
4848:
4840:
4834:
4830:
4824:
4820:
4807:
4801:
4797:
4795:
4793:
4787:
4783:
4777:
4773:
4760:
4754:
4750:
4748:
4746:
4740:
4736:
4727:
4726:
4720:
4716:
4710:
4706:
4693:
4687:
4683:
4681:
4679:
4673:
4668:
4651:
4645:
4641:
4639:
4631:
4625:
4621:
4615:
4611:
4598:
4592:
4588:
4586:
4584:
4578:
4574:
4565:
4564:
4558:
4554:
4548:
4544:
4531:
4525:
4521:
4519:
4517:
4511:
4507:
4501:
4497:
4484:
4478:
4474:
4472:
4470:
4464:
4459:
4442:
4436:
4432:
4430:
4422:
4416:
4412:
4403:
4402:
4396:
4392:
4384:
4378:
4374:
4366:
4360:
4356:
4348:
4338:
4337:
4327:
4326:
4318:
4315:
4312:
4309:
4308:
4300:
4297:
4294:
4291:
4288:
4287:
4279:
4276:
4273:
4270:
4267:
4266:
4258:
4248:
4247:
4245:
4242:
4241:
4220:
4215:
4207:
4205:
4202:
4201:
4185:
4183:
4180:
4179:
4170:
4164:
4156:
4150:
4144:
4130:
4129:
4104:
4100:
4090:
4082:
4079:
4078:
4053:
4049:
4039:
4031:
4024:
4022:
4019:
4018:
4012:
3994:
3984:
3981:
3967:
3966:
3955:
3948:
3942:
3941:
3933:
3926:
3920:
3919:
3906:
3892:
3891:
3887:
3877:
3871:
3870:
3857:
3853:
3844:
3840:
3838:
3827:
3826:
3822:
3812:
3805:
3803:
3800:
3799:
3789:
3773:
3754:
3753:notes an event
3747:
3740:
3733:
3727:
3720:
3714:
3708:
3701:
3695:
3688:
3669:
3651:
3637:
3626:
3619:
3602:
3592:
3582:
3572:
3566:
3555:
3549:
3534:
3528:
3522:
3515:
3508:
3501:
3494:
3487:
3475:
3442:
3438:
3432:
3428:
3426:
3414:
3406:
3403:
3402:
3392:
3386:
3380:
3377:
3363:
3362:
3352:
3344:
3341:
3340:
3330:
3322:
3319:
3318:
3300:
3296:
3286:
3278:
3275:
3274:
3261:
3257:
3249:
3247:
3240:
3236:
3226:
3218:
3214:
3212:
3209:
3208:
3198:
3181:
3163:
3157:
3126:
3115:
3109:
3102:
3096:
3089:
3083:
3076:
3070:
3048:
3041:
3035:
3029:
3022:
3004:
2998:
2987:
2986:-axis of frame
2980:
2974:
2968:
2963:
2957:
2956:-axis of frame
2951:
2940:
2935:
2926:
2921:
2915:
2886:
2852:linear function
2844:
2813:
2763:
2733:
2712:
2701:
2654:
2651:
2650:
2641:
2629:
2623:
2621:Minkowski space
2614:
2598:
2594:
2587:
2549:
2538:
2520:
2508:
2497:
2474:
2471:
2470:
2417:
2416:
2407:
2394:
2378:
2365:
2349:
2336:
2320:
2307:
2297:
2293:
2284:
2280:
2274:
2270:
2261:
2257:
2251:
2247:
2238:
2234:
2228:
2224:
2215:
2211:
2205:
2201:
2195:
2191:
2189:
2183:
2180:
2179:
2172:
2168:
2164:
2154:
2150:
2146:
2136:
2132:
2128:
2118:
2114:
2110:
2104:
2100:
2091:
2087:
2078:
2074:
2065:
2061:
2052:
2048:
2042:
2038:
2033:
2031:
2028:
2027:
2011:
2005:
2002:
1996:
1990:
1986:
1960:
1940:
1910:
1909:
1901:
1894:
1890:
1881:
1865:
1849:
1845:
1836:
1820:
1804:
1800:
1791:
1775:
1759:
1755:
1746:
1730:
1717:
1713:
1711:
1709:
1703:
1702:
1696:
1692:
1686:
1682:
1673:
1669:
1657:
1653:
1647:
1643:
1634:
1630:
1618:
1614:
1608:
1604:
1595:
1591:
1579:
1575:
1569:
1565:
1556:
1552:
1543:
1539:
1534:
1532:
1529:
1528:
1511:
1504:
1497:
1490:
1483:
1477:
1473:
1466:
1459:
1452:
1445:
1439:
1438:between events
1399:
1386:
1382:
1376:
1372:
1363:
1359:
1347:
1343:
1337:
1333:
1324:
1320:
1308:
1304:
1298:
1294:
1285:
1281:
1269:
1265:
1259:
1255:
1246:
1242:
1233:
1229:
1227:
1224:
1223:
1211:(invariance of
1191:
1175:
1167:Main articles:
1165:
1145:Albert Einstein
1081:had shown from
1075:Hendrik Lorentz
1059:
1053:
1032:Minkowski space
1009:reference frame
939:reference frame
903:
902:
889:
881:
878:
877:
867:
859:
856:
855:
834:
830:
820:
812:
809:
808:
787:
783:
773:
765:
758:
756:
753:
752:
728:
720:
717:
716:
707:
701:
685:
682:
681:
675:
669:
663:
657:
633:
619:
615:
609:
605:
603:
595:
591:
590:
582:
579:
578:
568:
564:
558:
551:
545:
525:
505:
491:
490:
480:
472:
469:
468:
458:
450:
447:
446:
428:
424:
414:
406:
403:
402:
389:
385:
377:
375:
368:
364:
354:
346:
342:
340:
337:
336:
332:
313:
310:
309:
303:Hendrik Lorentz
284:transformations
265:
236:
223:
210:
209:
201:
200:
199:
154:
146:
145:
144:
129:
121:
120:
119:
99:
91:
90:
89:
85:Minkowski space
64:
56:
17:
12:
11:
5:
28204:
28194:
28193:
28188:
28183:
28178:
28173:
28156:
28155:
28145:
28142:
28141:
28139:
28138:
28131:
28126:
28121:
28116:
28111:
28106:
28101:
28096:
28091:
28086:
28081:
28076:
28071:
28066:
28061:
28059:Choquet-Bruhat
28056:
28051:
28046:
28041:
28036:
28031:
28026:
28021:
28016:
28011:
28006:
28001:
27996:
27991:
27986:
27980:
27978:
27974:
27973:
27970:
27969:
27967:
27966:
27959:
27958:
27953:
27948:
27941:
27940:
27935:
27930:
27925:
27920:
27911:Axisymmetric:
27908:
27907:
27902:
27896:
27885:
27884:
27879:
27874:
27869:
27864:
27859:
27850:Cosmological:
27847:
27845:
27839:
27838:
27836:
27835:
27830:
27825:
27819:
27817:
27811:
27810:
27808:
27807:
27802:
27791:frame-dragging
27788:
27783:
27778:
27775:Einstein rings
27771:Einstein cross
27764:
27753:
27752:
27747:
27741:
27736:
27731:
27718:
27708:
27707:
27702:
27697:
27692:
27686:
27684:
27680:
27679:
27677:
27676:
27674:Ernst equation
27671:
27666:
27661:
27656:
27651:
27646:
27644:BSSN formalism
27641:
27635:
27633:
27629:
27628:
27626:
27625:
27620:
27615:
27610:
27605:
27599:
27597:
27591:
27590:
27588:
27587:
27582:
27576:
27574:
27567:
27559:
27558:
27555:
27554:
27552:
27551:
27546:
27541:
27536:
27531:
27525:
27523:
27517:
27516:
27514:
27513:
27508:
27503:
27501:Ladder paradox
27498:
27493:
27488:
27483:
27478:
27473:
27467:
27465:
27461:
27460:
27458:
27457:
27451:
27449:
27445:
27444:
27442:
27441:
27436:
27431:
27426:
27421:
27416:
27411:
27406:
27404:Speed of light
27401:
27395:
27393:
27387:
27386:
27384:
27383:
27378:
27373:
27367:
27357:
27355:
27348:
27340:
27339:
27332:
27331:
27324:
27317:
27309:
27303:
27302:
27289:
27283:
27273:
27263:
27255:Animation clip
27251:
27245:
27232:
27226:
27218:
27217:External links
27215:
27214:
27213:
27197:
27192:
27179:
27162:(3): 211–230,
27142:
27139:
27137:
27136:
27130:
27114:Lifshitz, E.M.
27106:
27100:
27085:
27079:
27064:
27059:978-3211834435
27058:
27043:
27038:978-0805384918
27037:
27020:
27015:978-0521478144
27014:
26994:
26988:
26968:
26962:
26942:Thorne, Kip S.
26934:
26928:
26908:Feynman, R. P.
26904:
26898:
26878:Feynman, R. P.
26874:
26868:
26840:
26834:
26810:Jackson, J. D.
26806:
26800:
26792:Addison-Wesley
26780:
26775:
26759:
26754:
26738:
26733:
26717:
26711:
26696:
26690:
26673:
26667:
26651:
26645:
26632:
26626:
26612:Carroll, S. M.
26608:
26602:
26582:Wheeler, J. A.
26578:
26572:
26556:Wheeler, J. A.
26552:
26546:
26533:
26527:
26514:
26508:
26495:
26489:
26474:
26468:
26453:
26447:
26431:
26429:
26426:
26424:
26423:
26417:
26395:
26369:(5): 443–456.
26356:
26327:
26306:10.1.1.35.1131
26299:(2): 331–342.
26286:
26247:
26201:
26195:
26159:
26140:
26098:
26076:
26031:
26026:
25986:
25966:
25925:
25907:(9): 858–862.
25889:
25887:
25884:
25883:
25882:
25872:
25860:
25857:
25855:
25852:
25850:
25849:
25838:inspirehep.net
25825:
25813:
25802:
25790:
25787:Griffiths 2007
25779:
25768:
25756:
25744:
25729:
25717:
25705:
25694:
25667:(8): 517–525.
25647:
25635:
25623:
25612:
25602:
25582:
25571:
25560:
25548:
25535:
25531:Macrossan 1986
25523:
25511:
25499:
25488:
25477:
25466:
25451:
25440:
25430:
25409:
25407:
25404:
25401:
25400:
25386:
25377:
25368:
25357:
25345:
25333:
25319:position space
25310:
25286:linear algebra
25261:
25227:
25213:
25209:
25203:
25199:
25195:
25190:
25186:
25180:
25176:
25172:
25167:
25163:
25157:
25153:
25149:
25145:
25141:
25137:
25115:
25111:
25105:
25101:
25097:
25092:
25088:
25082:
25078:
25074:
25069:
25065:
25059:
25055:
25051:
25047:
25043:
25039:
25022:
24986:
24957:
24945:Poincaré group
24936:
24917:
24916:
24914:
24911:
24909:
24908:
24903:
24898:
24893:
24888:
24883:
24878:
24873:
24868:
24863:
24858:
24853:
24848:
24846:Ricci calculus
24842:
24840:
24837:
24795:
24794:
24785:
24783:
24768:
24763:
24760:
24755:
24751:
24746:
24742:
24738:
24732:
24728:
24724:
24721:
24716:
24712:
24707:
24703:
24699:
24693:
24689:
24685:
24681:
24676:
24672:
24668:
24664:
24659:
24655:
24651:
24648:
24645:
24642:
24638:
24632:
24627:
24623:
24619:
24612:
24608:
24603:
24599:
24595:
24590:
24585:
24581:
24576:
24572:
24568:
24565:
24562:
24559:
24555:
24549:
24544:
24540:
24536:
24529:
24525:
24520:
24516:
24512:
24507:
24501:
24497:
24493:
24489:
24484:
24480:
24476:
24471:
24466:
24458:
24453:
24448:
24444:
24438:
24433:
24429:
24423:
24418:
24414:
24408:
24404:
24400:
24397:
24392:
24388:
24382:
24378:
24374:
24371:
24361:
24357:
24354:
24349:
24345:
24339:
24335:
24331:
24328:
24325:
24320:
24316:
24310:
24306:
24302:
24299:
24295:
24289:
24285:
24281:
24278:
24274:
24270:
24266:
24263:
24262:
24257:
24254:
24249:
24245:
24239:
24235:
24229:
24225:
24221:
24216:
24212:
24206:
24202:
24196:
24192:
24187:
24183:
24180:
24177:
24174:
24171:
24168:
24165:
24163:
24143:noninteracting
24138:
24135:
24103:
24100:
24096:
24090:
24083:
24079:
24075:
24072:
24069:
24062:
24055:
24051:
24047:
24044:
24041:
24036:
24033:
24031:
24029:
24024:
24020:
24016:
24011:
24007:
24001:
23994:
23990:
23986:
23983:
23980:
23973:
23966:
23962:
23958:
23955:
23952:
23947:
23944:
23942:
23940:
23935:
23931:
23925:
23918:
23914:
23910:
23907:
23904:
23899:
23894:
23890:
23884:
23877:
23873:
23869:
23866:
23863:
23858:
23855:
23853:
23851:
23848:
23845:
23842:
23839:
23836:
23833:
23830:
23827:
23824:
23821:
23818:
23815:
23812:
23809:
23806:
23805:
23795:
23766:
23763:
23741:
23736:
23732:
23726:
23718:
23715:
23710:
23704:
23698:
23695:
23690:
23663:
23659:
23651:
23647:
23641:
23637:
23631:
23627:
23623:
23620:
23616:
23612:
23609:
23606:
23604:
23601:
23598:
23594:
23593:
23589:
23585:
23581:
23577:
23573:
23569:
23565:
23561:
23558:
23555:
23551:
23547:
23543:
23539:
23536:
23533:
23530:
23526:
23522:
23519:
23517:
23514:
23510:
23505:
23504:
23479:charge density
23464:
23461:
23458:
23455:
23450:
23447:
23443:
23437:
23429:
23426:
23421:
23413:
23405:
23402:
23397:
23391:
23387:
23382:
23379:
23373:
23370:
23366:
23361:
23355:
23352:
23348:
23342:
23334:
23331:
23326:
23318:
23310:
23307:
23302:
23296:
23292:
23288:
23285:
23281:
23274:
23271:
23266:
23263:
23258:
23183:
23178:
23175:
23171:
23165:
23157:
23154:
23149:
23141:
23133:
23130:
23125:
23119:
23113:
23110:
23105:
23102:
23097:
23085:geometric view
23040:
23035:
23030:
23025:
23021:
23017:
23013:
23009:
23004:
22999:
22996:
22992:
22986:
22981:
22976:
22972:
22968:
22963:
22958:
22952:
22948:
22945:
22942:
22940:
22935:
22932:
22926:
22921:
22920:
22917:
22912:
22907:
22902:
22898:
22894:
22890:
22886:
22881:
22876:
22873:
22869:
22863:
22858:
22853:
22849:
22845:
22840:
22835:
22829:
22825:
22822:
22819:
22817:
22812:
22809:
22803:
22798:
22797:
22792:
22787:
22782:
22779:
22777:
22772:
22769:
22763:
22758:
22757:
22752:
22747:
22742:
22739:
22737:
22732:
22729:
22723:
22718:
22717:
22678:
22673:
22668:
22663:
22659:
22655:
22651:
22647:
22642:
22637:
22634:
22631:
22629:
22627:
22622:
22618:
22614:
22611:
22608:
22603:
22599:
22595:
22592:
22589:
22584:
22580:
22576:
22573:
22570:
22567:
22564:
22561:
22558:
22555:
22550:
22546:
22542:
22539:
22536:
22533:
22530:
22527:
22525:
22523:
22518:
22514:
22508:
22501:
22497:
22489:
22482:
22478:
22472:
22467:
22463:
22457:
22450:
22446:
22438:
22431:
22427:
22421:
22416:
22413:
22409:
22403:
22396:
22392:
22384:
22377:
22373:
22367:
22362:
22359:
22355:
22349:
22342:
22338:
22330:
22323:
22319:
22313:
22307:
22304:
22299:
22296:
22291:
22287:
22284:
22282:
22277:
22274:
22269:
22265:
22264:
22259:
22254:
22249:
22245:
22241:
22237:
22233:
22228:
22223:
22220:
22217:
22215:
22213:
22208:
22204:
22200:
22197:
22194:
22189:
22185:
22181:
22178:
22173:
22169:
22165:
22162:
22159:
22156:
22153:
22150:
22147:
22144:
22141:
22136:
22132:
22128:
22125:
22122:
22119:
22116:
22113:
22111:
22109:
22104:
22100:
22094:
22087:
22083:
22075:
22068:
22064:
22058:
22053:
22049:
22043:
22036:
22032:
22024:
22017:
22013:
22007:
22002:
21999:
21995:
21989:
21982:
21978:
21970:
21963:
21959:
21953:
21948:
21945:
21941:
21935:
21928:
21924:
21916:
21909:
21905:
21899:
21893:
21890:
21885:
21882:
21877:
21873:
21870:
21868:
21863:
21860:
21855:
21851:
21850:
21847:
21842:
21838:
21834:
21831:
21829:
21827:
21822:
21818:
21814:
21809:
21805:
21801:
21798:
21795:
21790:
21786:
21782:
21777:
21773:
21767:
21763:
21759:
21756:
21751:
21747:
21743:
21738:
21734:
21728:
21724:
21720:
21717:
21712:
21708:
21704:
21701:
21698:
21695:
21690:
21686:
21682:
21679:
21676:
21673:
21670:
21667:
21664:
21661:
21658:
21655:
21652:
21649:
21646:
21643:
21641:
21639:
21634:
21630:
21624:
21617:
21613:
21605:
21598:
21594:
21588:
21583:
21579:
21573:
21566:
21562:
21554:
21547:
21543:
21537:
21532:
21529:
21525:
21519:
21512:
21508:
21500:
21493:
21489:
21483:
21477:
21474:
21469:
21466:
21461:
21457:
21454:
21452:
21447:
21444:
21439:
21435:
21434:
21406:
21401:
21396:
21392:
21388:
21384:
21380:
21375:
21370:
21367:
21364:
21362:
21360:
21355:
21351:
21347:
21344:
21341:
21336:
21332:
21328:
21325:
21320:
21316:
21312:
21309:
21306:
21303:
21300:
21295:
21291:
21287:
21284:
21281:
21278:
21275:
21272:
21269:
21266:
21263:
21260:
21258:
21256:
21251:
21247:
21241:
21234:
21230:
21222:
21215:
21211:
21205:
21200:
21196:
21190:
21183:
21179:
21171:
21164:
21160:
21154:
21149:
21146:
21142:
21136:
21129:
21125:
21117:
21110:
21106:
21100:
21095:
21092:
21088:
21082:
21075:
21071:
21063:
21056:
21052:
21046:
21040:
21037:
21032:
21029:
21024:
21020:
21017:
21015:
21010:
21007:
21002:
20998:
20997:
20992:
20987:
20982:
20978:
20974:
20970:
20966:
20961:
20956:
20953:
20950:
20948:
20946:
20941:
20937:
20933:
20930:
20927:
20922:
20918:
20914:
20911:
20906:
20902:
20898:
20895:
20892:
20889:
20886:
20881:
20877:
20873:
20870:
20867:
20864:
20861:
20858:
20855:
20852:
20849:
20846:
20843:
20841:
20839:
20834:
