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20119:{\displaystyle {\begin{aligned}Vx^{j}&=V^{i}{\frac {\partial }{\partial u^{i}}}\left({\frac {2R^{2}u^{j}}{R^{2}-|u|^{2}}}\right)={\frac {2R^{2}V^{j}}{R^{2}-|u|^{2}}}-{\frac {4R^{2}u^{j}\langle \mathbf {V} ,\,\mathbf {u} \rangle }{\left(R^{2}-|u|^{2}\right)^{2}}},\quad \left({\text{here }}V|u|^{2}=2\sum _{k=1}^{n}V^{k}u^{k}\equiv 2\langle \mathbf {V} ,\,\mathbf {u} \rangle \right)\\V\tau &=V\left(R{\frac {R^{2}+|u|^{2}}{R^{2}-|u|^{2}}}\right)={\frac {4R^{3}\langle \mathbf {V} ,\,\mathbf {u} \rangle }{\left(R^{2}-|u|^{2}\right)^{2}}}.\end{aligned}}}
17798:{\displaystyle {\begin{aligned}&\left(dx^{1}(\mathbf {u} )\right)^{2}+\cdots +\left(dx^{n}(\mathbf {u} )\right)^{2}\\={}&{\frac {4R^{2}\left(R^{2}-|u|^{2}\right)^{2}\left+16R^{4}\left(R^{2}-|u|^{2}\right)(\mathbf {u} \cdot d\mathbf {u} )(\mathbf {u} \cdot d\mathbf {u} )+16R^{4}|u|^{2}(\mathbf {u} \cdot d\mathbf {u} )^{2}}{\left(R^{2}-|u|^{2}\right)^{4}}}\\={}&{\frac {4R^{2}\left(R^{2}-|u|^{2}\right)^{2}\left}{\left(R^{2}-|u|^{2}\right)^{4}}}+R^{2}{\frac {16R^{4}(\mathbf {u} \cdot d\mathbf {u} )}{\left(R^{2}-|u|^{2}\right)^{4}}}.\end{aligned}}}
15488:{\displaystyle {\begin{aligned}dx^{1}(\mathbf {u} )&=d\left({\frac {2R^{2}u^{1}}{R^{2}-|u|^{2}}}\right)={\frac {\partial }{\partial u^{1}}}{\frac {2R^{2}u^{1}}{R^{2}-|u|^{2}}}du^{1}+\cdots +{\frac {\partial }{\partial u^{n}}}{\frac {2R^{2}u^{1}}{R^{2}-|u|^{2}}}du^{n}+{\frac {\partial }{\partial \tau }}{\frac {2R^{2}u^{1}}{R^{2}-|u|^{2}}}d\tau ,\\&\ \ \vdots \\dx^{n}(\mathbf {u} )&=d\left({\frac {2R^{2}u^{n}}{R^{2}-|u|^{2}}}\right)=\cdots ,\\d\tau (\mathbf {u} )&=d\left(R{\frac {R^{2}+|u|^{2}}{R^{2}-|u|^{2}}}\right)=\cdots ,\end{aligned}}}
2500:
19477:{\displaystyle \left(\sigma ^{-1}\right)_{*}V=\left(\sigma ^{-1}\right)_{*}V^{i}{\frac {\partial }{\partial u^{i}}}=V^{i}{\frac {\partial x^{j}}{\partial u^{i}}}{\frac {\partial }{\partial x^{j}}}+V^{i}{\frac {\partial \tau }{\partial u^{i}}}{\frac {\partial }{\partial \tau }}=V^{i}{\frac {\partial }{x}}^{j}{\partial u^{i}}{\frac {\partial }{\partial x^{j}}}+V^{i}{\frac {\partial }{\tau }}{\partial u^{i}}{\frac {\partial }{\partial \tau }}=Vx^{j}{\frac {\partial }{\partial x^{j}}}+V\tau {\frac {\partial }{\partial \tau }}.}
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489:
2925:{\displaystyle {\begin{aligned}\left(x^{0},\,x^{1},\,x^{2},\,x^{3}\right)\ &\leftrightarrow \ \left.x^{0}\mathbf {e} _{0}\right|_{p}+\left.x^{1}\mathbf {e} _{1}\right|_{p}+\left.x^{2}\mathbf {e} _{2}\right|_{p}+\left.x^{3}\mathbf {e} _{3}\right|_{p}\\&\leftrightarrow \ \left.x^{0}\mathbf {e} _{0}\right|_{q}+\left.x^{1}\mathbf {e} _{1}\right|_{q}+\left.x^{2}\mathbf {e} _{2}\right|_{q}+\left.x^{3}\mathbf {e} _{3}\right|_{q}\end{aligned}}}
299:
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2384:
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16254:{\displaystyle {\begin{aligned}{\frac {\partial }{\partial u^{1}}}{\frac {2R^{2}u^{1}}{R^{2}-|u|^{2}}}du^{1}&={\frac {2\left(R^{2}-|u|^{2}\right)+4R^{2}\left(u^{1}\right)^{2}}{\left(R^{2}-|u|^{2}\right)^{2}}}du^{1},\\{\frac {\partial }{\partial u^{2}}}{\frac {2R^{2}u^{1}}{R^{2}-|u|^{2}}}du^{2}&={\frac {4R^{2}u^{1}u^{2}}{\left(R^{2}-|u|^{2}\right)^{2}}}du^{2},\end{aligned}}}
18462:
3112:. The first identification is the canonical identification of vectors in the tangent space at any point with vectors in the space itself. The appearance of basis vectors in tangent spaces as first-order differential operators is due to this identification. It is motivated by the observation that a geometrical tangent vector can be associated in a one-to-one manner with a
11822:
17814:
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2142:
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10797:. The subscript star denotes the pushforward (to be introduced later), and it is in this special case simply the identity map (as is the inclusion map). The latter equality holds because a tangent space to a submanifold at a point is in a canonical way a subspace of the tangent space of the manifold itself at the point in question. One may simply write
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3092:
14280:
18757:
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12365:
2045:
1119:, as well, which depends on the signature. The classification of any vector will be the same in all frames of reference that are related by a Lorentz transformation (but not by a general Poincaré transformation because the origin may then be displaced) because of the invariance of the spacetime interval under Lorentz transformation.
10019:
12901:
16986:{\displaystyle \left(dx^{1}(\mathbf {u} )\right)^{2}={\frac {4R^{2}\left(r^{2}-|u|^{2}\right)^{2}\left(du^{1}\right)^{2}+16R^{4}\left(R^{2}-|u|^{2}\right)\left(\mathbf {u} \cdot d\mathbf {u} \right)u^{1}du^{1}+16R^{4}\left(u^{1}\right)^{2}\left(\mathbf {u} \cdot d\mathbf {u} \right)^{2}}{\left(R^{2}-|u|^{2}\right)^{4}}}.}
18196:
11682:
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passing through the origin, and its norm. Geometrically thus, covariant vectors should be viewed as a set of hyperplanes, with spacing depending on the norm (bigger = smaller spacing), with one of them (the kernel) passing through the origin. The mathematical term for a covariant vector is 1-covector
1179:
basis for
Minkowski space necessarily consists of one timelike and three spacelike unit vectors. If one wishes to work with non-orthonormal bases, it is possible to have other combinations of vectors. For example, one can easily construct a (non-orthonormal) basis consisting entirely of null vectors,
265:
between two events when given their coordinate difference vector as an argument. Equipped with this inner product, the mathematical model of spacetime is called
Minkowski space. The group of transformations for Minkowski space that preserves the spacetime interval (as opposed to the spatial Euclidean
8811:
As mentioned, in a vector space, such as modeling the spacetime of special relativity, tangent vectors can be canonically identified with vectors in the space itself, and vice versa. This means that the tangent spaces at each point are canonically identified with each other and with the vector space
7535:
Non-degeneracy is then equivalent to injectivity of the partial evaluation map, or equivalently non-degeneracy indicates that the kernel of the map is trivial. In finite dimension, as is the case here, and noting that the dimension of a finite-dimensional space is equal to the dimension of the dual,
7238:
to illustrate how one could imagine a covariant version of a contravariant vector. The inner product of two contravariant vectors could equally well be thought of as the action of the covariant version of one of them on the contravariant version of the other. The inner product is then how many times
858:
The views of space and time which I wish to lay before you have sprung from the soil of experimental physics, and therein lies their strength. They are radical. Henceforth, space by itself and time by itself are doomed to fade away into mere shadows, and only a kind of union of the two will preserve
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below. It follows that if the scalar product of two vectors is zero, then one of these, at least, must be space-like. The scalar product of two space-like vectors can be positive or negative as can be seen by considering the product of two space-like vectors having orthogonal spatial components and
751:
combined with redefined vector variables for electromagnetic quantities, and he was able to show directly and very simply their invariance under
Lorentz transformation. He also made other important contributions and used matrix notation for the first time in this context. From his reformulation, he
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10533:
4811:, an orthonormal basis in space, combined with the unit time vector, forms an orthonormal basis in Minkowski space. The number of positive and negative unit vectors in any such basis is a fixed pair of numbers equal to the signature of the bilinear form associated with the inner product. This is
3331:
707:
Rotations in planes spanned by two space unit vectors appear in coordinate space as well as in physical spacetime as
Euclidean rotations and are interpreted in the ordinary sense. The "rotation" in a plane spanned by a space unit vector and a time unit vector, while formally still a rotation in
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2935:
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one may apply the
Minkowski metric to two vectors and obtain a real number, i.e. to display the role of the differentials and how they disappear in a calculation. The setting is that of smooth manifold theory, and concepts such as convector fields and exterior derivatives are introduced.