20830:
20824:
20817:
20813:
20805:
20798:
20794:
20788:
20783:
20779:
20773:
20766:
20762:
20754:
20747:
20743:
20737:
20732:
20729:
20725:
20719:
20712:
20708:
20700:
20693:
20689:
20683:
20678:
20675:
20671:
20665:
20658:
20654:
20646:
20639:
20635:
20629:
20623:
20620:
20615:
20612:
20607:
20603:
20600:
20598:
20593:
20590:
20585:
20581:
20580:
20577:
20572:
20568:
20564:
20561:
20559:
20557:
20552:
20548:
20544:
20541:
20538:
20535:
20532:
20527:
20523:
20517:
20510:
20506:
20498:
20491:
20487:
20481:
20476:
20473:
20469:
20463:
20456:
20452:
20444:
20437:
20433:
20427:
20421:
20418:
20413:
20410:
20405:
20401:
20398:
20396:
20391:
20388:
20383:
20379:
20378:
20356:
20351:
20348:
20344:
20338:
20330:
20327:
20322:
20314:
20306:
20303:
20298:
20292:
20286:
20283:
20278:
20275:
20270:
20240:
20232:
20229:
20226:
20223:
20220:
20217:
20214:
20211:
20208:
20198:
20192:
20189:
20185:
20181:
20177:
20174:
20170:
20166:
20162:
20158:
20154:
20150:
20147:
20146:
20141:
20137:
20133:
20131:
20128:
20124:
20120:
20116:
20113:
20109:
20105:
20101:
20098:
20097:
20092:
20088:
20084:
20081:
20077:
20073:
20069:
20067:
20064:
20060:
20056:
20052:
20049:
20048:
20043:
20039:
20035:
20031:
20027:
20023:
20019:
20015:
20011:
20009:
20006:
20005:
20003:
19998:
19993:
19990:
19986:
19981:
19976:
19970:
19967:
19965:
19962:
19960:
19957:
19955:
19952:
19951:
19948:
19945:
19943:
19940:
19938:
19935:
19933:
19930:
19929:
19926:
19923:
19921:
19918:
19916:
19913:
19911:
19908:
19905:
19902:
19901:
19898:
19895:
19893:
19890:
19888:
19885:
19882:
19879:
19877:
19874:
19873:
19871:
19866:
19861:
19854:
19850:
19797:
19789:
19786:
19783:
19780:
19777:
19774:
19771:
19768:
19765:
19755:
19749:
19746:
19742:
19738:
19734:
19730:
19726:
19722:
19719:
19715:
19711:
19705:
19702:
19697:
19696:
19691:
19687:
19683:
19680:
19678:
19675:
19671:
19667:
19663:
19659:
19655:
19649:
19646:
19641:
19640:
19635:
19631:
19627:
19623:
19619:
19615:
19612:
19610:
19607:
19603:
19599:
19593:
19590:
19585:
19584:
19579:
19575:
19569:
19566:
19561:
19558:
19554:
19550:
19544:
19541:
19536:
19533:
19529:
19525:
19519:
19516:
19511:
19508:
19506:
19503:
19502:
19500:
19495:
19490:
19487:
19483:
19468:
19467:
19443:
19422:electric field
19413:magnetic field
19399:Main article:
19388:
19385:
19368:
19344:
19339:
19336:
19333:
19330:
19325:
19322:
19319:
19316:
19312:
19306:
19298:
19295:
19290:
19284:
19279:
19271:
19268:
19263:
19255:
19247:
19244:
19239:
19231:
19223:
19220:
19215:
19209:
19204:
19196:
19193:
19188:
19180:
19172:
19169:
19164:
19158:
19152:
19149:
19145:
19141:
19138:
19133:
19130:
19123:
19120:
19116:
19112:
19109:
19104:
19101:
19096:
19085:
19034:
19029:
19026:
19022:
19016:
19009:
19005:
18997:
18990:
18986:
18980:
18975:
18971:
18967:
18962:
18958:
18952:
18945:
18941:
18933:
18926:
18922:
18916:
18911:
18907:
18901:
18894:
18890:
18884:
18879:
18875:
18869:
18862:
18858:
18852:
18849:
18846:
18843:
18840:
18837:
18834:
18831:
18828:
18825:
18815:
18813:transforms as
18798:
18753:
18750:
18747:
18744:
18741:
18738:
18735:
18732:
18729:
18726:
18723:
18720:
18717:
18714:
18711:
18708:
18704:
18701:
18698:
18695:
18692:
18689:
18686:
18683:
18680:
18677:
18674:
18671:
18668:
18665:
18662:
18659:
18656:
18646:
18624:tensor product
18593:
18590:
18563:
18560:
18554:
18548:
18543:
18540:
18536:
18532:
18527:
18523:
18520:
18504:
18470:
18466:
18460:
18453:
18448:
18443:
18440:
18436:
18432:
18425:
18420:
18416:
18410:
18403:
18399:
18393:
18388:
18382:
18379:
18354:
18349:
18345:
18339:
18332:
18328:
18322:
18317:
18311:
18308:
18285:
18280:
18273:
18268:
18263:
18260:
18256:
18252:
18245:
18240:
18233:
18229:
18188:
18183:
18176:
18171:
18166:
18163:
18159:
18155:
18148:
18143:
18140:
18136:
18130:
18123:
18119:
18111:
18108:
18104:
18081:
18076:
18072:
18066:
18063:
18059:
18053:
18046:
18042:
18034:
18031:
18027:
18023:
18018:
18012:
18009:
17987:
17971:
17955:
17933:
17928:
17924:
17918:
17915:
17911:
17907:
17902:
17898:
17867:
17862:
17858:
17852:
17849:
17845:
17841:
17836:
17832:
17815:
17812:
17751:
17745:
17741:
17735:
17728:
17724:
17720:
17717:
17714:
17709:
17704:
17698:
17695:
17666:
17660:
17656:
17650:
17642:
17639:
17634:
17628:
17622:
17619:
17614:
17591:
17585:
17581:
17575:
17568:
17564:
17558:
17553:
17547:
17544:
17483:
17478:
17474:
17468:
17461:
17457:
17451:
17446:
17440:
17437:
17400:
17392:
17388:
17384:
17383:
17378:
17374:
17370:
17369:
17364:
17360:
17356:
17355:
17350:
17346:
17342:
17341:
17339:
17332:
17324:
17317:
17313:
17307:
17303:
17296:
17292:
17286:
17282:
17275:
17271:
17265:
17261:
17254:
17250:
17244:
17243:
17238:
17231:
17227:
17221:
17217:
17210:
17206:
17200:
17196:
17189:
17185:
17179:
17175:
17168:
17164:
17158:
17157:
17152:
17145:
17141:
17135:
17131:
17124:
17120:
17114:
17110:
17103:
17099:
17093:
17089:
17082:
17078:
17072:
17071:
17066:
17059:
17055:
17049:
17045:
17038:
17034:
17028:
17024:
17017:
17013:
17007:
17003:
16996:
16992:
16986:
16985:
16983:
16978:
16973:
16965:
16959:
16956:
16950:
16949:
16944:
16938:
16935:
16929:
16928:
16923:
16917:
16914:
16908:
16907:
16902:
16896:
16893:
16887:
16886:
16884:
16870:
16867:
16863:Ricci calculus
16855:Main article:
16852:
16849:
16810:
16807:
16804:
16801:
16798:
16794:
16791:
16777:
16774:
16713:
16706:
16702:
16700:
16697:
16696:
16693:
16690:
16688:
16685:
16682:
16681:
16679:
16674:
16671:
16646:
16639:
16635:
16632:
16630:
16627:
16626:
16623:
16620:
16618:
16615:
16614:
16612:
16607:
16604:
16589:
16586:
16564:
16561:
16558:
16555:
16552:
16549:
16545:
16542:
16538:
16535:
16532:
16529:
16526:
16523:
16518:
16515:
16509:
16505:
16439:
16436:
16433:
16430:
16427:
16422:
16419:
16370:
16364:
16360:
16356:
16353:
16348:
16344:
16340:
16335:
16331:
16327:
16323:
16317:
16313:
16309:
16306:
16303:
16298:
16294:
16290:
16285:
16281:
16277:
16273:
16267:
16263:
16259:
16256:
16251:
16247:
16243:
16238:
16234:
16230:
16205:
16201:
16197:
16193:
16189:
16185:
16168:
16164:
16160:
16156:
16152:
16148:
16121:
16117:
16113:
16109:
16105:
16101:
16097:
16093:
16089:
16086:
16083:
16067:
16064:
16023:
16017:
16013:
16009:
16004:
15997:
15993:
15989:
15985:
15981:
15977:
15971:
15967:
15963:
15959:
15955:
15951:
15947:
15943:
15939:
15916:
15910:
15906:
15902:
15898:
15894:
15890:
15886:
15882:
15878:
15874:
15871:
15867:
15863:
15859:
15855:
15852:
15788:
15785:
15781:
15777:
15773:
15769:
15765:
15761:
15757:
15753:
15750:
15747:
15744:
15742:
15740:
15737:
15734:
15731:
15727:
15723:
15719:
15715:
15712:
15709:
15706:
15703:
15700:
15696:
15692:
15688:
15684:
15681:
15678:
15675:
15672:
15670:
15668:
15665:
15662:
15659:
15655:
15651:
15647:
15643:
15640:
15637:
15634:
15631:
15628:
15624:
15620:
15616:
15612:
15609:
15606:
15603:
15600:
15598:
15596:
15593:
15592:
15565:
15559:
15555:
15550:
15546:
15542:
15538:
15535:
15532:
15529:
15526:
15523:
15520:
15517:
15514:
15511:
15507:
15503:
15499:
15495:
15492:
15489:
15486:
15483:
15480:
15477:
15474:
15470:
15466:
15463:
15442:
15438:
15433:
15429:
15425:
15421:
15418:
15415:
15412:
15409:
15406:
15403:
15400:
15397:
15394:
15390:
15386:
15382:
15378:
15375:
15372:
15369:
15366:
15363:
15360:
15357:
15353:
15349:
15346:
15229:
15206:
15200:
15197:
15195:
15192:
15190:
15187:
15185:
15182:
15181:
15178:
15175:
15173:
15170:
15168:
15165:
15163:
15160:
15159:
15156:
15153:
15151:
15148:
15145:
15143:
15140:
15138:
15135:
15134:
15131:
15128:
15126:
15123:
15121:
15118:
15116:
15113:
15112:
15110:
15105:
15102:
15100:
15096:
15092:
15088:
15085:
15079:
15073:
15070:
15068:
15065:
15063:
15060:
15057:
15055:
15052:
15051:
15048:
15045:
15043:
15040:
15038:
15035:
15033:
15030:
15029:
15026:
15023:
15021:
15018:
15016:
15013:
15011:
15008:
15007:
15004:
15001:
14999:
14996:
14994:
14991:
14989:
14986:
14985:
14983:
14978:
14975:
14973:
14969:
14965:
14961:
14958:
14952:
14946:
14943:
14941:
14938:
14936:
14933:
14931:
14928:
14927:
14924:
14921:
14918:
14916:
14913:
14911:
14908:
14906:
14903:
14902:
14899:
14896:
14894:
14891:
14889:
14886:
14884:
14881:
14880:
14877:
14874:
14872:
14869:
14867:
14864:
14862:
14859:
14858:
14856:
14851:
14848:
14846:
14842:
14838:
14834:
14833:
14828:
14822:
14819:
14817:
14814:
14812:
14809:
14807:
14804:
14803:
14800:
14797:
14795:
14792:
14790:
14787:
14785:
14782:
14781:
14778:
14775:
14773:
14770:
14768:
14765:
14763:
14760:
14759:
14756:
14753:
14751:
14748:
14746:
14743:
14741:
14738:
14737:
14735:
14730:
14727:
14725:
14721:
14717:
14713:
14710:
14704:
14698:
14695:
14693:
14690:
14688:
14685:
14683:
14680:
14679:
14676:
14673:
14671:
14668:
14666:
14663:
14661:
14658:
14657:
14654:
14651:
14649:
14646:
14644:
14641:
14639:
14636:
14635:
14632:
14629:
14627:
14624:
14622:
14619:
14617:
14614:
14613:
14611:
14606:
14603:
14601:
14597:
14593:
14589:
14586:
14580:
14574:
14571:
14569:
14566:
14564:
14561:
14559:
14556:
14555:
14552:
14549:
14547:
14544:
14542:
14539:
14537:
14534:
14533:
14530:
14527:
14525:
14522:
14520:
14517:
14515:
14512:
14511:
14508:
14505:
14503:
14500:
14498:
14495:
14493:
14490:
14489:
14487:
14482:
14479:
14477:
14473:
14469:
14465:
14464:
14447:
14443:
14439:
14425:
14421:
14417:
14384:
14377:
14373:
14369:
14364:
14360:
14357:
14353:
14349:
14346:
14342:
14335:
14331:
14327:
14323:
14319:
14315:
14312:
14308:
14304:
14301:
14267:
14263:
14259:
14256:
14252:
14248:
14243:
14238:
14232:
14228:
14222:
14219:
14214:
14211:
14207:
14200:
14197:
14194:
14190:
14186:
14181:
14177:
14158:
14141:
14135:
14131:
14127:
14124:
14119:
14116:
14113:
14108:
14102:
14099:
14092:
14088:
14084:
14078:
14048:
14044:is small, and
14027:
14024:
14019:
14016:
14013:
14008:
14002:
13999:
13992:
13988:
13984:
13978:
13974:
13971:
13968:
13965:
13960:
13956:
13916:
13913:
13909:
13905:
13901:
13897:
13894:
13891:
13888:
13884:
13880:
13877:
13874:
13871:
13867:
13863:
13860:
13857:
13845:
13842:
13763:
13762:
13722:
13703:
13676:
13614:
13610:
13607:
13603:
13455:
13449:
13443:
13439:
13435:
13431:
13427:
13425:
13422:
13421:
13418:
13415:
13413:
13410:
13409:
13407:
13402:
13399:
13395:
13391:
13388:
13139:
13135:
13132:
13128:
13124:
13121:
13118:
13114:
13110:
13107:
13104:
13100:
13097:
13063:
13060:
13056:
13052:
13049:
13046:
13042:
13039:
13034:
13029:
13026:
13022:
13018:
13014:
13011:
13008:
13004:
13001:
12938:
12934:
12930:
12927:
12924:
12896:
12892:
12886:
12882:
12876:
12873:
12869:
12864:
12861:
12837:
12832:
12828:
12824:
12819:
12814:
12810:
12806:
12801:
12796:
12792:
12786:
12783:
12761:
12756:
12745:
12741:
12734:
12727:
12724:
12714:
12711:
12701:
12698:
12695:
12692:
12689:
12686:
12683:
12680:
12675:
12672:
12666:
12663:
12660:
12659:
12654:
12647:
12644:
12637:
12634:
12631:
12629:
12626:
12625:
12623:
12618:
12613:
12602:
12598:
12592:
12587:
12583:
12576:
12573:
12570:
12567:
12564:
12561:
12558:
12555:
12548:
12544:
12537:
12533:
12527:
12523:
12515:
12512:
12509:
12506:
12503:
12500:
12493:
12489:
12482:
12478:
12472:
12468:
12460:
12457:
12454:
12451:
12448:
12445:
12443:
12439:
12433:
12429:
12425:
12422:
12419:
12418:
12410:
12406:
12399:
12395:
12389:
12385:
12377:
12374:
12371:
12368:
12365:
12362:
12355:
12351:
12345:
12340:
12336:
12329:
12326:
12323:
12320:
12317:
12314:
12311:
12308:
12301:
12297:
12290:
12286:
12280:
12276:
12268:
12265:
12262:
12259:
12256:
12253:
12251:
12247:
12241:
12237:
12233:
12230:
12227:
12226:
12218:
12214:
12207:
12203:
12197:
12193:
12185:
12182:
12179:
12176:
12173:
12170:
12163:
12159:
12152:
12148:
12142:
12138:
12130:
12127:
12124:
12121:
12118:
12115:
12108:
12104:
12098:
12093:
12089:
12082:
12079:
12076:
12073:
12070:
12067:
12064:
12061:
12059:
12055:
12049:
12045:
12041:
12038:
12035:
12034:
12031:
12027:
12021:
12017:
12013:
12010:
12007:
12005:
12001:
11995:
11991:
11987:
11984:
11981:
11979:
11975:
11969:
11965:
11961:
11958:
11955:
11953:
11950:
11949:
11947:
11942:
11939:
11935:
11931:
11928:
11907:
11886:
11882:
11878:
11875:
11855:
11852:
11848:
11844:
11841:
11838:
11834:
11831:
11809:
11806:
11785:
11773:
11770:
11755:
11750:
11744:
11719:
11714:
11708:
11683:
11678:
11672:
11647:
11642:
11636:
11630:
11625:
11620:
11614:
11608:
11603:
11597:
11572:
11566:
11541:
11535:
11510:
11505:
11499:
11430:
11425:
11419:
11413:
11408:
11403:
11397:
11391:
11386:
11381:
11375:
11369:
11364:
11359:
11353:
11347:
11342:
11317:
11316:
11315:
11314:
11301:
11295:
11289:
11284:
11278:
11272:
11267:
11262:
11256:
11237:
11236:
11235:
11222:
11216:
11210:
11205:
11199:
11193:
11188:
11183:
11177:
11158:
11157:
11156:
11145:
11142:
11139:
11136:
11133:
11130:
11127:
11124:
11120:
11116:
11113:
11110:
11105:
11099:
11079:
11078:
11077:
11076:
11063:
11057:
11051:
11046:
11040:
11034:
11029:
11024:
11018:
10999:
10998:
10997:
10984:
10978:
10972:
10967:
10961:
10955:
10950:
10945:
10939:
10920:
10919:
10918:
10907:
10904:
10901:
10898:
10895:
10892:
10889:
10886:
10882:
10878:
10875:
10872:
10867:
10861:
10841:
10840:
10839:
10838:
10827:
10824:
10821:
10818:
10814:
10810:
10807:
10804:
10799:
10793:
10774:
10773:
10772:
10761:
10758:
10755:
10752:
10749:
10745:
10741:
10738:
10735:
10730:
10724:
10705:
10646:
10643:
10640:
10636:
10632:
10629:
10626:
10623:
10619:
10615:
10612:
10607:
10603:
10571:
10564:
10558:
10553:
10550:
10547:
10542:
10538:
10502:
10496:
10489:
10485:
10482:
10478:
10475:
10474:
10468:
10462:
10457:
10454:
10452:
10449:
10448:
10446:
10441:
10438:
10434:
10428:
10421:
10417:
10415:
10412:
10411:
10408:
10405:
10403:
10400:
10397:
10396:
10394:
10389:
10386:
10364:
10361:
10358:
10355:
10352:
10349:
10346:
10342:
10338:
10335:
10330:
10325:
10321:
10318:
10315:
10312:
10308:
10282:
10279:
10273:
10268:
10264:
10261:
10231:quadratic form
10196:
10174:
10143:
10140:
10137:
10133:
10130:
10103:
10100:
10095:
10089:
10082:
10079:
10073:
10070:
10067:
10061:
10056:
10052:
10049:
10046:
10043:
10013:
10007:
10004:
10003:
10000:
9997:
9996:
9993:
9990:
9989:
9986:
9982:
9979:
9978:
9976:
9971:
9968:
9964:
9958:
9952:
9949:
9947:
9944:
9942:
9939:
9937:
9934:
9933:
9930:
9927:
9925:
9922:
9920:
9917:
9915:
9912:
9911:
9908:
9905:
9903:
9900:
9898:
9895:
9893:
9890:
9889:
9886:
9883:
9881:
9878:
9876:
9873:
9871:
9868:
9865:
9864:
9862:
9857:
9854:
9850:
9844:
9837:
9834:
9830:
9829:
9825:
9822:
9818:
9817:
9813:
9810:
9806:
9805:
9801:
9798:
9793:
9790:
9789:
9787:
9782:
9778:
9775:
9753:
9750:
9740:linear algebra
9736:matrix product
9724:Main article:
9721:
9718:
9566:magnetic field
9557:electric field
9538:
9503:charge density
9485:
9484:
9473:
9452:
9446:
9445:
9434:
9420:Charge density
9417:
9411:
9410:
9399:
9395:
9388:
9382:
9381:
9370:
9349:
9343:
9342:
9331:
9310:
9304:
9303:
9292:
9275:
9268:
9267:
9260:
9253:
9131:
9125:
9120:
9116:
9113:
9110:
9104:
9100:
9096:
9092:
9088:
9084:
9080:
9077:
9074:
9071:
9068:
9065:
9062:
9058:
9054:
9051:
9049:
9046:
9042:
9037:
9036:
9033:
9028:
9022:
9017:
9013:
9009:
9005:
8999:
8996:
8992:
8988:
8985:
8982:
8980:
8977:
8974:
8970:
8969:
8948:
8944:
8939:
8935:
8931:
8926:
8921:
8915:
8912:
8906:
8902:
8898:
8894:
8890:
8885:
8881:
8866:
8858:
8850:
8828:
8821:
8814:
8796:
8793:
8701:
8700:
8688:
8683:
8678:
8673:
8669:
8665:
8660:
8653:
8650:
8644:
8639:
8631:
8626:
8616:
8612:
8608:
8603:
8599:
8595:
8587:
8582:
8577:
8570:
8559:
8555:
8549:
8545:
8541:
8534:
8531:
8527:
8522:
8518:
8514:
8498:
8497:
8478:
8474:
8468:
8464:
8460:
8452:
8449:
8445:
8440:
8434:
8429:
8424:
8416:
8413:
8409:
8403:
8399:
8394:
8388:
8384:
8380:
8374:
8367:
8364:
8358:
8354:
8348:
8344:
8275:
8272:
8237:
8233:
8230:
8225:
8222:
8218:
8213:
8209:
8205:
8201:
8198:
8194:
8173:
8169:
8166:
8163:
8159:
8155:
8151:
8147:
8144:
8140:
8085:
8080:
8076:
8072:
8069:
8065:
8062:
8058:
8054:
8050:
8046:
8042:
8038:
8033:
8030:
8027:
8024:
8021:
8018:
8015:
8011:
8007:
8002:
7999:
7997:
7994:
7990:
7989:
7986:
7981:
7973:
7969:
7963:
7959:
7956:
7952:
7948:
7940:
7936:
7933:
7928:
7924:
7921:
7918:
7916:
7914:
7911:
7910:
7879:
7813:
7808:
7804:
7801:
7798:
7795:
7791:
7787:
7783:
7779:
7775:
7771:
7768:
7765:
7762:
7759:
7756:
7753:
7749:
7745:
7742:
7740:
7737:
7733:
7728:
7727:
7724:
7719:
7711:
7707:
7701:
7697:
7693:
7689:
7683:
7680:
7676:
7672:
7669:
7666:
7664:
7661:
7658:
7654:
7653:
7622:
7606:
7602:
7598:
7594:
7590:
7586:
7583:
7579:
7575:
7570:
7565:
7559:
7554:
7550:
7546:
7542:
7538:
7534:
7531:
7526:
7521:
7500:and direction
7456:Introducing a
7400:
7395:
7390:
7387:
7385:
7382:
7378:
7373:
7368:
7367:
7364:
7361:
7357:
7353:
7348:
7343:
7338:
7335:
7332:
7329:
7327:
7324:
7320:
7315:
7310:
7309:
7305:
7297:
7293:
7287:
7283:
7278:
7273:
7265:
7262:
7258:
7254:
7251:
7248:
7246:
7243:
7240:
7236:
7235:
7215:
7210:
7206:
7201:
7196:
7192:
7188:
7183:
7178:
7174:
7170:
7164:
7158:
7153:
7148:
7143:
7138:
7133:
7129:
6964:
6961:
6960:
6959:
6885:
6880:
6804:
6799:
6783:
6779:
6774:
6771:
6767:
6764:
6758:
6755:
6751:
6748:
6744:
6705:
6676:
6673:
6670:
6667:
6664:
6661:
6659:
6656:
6653:
6649:
6648:
6645:
6642:
6639:
6637:
6634:
6631:
6627:
6626:
6553:
6550:
6549:
6548:
6542:
6524:
6498:
6491:
6484:
6477:
6462:
6455:
6448:
6441:
6434:
6431:
6408:
6401:
6366:
6361:
6356:
6353:
6349:
6345:
6342:
6338:
6335:
6331:
6327:
6323:
6320:
6317:
6315:
6313:
6310:
6307:
6306:
6303:
6298:
6290:
6286:
6280:
6277:
6273:
6269:
6263:
6259:
6256:
6252:
6248:
6244:
6241:
6238:
6236:
6234:
6231:
6228:
6227:
6203:
6198:
6194:
6191:
6187:
6184:
6181:
6178:
6174:
6170:
6167:
6164:
6162:
6159:
6156:
6152:
6149:
6148:
6145:
6140:
6132:
6128:
6123:
6120:
6116:
6110:
6107:
6104:
6100:
6096:
6093:
6090:
6088:
6085:
6082:
6078:
6075:
6074:
6018:
6015:
6011:
6008:
6006:
6004:
6001:
6000:
5996:
5993:
5989:
5986:
5984:
5982:
5979:
5978:
5975:
5972:
5969:
5965:
5962:
5958:
5955:
5952:
5949:
5946:
5942:
5939:
5935:
5932:
5930:
5928:
5925:
5924:
5921:
5918:
5915:
5911:
5908:
5904:
5901:
5898:
5895:
5891:
5888:
5884:
5881:
5878:
5876:
5874:
5871:
5868:
5867:
5839:
5748:
5744:
5741:
5736:
5733:
5729:
5725:
5722:
5696:
5692:
5689:
5686:
5683:
5680:
5678:
5676:
5673:
5670:
5669:
5666:
5662:
5659:
5656:
5653:
5650:
5648:
5646:
5643:
5642:
5639:
5635:
5632:
5629:
5626:
5623:
5621:
5619:
5616:
5615:
5575:
5568:
5565:
5560:
5556:
5552:
5549:
5545:
5540:
5537:
5534:
5531:
5501:
5494:
5491:
5488:
5483:
5480:
5477:
5471:
5468:
5465:
5462:
5422:
5418:
5415:
5412:
5409:
5404:
5400:
5396:
5393:
5390:
5385:
5381:
5282:
5279:
5276:
5274:
5271:
5268:
5264:
5263:
5260:
5257:
5254:
5252:
5249:
5246:
5242:
5241:
5238:
5235:
5232:
5229:
5226:
5223:
5220:
5217:
5214:
5211:
5208:
5205:
5203:
5200:
5197:
5193:
5192:
5189:
5186:
5183:
5180:
5177:
5174:
5171:
5168:
5165:
5162:
5159:
5156:
5154:
5151:
5148:
5144:
5141:
5140:
5112:
5085:
5064:
5061:
5058:
5055:
5052:
5049:
5017:
5012:
5007:
5004:
4998:
4994:
4991:
4988:
4965:
4960:
4954:
4951:
4950:
4947:
4944:
4941:
4938:
4935:
4934:
4931:
4928:
4925:
4922:
4919:
4918:
4915:
4912:
4909:
4906:
4903:
4902:
4900:
4893:
4885:
4880:
4876:
4869:
4866:
4863:
4857:
4853:
4847:
4844:
4841:
4837:
4833:
4827:
4823:
4816:
4813:
4810:
4804:
4800:
4794:
4790:
4786:
4780:
4776:
4769:
4766:
4763:
4757:
4753:
4747:
4743:
4739:
4735:
4732:
4729:
4728:
4723:
4719:
4713:
4709:
4702:
4699:
4696:
4690:
4686:
4680:
4676:
4671:
4667:
4660:
4657:
4654:
4648:
4644:
4638:
4635:
4632:
4628:
4624:
4618:
4614:
4607:
4604:
4601:
4595:
4591:
4585:
4581:
4577:
4573:
4570:
4567:
4566:
4561:
4557:
4551:
4547:
4540:
4537:
4534:
4528:
4524:
4518:
4514:
4510:
4504:
4500:
4493:
4490:
4487:
4481:
4477:
4471:
4467:
4462:
4458:
4451:
4448:
4445:
4439:
4435:
4429:
4426:
4423:
4419:
4415:
4411:
4408:
4405:
4404:
4399:
4395:
4391:
4388:
4385:
4381:
4377:
4373:
4370:
4367:
4363:
4359:
4355:
4352:
4349:
4347:
4344:
4343:
4341:
4336:
4331:
4324:
4321:
4317:
4316:
4313:
4310:
4306:
4303:
4299:
4298:
4295:
4292:
4289:
4285:
4282:
4278:
4277:
4274:
4271:
4268:
4264:
4261:
4257:
4254:
4253:
4251:
4227:
4223:
4218:
4214:
4210:
4188:
4128:
4123:
4119:
4116:
4113:
4110:
4107:
4103:
4099:
4096:
4093:
4091:
4088:
4085:
4081:
4080:
4077:
4072:
4068:
4065:
4062:
4059:
4056:
4052:
4048:
4045:
4042:
4040:
4037:
4034:
4030:
4027:
4026:
3965:
3961:
3958:
3954:
3951:
3949:
3947:
3944:
3943:
3939:
3936:
3932:
3929:
3927:
3925:
3922:
3921:
3917:
3912:
3909:
3905:
3902:
3898:
3895:
3890:
3886:
3883:
3880:
3878:
3876:
3873:
3872:
3868:
3860:
3856:
3850:
3847:
3843:
3837:
3833:
3830:
3825:
3821:
3818:
3815:
3813:
3811:
3808:
3807:
3784:
3609:complex number
3469:Lorentz factor
3445:
3441:
3435:
3431:
3425:
3422:
3418:
3413:
3410:
3399:speed of light
3361:
3358:
3355:
3353:
3350:
3347:
3343:
3342:
3339:
3336:
3333:
3331:
3328:
3325:
3321:
3320:
3316:
3312:
3309:
3306:
3303:
3299:
3295:
3292:
3289:
3287:
3284:
3281:
3277:
3276:
3272:
3264:
3260:
3255:
3252:
3246:
3243:
3239:
3235:
3232:
3229:
3227:
3224:
3221:
3217:
3216:
3193:
3143:′) = (0, 0, 0)
2925:
2922:
2914:
2911:
2864:Lorentz boosts
2843:
2840:
2836:Poincaré group
2802:
2801:
2792:
2790:
2779:
2776:
2773:
2769:
2766:
2762:
2759:
2755:
2752:
2749:
2746:
2743:
2740:
2736:
2732:
2729:
2725:
2722:
2718:
2715:
2711:
2707:
2704:
2700:
2697:
2694:
2691:
2688:
2685:
2682:
2679:
2676:
2673:
2670:
2667:
2664:
2661:
2658:
2582:
2581:
2572:
2570:
2559:
2555:
2552:
2548:
2544:
2541:
2537:
2534:
2531:
2528:
2518:
2514:
2511:
2507:
2503:
2500:
2496:
2493:
2490:
2487:
2484:
2481:
2478:
2456:bilinear forms
2454:that preserve
2442:
2441:
2432:
2430:
2414:
2410:
2406:
2401:
2397:
2393:
2389:
2385:
2381:
2377:
2372:
2368:
2364:
2360:
2356:
2352:
2348:
2343:
2339:
2335:
2331:
2327:
2323:
2319:
2314:
2310:
2306:
2300:
2296:
2292:
2287:
2283:
2277:
2273:
2269:
2264:
2260:
2254:
2250:
2246:
2241:
2237:
2231:
2227:
2223:
2218:
2214:
2208:
2204:
2198:
2194:
2190:
2182:
2181:
2175:
2171:
2167:
2163:
2157:
2153:
2149:
2145:
2139:
2135:
2131:
2127:
2121:
2117:
2113:
2107:
2103:
2099:
2094:
2090:
2086:
2081:
2077:
2073:
2068:
2064:
2060:
2055:
2051:
2045:
2041:
2037:
2035:
2009:
2000:
1935:
1934:
1925:
1923:
1908:
1897:
1893:
1888:
1884:
1880:
1876:
1872:
1868:
1864:
1860:
1857:
1852:
1848:
1843:
1839:
1835:
1831:
1827:
1823:
1819:
1815:
1812:
1807:
1803:
1798:
1794:
1790:
1786:
1782:
1778:
1774:
1770:
1767:
1762:
1758:
1753:
1749:
1745:
1741:
1737:
1733:
1729:
1725:
1720:
1716:
1712:
1708:
1705:
1704:
1699:
1695:
1689:
1685:
1681:
1676:
1672:
1668:
1665:
1660:
1656:
1650:
1646:
1642:
1637:
1633:
1629:
1626:
1621:
1617:
1611:
1607:
1603:
1598:
1594:
1590:
1587:
1582:
1578:
1572:
1568:
1564:
1559:
1555:
1551:
1546:
1542:
1538:
1536:
1509:
1502:
1495:
1488:
1481:
1471:
1464:
1457:
1450:
1443:
1426:
1425:
1416:
1414:
1397:
1394:
1389:
1385:
1379:
1375:
1371:
1366:
1362:
1358:
1355:
1350:
1346:
1340:
1336:
1332:
1327:
1323:
1319:
1316:
1311:
1307:
1301:
1297:
1293:
1288:
1284:
1280:
1277:
1272:
1268:
1262:
1258:
1254:
1249:
1245:
1241:
1236:
1232:
1164:
1161:
1122:Henri Poincaré
1087:electric field
1063:Woldemar Voigt
1055:Main article:
1052:
1049:
1045:Poincaré group
994:speed of light
901:
898:
895:
892:
890:
887:
884:
880:
879:
876:
873:
870:
868:
865:
862:
858:
857:
853:
849:
846:
843:
840:
837:
833:
829:
826:
823:
821:
818:
815:
811:
810:
806:
802:
799:
796:
793:
790:
786:
782:
779:
776:
774:
771:
768:
764:
761:
760:
740:
735:
732:
727:
724:
689:
654:Lorentz factor
639:
636:
631:
622:
618:
612:
608:
602:
599:
594:
589:
586:
575:speed of light
489:
486:
483:
481:
478:
475:
471:
470:
467:
464:
461:
459:
456:
453:
449:
448:
444:
440:
437:
434:
431:
427:
423:
420:
417:
415:
412:
409:
405:
404:
400:
392:
388:
383:
380:
374:
371:
367:
363:
360:
357:
355:
352:
349:
345:
344:
320:
317:
267:
266:
264:
263:
256:
249:
241:
238:
237:
235:
234:
221:
206:
203:
202:
198:
197:
192:
187:
182:
177:
172:
167:
162:
156:
155:
152:
151:
148:
147:
143:
142:
137:
131:
130:
127:
126:
123:
122:
118:
117:
112:
107:
101:
100:
97:
96:
93:
92:
88:
87:
82:
77:
72:
66:
65:
62:
61:
58:
57:
55:
54:
49:
43:
40:
39:
31:
30:
24:
23:
15:
9:
6:
4:
3:
2:
28203:
28192:
28189:
28187:
28184:
28182:
28179:
28177:
28174:
28172:
28169:
28168:
28166:
28153:
28143:
28137:
28136:
28132:
28130:
28127:
28125:
28122:
28120:
28117:
28115:
28112:
28110:
28107:
28105:
28102:
28100:
28097:
28095:
28092:
28090:
28087:
28085:
28082:
28080:
28077:
28075:
28072:
28070:
28067:
28065:
28062:
28060:
28057:
28055:
28052:
28050:
28047:
28045:
28044:Chandrasekhar
28042:
28040:
28037:
28035:
28032:
28030:
28027:
28025:
28022:
28020:
28017:
28015:
28012:
28010:
28007:
28005:
28004:Schwarzschild
28002:
28000:
27997:
27995:
27992:
27990:
27987:
27985:
27982:
27981:
27979:
27975:
27965:
27961:
27960:
27957:
27954:
27952:
27949:
27947:
27943:
27942:
27939:
27936:
27934:
27931:
27929:
27926:
27924:
27921:
27918:
27914:
27910:
27909:
27906:
27903:
27900:
27897:
27895:
27891:
27890:Schwarzschild
27887:
27886:
27883:
27880:
27878:
27875:
27873:
27870:
27868:
27865:
27863:
27860:
27857:
27853:
27849:
27848:
27846:
27844:
27840:
27834:
27831:
27829:
27826:
27824:
27821:
27820:
27818:
27812:
27806:
27803:
27800:
27796:
27792:
27789:
27787:
27786:Shapiro delay
27784:
27782:
27779:
27776:
27772:
27768:
27765:
27762:
27758:
27755:
27754:
27751:
27748:
27745:
27742:
27740:
27737:
27735:
27732:
27730:
27729:collaboration
27726:
27722:
27719:
27717:
27713:
27710:
27709:
27706:
27703:
27701:
27698:
27696:
27695:Event horizon
27693:
27691:
27688:
27687:
27685:
27681:
27675:
27672:
27670:
27667:
27665:
27662:
27660:
27657:
27655:
27652:
27650:
27647:
27645:
27642:
27640:
27639:ADM formalism
27637:
27636:
27634:
27630:
27624:
27621:
27619:
27616:
27614:
27611:
27609:
27606:
27604:
27601:
27600:
27598:
27592:
27586:
27583:
27581:
27578:
27577:
27575:
27571:
27568:
27566:
27560:
27550:
27547:
27545:
27544:Biquaternions
27542:
27540:
27537:
27535:
27532:
27530:
27527:
27526:
27524:
27522:
27518:
27512:
27509:
27507:
27504:
27502:
27499:
27497:
27494:
27492:
27489:
27487:
27484:
27482:
27479:
27477:
27474:
27472:
27471:Time dilation
27469:
27468:
27466:
27462:
27456:
27453:
27452:
27450:
27446:
27440:
27437:
27435:
27432:
27430:
27427:
27425:
27424:Proper length
27422:
27420:
27417:
27415:
27412:
27410:
27407:
27405:
27402:
27400:
27397:
27396:
27394:
27388:
27382:
27379:
27377:
27374:
27371:
27368:
27366:
27362:
27359:
27358:
27356:
27352:
27349:
27347:
27341:
27337:
27330:
27325:
27323:
27318:
27316:
27311:
27310:
27307:
27300:
27296:
27293:
27290:
27287:
27284:
27281:
27277:
27274:
27271:
27267:
27264:
27261:
27257:
27252:
27249:
27246:
27243:
27239:
27236:
27233:
27230:
27227:
27224:
27221:
27220:
27210:
27206:
27202:
27198:
27195:
27189:
27185:
27180:
27177:on 2011-07-16
27173:
27169:
27165:
27161:
27157:
27150:
27145:
27144:
27133:
27131:0-7506-2768-9
27127:
27123:
27119:
27115:
27111:
27107:
27103:
27097:
27093:
27092:
27086:
27082:
27076:
27072:
27071:
27065:
27061:
27055:
27051:
27050:
27044:
27040:
27034:
27029:
27028:
27021:
27017:
27011:
27007:
27003:
26999:
26995:
26991:
26985:
26981:
26977:
26973:
26969:
26965:
26959:
26955:
26954:W. H. Freeman
26951:
26947:
26943:
26939:
26935:
26931:
26925:
26921:
26917:
26913:
26909:
26905:
26901:
26895:
26891:
26887:
26883:
26879:
26875:
26871:
26869:0-7506-2768-9
26865:
26861:
26857:
26853:
26849:
26845:
26844:Landau, L. D.