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12107:{\displaystyle {\begin{aligned}\sigma (\tau ,\mathbf {x} )=\mathbf {u} &={\frac {R\mathbf {x} }{R+\tau }},\\\sigma ^{-1}(\mathbf {u} )=(\tau ,\mathbf {x} )&=\left(R{\frac {R^{2}+|u|^{2}}{R^{2}-|u|^{2}}},{\frac {2R^{2}\mathbf {u} }{R^{2}-|u|^{2}}}\right),\end{aligned}}}
18186:{\displaystyle d\tau =\sum _{i=1}^{n}{\frac {\partial }{\partial u^{i}}}R{\frac {R^{2}+|u|^{2}}{R^{2}+|u|^{2}}}du^{i}+{\frac {\partial }{\partial \tau }}R{\frac {R^{2}+|u|^{2}}{R^{2}+|u|^{2}}}d\tau =\sum _{i=1}^{n}R^{4}{\frac {4R^{2}u^{i}du^{i}}{\left(R^{2}-|u|^{2}\right)}},}
8558:
9792:
18714:
15743:
12752:
2379:{\displaystyle {\begin{aligned}\left\|u+w\right\|^{2}&=\left\|u\right\|^{2}+2\left(u,w\right)+\left\|w\right\|^{2}\\&\geq \left\|u\right\|^{2}+2\left\|u\right\|\left\|w\right\|+\left\|w\right\|^{2}=\left(\left\|u\right\|+\left\|w\right\|\right)^{2}.\end{aligned}}}
14275:
13477:
14571:{\displaystyle \sigma ^{-1}:\mathbf {R} ^{n}\rightarrow \mathbf {H} _{R}^{1(n)};\quad \sigma ^{-1}(\mathbf {u} )=(\tau (\mathbf {u} ),\,\mathbf {x} (\mathbf {u} ))=\left(R{\frac {R^{2}+|u|^{2}}{R^{2}-|u|^{2}}},\,{\frac {2R^{2}\mathbf {u} }{R^{2}-|u|^{2}}}\right).}
8977:
6466:
2488:
In practice, one need not be concerned with the tangent spaces. The vector space structure of
Minkowski space allows for the canonical identification of vectors in tangent spaces at points (events) with vectors (points, events) in Minkowski space itself. See e.g.
19055:{\displaystyle \left(\sigma ^{-1}\right)^{*}h_{R}^{1(n)}(V,\,V)=h_{R}^{1(n)}\left(\left(\sigma ^{-1}\right)_{*}V,\,\left(\sigma ^{-1}\right)_{*}V\right)=\eta |_{\mathbf {H} _{R}^{1(n)}}\left(\left(\sigma ^{-1}\right)_{*}V,\,\left(\sigma ^{-1}\right)_{*}V\right).}
1484:
Using the positivity property of time-like vectors, it is easy to verify that a linear sum with positive coefficients of similarly directed time-like vectors is also similarly directed time-like (the sum remains within the light cone because of convexity).
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Consistent use of the terms "Minkowski inner product", "Minkowski norm" or "Minkowski metric" is intended for the bilinear form here, since it is in widespread use. It is by no means "standard" in the literature, but no standard terminology seems to
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11077:
3352:, which is, of course, admissible by essentially the same canonical identification. The identifications of vectors referred to above in the mathematical setting can correspondingly be found in a more physical and explicitly geometrical setting in
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1469:
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8778:
1203:, i.e. all either in the forward or in the backward cones. Such vectors have several properties not shared by space-like vectors. These arise because both forward and backward cones are convex, whereas the space-like region is not convex.
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1198:
Time-like vectors have special importance in the theory of relativity as they correspond to events that are accessible to the observer at (0, 0, 0, 0) with a speed less than that of light. Of most interest are time-like vectors that are
1633:
18457:{\displaystyle -d\tau ^{2}=-\left(R{\frac {4R^{4}\left(\mathbf {u} \cdot d\mathbf {u} \right)}{\left(R^{2}-|u|^{2}\right)^{2}}}\right)^{2}=-R^{2}{\frac {16R^{4}(\mathbf {u} \cdot d\mathbf {u} )^{2}}{\left(R^{2}-|u|^{2}\right)^{4}}}.}
13345:
8283:
13482:
3199:
5569:) with the symmetry group preserving the other choice of signature. This means that both choices are in accord with the two postulates of relativity. Switching between the two conventions is straightforward. If the metric tensor
4676:. In fact, it can be taken as the defining property of a Lorentz transformation in that it preserves the inner product (i.e. the value of the corresponding bilinear form on two vectors). This approach is taken more generally for
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588:
can then be thought of as rotations in this four-dimensional space, where the rotation axis corresponds to the direction of relative motion between the two observers and the rotation angle is related to their relative velocity.
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3372:) as arguments. Further discussion about this theoretically inconsequential but practically necessary choice for purposes of internal consistency and convenience is deferred to the hide box below. See also the page treating
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itself. This explains how the right-hand side of the above equation can be employed directly, without regard to the spacetime point the metric is to be evaluated and from where (which tangent space) the vectors come from.
3509:. The Minkowski metric, the bilinear form, and the Minkowski inner product are all the same object; it is a bilinear function that accepts two (contravariant) vectors and returns a real number. In coordinates, this is the
7855:
7269:, ensures that moving vectors around and changing from contravariant to covariant vectors and vice versa (as well as higher order tensors) is mathematically sound. Incorrect expressions tend to reveal themselves quickly.
1744:
13176:
13104:{\displaystyle {\begin{aligned}{\overrightarrow {SU}}&=(0,\mathbf {u} )-(-R,\mathbf {0} )=(R,\mathbf {u} ),\\{\overrightarrow {SP}}&=(\tau ,\mathbf {x} )-(-R,\mathbf {0} )=(\tau +R,\mathbf {x} ).\end{aligned}}.}
10441:
8452:
7419:
9413:
Minkowski space refers to a mathematical formulation in four dimensions. However, the mathematics can easily be extended or simplified to create an analogous generalized
Minkowski space in any number of dimensions. If
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15498:
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for the
Minkowski space depends heavily on the context. Minkowski space is not endowed with Euclidean geometry, and not with any of the generalized Riemannian geometries with intrinsic curvature, those exposed by the
5273:
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can be visualized as ordinary rotations of the four-dimensional
Euclidean sphere. The four-dimensional spacetime can be visualized as a four-dimensional space, with each point representing an event in spacetime. The
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14794:{\displaystyle \left.\left(\sigma ^{-1}\right)^{*}\eta \right|_{\mathbf {H} _{R}^{1(n)}}=\left(dx^{1}(\mathbf {u} )\right)^{2}+\cdots +\left(dx^{n}(\mathbf {u} )\right)^{2}-\left(d\tau (\mathbf {u} )\right)^{2}.}
10763:{\displaystyle \iota ^{*}\alpha \left(X_{1},\,X_{2},\,\ldots ,\,X_{k}\right)=\alpha \left(\iota _{*}X_{1},\,\iota _{*}X_{2},\,\ldots ,\,\iota _{*}X_{k}\right)=\alpha \left(X_{1},\,X_{2},\,\ldots ,\,X_{k}\right),}
764:
In a further development in his 1908 "Space and Time" lecture, Minkowski gave an alternative formulation of this idea that used a real time coordinate instead of an imaginary one, representing the four variables
5564:
is adopted below. This choice (or the other possible choice) has no (known) physical implications. The symmetry group preserving the bilinear form with one choice of signature is isomorphic (under the map given
4637:
3087:{\displaystyle \left.\mathbf {e} _{\mu }\right|_{p}=\left.{\frac {\partial }{\partial x^{\mu }}}\right|_{p}{\text{ or }}\mathbf {e} _{0}|_{p}=\left({\begin{matrix}1\\0\\0\\0\end{matrix}}\right){\text{, etc}}.}