26841:
26837:
26831:
26827:
26823:
26819:
26815:
26811:
26807:
26803:
26797:
26793:
26789:
26785:
26784:Goldstein, H.
26781:
26778:
26772:
26768:
26764:
26760:
26757:
26751:
26747:
26743:
26739:
26736:
26730:
26726:
26722:
26718:
26714:
26708:
26704:
26703:
26697:
26693:
26687:
26683:
26679:
26674:
26670:
26664:
26660:
26656:
26652:
26648:
26642:
26638:
26633:
26629:
26623:
26619:
26618:
26613:
26609:
26605:
26599:
26595:
26591:
26590:Misner, C. W.
26587:
26586:Thorne, K. S.
26583:
26579:
26575:
26569:
26565:
26561:
26557:
26553:
26549:
26543:
26539:
26534:
26530:
26524:
26520:
26515:
26511:
26505:
26501:
26496:
26492:
26486:
26482:
26481:
26475:
26471:
26465:
26461:
26460:
26454:
26450:
26444:
26441:. Macmillan.
26440:
26439:
26433:
26432:
26420:
26414:
26410:
26406:
26405:
26400:
26396:
26392:
26388:
26384:
26380:
26376:
26372:
26368:
26364:
26363:
26357:
26353:
26349:
26345:
26341:
26338:(2): 97–110.
26337:
26333:
26328:
26324:
26320:
26316:
26312:
26307:
26302:
26298:
26294:
26293:
26287:
26283:
26279:
26275:
26271:
26267:
26263:
26259:
26255:
26254:
26248:
26243:
26239:
26235:
26231:
26227:
26223:
26219:
26215:
26211:
26207:
26202:
26198:
26192:
26188:
26187:
26182:
26170:
26169:
26164:
26160:
26156:
26152:
26147:
26141:
26138:
26132:
26127:
26123:
26119:
26115:
26111:
26104:
26099:
26095:
26091:
26086:
26081:
26077:
26068:on 2013-10-29
26067:
26063:
26059:
26054:
26049:
26046:(2): 232–34,
26045:
26041:
26037:
26032:
26029:
26023:
26019:
26015:
26011:
26007:
26003:
25999:
25992:
25987:
25983:
25979:
25972:
25967:
25962:
25957:
25953:
25949:
25945:
25941:
25937:
25933:
25932:
25926:
25922:
25918:
25914:
25910:
25906:
25902:
25901:
25896:
25891:
25890:
25880:
25879:
25873:
25870:
25869:
25863:
25862:
25839:
25835:
25829:
25822:
25821:Weinberg 2002
25817:
25811:
25806:
25799:
25794:
25788:
25783:
25777:
25772:
25765:
25760:
25753:
25748:
25742:
25738:
25733:
25727:, p. 3–9
25726:
25721:
25714:
25713:Weinberg 2005
25709:
25703:
25702:Weinberg 1972
25698:
25690:
25686:
25682:
25678:
25674:
25670:
25666:
25662:
25658:
25651:
25645:, p. 239
25644:
25639:
25632:
25627:
25621:
25620:Einstein 1916
25616:
25610:
25605:
25599:
25595:
25594:
25586:
25580:
25575:
25569:
25564:
25557:
25556:Einstein 1905
25552:
25545:
25544:Poincaré 1905
25539:
25532:
25527:
25520:
25519:Darrigol 2005
25515:
25508:
25503:
25497:
25492:
25486:
25481:
25475:
25470:
25464:
25460:
25455:
25449:
25444:
25438:
25433:
25427:
25423:
25422:
25414:
25410:
25396:
25389:
25385:
25380:
25376:
25371:
25367:
25360:
25356:
25353:
25348:
25344:
25341:
25336:
25332:
25329:
25325:
25320:
25314:
25307:
25303:
25299:
25295:
25291:
25287:
25282:
25276:
25271:
25265:
25250:
25245:
25241:
25237:
25231:
25211:
25207:
25201:
25197:
25193:
25188:
25184:
25178:
25174:
25170:
25165:
25161:
25155:
25151:
25147:
25139:
25113:
25109:
25103:
25099:
25095:
25090:
25086:
25080:
25076:
25072:
25067:
25063:
25057:
25053:
25049:
25041:
25026:
25018:
25014:
25010:
25003:
24997:
24990:
24983:
24961:
24955:of spacetime.
24954:
24950:
24946:
24940:
24933:
24929:
24922:
24918:
24907:
24904:
24902:
24899:
24897:
24894:
24892:
24889:
24887:
24884:
24882:
24879:
24877:
24874:
24872:
24869:
24867:
24866:Lorentz group
24864:
24862:
24859:
24857:
24854:
24852:
24849:
24847:
24844:
24843:
24836:
24826:
24818:
24813:
24807:
24803:
24793:
24786:
24784:
24766:
24761:
24758:
24753:
24749:
24744:
24740:
24736:
24730:
24726:
24719:
24714:
24710:
24705:
24701:
24697:
24691:
24687:
24674:
24670:
24666:
24657:
24653:
24649:
24640:
24636:
24625:
24621:
24610:
24606:
24601:
24597:
24593:
24588:
24583:
24574:
24570:
24566:
24557:
24553:
24542:
24538:
24527:
24523:
24518:
24514:
24510:
24505:
24499:
24495:
24491:
24487:
24482:
24478:
24474:
24469:
24464:
24456:
24451:
24446:
24442:
24436:
24431:
24427:
24421:
24416:
24406:
24402:
24390:
24380:
24376:
24359:
24355:
24352:
24347:
24337:
24333:
24323:
24318:
24308:
24304:
24293:
24287:
24283:
24279:
24276:
24272:
24264:
24255:
24252:
24247:
24243:
24237:
24233:
24227:
24223:
24219:
24214:
24210:
24204:
24200:
24194:
24190:
24178:
24175:
24166:
24154:
24153:
24150:
24148:
24144:
24134:
24132:
24126:
24101:
24098:
24094:
24088:
24081:
24060:
24053:
24034:
24032:
24022:
24018:
24014:
24009:
24005:
23999:
23992:
23971:
23964:
23945:
23943:
23933:
23929:
23923:
23916:
23897:
23892:
23888:
23882:
23875:
23856:
23854:
23849:
23834:
23831:
23813:
23810:
23807:
23794:
23784:
23783:
23778:
23774:
23773:
23762:
23760:
23755:
23752:
23739:
23734:
23730:
23724:
23716:
23713:
23702:
23696:
23693:
23688:
23678:
23661:
23657:
23649:
23645:
23635:
23629:
23621:
23618:
23614:
23610:
23607:
23605:
23599:
23596:
23575:
23563:
23559:
23556:
23553:
23549:
23545:
23537:
23534:
23531:
23528:
23520:
23518:
23512:
23493:
23489:
23484:
23480:
23475:
23462:
23456:
23448:
23445:
23441:
23435:
23427:
23424:
23411:
23403:
23400:
23389:
23385:
23380:
23377:
23371:
23368:
23359:
23353:
23350:
23346:
23340:
23332:
23329:
23316:
23308:
23305:
23294:
23290:
23286:
23283:
23279:
23272:
23269:
23264:
23261:
23256:
23247:
23243:
23242:
23237:
23236:
23231:
23230:
23224:
23218:
23214:. The fields
23213:
23209:
23205:
23201:
23181:
23176:
23173:
23169:
23163:
23155:
23152:
23139:
23131:
23128:
23117:
23111:
23108:
23103:
23100:
23095:
23086:
23078:
23071:
23065:
23059:
23038:
23028:
23019:
23011:
23002:
22997:
22994:
22990:
22974:
22966:
22950:
22946:
22943:
22941:
22933:
22915:
22905:
22896:
22888:
22879:
22874:
22871:
22867:
22851:
22843:
22827:
22823:
22820:
22818:
22810:
22790:
22780:
22778:
22770:
22767:
22750:
22740:
22738:
22730:
22727:
22705:
22701:
22700:
22693:
22676:
22671:
22666:
22657:
22649:
22640:
22635:
22632:
22630:
22620:
22616:
22612:
22609:
22606:
22601:
22597:
22593:
22590:
22582:
22578:
22574:
22568:
22565:
22562:
22559:
22556:
22553:
22548:
22544:
22540:
22537:
22534:
22531:
22528:
22526:
22516:
22512:
22506:
22499:
22487:
22480:
22470:
22465:
22461:
22455:
22448:
22436:
22429:
22419:
22414:
22411:
22407:
22401:
22394:
22382:
22375:
22365:
22360:
22357:
22353:
22347:
22340:
22328:
22321:
22311:
22305:
22302:
22297:
22294:
22289:
22285:
22283:
22275:
22272:
22267:
22257:
22252:
22243:
22235:
22226:
22221:
22218:
22216:
22206:
22202:
22198:
22195:
22192:
22187:
22183:
22179:
22176:
22171:
22167:
22163:
22160:
22157:
22151:
22148:
22145:
22139:
22134:
22130:
22126:
22123:
22120:
22117:
22114:
22112:
22102:
22098:
22092:
22085:
22073:
22066:
22056:
22051:
22047:
22041:
22034:
22022:
22015:
22005:
22000:
21997:
21993:
21987:
21980:
21968:
21961:
21951:
21946:
21943:
21939:
21933:
21926:
21914:
21907:
21897:
21891:
21888:
21883:
21880:
21875:
21871:
21869:
21861:
21858:
21853:
21845:
21840:
21836:
21832:
21830:
21820:
21816:
21807:
21803:
21799:
21796:
21788:
21784:
21780:
21775:
21771:
21765:
21761:
21757:
21749:
21745:
21736:
21732:
21726:
21722:
21718:
21715:
21710:
21706:
21702:
21699:
21696:
21688:
21684:
21680:
21671:
21668:
21665:
21656:
21653:
21650:
21644:
21642:
21632:
21628:
21622:
21615:
21603:
21596:
21586:
21581:
21577:
21571:
21564:
21552:
21545:
21535:
21530:
21527:
21523:
21517:
21510:
21498:
21491:
21481:
21475:
21472:
21467:
21464:
21459:
21455:
21453:
21445:
21442:
21437:
21423:
21404:
21399:
21390:
21382:
21373:
21368:
21365:
21363:
21353:
21349:
21345:
21342:
21339:
21334:
21330:
21326:
21323:
21318:
21314:
21310:
21307:
21304:
21301:
21298:
21293:
21289:
21285:
21282:
21279:
21273:
21270:
21267:
21261:
21259:
21249:
21245:
21239:
21232:
21220:
21213:
21203:
21198:
21194:
21188:
21181:
21169:
21162:
21152:
21147:
21144:
21140:
21134:
21127:
21115:
21108:
21098:
21093:
21090:
21086:
21080:
21073:
21061:
21054:
21044:
21038:
21035:
21030:
21027:
21022:
21018:
21016:
21008:
21005:
21000:
20990:
20985:
20976:
20968:
20959:
20954:
20951:
20949:
20939:
20935:
20931:
20928:
20925:
20920:
20916:
20912:
20909:
20904:
20900:
20896:
20893:
20890:
20887:
20879:
20875:
20871:
20862:
20859:
20856:
20850:
20847:
20844:
20842:
20832:
20828:
20822:
20815:
20803:
20796:
20786:
20781:
20777:
20771:
20764:
20752:
20745:
20735:
20730:
20727:
20723:
20717:
20710:
20698:
20691:
20681:
20676:
20673:
20669:
20663:
20656:
20644:
20637:
20627:
20621:
20618:
20613:
20610:
20605:
20601:
20599:
20591:
20588:
20583:
20575:
20570:
20566:
20562:
20560:
20550:
20546:
20542:
20539:
20536:
20533:
20530:
20525:
20521:
20515:
20508:
20496:
20489:
20479:
20474:
20471:
20467:
20461:
20454:
20442:
20435:
20425:
20419:
20416:
20411:
20408:
20403:
20399:
20397:
20389:
20386:
20381:
20367:
20354:
20349:
20346:
20342:
20336:
20328:
20325:
20312:
20304:
20301:
20290:
20284:
20281:
20276:
20273:
20268:
20259:
20258:
20252:
20238:
20227:
20224:
20221:
20218:
20215:
20212:
20209:
20196:
20190:
20183:
20179:
20175:
20168:
20164:
20156:
20152:
20148:
20139:
20135:
20129:
20122:
20118:
20114:
20107:
20103:
20099:
20090:
20086:
20082:
20075:
20071:
20065:
20058:
20054:
20050:
20041:
20037:
20029:
20025:
20017:
20013:
20007:
20001:
19996:
19991:
19988:
19984:
19979:
19974:
19968:
19963:
19958:
19953:
19946:
19941:
19936:
19931:
19924:
19919:
19914:
19909:
19906:
19903:
19896:
19891:
19886:
19883:
19880:
19875:
19869:
19864:
19859:
19852:
19836:
19831:
19826:
19820:
19815:
19811:
19795:
19784:
19781:
19778:
19775:
19772:
19769:
19766:
19753:
19747:
19740:
19736:
19728:
19724:
19720:
19713:
19709:
19703:
19700:
19689:
19685:
19681:
19676:
19669:
19665:
19657:
19653:
19647:
19644:
19633:
19629:
19621:
19617:
19613:
19608:
19601:
19597:
19591:
19588:
19577:
19573:
19567:
19564:
19559:
19552:
19548:
19542:
19539:
19534:
19527:
19523:
19517:
19514:
19509:
19504:
19498:
19493:
19488:
19485:
19481:
19471:
19465:
19460:
19454:
19449:
19444:
19440:
19439:
19438:
19436:
19432:
19427:
19423:
19418:
19414:
19408:
19402:
19393:
19384:
19377:
19371:
19361:
19342:
19337:
19334:
19331:
19328:
19323:
19320:
19317:
19314:
19310:
19304:
19296:
19293:
19282:
19277:
19269:
19266:
19253:
19245:
19242:
19229:
19221:
19218:
19207:
19202:
19194:
19191:
19178:
19170:
19167:
19156:
19150:
19147:
19143:
19139:
19136:
19131:
19128:
19121:
19118:
19114:
19110:
19107:
19102:
19099:
19094:
19084:
19081:
19076:
19071:
19065:
19062:
19058:
19051:
19032:
19027:
19024:
19020:
19014:
19007:
18995:
18988:
18978:
18973:
18969:
18965:
18960:
18956:
18950:
18943:
18931:
18924:
18914:
18909:
18905:
18899:
18892:
18882:
18877:
18873:
18867:
18860:
18850:
18847:
18841:
18838:
18829:
18826:
18823:
18814:
18811:
18807:
18803:
18797:
18793:
18789:
18770:
18751:
18748:
18745:
18742:
18739:
18736:
18733:
18730:
18727:
18724:
18721:
18718:
18715:
18712:
18709:
18706:
18702:
18699:
18696:
18693:
18690:
18687:
18684:
18678:
18675:
18672:
18663:
18660:
18657:
18645:
18644:according to
18642:
18638:
18625:
18620:
18616:
18589:
18579:
18574:
18561:
18558:
18546:
18541:
18538:
18530:
18525:
18521:
18518:
18507:
18503:
18494:
18486:
18468:
18464:
18458:
18451:
18446:
18441:
18438:
18430:
18423:
18418:
18414:
18408:
18401:
18391:
18386:
18380:
18377:
18365:
18352:
18347:
18343:
18337:
18330:
18320:
18315:
18309:
18306:
18283:
18278:
18271:
18266:
18261:
18258:
18250:
18243:
18238:
18231:
18216:
18210:
18206:
18199:
18186:
18181:
18174:
18169:
18164:
18161:
18153:
18146:
18141:
18138:
18134:
18128:
18121:
18109:
18106:
18102:
18092:
18079:
18074:
18070:
18064:
18061:
18057:
18051:
18044:
18032:
18029:
18025:
18021:
18016:
18010:
18007:
17990:
17986:
17981:
17974:
17970:
17965:
17958:
17954:
17948:
17931:
17926:
17922:
17916:
17913:
17909:
17905:
17900:
17896:
17887:
17886:metric tensor
17882:
17865:
17860:
17856:
17850:
17847:
17843:
17839:
17834:
17830:
17821:
17811:
17809:
17808:Dirac indices
17805:
17796:
17787:
17778:
17774:
17769:
17749:
17743:
17739:
17733:
17726:
17707:
17702:
17696:
17693:
17681:
17664:
17658:
17654:
17648:
17640:
17637:
17626:
17620:
17617:
17612:
17602:
17589:
17583:
17579:
17573:
17566:
17556:
17551:
17545:
17542:
17532:
17527:
17522:
17517:
17515:
17511:
17507:
17503:
17499:
17494:
17481:
17476:
17472:
17466:
17459:
17449:
17444:
17438:
17435:
17424:
17420:
17416:
17398:
17390:
17386:
17376:
17372:
17362:
17358:
17348:
17344:
17337:
17330:
17322:
17315:
17301:
17294:
17280:
17273:
17259:
17252:
17236:
17229:
17215:
17208:
17194:
17187:
17173:
17166:
17150:
17143:
17129:
17122:
17108:
17101:
17087:
17080:
17064:
17057:
17043:
17036:
17022:
17015:
17001:
16994:
16981:
16976:
16971:
16963:
16957:
16954:
16942:
16936:
16933:
16921:
16915:
16912:
16900:
16894:
16891:
16882:
16864:
16858:
16848:
16846:
16842:
16838:
16837:
16832:
16828:
16824:
16808:
16805:
16802:
16796:
16792:
16789:
16773:
16769:
16765:
16758:
16751:
16741:
16739:
16735:
16730:
16711:
16698:
16691:
16686:
16683:
16677:
16672:
16669:
16662:
16661:time reversal
16644:
16633:
16628:
16621:
16616:
16610:
16605:
16602:
16595:
16585:
16583:
16579:
16562:
16556:
16553:
16550:
16530:
16527:
16524:
16507:
16503:
16495:
16490:
16486:
16484:
16480:
16476:
16472:
16468:
16464:
16460:
16456:
16453:
16434:
16431:
16428:
16408:
16403:
16401:
16397:
16396:
16390:
16386:
16368:
16362:
16358:
16354:
16346:
16342:
16338:
16333:
16329:
16321:
16315:
16311:
16307:
16304:
16296:
16292:
16288:
16283:
16279:
16271:
16265:
16261:
16257:
16249:
16245:
16241:
16236:
16232:
16220:
16217:Three of the
16215:
16213:
16208:
16180:
16176:
16171:
16143:
16139:
16135:
16111:
16103:
16095:
16084:
16081:
16073:
16063:
16061:
16057:
16053:
16048:
16042:
16037:
16021:
16011:
16002:
15991:
15983:
15979:
15975:
15965:
15957:
15949:
15941:
15937:
15927:
15914:
15904:
15896:
15888:
15880:
15876:
15872:
15861:
15840:
15836:
15832:
15828:
15820:
15816:
15812:
15808:
15786:
15783:
15775:
15767:
15759:
15751:
15748:
15745:
15743:
15732:
15729:
15721:
15713:
15710:
15701:
15698:
15690:
15682:
15679:
15673:
15671:
15660:
15657:
15649:
15641:
15638:
15629:
15626:
15618:
15610:
15607:
15601:
15599:
15582:
15581:infinitesimal
15577:
15563:
15557:
15544:
15530:
15527:
15524:
15521:
15518:
15512:
15501:
15490:
15487:
15484:
15481:
15478:
15475:
15461:
15440:
15427:
15413:
15410:
15407:
15404:
15401:
15395:
15384:
15373:
15370:
15367:
15364:
15361:
15358:
15344:
15335:
15331:
15325:
15319:
15312:
15308:
15302:
15296:
15289:
15283:
15277:
15273:
15266:
15262:
15257:
15252:
15247:
15243:
15238:
15232:
15223:
15204:
15198:
15193:
15188:
15183:
15176:
15171:
15166:
15161:
15154:
15149:
15146:
15141:
15136:
15129:
15124:
15119:
15114:
15108:
15103:
15101:
15094:
15090:
15083:
15077:
15071:
15066:
15061:
15058:
15053:
15046:
15041:
15036:
15031:
15024:
15019:
15014:
15009:
15002:
14997:
14992:
14987:
14981:
14976:
14974:
14967:
14963:
14956:
14950:
14944:
14939:
14934:
14929:
14922:
14919:
14914:
14909:
14904:
14897:
14892:
14887:
14882:
14875:
14870:
14865:
14860:
14854:
14849:
14847:
14840:
14836:
14826:
14820:
14815:
14810:
14805:
14798:
14793:
14788:
14783:
14776:
14771:
14766:
14761:
14754:
14749:
14744:
14739:
14733:
14728:
14726:
14719:
14715:
14708:
14702:
14696:
14691:
14686:
14681:
14674:
14669:
14664:
14659:
14652:
14647:
14642:
14637:
14630:
14625:
14620:
14615:
14609:
14604:
14602:
14595:
14591:
14584:
14578:
14572:
14567:
14562:
14557:
14550:
14545:
14540:
14535:
14528:
14523:
14518:
14513:
14506:
14501:
14496:
14491:
14485:
14480:
14478:
14471:
14467:
14454:
14450:
14435:
14428:
14413:
14407:
14401:
14395:
14382:
14371:
14362:
14358:
14344:
14340:
14329:
14321:
14317:
14313:
14299:
14291:
14289:
14285:
14265:
14261:
14257:
14254:
14250:
14246:
14241:
14236:
14230:
14226:
14220:
14217:
14212:
14209:
14205:
14192:
14184:
14179:
14175:
14166:
14161:
14152:
14139:
14133:
14129:
14125:
14122:
14117:
14114:
14111:
14106:
14100:
14090:
14086:
14065:
14060:
14056:
14051:
14042:
14025:
14022:
14017:
14014:
14011:
14006:
14000:
13990:
13986:
13972:
13969:
13966:
13963:
13958:
13954:
13943:
13938:
13932:
13930:
13903:
13889:
13875:
13872:
13858:
13841:
13837:
13833:
13826:
13822:
13814:
13810:
13806:
13802:
13794:
13790:
13786:
13782:
13774:
13770:
13759:
13755:
13750:
13746:
13741:
13736:
13731:
13726:
13723:
13718:
13714:
13708:
13704:
13701:
13697:
13693:
13689:
13685:
13680:
13677:
13672:
13668:
13664:
13660:
13653:
13649:
13645:
13641:
13636:
13633:
13632:
13631:
13627:
13608:
13605:
13592:
13587:
13581:
13577:
13572:
13567:
13562:
13558:
13552:
13547:
13543:
13538:
13534:
13529:
13523:
13517:
13510:
13502:
13495:
13489:
13484:
13480:
13474:
13470:
13453:
13447:
13423:
13416:
13411:
13405:
13400:
13386:
13377:
13373:
13369:
13363:
13356:
13351:
13345:
13338:
13330:
13325:
13320:
13315:
13308:
13304:
13300:
13296:
13290:
13284:
13280:
13276:
13272:
13265:
13261:
13257:
13253:
13247:
13241:
13235:
13232:
13226:
13220:
13213:
13209:
13202:
13198:
13194:
13190:
13186:
13182:
13178:
13174:
13169:
13165:
13160:
13154:
13137:
13133:
13119:
13105:
13102:
13098:
13095:
13085:
13078:
13061:
13047:
13044:
13040:
13037:
13032:
13027:
13024:
13009:
13006:
13002:
12999:
12988:
12982:
12975:
12969:
12963:
12956:
12950:
12928:
12922:
12894:
12890:
12884:
12880:
12874:
12871:
12867:
12862:
12859:
12835:
12830:
12826:
12822:
12817:
12812:
12808:
12804:
12799:
12794:
12790:
12784:
12781:
12772:
12759:
12754:
12743:
12739:
12732:
12722:
12709:
12696:
12693:
12690:
12684:
12681:
12670:
12664:
12661:
12652:
12642:
12635:
12632:
12627:
12621:
12616:
12611:
12600:
12596:
12590:
12585:
12581:
12571:
12568:
12565:
12559:
12556:
12546:
12542:
12535:
12531:
12525:
12521:
12510:
12507:
12504:
12491:
12487:
12480:
12476:
12470:
12466:
12455:
12452:
12449:
12441:
12437:
12431:
12427:
12423:
12420:
12408:
12404:
12397:
12393:
12387:
12383:
12372:
12369:
12366:
12353:
12349:
12343:
12338:
12334:
12324:
12321:
12318:
12312:
12309:
12299:
12295:
12288:
12284:
12278:
12274:
12263:
12260:
12257:
12249:
12245:
12239:
12235:
12231:
12228:
12216:
12212:
12205:
12201:
12195:
12191:
12180:
12177:
12174:
12161:
12157:
12150:
12146:
12140:
12136:
12125:
12122:
12119:
12106:
12102:
12096:
12091:
12087:
12077:
12074:
12071:
12065:
12062:
12057:
12053:
12047:
12043:
12039:
12036:
12029:
12025:
12019:
12015:
12011:
12008:
12003:
11999:
11993:
11989:
11985:
11982:
11977:
11973:
11967:
11963:
11959:
11956:
11951:
11945:
11940:
11926:
11873:
11853:
11839:
11836:
11832:
11829:
11807:
11804:
11783:
11769:
11748:
11712:
11676:
11640:
11628:
11618:
11606:
11601:
11539:
11503:
11485:
11474:
11468:
11461:
11452:
11448:
11443:
11423:
11411:
11401:
11389:
11379:
11367:
11357:
11345:
11331:
11330:disjoint sets
11327:
11322:
11287:
11282:
11270:
11260:
11244:
11243:
11241:
11238:
11208:
11203:
11191:
11181:
11165:
11164:
11162:
11159:
11140:
11137:
11134:
11118:
11108:
11103:
11087:
11086:
11084:
11081:
11080:
11049:
11044:
11032:
11022:
11006:
11005:
11003:
11000:
10970:
10965:
10953:
10943:
10927:
10926:
10924:
10921:
10902:
10899:
10896:
10880:
10870:
10865:
10849:
10848:
10846:
10843:
10842:
10822:
10819:
10812:
10802:
10781:
10780:
10778:
10777:Orthochronous
10775:
10756:
10753:
10750:
10743:
10733:
10712:
10711:
10709:
10706:
10703:
10702:
10699:
10697:
10696:intersections
10693:
10689:
10688:s for brevity
10687:
10681:
10677:
10672:
10666:
10665:time symmetry
10657:
10644:
10641:
10634:
10630:
10627:
10624:
10613:
10610:
10605:
10590:
10587:
10551:
10548:
10545:
10540:
10526:
10522:
10518:
10500:
10494:
10476:
10455:
10444:
10439:
10432:
10426:
10413:
10406:
10401:
10398:
10392:
10387:
10384:
10375:
10362:
10359:
10356:
10336:
10333:
10328:
10323:
10306:
10296:
10277:
10262:
10259:
10250:
10248:
10244:
10240:
10236:
10232:
10227:
10223:
10218:
10217:
10216:Lorentz group
10212:
10164:
10159:
10141:
10135:
10131:
10128:
10119:
10101:
10098:
10093:
10080:
10077:
10071:
10068:
10065:
10054:
10050:
10047:
10044:
10041:
10033:
10011:
10005:
9998:
9991:
9984:
9980:
9974:
9969:
9966:
9962:
9956:
9950:
9945:
9940:
9935:
9928:
9923:
9918:
9913:
9906:
9901:
9896:
9891:
9884:
9879:
9874:
9869:
9866:
9860:
9855:
9852:
9848:
9842:
9835:
9832:
9823:
9820:
9811:
9808:
9799:
9796:
9791:
9785:
9780:
9776:
9773:
9763:
9759:
9749:
9745:
9741:
9737:
9733:
9727:
9726:Lorentz group
9717:
9715:
9709:
9705:
9701:
9697:
9693:
9686:
9679:
9675:
9671:
9667:
9663:
9657:
9652:
9651:Lorentz force
9647:
9641:
9636:
9631:
9628:
9624:
9620:
9616:
9612:
9606:
9599:
9595:
9591:
9585:
9581:
9577:
9571:
9567:
9562:
9558:
9553:
9549:
9544:
9541:
9532:
9527:
9523:
9517:
9512:
9508:
9504:
9499:
9493:
9482:
9477:
9474:
9471:
9467:
9461:
9456:
9453:
9451:
9448:
9447:
9443:
9438:
9435:
9432:
9426:
9421:
9418:
9416:
9413:
9412:
9408:
9403:
9400:
9394:
9389:
9387:
9384:
9383:
9379:
9374:
9371:
9368:
9364:
9358:
9353:
9350:
9348:
9345:
9344:
9340:
9335:
9332:
9329:
9325:
9319:
9314:
9311:
9309:
9308:Four-momentum
9306:
9305:
9301:
9296:
9293:
9290:
9284:
9279:
9276:
9274:
9270:
9269:
9265:
9261:
9258:
9254:
9251:
9250:
9247:
9244:
9238:
9232:
9226:
9221:
9220:
9213:
9209:
9202:
9199:
9195:
9188:
9184:
9176:
9172:
9165:
9158:
9152:
9146:
9129:
9123:
9114:
9111:
9108:
9102:
9086:
9072:
9069:
9066:
9060:
9052:
9050:
9044:
9031:
9026:
9020:
9011:
9003:
8997:
8994:
8990:
8986:
8983:
8981:
8975:
8972:
8946:
8937:
8933:
8924:
8919:
8913:
8910:
8904:
8896:
8888:
8883:
8879:
8864:
8856:
8848:
8844:
8837:
8827:
8820:
8813:
8809:
8803:
8792:
8790:
8785:
8783:
8779:
8775:
8770:
8767:
8760:
8756:, and change
8753:
8747:
8741:
8734:
8728:
8723:
8719:
8713:
8707:
8686:
8676:
8667:
8658:
8651:
8648:
8637:
8624:
8614:
8610:
8606:
8601:
8593:
8580:
8568:
8557:
8553:
8543:
8532:
8529:
8525:
8520:
8516:
8503:
8502:
8501:
8476:
8472:
8462:
8450:
8447:
8443:
8438:
8427:
8422:
8414:
8411:
8407:
8401:
8392:
8386:
8382:
8372:
8365:
8362:
8352:
8346:
8334:
8333:
8332:
8324:
8320:
8316:
8309:
8303:
8295:
8290:
8285:
8281:
8271:
8267:
8259:
8252:
8235:
8231:
8228:
8223:
8220:
8216:
8207:
8199:
8196:
8171:
8167:
8164:
8161:
8153:
8145:
8142:
8128:
8125:
8119:
8113:
8107:
8100:
8083:
8074:
8070:
8067:
8063:
8060:
8044:
8040:
8025:
8022:
8019:
8013:
8009:
8000:
7998:
7984:
7979:
7971:
7967:
7957:
7954:
7950:
7938:
7934:
7931:
7926:
7922:
7919:
7917:
7912:
7898:
7894:
7890:
7886:
7885:in direction
7882:
7878:
7875:
7869:
7865:
7859:
7855:
7848:
7842:
7835:
7828:
7811:
7802:
7799:
7796:
7793:
7777:
7763:
7760:
7757:
7751:
7743:
7741:
7735:
7722:
7717:
7709:
7705:
7695:
7687:
7681:
7678:
7674:
7670:
7667:
7665:
7659:
7656:
7641:
7637:
7633:
7629:
7628:in direction
7625:
7624:Lorentz boost
7621:
7618:
7592:
7581:
7573:
7568:
7557:
7540:
7529:
7524:
7509:
7504:
7498:
7492:
7489:
7485:
7479:
7475:
7471:
7467:
7463:
7459:
7454:
7450:
7443:
7439:
7435:
7429:
7424:
7398:
7388:
7386:
7380:
7376:
7359:
7351:
7333:
7330:
7328:
7322:
7303:
7295:
7291:
7281:
7276:
7263:
7260:
7256:
7252:
7249:
7247:
7241:
7238:
7213:
7208:
7194:
7190:
7186:
7176:
7172:
7162:
7146:
7141:
7131:
7117:
7111:
7104:
7098:
7092:
7088:
7083:
7080:
7076:
7069:
7063:
7058:
7050:
7046:
7038:
7034:
7028:
7018:
7011:
7004:
6998:
6991:
6985:
6979:
6974:
6970:
6955:
6948:
6942:
6935:
6931:
6927:
6919:
6912:
6904:
6898:
6891:
6886:
6884:
6881:
6877:
6870:
6866:
6862:
6854:
6846:
6840:
6836:
6832:
6824:
6817:
6810:
6805:
6803:
6802:Time dilation
6800:
6781:
6777:
6772:
6765:
6762:
6756:
6753:
6749:
6746:
6732:
6726:
6719:
6712:
6706:
6704:
6701:
6700:
6699:
6696:
6694:
6674:
6671:
6668:
6665:
6662:
6660:
6654:
6651:
6643:
6640:
6638:
6632:
6629:
6617:
6612:
6609:
6605:
6601:
6593:
6589:
6582:
6576:
6572:
6568:direction is
6566:
6560:
6541:
6537:
6533:
6523:
6519:
6515:
6507:
6497:
6490:
6483:
6476:
6470:
6461:
6454:
6447:
6440:
6435:
6432:
6428:
6427:
6426:
6424:
6419:
6416:
6407:
6400:
6396:
6390:
6381:
6364:
6359:
6354:
6351:
6343:
6340:
6336:
6333:
6325:
6321:
6318:
6316:
6311:
6301:
6296:
6288:
6284:
6278:
6275:
6267:
6261:
6257:
6254:
6246:
6242:
6239:
6237:
6232:
6201:
6196:
6192:
6185:
6182:
6179:
6172:
6168:
6165:
6163:
6157:
6154:
6143:
6138:
6130:
6126:
6121:
6114:
6108:
6105:
6098:
6094:
6091:
6089:
6083:
6080:
6064:
6061:
6057:
6052:
6048:
6041:
6034:
6016:
6013:
6009:
6007:
6002:
5994:
5991:
5987:
5985:
5980:
5973:
5970:
5967:
5963:
5960:
5956:
5953:
5950:
5947:
5944:
5940:
5937:
5933:
5931:
5926:
5919:
5916:
5913:
5909:
5906:
5902:
5899:
5896:
5893:
5889:
5886:
5882:
5879:
5877:
5872:
5869:
5855:
5851:
5847:
5842:
5838:
5835:
5831:
5825:
5821:
5814:
5807:
5800:
5793:
5787:
5781:
5774:
5769:, it follows
5766:
5759:
5746:
5742:
5739:
5734:
5731:
5727:
5723:
5720:
5711:
5694:
5690:
5687:
5684:
5681:
5679:
5674:
5671:
5664:
5660:
5657:
5654:
5651:
5649:
5644:
5637:
5633:
5630:
5627:
5624:
5622:
5617:
5604:
5598:
5592:
5586:
5573:
5566:
5563:
5558:
5554:
5550:
5547:
5543:
5538:
5535:
5532:
5529:
5520:
5515:
5499:
5492:
5489:
5486:
5481:
5478:
5475:
5469:
5466:
5463:
5460:
5451:
5445:
5439:
5433:
5420:
5416:
5413:
5410:
5407:
5402:
5398:
5394:
5391:
5388:
5383:
5379:
5369:
5362:
5355:
5350:
5345:
5343:
5339:
5334:
5329:
5325:
5320:
5315:
5314:
5309:
5304:
5297:
5280:
5277:
5275:
5269:
5266:
5258:
5255:
5253:
5247:
5244:
5236:
5233:
5230:
5227:
5224:
5221:
5218:
5215:
5212:
5209:
5206:
5204:
5198:
5195:
5187:
5184:
5181:
5178:
5175:
5172:
5169:
5166:
5163:
5160:
5157:
5155:
5149:
5146:
5142:
5128:
5124:
5120:
5115:
5114:Lorentz boost
5111:
5108:
5103:
5098:
5059:
5056:
5053:
5047:
5039:
5035:
5015:
5005:
5002:
4996:
4992:
4989:
4986:
4976:
4963:
4958:
4952:
4936:
4920:
4907:
4904:
4898:
4891:
4883:
4878:
4874:
4867:
4864:
4861:
4855:
4851:
4845:
4842:
4835:
4831:
4825:
4821:
4814:
4811:
4808:
4802:
4798:
4788:
4784:
4778:
4774:
4767:
4764:
4761:
4755:
4751:
4741:
4737:
4733:
4730:
4721:
4717:
4711:
4707:
4700:
4697:
4694:
4688:
4684:
4674:
4669:
4665:
4658:
4655:
4652:
4646:
4642:
4636:
4633:
4626:
4622:
4616:
4612:
4605:
4602:
4599:
4593:
4589:
4579:
4575:
4571:
4568:
4559:
4555:
4549:
4545:
4538:
4535:
4532:
4526:
4522:
4512:
4508:
4502:
4498:
4491:
4488:
4485:
4479:
4475:
4465:
4460:
4456:
4449:
4446:
4443:
4437:
4433:
4427:
4424:
4417:
4413:
4409:
4406:
4397:
4393:
4389:
4386:
4379:
4375:
4371:
4368:
4361:
4357:
4353:
4350:
4345:
4339:
4334:
4329:
4322:
4319:
4304:
4301:
4283:
4280:
4262:
4259:
4255:
4249:
4239:
4225:
4221:
4212:
4176:
4173:
4167:
4163:. The use of
4160:
4155:, it follows
4153:
4147:
4126:
4121:
4117:
4114:
4111:
4108:
4105:
4101:
4097:
4094:
4092:
4086:
4083:
4075:
4070:
4066:
4063:
4060:
4057:
4054:
4050:
4046:
4043:
4041:
4035:
4032:
4028:
4015:
4011:) instead of
4010:
4005:
4001:
3997:
3991:
3987:
3980:
3963:
3959:
3956:
3952:
3950:
3945:
3937:
3934:
3930:
3928:
3923:
3915:
3910:
3907:
3903:
3900:
3896:
3893:
3888:
3884:
3881:
3879:
3874:
3866:
3858:
3854:
3848:
3845:
3841:
3835:
3831:
3828:
3823:
3819:
3816:
3814:
3809:
3796:
3792:
3787:
3783:
3781:
3776:
3769:
3765:
3761:
3757:
3750:
3743:
3737:
3730:
3723:
3717:
3711:
3704:
3698:
3691:
3684:
3680:
3676:
3672:
3666:
3662:
3658:
3654:
3648:
3646:
3640:
3635:
3630:
3622:
3617:
3612:
3610:
3605:
3599:
3595:
3590:
3585:
3579:
3575:
3569:
3563:
3559:
3552:
3546:
3542:
3538:
3531:
3525:
3518:
3511:
3504:
3497:
3490:
3485:
3484:
3478:
3472:
3470:
3466:
3443:
3439:
3433:
3429:
3423:
3420:
3416:
3411:
3408:
3400:
3395:
3389:
3383:
3376:
3359:
3356:
3354:
3348:
3345:
3337:
3334:
3332:
3326:
3323:
3314:
3310:
3307:
3304:
3301:
3297:
3293:
3290:
3288:
3282:
3279:
3270:
3262:
3258:
3253:
3250:
3244:
3241:
3237:
3233:
3230:
3228:
3222:
3219:
3205:
3201:
3196:
3195:Lorentz boost
3192:
3190:
3184:
3178:
3174:
3170:
3166:
3160:
3154:
3152:
3148:
3142:
3138:
3134:
3130:
3122:
3118:
3112:
3105:
3099:
3092:
3086:
3079:
3073:
3067:
3063:
3059:
3055:
3051:
3044:
3038:
3032:
3025:
3019:
3015:
3011:
3007:
3001:
2990:
2983:
2977:
2971:
2966:
2960:
2954:
2949:
2943:
2938:
2932:
2928:
2920:
2910:
2908:
2902:
2900:
2896:
2892:
2883:
2881:
2877:
2873:
2869:
2865:
2860:
2857:
2853:
2849:
2839:
2837:
2833:
2829:
2825:
2821:
2820:
2811:
2810:
2800:
2793:
2791:
2777:
2774:
2771:
2767:
2764:
2760:
2757:
2753:
2747:
2744:
2741:
2730:
2723:
2716:
2713:
2709:
2705:
2702:
2695:
2689:
2683:
2680:
2671:
2665:
2662:
2659:
2649:
2648:
2645:
2639:
2635:
2634:Lorentz group
2626:
2622:
2617:
2612:
2608:
2607:
2601:
2593:
2580:
2573:
2571:
2557:
2553:
2550:
2546:
2542:
2539:
2535:
2532:
2529:
2526:
2512:
2509:
2505:
2501:
2498:
2491:
2485:
2482:
2479:
2469:
2468:
2465:
2463:
2462:
2457:
2453:
2449:
2440:
2433:
2431:
2412:
2408:
2404:
2399:
2395:
2391:
2387:
2383:
2379:
2375:
2370:
2366:
2362:
2358:
2354:
2350:
2346:
2341:
2337:
2333:
2329:
2325:
2321:
2317:
2312:
2308:
2304:
2298:
2294:
2290:
2285:
2281:
2275:
2271:
2267:
2262:
2258:
2252:
2248:
2244:
2239:
2235:
2229:
2225:
2221:
2216:
2212:
2206:
2202:
2196:
2192:
2173:
2169:
2165:
2161:
2155:
2151:
2147:
2143:
2137:
2133:
2129:
2125:
2119:
2115:
2111:
2105:
2101:
2097:
2092:
2088:
2084:
2079:
2075:
2071:
2066:
2062:
2058:
2053:
2049:
2043:
2039:
2026:
2025:
2022:
2020:
2016:
2008:
1999:
1993:
1984:
1983:
1976:
1972:
1968:
1964:
1956:
1952:
1948:
1944:
1933:
1926:
1924:
1906:
1895:
1886:
1882:
1878:
1874:
1870:
1866:
1862:
1855:
1850:
1841:
1837:
1833:
1829:
1825:
1821:
1817:
1810:
1805:
1796:
1792:
1788:
1784:
1780:
1776:
1772:
1765:
1760:
1751:
1747:
1743:
1739:
1735:
1731:
1727:
1718:
1714:
1706:
1697:
1687:
1683:
1679:
1674:
1670:
1663:
1658:
1648:
1644:
1640:
1635:
1631:
1624:
1619:
1609:
1605:
1601:
1596:
1592:
1585:
1580:
1570:
1566:
1562:
1557:
1553:
1544:
1540:
1527:
1526:
1523:
1521:
1517:
1508:
1501:
1494:
1487:
1480:
1470:
1463:
1456:
1449:
1442:
1437:
1433:
1432:light signals
1424:
1417:
1415:
1395:
1392:
1387:
1377:
1373:
1369:
1364:
1360:
1353:
1348:
1338:
1334:
1330:
1325:
1321:
1314:
1309:
1299:
1295:
1291:
1286:
1282:
1275:
1270:
1260:
1256:
1252:
1247:
1243:
1234:
1230:
1222:
1221:
1218:
1216:
1215:
1210:
1205:
1202:
1198:
1194:
1190:
1187:and a set of
1186:
1182:
1181:
1174:
1173:Lorentz group
1170:
1160:
1158:
1154:
1150:
1146:
1142:
1137:
1135:
1134:time dilation
1131:
1127:
1123:
1119:
1115:
1111:
1106:
1104:
1100:
1096:
1092:
1088:
1084:
1080:
1076:
1072:
1071:Joseph Larmor
1068:
1064:
1058:
1048:
1046:
1042:
1037:
1033:
1029:
1028:Lorentz boost
1025:
1020:
1018:
1014:
1010:
1006:
1001:
999:
995:
991:
987:
986:elapsed times
983:
979:
975:
971:
967:
963:
958:
956:
952:
948:
944:
940:
935:
933:
929:
925:
921:
916:
899:
896:
893:
891:
885:
882:
874:
871:
869:
863:
860:
851:
847:
844:
841:
838:
835:
831:
827:
824:
822:
816:
813:
804:
800:
797:
794:
791:
788:
784:
780:
777:
775:
769:
766:
762:
738:
733:
730:
725:
722:
713:
710:
704:
687:
678:
672:
666:
660:
656:. When speed
655:
637:
634:
629:
620:
616:
610:
606:
600:
597:
592:
587:
584:
576:
571:
567:-axis, where
561:
554:
548:
541:
537:
533:
529:
521:
517:
513:
509:
487:
484:
482:
476:
473:
465:
462:
460:
454:
451:
442:
438:
435:
432:
429:
425:
421:
418:
416:
410:
407:
398:
390:
386:
381:
378:
372:
369:
365:
361:
358:
356:
350:
347:
318:
315:
306:
304:
301:
297:
293:
289:
285:
282:
278:
274:
262:
257:
255:
250:
248:
243:
242:
240:
239:
232:
222:
219:
214:
208:
207:
205:
204:
196:
193:
191:
188:
186:
183:
181:
178:
176:
173:
171:
168:
166:
163:
161:
158:
157:
150:
149:
141:
138:
136:
133:
132:
125:
124:
116:
113:
111:
108:
106:
103:
102:
95:
94:
86:
83:
81:
78:
76:
73:
71:
68:
67:
60:
59:
53:
50:
48:
45:
44:
42:
41:
37:
33:
32:
29:
26:
25:
21:
20:
28134:
27828:Kaluza–Klein
27580:Introduction
27506:Twin paradox
27454:
27298:
27294:
27208:
27204:
27183:
27172:the original
27159:
27155:
27117:
27110:Landau, L.D.
27090:
27069:
27052:. Springer.
27048:
27026:
27001:
26998:Ryder, L. H.
26975:
26949:
26919:
26889:
26851:
26817:
26814:"Chapter 11"
26787:
26766:
26745:
26742:Weinberg, S.
26724:
26721:Weinberg, S.
26701:
26677:
26658:
26636:
26616:
26593:
26563:
26560:Taylor, E. F
26537:
26518:
26499:
26479:
26458:
26437:
26403:
26399:Weinberg, S.
26366:
26360:
26335:
26331:
26296:
26290:
26257:
26251:
26212:(1): 55–89.
26209:
26205:
26185:
26181:Einstein, A.
26172:. Retrieved
26167:
26163:Einstein, A.
26154:
26150:
26135:. See also:
26113:
26109:
26093:
26089:
26070:, retrieved
26066:the original
26043:
26039:
26001:
25997:
25981:
25977:
25935:
25929:
25904:
25898:
25877:
25867:
25841:. Retrieved
25837:
25828:
25816:
25805:
25798:Jackson 1975
25793:
25782:
25771:
25766:, p. 22
25764:Carroll 2004
25759:
25747:
25732:
25725:Ohlsson 2011
25720:
25708:
25697:
25664:
25660:
25650:
25638:
25626:
25615:
25592:
25585:
25574:
25563:
25551:
25538:
25526:
25514:
25507:Rothman 2006
25502:
25491:
25480:
25474:Lorentz 1904
25469:
25454:
25443:
25420:
25413:
25394:
25387:
25383:
25378:
25374:
25369:
25365:
25358:
25354:
25351:
25346:
25342:
25339:
25334:
25330:
25327:
25323:
25313:
25298:real numbers
25290:vector space
25280:
25274:
25264:
25248:
25230:
25029:Explicitly,
25025:
25016:
25012:
25008:
25001:
24995:
24989:
24960:
24948:
24939:
24927:
24921:
24829:-dimensional
24824:
24816:
24805:
24801:
24798:
24787:
24142:
24140:
24128:
23796:
23780:
23770:
23768:
23756:
23753:
23679:
23491:
23482:
23476:
23245:
23239:
23233:
23227:
23222:
23220:(alone) and
23216:
23211:
23207:
23203:
23199:
23084:
23069:
23063:
23057:
22703:
22698:
22697:
22694:
21424:
20368:
20255:
20253:
19834:
19824:
19818:
19472:
19469:
19458:
19452:
19447:
19425:
19416:
19410:
19375:
19366:
19363:
19086:
19079:
19069:
19066:
19060:
19056:
19053:
18816:
18809:
18805:
18801:
18795:
18791:
18787:
18772:
18647:
18640:
18636:
18618:
18614:
18595:
18575:
18505:
18501:
18492:
18484:
18366:
18214:
18208:
18204:
18200:
18093:
17988:
17984:
17979:
17972:
17968:
17963:
17956:
17952:
17946:
17880:
17819:
17817:
17807:
17794:
17785:
17776:
17772:
17679:
17603:
17525:
17521:four-vectors
17518:
17513:
17495:
16872:
16844:
16840:
16834:
16830:
16826:
16822:
16779:
16767:
16763:
16756:
16749:
16742:
16728:
16591:
16491:
16487:
16482:
16454:
16452:vector space
16404:
16393:
16388:
16384:
16216:
16183:
16178:
16146:
16142:vector space
16069:
16055:
16046:
16040:
16036:in principle
16035:
15928:
15838:
15834:
15830:
15826:
15818:
15814:
15810:
15806:
15580:
15578:
15336:
15329:
15323:
15317:
15310:
15306:
15300:
15294:
15287:
15281:
15275:
15271:
15264:
15260:
15255:
15250:
15241:
15236:
15227:
15224:
14455:
14437:
14433:
14415:
14411:
14405:
14399:
14396:
14292:
14167:is obtained
14156:
14153:
14063:
14054:
14046:
14040:
13941:
13933:
13847:
13835:
13831:
13824:
13820:
13812:
13808:
13804:
13800:
13792:
13788:
13784:
13780:
13772:
13768:
13764:
13757:
13753:
13739:
13729:
13716:
13712:
13699:
13695:
13691:
13687:
13683:
13670:
13666:
13662:
13658:
13651:
13647:
13643:
13639:
13628:
13585:
13579:
13570:
13568:is used for
13563:
13556:
13550:
13545:
13541:
13527:
13521:
13515:
13508:
13500:
13493:
13487:
13482:
13472:
13468:
13376:block matrix
13372:Euler angles
13361:
13354:
13343:
13336:
13328:
13323:
13318:
13313:
13306:
13302:
13298:
13294:
13288:
13282:
13278:
13274:
13270:
13263:
13259:
13255:
13251:
13245:
13239:
13236:
13230:
13224:
13218:
13211:
13207:
13200:
13196:
13192:
13188:
13184:
13180:
13176:
13172:
13158:
13152:
13083:
13076:
12986:
12984:relative to
12980:
12973:
12967:
12961:
12954:
12951:
12773:
11775:
11483:
11472:
11466:
11459:
11444:
11323:
11320:
11239:
11160:
11082:
11001:
10922:
10844:
10776:
10708:Antichronous
10707:
10691:
10685:
10683:
10679:
10675:
10673:
10658:
10588:
10585:
10524:
10520:
10516:
10376:
10251:
10225:
10221:
10214:
10160:
9761:
9755:
9747:
9707:
9703:
9699:
9695:
9691:
9684:
9677:
9673:
9669:
9665:
9661:
9655:
9645:
9639:
9629:
9626:
9622:
9618:
9614:
9610:
9604:
9597:
9593:
9589:
9583:
9579:
9575:
9569:
9560:
9551:
9545:
9536:
9530:
9515:
9507:mass density
9497:
9491:
9488:
9480:
9469:
9465:
9459:
9457:(divided by
9441:
9430:
9424:
9415:Four-current
9406:
9392:
9377:
9366:
9362:
9356:
9354:(divided by
9338:
9327:
9323:
9317:
9315:(divided by
9299:
9288:
9282:
9263:
9256:
9252:Four-vector
9242:
9236:
9230:
9224:
9217:
9211:
9207:
9203:
9197:
9193:
9186:
9182:
9174:
9170:
9163:
9156:
9150:
9147:
8862:
8854:
8846:
8842:
8835:
8825:
8818:
8811:
8807:
8801:
8798:
8786:
8771:
8765:
8758:
8751:
8745:
8739:
8732:
8726:
8721:
8717:
8711:
8705:
8702:
8499:
8330:
8322:
8318:
8314:
8307:
8301:
8265:
8257:
8250:
8129:
8123:
8117:
8111:
8105:
8102:
7901:
7896:
7892:
7888:
7884:
7880:
7873:
7867:
7863:
7857:
7853:
7846:
7840:
7833:
7830:
7644:
7639:
7635:
7631:
7627:
7623:
7619:
7502:
7496:
7490:
7487:
7483:
7477:
7473:
7469:
7465:
7461:
7455:
7448:
7441:
7437:
7433:
7427:
7115:
7109:
7102:
7096:
7090:
7084:
7078:
7074:
7067:
7061:
7056:
7048:
7042:
7036:
7032:
7026:
7016:
7009:
7002:
6996:
6989:
6983:
6953:
6946:
6940:
6933:
6929:
6925:
6917:
6910:
6902:
6896:
6889:
6875:
6868:
6864:
6860:
6852:
6844:
6838:
6834:
6830:
6822:
6815:
6808:
6730:
6724:
6717:
6710:
6697:
6613:
6607:
6603:
6599:
6591:
6587:
6580:
6574:
6570:
6564:
6558:
6555:
6539:
6535:
6531:
6521:
6517:
6513:
6505:
6495:
6488:
6481:
6474:
6468:
6459:
6452:
6445:
6438:
6422:
6420:
6414:
6405:
6398:
6394:
6382:
6065:
6055:
6053:
6046:
6039:
6036:
5858:
5853:
5849:
5845:
5840:
5833:
5829:
5826:
5819:
5812:
5805:
5798:
5791:
5785:
5779:
5772:
5764:
5760:
5712:
5602:
5596:
5590:
5587:
5518:
5449:
5443:
5437:
5434:
5367:
5360:
5353:
5348:
5346:
5332:
5318:
5311:
5302:
5299:
5131:
5126:
5122:
5118:
5113:
5106:
5099:
4977:
4240:
4177:
4171:
4165:
4158:
4151:
4145:
4013:
4003:
3999:
3995:
3992:
3985:
3982:
3798:
3794:
3790:
3785:
3779:
3774:
3767:
3763:
3759:
3755:
3748:
3741:
3739:relative to
3735:
3728:
3721:
3715:
3709:
3702:
3696:
3689:
3682:
3678:
3674:
3670:
3668:in terms of
3664:
3660:
3656:
3652:
3649:
3638:
3633:
3628:
3620:
3613:
3603:
3597:
3593:
3583:
3577:
3573:
3567:
3564:
3557:
3550:
3544:
3540:
3536:
3529:
3523:
3516:
3509:
3502:
3495:
3488:
3481:
3476:
3473:
3393:
3391:-direction,
3387:
3381:
3378:
3207:
3203:
3199:
3194:
3188:
3187:records the
3182:
3176:
3172:
3168:
3164:
3158:
3155:
3151:synchronized
3150:
3146:
3140:
3136:
3132:
3128:
3120:
3116:
3110:
3103:
3097:
3090:
3084:
3077:
3071:
3068:
3061:
3057:
3053:
3049:
3042:
3036:
3034:relative to
3030:
3023:
3017:
3013:
3009:
3005:
2999:
2996:
2988:
2981:
2975:
2969:
2964:
2958:
2952:
2947:
2941:
2936:
2927:
2906:
2903:
2884:
2879:
2876:Euler angles
2867:
2863:
2861:
2847:
2845:
2842:Generalities
2831:
2827:
2817:
2807:
2805:
2794:
2624:
2615:
2610:
2604:
2599:
2585:
2574:
2459:
2445:
2434:
2018:
2014:
2006:
1997:
1991:
1980:
1974:
1970:
1966:
1962:
1954:
1950:
1946:
1942:
1938:
1927:
1515:
1506:
1499:
1492:
1485:
1478:
1468:
1461:
1454:
1447:
1440:
1435:
1431:
1429:
1418:
1213:
1206:
1200:
1196:
1192:
1184:
1178:
1176:
1138:
1129:
1125:
1107:
1060:
1027:
1021:
1002:
959:
936:
931:
924:non-inertial
917:
714:
708:
702:
676:
670:
664:
658:
569:
559:
552:
546:
539:
535:
531:
527:
519:
515:
511:
507:
307:
276:
270:
180:Curved space
79:
27917:Kerr–Newman
27888:Spherical:
27757:Other tests
27700:Singularity
27632:Formulation
27594:Fundamental
27448:Formulation
27429:Proper time
27390:Fundamental
26972:Rindler, W.
26950:Gravitation
26824:. pp.
26763:Ohlsson, T.
26596:. Freeman.
26594:Gravitation
26566:. Freeman.
26096:: 1504–1508
25823:, Chapter 3
25754:, Chapter 4
25304:), since a
25278:, velocity
24964:The groups
24932:Sard (1970)
18485:a row index
17798:. E.g., if
16471:bilinearity
16467:Lie bracket
16407:Lie algebra
16212:coordinates
13707:determinant
13576:unit vector
13370:variables,
12952:If a frame
10583:is useful;
10295:determinant
9714:given below
9522:rest energy
9390:(No name),
9373:wave vector
9273:four-vector
9219:four-vector
7458:unit vector
7423:dot product
6728:. Then in
6423:differences
6387:(uppercase
5306:(lowercase
5034:determinant
4007:(lowercase
3463:(lowercase
2891:reflections
674:approaches
160:Four-vector
28165:Categories
28069:Zel'dovich
27977:Scientists
27956:Alcubierre
27763:of Mercury
27761:precession
27690:Black hole
27573:Background
27565:relativity
27534:World line
27529:Light cone
27354:Background
27346:relativity
27336:Relativity
27235:Relativity
26174:2012-01-23
26157:: 809–831.
26072:2007-04-02
25854:References
25843:2024-09-04
25800:, p.
25739:, p.
25631:Barut 1964
25496:Brown 2003
25461:, p.
24928:particular
24141:A general
24131:Fock space
18634:, denoted
16732:is the 3d
16582:surjective
16477:, and the
16465:(called a
16395:commutator
16210:, are the
16140:, forms a
14282:where the
13745:orthogonal
13368:axis-angle
10237:O(3,1), a
8780:, and the
7071:, so that
6578:, then in
5349:difference
4017:, so that
3778:notes the
2979:along the
2950:along the
2866:or simply
1118:local time
978:velocities
563:along the
28181:Spacetime
28039:Robertson
28024:Friedmann
28019:Eddington
28009:de Sitter
27843:Solutions
27721:detectors
27716:astronomy
27683:Phenomena
27618:Geodesics
27521:Spacetime
27464:Phenomena
27116:(2002) .
27000:(1996) .
26916:Sands, M.
26886:Sands, M.
26850:(2002) .
26812:(1975) .
26786:(1980) .
26727:, Wiley,
26725:Cosmology
26705:. Wiley.
26391:122472788
26352:123543303
26323:118634052
26301:CiteSeerX
26246:eqn (55).
26242:121240925
26234:0894-9875
26048:CiteSeerX
25984:(2): 112f
25834:"INSPIRE"
25752:Hall 2003
25689:0002-9505
25198:θ
25175:θ
25152:θ
25140:⋅
25136:θ
25100:ζ
25077:ζ
25054:ζ
25042:⋅
25038:ζ
24913:Footnotes
24762:⋯
24737:σ
24723:Λ
24698:σ
24684:Λ
24680:Ψ
24671:⋯
24647:Λ
24607:σ
24594:σ
24564:Λ
24524:σ
24511:σ
24500:⋯
24488:σ
24475:σ
24470:∑
24457:⋯
24422:⋯
24399:Λ
24373:Λ
24356:⋯
24348:μ
24330:Λ
24319:μ
24301:Λ
24288:μ
24277:−
24256:⋯
24234:σ
24201:σ
24186:Ψ
24173:Λ
24102:σ
24099:β
24089:σ
24082:ρ
24074:Λ
24068:Π
24061:β
24054:α
24046:Λ
24040:Π
24035:≡
24023:σ
24015:⊗
24010:β
24000:σ
23993:ρ
23985:Λ
23979:Π
23972:β
23965:α
23957:Λ
23951:Π
23934:σ
23924:σ
23917:ρ
23909:Λ
23903:Π
23898:⊗
23893:β
23883:β
23876:α
23868:Λ
23862:Π
23844:Λ
23838:Π
23835:⊗
23826:Λ
23820:Π
23817:→
23811:⊗
23769:Equation
23735:μ
23725:μ
23714:μ
23709:Λ
23694:μ
23630:⋅
23622:−
23619:ρ
23611:γ
23597:ρ
23576:⋅
23557:−
23554:γ
23535:ρ
23532:γ
23529:−
23449:ν
23446:μ
23436:ν
23425:ν
23420:Λ
23412:μ
23401:μ
23396:Λ
23369:−
23365:Λ
23354:ν
23351:μ
23341:ν
23330:ν
23325:Λ
23317:μ
23306:μ
23301:Λ
23270:ν
23262:μ
23204:spacetime
23177:ν
23174:μ
23164:ν
23153:ν
23148:Λ
23140:μ
23129:μ
23124:Λ
23109:ν
23101:μ
23034:⊥
23020:×
23016:β
23012:−
22998:γ
22985:⊥
22975:×
22971:β
22967:−
22962:⊥
22947:γ
22931:⊥
22911:⊥
22897:×
22893:β
22875:γ
22862:⊥
22852:×
22848:β
22839:⊥
22824:γ
22808:⊥
22791:∥
22768:∥
22751:∥
22728:∥
22658:×
22654:β
22636:γ
22613:γ
22610:β
22594:γ
22575:−
22569:×
22563:×
22560:γ
22557:β
22554:−
22541:×
22535:×
22532:γ
22496:Λ
22477:Λ
22445:Λ
22426:Λ
22412:μ
22391:Λ
22383:μ
22372:Λ
22361:ν
22358:μ
22348:ν
22337:Λ
22329:μ
22318:Λ
22244:×
22240:β
22222:γ
22199:γ
22196:β
22193:−
22180:γ
22164:×
22158:×
22152:γ
22149:β
22146:−
22127:×
22121:×
22118:γ
22082:Λ
22063:Λ
22031:Λ
22012:Λ
21998:μ
21977:Λ
21969:μ
21958:Λ
21947:ν
21944:μ
21934:ν
21923:Λ
21915:μ
21904:Λ
21817:γ
21804:β
21800:−
21762:γ
21733:β
21723:γ
21719:−
21703:γ
21700:γ
21681:−
21672:β
21669:γ
21666:−
21657:β
21654:γ
21651:−
21612:Λ
21593:Λ
21561:Λ
21542:Λ
21531:ν
21528:μ
21518:ν
21507:Λ
21499:μ
21488:Λ
21391:×
21387:β
21383:−
21369:γ
21346:γ
21343:β
21340:−
21327:γ
21311:×
21305:×
21302:γ
21286:×
21280:×
21274:β
21271:γ
21268:−
21229:Λ
21210:Λ
21178:Λ
21159:Λ
21145:μ
21124:Λ
21116:μ
21105:Λ
21094:ν
21091:μ
21081:ν
21070:Λ
21062:μ
21051:Λ
20977:×
20973:β
20969:−
20955:γ
20932:γ
20929:β
20913:γ
20897:γ
20894:×
20872:−
20863:γ
20860:β
20857:−
20851:×
20812:Λ
20793:Λ
20761:Λ
20742:Λ
20731:ν
20718:ν
20707:Λ
20688:Λ
20677:ν
20674:μ
20664:ν
20653:Λ
20645:μ
20634:Λ
20543:×
20537:×
20505:Λ
20486:Λ
20475:ν
20472:μ
20462:ν
20451:Λ
20443:μ
20432:Λ
20350:ν
20347:μ
20337:ν
20326:ν
20321:Λ
20313:μ
20302:μ
20297:Λ
20282:ν
20274:μ
20210:−
20176:−
20149:−
20115:−
20100:−
20083:−
20051:−
19992:ν
19989:μ
19915:γ
19910:β
19907:γ
19904:−
19887:β
19884:γ
19881:−
19876:γ
19860:ν
19853:μ
19849:Λ
19785:−
19779:−
19773:−
19721:−
19682:−
19614:−
19560:−
19535:−
19510:−
19489:ν
19486:μ
19338:ρ
19335:⋯
19332:ν
19329:μ
19324:ζ
19321:⋯
19318:υ
19315:σ
19305:ζ
19294:κ
19289:Λ
19283:⋯
19278:υ
19267:ι
19262:Λ
19254:σ
19243:θ
19238:Λ
19230:ρ
19219:ζ
19214:Λ
19208:⋯
19203:ν
19192:β
19187:Λ
19179:μ
19168:α
19163:Λ
19148:ζ
19144:⋯
19137:β
19129:α
19119:κ
19115:⋯
19108:ι
19100:θ
19077:quantity
19028:σ
19025:ν
19015:σ
19008:ρ
19004:Λ
18996:ν
18989:μ
18985:Λ
18979:≡
18974:σ
18966:⊗
18961:ν
18951:σ
18944:ρ
18940:Λ
18932:ν
18925:μ
18921:Λ
18910:σ
18900:σ
18893:ρ
18889:Λ
18883:⊗
18878:ν
18868:ν
18861:μ
18857:Λ
18845:Λ
18842:⊗
18836:Λ
18833:→
18827:⊗
18746:⊗
18740:∈
18734:⊗
18722:∈
18710:∈
18694:⊗
18676:⊗
18661:⊗
18539:−
18535:Λ
18469:μ
18459:ν
18452:μ
18439:−
18435:Λ
18419:μ
18409:μ
18402:ν
18398:Λ
18387:ν
18348:μ
18338:μ
18331:ν
18327:Λ
18316:ν
18279:ν
18272:μ
18259:−
18255:Λ
18244:≡
18239:μ
18232:ν
18228:Λ
18182:ν
18175:μ
18162:−
18158:Λ
18142:σ
18139:μ
18135:η
18129:σ
18122:ρ
18118:Λ
18110:ν
18107:ρ
18103:η
18075:μ
18065:σ
18062:μ
18058:η
18052:σ
18045:ρ
18041:Λ
18033:ν
18030:ρ
18026:η
18017:ν
17927:ν
17917:ν
17914:μ
17910:η
17901:μ
17861:μ
17851:ν
17848:μ
17844:η
17835:ν
17744:β
17734:β
17727:α
17719:Λ
17713:Π
17703:α
17659:μ
17649:μ
17638:ν
17633:Λ
17618:ν
17584:μ
17574:μ
17567:ν
17563:Λ
17552:ν
17514:row index
17477:μ
17467:μ
17460:ν
17456:Λ
17445:ν
17312:Λ
17291:Λ
17270:Λ
17249:Λ
17226:Λ
17205:Λ
17184:Λ
17163:Λ
17140:Λ
17119:Λ
17098:Λ
17077:Λ
17054:Λ
17033:Λ
17012:Λ
16991:Λ
16800:Λ
16684:−
16634:−
16537:→
16308:−
16112:⋅
16108:θ
16096:⋅
16092:ζ
16062:applies.
16012:⋅
16008:θ
15992:⋅
15988:ζ
15984:−
15976:≠
15966:⋅
15962:θ
15950:⋅
15946:ζ
15942:−
15905:⋅
15901:θ
15889:⋅
15885:ζ
15881:−
15866:θ
15858:ζ
15851:Λ
15787:⋯
15776:⋅
15772:θ
15760:⋅
15756:ζ
15752:−
15733:⋯
15722:⋅
15718:ζ
15714:−
15702:⋯
15691:⋅
15687:θ
15661:⋯
15650:⋅
15646:θ
15630:⋯
15619:⋅
15615:ζ
15611:−
15595:Λ
15545:⋅
15531:θ
15528:
15522:−
15502:⋅
15491:θ
15488:
15469:θ
15428:⋅
15411:−
15408:ζ
15405:
15385:⋅
15374:ζ
15371:
15365:−
15352:ζ
15147:−
15059:−
14920:−
14372:⋅
14368:θ
14352:θ
14330:⋅
14326:ζ
14322:−
14307:ζ
14258:ζ
14255:−
14218:ζ
14213:−
14199:∞
14196:→
14154:For now,
14126:−
14112:ζ
14101:ζ
14098:∂
14083:∂
14026:⋯
14012:ζ
14001:ζ
13998:∂
13983:∂
13973:ζ
13908:θ
13900:ζ
13893:Λ
13883:θ
13866:ζ
13737:), while
13735:transpose
13609:θ
13602:θ
13438:ρ
13394:ρ
13164:collinear
12929:−
12875:−
12860:γ
12740:β
12726:→
12723:β
12713:→
12710:β
12694:−
12691:γ
12674:→
12671:β
12665:γ
12662:−
12646:→
12643:β
12636:γ
12633:−
12628:γ
12569:−
12566:γ
12508:−
12505:γ
12453:−
12450:γ
12424:γ
12421:−
12370:−
12367:γ
12322:−
12319:γ
12261:−
12258:γ
12232:γ
12229:−
12178:−
12175:γ
12123:−
12120:γ
12075:−
12072:γ
12040:γ
12037:−
12012:γ
12009:−
11986:γ
11983:−
11960:γ
11957:−
11952:γ
11754:↑
11749:−
11718:↓
11713:−
11682:↓
11646:↓
11641:−
11629:∪
11624:↑
11571:↑
11509:↑
11429:↓
11424:−
11412:∪
11407:↓
11390:∪
11385:↑
11380:−
11368:∪
11363:↑
11300:↑
11288:∩
11283:−
11266:↑
11261:−
11221:↓
11209:∩
11204:−
11187:↓
11182:−
11138:−
11129:Λ
11115:Λ
11104:−
11062:↑
11050:∩
11028:↑
10983:↓
10971:∩
10949:↓
10891:Λ
10877:Λ
10820:≥
10817:Γ
10809:Λ
10798:↑
10754:−
10751:≤
10748:Γ
10740:Λ
10729:↓
10642:≥
10639:Γ
10628:−
10625:≤
10622:Γ
10618:⇒
10611:≥
10602:Γ
10537:Γ
10477:−
10456:−
10451:Γ
10437:Λ
10399:−
10385:η
10360:±
10351:Λ
10341:⇒
10317:Λ
10281:Λ
10278:η
10267:Λ
10260:η
10239:Lie group
10173:Λ
10139:Λ
10118:invariant
10094:η
10066:η
10045:⋅
10032:transpose
9867:−
9853:η
9682:while in
9386:Four-spin
9271:Position
9109:γ
9103:−
9087:⋅
9070:−
9067:γ
9012:⋅
8998:−
8987:γ
8938:⋅
8925:−
8897:⋅
8889:−
8668:⋅
8638:γ
8625:γ
8594:−
8581:γ
8544:⋅
8533:−
8463:⋅
8451:−
8428:γ
8232:β
8229:
8221:−
8200:ζ
8193:ζ
8168:ζ
8165:
8146:β
8139:β
8064:γ
8045:⋅
8023:−
8020:γ
7955:⋅
7923:γ
7797:γ
7794:−
7778:⋅
7761:−
7758:γ
7696:⋅
7682:−
7671:γ
7593:⋅
7582:−
7569:⊥
7541:⋅
7525:∥
7399:⊥
7377:⊥
7352:−
7347:‖
7334:γ
7319:‖
7282:⋅
7277:∥
7264:−
7253:γ
7205:‖
7187:⊥
7157:‖
7142:⊥
7060:| =
7007:observes
6987:observes
6770:Δ
6763:−
6757:γ
6743:Δ
6669:−
6663:≈
6641:≈
6430:another),
6348:Δ
6330:Δ
6322:γ
6309:Δ
6272:Δ
6251:Δ
6243:γ
6230:Δ
6190:Δ
6183:−
6177:Δ
6169:γ
6151:Δ
6119:Δ
6109:−
6103:Δ
6095:γ
6077:Δ
6060:linearity
6056:one event
5974:ζ
5971:
5951:ζ
5948:
5920:ζ
5917:
5900:ζ
5897:
5743:β
5740:
5732:−
5721:ζ
5691:ζ
5688:
5675:γ
5672:β
5661:ζ
5658:
5645:γ
5634:ζ
5631:
5618:β
5567:ζ
5564:
5551:−
5536:ζ
5533:
5493:ζ
5490:
5482:ζ
5479:
5467:ζ
5464:
5411:ζ
5408:
5395:−
5392:ζ
5389:
5237:ζ
5234:
5222:−
5219:ζ
5216:
5188:ζ
5185:
5176:−
5173:ζ
5170:
5084:β
5060:γ
5011:β
5006:−
4987:γ
4875:β
4868:γ
4852:γ
4832:β
4822:β
4815:γ
4799:γ
4785:β
4775:β
4768:γ
4752:γ
4738:β
4734:γ
4731:−
4718:β
4708:β
4701:γ
4685:γ
4666:β
4659:γ
4643:γ
4623:β
4613:β
4606:γ
4590:γ
4576:β
4572:γ
4569:−
4556:β
4546:β
4539:γ
4523:γ
4509:β
4499:β
4492:γ
4476:γ
4457:β
4450:γ
4434:γ
4414:β
4410:γ
4407:−
4394:β
4390:γ
4387:−
4376:β
4372:γ
4369:−
4358:β
4354:γ
4351:−
4346:γ
4209:β
4112:β
4109:−
4098:γ
4064:β
4061:−
4047:γ
3885:γ
3820:γ
3795:direction
3483:parameter
3467:) is the
3424:−
3409:γ
3305:−
3294:γ
3245:−
3234:γ
3204:direction
3114:axes. At
2824:rotations
2772:∈
2731:∈
2728:Λ
2687:Λ
2678:Λ
2592:signature
2547:⋅
2530:⋅
2388:−
2359:−
2330:−
2268:−
2245:−
2222:−
2162:−
2144:−
2126:−
2085:−
2072:−
2059:−
1875:−
1856:−
1830:−
1811:−
1785:−
1766:−
1740:−
1680:−
1664:−
1641:−
1625:−
1602:−
1586:−
1563:−
1370:−
1354:−
1331:−
1315:−
1292:−
1276:−
1253:−
1085:that the
982:distances
951:spacetime
842:β
839:−
828:γ
798:β
795:−
781:γ
723:β
688:γ
635:−
601:−
585:γ
433:−
422:γ
373:−
362:γ
300:physicist
292:spacetime
28:Spacetime
28152:Category
28029:Lemaître
27994:Einstein
27984:Poincaré
27944:Others:
27928:Taub–NUT
27894:interior
27816:theories
27814:Advanced
27781:redshift
27596:concepts
27414:Rapidity
27392:concepts
27238:Archived
26948:(1973).
26765:(2011),
26744:(2005),
26723:(2008),
26682:Springer
26657:(2007).
26614:(2004).
26592:(1973).
26562:(1971).
26401:(2002).
26282:55561589
26183:(1916).
26165:(1916).
26082:(1905),
26004:: 1–22,
25859:Websites
25011:) = det(
24839:See also
24745:′
24706:′
24602:′
24519:′
24496:′
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23777:bispinor
23717:′
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20420:′
20412:′
20390:′
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20305:′
20285:′
20277:′
20260:becomes
19810:SI units
19297:′
19270:′
19246:′
19222:′
19195:′
19171:′
19151:′
19140:′
19132:′
19122:′
19111:′
19103:′
18522:′
18381:′
18310:′
18011:′
17822:; e.g.,
17804:bispinor
17697:′
17641:′
17621:′
17546:′
17439:′
16958:′
16937:′
16916:′
16895:′
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15315:for all
15254:are the
15240:are the
13715:) = det(
13635:inverses
13477:is a 3d
13352:, while
13216:, where
13099:″
13041:′
13028:′
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11833:′
11808:′
11447:subgroup
11083:Improper
10132:′
10102:′
10081:′
10030:denotes
9836:′
9824:′
9812:′
9800:′
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9564:nor the
9334:Momentum
9222:, where
9045:′
8976:′
8947:′
8934:′
8914:′
8722:constant
8517:′
8415:′
8402:′
8383:′
8071:′
8041:′
8010:′
7951:′
7935:′
7736:′
7660:′
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6750:′
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6633:′
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5964:′
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5771:−∞ <
5763:−1 <
5313:rapidity
5270:′
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4087:′
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2706:′
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2400:′
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1826:′
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1781:′
1752:′
1736:′
937:In each
932:inertial
920:inertial
886:′
864:′
817:′
770:′
477:′
455:′
411:′
351:′
296:velocity
231:Category
28094:Hawking
28089:Penrose
28074:Novikov
28054:Wheeler
27999:Hilbert
27989:Lorentz
27946:pp-wave
27767:lensing
27563:General
27344:Special
27270:YouTube
27260:YouTube
27211:: 41–51
27164:Bibcode
26826:542–545
26371:Bibcode
26262:Bibcode
26214:Bibcode
26118:Bibcode
26006:Bibcode
25940:Bibcode
25909:Bibcode
25669:Bibcode
25395:x, y, z
25253:√
24978:O(1, 3)
24974:O(3, 1)
24970:O(1, 3)
24966:O(3, 1)
24947:or the
24820:is the
24810:is the
23765:Spinors
23067:⁄
22706:, 0, 0)
18785:, then
18592:Tensors
18215:inverse
17884:is the
17419:spinors
17415:tensors
16457:over a
16177:set of
16173:form a
16050:), see
13749:inverse
13168:commute
10684:herein
10678:orentz
10116:and is
7421:is the
6958:motion.
5516:of the
5336:is the
3480:is the
3397:is the
3129:x, y, z
2965:Bottom:
2887:O(3, 1)
2834:or the
2642:O(3, 1)
2630:O(1, 3)
1989:-tuple
1516:any two
1051:History
652:is the
573:is the
286:from a
273:physics
28135:others
28124:Thorne
28114:Misner
28099:Taylor
28084:Geroch
28079:Ehlers
28049:Zwicky
27867:Kasner
27190:
27128:
27098:
27077:
27056:
27035:
27012:
26986:
26960:
26926:
26896:
26866:
26832:
26798:
26773:
26752:
26731:
26709:
26688:
26665:
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26624:
26600:
26570:
26544:
26525:
26506:
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26466:
26445:
26415:
26389:
26350:
26321:
26303:
26280:
26240:
26232:
26193:
26050:
26024:
25886:Papers
25687:
25600:
25428:
25242:, and
24799:where
22695:Here,
19442:field.
19364:where
19075:tensor
17878:where
17762:where
16821:where
15309:→ exp(
15274:(0) =
15248:, and
13719:) = +1
13466:where
13334:. The
12774:where
11585:, and
11451:closed
10845:Proper
10154:where
9742:, and
9689:it is
9659:it is
9313:Energy
8776:, the
8268:< 1
8260:< ∞
7506:, and
7451:< 1
7417:where
7100:, and
7054:|
7037:Right:
6900:. In
6547:, etc.
6383:where
5824:axes.
5815:< 0
5794:> 0
5775:< ∞
5767:< 1
5761:Since
5600:, and
5300:where
5032:. The
4161:< 1
3614:As an
3560:< ∞
3512:< 0
3491:> 0
3474:Here,
3401:, and
3379:where
2967:frame
2939:frame
2868:boosts
2828:boosts
2632:, the
2595:(1, 3)
2588:(·, ·)
2586:where
2019:linear
1939:where
1073:, and
577:, and
504:where
281:linear
275:, the
229:
28129:Weiss
28109:Bondi
28104:Hulse
28034:Milne
27938:discs
27882:Milne
27877:Gödel
27734:Virgo
27175:(PDF)
27152:(PDF)
26428:Books
26387:S2CID
26348:S2CID
26319:S2CID
26278:S2CID
26238:S2CID
26106:(PDF)
25994:(PDF)
25974:(PDF)
25406:Notes
25294:field
25015:)det(
24833:SO(3)
23208:space
23196:3 + 1
23081:3 + 1
18584:with
17802:is a
17770:, an
17510:Latin
17506:Greek
16839:. If
16459:field
16175:basis
15269:with
13705:unit
11326:union
10229:is a
10211:group
9528:spin
9154:(and
8845:′ = (
7077:<
7033:Left:
6938:. In
6920:′ = 0
6855:′ = 0
6715:) in
6606:<
6602:<
6389:delta
6049:′ = 0
6042:′ = 0
5514:slope
5038:trace
3607:is a
3596:>
3543:<
3539:<
3465:gamma
3149:, or
3131:) = (
3123:′ = 0
2874:, or
1180:event
1030:. In
1005:light
972:(see
955:event
947:event
28064:Kerr
28014:Weyl
27913:Kerr
27773:and
27727:and
27725:LIGO
27188:ISBN
27126:ISBN
27096:ISBN
27075:ISBN
27054:ISBN
27033:ISBN
27010:ISBN
26984:ISBN
26958:ISBN
26924:ISBN
26894:ISBN
26864:ISBN
26830:ISBN
26796:ISBN
26771:ISBN
26750:ISBN
26729:ISBN
26707:ISBN
26686:ISBN
26663:ISBN
26641:ISBN
26622:ISBN
26598:ISBN
26568:ISBN
26542:ISBN
26523:ISBN
26504:ISBN
26485:ISBN
26464:ISBN
26443:ISBN
26413:ISBN
26230:ISSN
26191:ISBN
26022:ISBN
25685:ISSN
25598:ISBN
25426:ISBN
25288:and
25007:det(
24999:and
24976:and
24968:and
24827:+ 1)
24814:and
24804:(Λ,
24124:(T4)
23791:Π(Λ)
23782:(T2)
23772:(T1)
23757:The
23486:and
23241:(T3)
23235:(T2)
23232:and
23229:(T1)
23212:time
23210:and
20257:(T3)
19453:This
19420:and
19381:Π(Λ)
19359:(T3)
19049:(T2)
18777:and
18768:(T1)
18630:and
18608:and
18600:and
18586:Π(Λ)
18094:But
16766:Λ =
16492:The
16136:and
16044:and
15823:and
15402:cosh
15368:sinh
14431:and
13829:and
13807:) =
13797:and
13787:) =
13711:det(
13698:) =
13690:) =
13665:) =
13646:) =
13519:and
13505:and
13497:(or
13491:and
13358:and
13348:are
13340:and
13268:and
13243:and
13228:and
13187:) =
13162:are
13156:and
13081:and
11481:and
11470:and
11457:and
11242:LTs
11163:LTs
11085:LTs
11004:LTs
10925:LTs
10847:LTs
10779:LTs
10161:The
9706:′ ×
9702:′ +
9694:′ =
9643:and
9596:′ ×
9592:′ =
9511:spin
9402:Spin
9278:Time
9240:and
9179:and
8840:and
8805:and
8787:The
8749:and
8709:and
8282:and
8264:0 ≤
8256:0 ≤
8217:tanh
8162:tanh
8121:and
7844:and
7447:0 ≤
7073:0 ≤
6971:and
6833:′ =
6590:′ =
6538:′ −
6534:′ =
6472:and
6044:and
5968:sinh
5945:cosh
5914:sinh
5894:cosh
5783:and
5728:tanh
5685:sinh
5655:cosh
5628:tanh
5606:are
5555:tanh
5530:cosh
5487:cosh
5476:sinh
5461:tanh
5441:and
5399:sinh
5380:cosh
5308:zeta
5231:sinh
5213:cosh
5182:sinh
5167:cosh
4169:and
4009:beta
3780:same
3556:1 ≤
3189:same
3101:and
3088:and
3075:and
2937:Top:
2826:and
2004:and
1476:and
1171:and
1120:").
524:and
28119:Yau
27744:GEO
27299:etc
27278:on
27268:on
27258:on
26379:doi
26340:doi
26311:doi
26270:doi
26222:doi
26126:doi
26114:322
26094:140
26058:doi
26014:doi
25956:hdl
25948:doi
25917:doi
25741:138
25677:doi
25234:In
23200:any
22702:= (
19808:in
18626:of
18596:If
18495:of
17792:to
17786:not
17417:or
16833:or
16743:If
16504:exp
16204:, ζ
16200:, ζ
16196:, ζ
16192:, θ
16188:, θ
16167:, K
16163:, K
16159:, K
16155:, J
16151:, J
16072:set
15525:cos
15485:sin
15332:= 0
15290:= 0
14446:, J
14442:, J
14436:= (
14424:, K
14420:, K
14414:= (
14189:lim
14066:= 0
13944:= 0
13483:not
13311:or
13289:not
13237:If
11123:det
10885:det
10692:and
10591:≥ 0
10515:Γ,
10345:det
10311:det
10163:set
9613:= (
9495:or
9463:),
9428:),
9360:),
9321:),
9286:),
9201:).
9196:↦ −
9185:′,
8810:= (
8762:to
8109:or
7866:→ −
7856:→ −
6827:by
6818:= 0
6713:= 0
6503:in
6494:′,
6487:′,
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6466:in
5832:→ −
5808:= 0
5363:= 0
5358:or
5356:= 0
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3766:′,
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3719:to
3707:to
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3505:= 0
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1973:′,
1969:′,
1965:′,
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1446:= (
1177:An
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538:′,
534:′,
530:′,
290:in
271:In
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25255:−1
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25005:,
24835:.
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23075:.
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21250:12
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20833:31
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18617:⊗
18588:.
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17973:μν
17966:=
17957:μν
17810:.
16768:PT
16764:TP
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15813:)(
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15311:tG
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13946:,
13840:.
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2809:D3
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2437:D3
2185:or
1982:D2
1977:′)
1953:,
1949:,
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1930:D2
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24817:D
24808:)
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16901:0
16892:x
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16823:C
16809:C
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16305:=
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15877:e
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15683:+
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15677:(
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15636:(
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15537:(
15534:)
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15513:+
15510:)
15506:J
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15482:+
15479:I
15476:=
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15465:(
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15424:n
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15417:)
15414:1
15399:(
15396:+
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15389:K
15381:n
15377:(
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15348:(
15345:B
15330:t
15324:G
15318:t
15313:)
15307:t
15301:G
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15288:t
15282:t
15276:I
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15104:=
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15084:,
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15037:0
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15020:0
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14734:[
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14716:K
14709:,
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14610:[
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14585:,
14579:]
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14568:0
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14546:0
14541:0
14536:0
14529:0
14524:0
14519:0
14514:1
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14492:0
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14440:x
14438:J
14434:J
14429:)
14426:z
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14418:x
14416:K
14412:K
14406:ζ
14400:θ
14383:.
14376:J
14363:e
14359:=
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14348:(
14345:R
14341:,
14334:K
14318:e
14314:=
14311:)
14303:(
14300:B
14266:x
14262:K
14251:e
14247:=
14242:N
14237:)
14231:x
14227:K
14221:N
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14206:(
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14185:=
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14140:.
14134:x
14130:K
14123:=
14118:0
14115:=
14107:|
14091:x
14087:B
14064:ζ
14055:x
14049:x
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14041:ζ
14023:+
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13970:+
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13942:ζ
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13912:)
13904:,
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13890:,
13887:)
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13873:,
13870:)
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13838:)
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13815:)
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13686:(
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13654:)
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13580:e
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13551:w
13546:ρ
13542:w
13528:R
13522:v
13516:u
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13454:,
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13434:(
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13362:ρ
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13332:)
13329:ρ
13326:(
13324:R
13322:)
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13316:(
13314:B
13309:)
13307:w
13305:(
13303:B
13301:)
13299:ρ
13297:(
13295:R
13285:)
13283:v
13281:(
13279:B
13277:)
13275:u
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13271:B
13266:)
13264:u
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13260:B
13258:)
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13252:B
13246:v
13240:u
13231:v
13225:u
13219:w
13214:)
13212:w
13210:(
13208:B
13203:)
13201:v
13199:(
13197:B
13195:)
13193:u
13191:(
13189:B
13185:u
13183:(
13181:B
13179:)
13177:v
13175:(
13173:B
13159:v
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6825:′
6823:F
6816:x
6814:Δ
6809:F
6782:2
6778:c
6773:x
6766:v
6754:=
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6733:′
6731:F
6725:x
6723:Δ
6718:F
6711:t
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6675:t
6672:v
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6630:t
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6581:F
6571:x
6565:x
6559:F
6545:′
6543:0
6540:x
6536:x
6532:x
6530:Δ
6525:0
6522:x
6518:x
6514:x
6512:Δ
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6442:0
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6393:Δ
6385:Δ
6365:.
6360:)
6352:t
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6334:x
6326:(
6319:=
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6302:,
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6289:2
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5961:t
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5954:+
5938:x
5934:=
5927:x
5907:x
5903:+
5887:t
5883:c
5880:=
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5870:c
5854:ζ
5846:x
5843:(
5834:ζ
5830:ζ
5822:′
5813:ζ
5806:ζ
5801:′
5792:ζ
5786:β
5780:ζ
5773:ζ
5765:β
5747:.
5735:1
5724:=
5695:.
5682:=
5665:,
5652:=
5638:,
5625:=
5603:ζ
5597:γ
5591:β
5574:.
5559:2
5548:1
5544:1
5539:=
5500:,
5470:=
5450:ζ
5444:x
5421:.
5417:1
5414:=
5403:2
5384:2
5368:ζ
5354:x
5333:ζ
5303:ζ
5281:z
5278:=
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5225:c
5210:x
5207:=
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5127:ζ
5119:x
5116:(
5107:x
5063:)
5057:+
5054:1
5051:(
5048:2
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5003:1
4997:/
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4990:=
4964:,
4959:]
4953:z
4937:y
4921:x
4908:t
4905:c
4899:[
4892:]
4884:2
4879:z
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4812:+
4809:1
4803:2
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4722:z
4712:y
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4637:+
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4617:x
4603:+
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4550:x
4536:+
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4513:y
4503:x
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4447:+
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4438:2
4428:+
4425:1
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4398:z
4380:y
4362:x
4340:[
4335:=
4330:]
4320:z
4302:y
4281:x
4260:t
4256:c
4250:[
4226:c
4222:/
4217:v
4213:=
4187:v
4172:γ
4166:β
4159:β
4152:β
4146:v
4127:,
4122:)
4118:t
4115:c
4106:x
4102:(
4095:=
4084:x
4076:,
4071:)
4067:x
4058:t
4055:c
4051:(
4044:=
4033:t
4029:c
4014:v
4004:c
4002:/
4000:v
3996:β
3986:γ
3964:,
3957:z
3953:=
3946:z
3935:y
3931:=
3924:y
3916:)
3908:t
3904:v
3901:+
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3889:(
3882:=
3875:x
3867:)
3859:2
3855:c
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3824:(
3817:=
3810:t
3791:x
3788:(
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3764:y
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3641:′
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3604:γ
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3591:(
3584:γ
3578:c
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3568:v
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3535:−
3530:c
3524:v
3519:′
3510:v
3503:v
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3489:v
3477:v
3444:2
3440:c
3434:2
3430:v
3421:1
3417:1
3412:=
3394:c
3388:x
3382:v
3360:z
3357:=
3346:z
3338:y
3335:=
3324:y
3315:)
3311:t
3308:v
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3291:=
3280:x
3271:)
3263:2
3259:c
3254:x
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3242:t
3238:(
3231:=
3220:t
3200:x
3197:(
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3173:y
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2696:=
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2666:a
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2351:2
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2338:1
2334:x
2322:2
2318:t
2309:1
2305:t
2299:2
2295:c
2291:=
2286:2
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2276:1
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2259:y
2253:1
2249:y
2240:2
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2230:1
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2207:1
2203:t
2197:2
2193:c
2174:2
2166:z
2156:2
2148:y
2138:2
2130:x
2120:2
2112:t
2106:2
2102:c
2098:=
2093:2
2089:z
2080:2
2076:y
2067:2
2063:x
2054:2
2050:t
2044:2
2040:c
2010:2
2007:a
2001:1
1998:a
1992:b
1987:4
1975:z
1971:y
1967:x
1963:t
1961:(
1957:)
1955:z
1951:y
1947:x
1943:t
1941:(
1932:)
1928:(
1907:.
1896:2
1892:)
1883:1
1879:z
1867:2
1863:z
1859:(
1851:2
1847:)
1838:1
1834:y
1822:2
1818:y
1814:(
1806:2
1802:)
1793:1
1789:x
1777:2
1773:x
1769:(
1761:2
1757:)
1748:1
1744:t
1732:2
1728:t
1724:(
1719:2
1715:c
1707:=
1698:2
1694:)
1688:1
1684:z
1675:2
1671:z
1667:(
1659:2
1655:)
1649:1
1645:y
1636:2
1632:y
1628:(
1620:2
1616:)
1610:1
1606:x
1597:2
1593:x
1589:(
1581:2
1577:)
1571:1
1567:t
1558:2
1554:t
1550:(
1545:2
1541:c
1512:)
1510:2
1507:z
1503:2
1500:y
1496:2
1493:x
1489:2
1486:t
1482:2
1479:a
1474:)
1472:1
1469:z
1465:1
1462:y
1458:1
1455:x
1451:1
1448:t
1444:1
1441:a
1423:)
1419:(
1396:0
1393:=
1388:2
1384:)
1378:1
1374:z
1365:2
1361:z
1357:(
1349:2
1345:)
1339:1
1335:y
1326:2
1322:y
1318:(
1310:2
1306:)
1300:1
1296:x
1287:2
1283:x
1279:(
1271:2
1267:)
1261:1
1257:t
1248:2
1244:t
1240:(
1235:2
1231:c
1214:c
1201:z
1197:y
1193:x
1130:c
1128:/
1126:v
900:.
897:z
894:=
883:z
875:y
872:=
861:y
852:)
848:t
845:c
836:x
832:(
825:=
814:x
805:)
801:x
792:t
789:c
785:(
778:=
767:t
763:c
739:,
734:c
731:v
726:=
709:c
703:v
677:c
671:v
665:c
659:v
638:1
630:)
621:2
617:c
611:2
607:v
598:1
593:(
588:=
570:c
565:x
560:v
555:′
553:t
550:=
547:t
540:z
536:y
532:x
528:t
526:(
522:)
520:z
516:y
512:x
508:t
506:(
488:z
485:=
474:z
466:y
463:=
452:y
443:)
439:t
436:v
430:x
426:(
419:=
408:x
399:)
391:2
387:c
382:x
379:v
370:t
366:(
359:=
348:t
333:x
319:,
316:v
260:e
253:t
246:v
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