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20416:{\displaystyle \eta \left(\sigma _{*}^{-1}V,\,\sigma _{*}^{-1}V\right)=\sum _{j=1}^{n}\left(Vx^{j}\right)^{2}-(V\tau )^{2}={\frac {4R^{4}|V|^{2}}{\left(R^{2}-|u|^{2}\right)^{2}}}=h_{R}^{2(n)}(V,z,V),}
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13670:
835:. For these special topics, see the referenced articles, as the presentation below will be principally confined to the mathematical structure (Minkowski metric and from it derived quantities and the
13183:
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or by juxtaposition) has been taken. The equality holds since, by definition, the
Minkowski metric is symmetric. The notation on the far right is also sometimes used for the related, but different,
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7160:. In this latter interpretation, the covariant vectors are (almost always implicitly) identified with vectors (linear functionals) in the dual of Minkowski space. The ones with upper indices are
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2505:
12360:{\displaystyle \mathbf {H} _{R}^{n}=\left\{\left(\tau ,x^{1},\ldots ,x^{n}\right)\subset \mathbf {M} :-\tau ^{2}+\left(x^{1}\right)^{2}+\cdots +\left(x^{n}\right)^{2}=-R^{2},\tau >0\right\}}
2040:{\displaystyle c^{2}t_{1}t_{2}-x_{1}x_{2}-y_{1}y_{2}-z_{1}z_{2}>{\sqrt {\left(c^{2}t_{1}^{2}-x_{1}^{2}-y_{1}^{2}-z_{1}^{2}\right)\left(c^{2}t_{2}^{2}-x_{2}^{2}-y_{2}^{2}-z_{2}^{2}\right)}}}
716:
inertial coordinates. The analogy with Euclidean rotations is only partial since the radius of the sphere is actually imaginary, which turns rotations into rotations in hyperbolic space (see
10983:
10846:
6971:
10014:{\displaystyle \mathbf {H} _{R}^{1(n)}=\left\{\left(ct,x^{1},\ldots ,x^{n}\right)\in \mathbf {M} ^{n}:c^{2}t^{2}-\left(x^{1}\right)^{2}-\cdots -\left(x^{n}\right)^{2}=R^{2},ct>0\right\}}
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that are otherwise immediately available and useful for geometrical description and calculation – even in the flat spacetime of special relativity, e.g. of the electromagnetic field.
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6584:
616:
that acts on the four-vector, changing its components. This matrix can be thought of as a rotation matrix in four-dimensional space, which rotates the four-vector around a particular axis.
12896:{\displaystyle {\begin{aligned}S+{\overrightarrow {SU}}&=U\Rightarrow {\overrightarrow {SU}}=U-S,\\S+{\overrightarrow {SP}}&=P\Rightarrow {\overrightarrow {SP}}=P-S\end{aligned}}}
7210:
Contravariant and covariant vectors are geometrically very different objects. The first can and should be thought of as arrows. A linear function can be characterized by two objects: its
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respectively) stick to one choice regardless of topic. Arguments for the former convention include "continuity" from the Euclidean case corresponding to the non-relativistic limit
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surrounding any point (barring gravitational singularities). More abstractly, it can be said that in the presence of gravity spacetime is described by a curved 4-dimensional
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and is the most common mathematical structure by which special relativity is formalized. While the individual components in Euclidean space and time might differ due to
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of the invariance of the spacetime interval. This structure provides the background setting of all present relativistic theories, barring general relativity for which
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10172:
9688:
9666:, preventing easy visualization. By comparison, model spaces with positive curvature are just spheres in Euclidean space of one higher dimension. Hyperbolic spaces
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7335:
1661:
16591:{\displaystyle dx^{1}(\mathbf {u} )={\frac {2R^{2}\left(R^{2}-|u|^{2}\right)du^{1}+4R^{2}u^{1}(\mathbf {u} \cdot d\mathbf {u} )}{\left(R^{2}-|u|^{2}\right)^{2}}},}
13116:
10337:
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7340:
3415:. Arguments for the latter include that minus signs, otherwise ubiquitous in particle physics, go away. Yet other authors, especially of introductory texts, e.g.
14092:{\displaystyle \sigma ^{-1}(\mathbf {u} )=(\tau ,\mathbf {x} )=\left(R{\frac {R^{2}+|u|^{2}}{R^{2}-|u|^{2}}},{\frac {2R^{2}\mathbf {u} }{R^{2}-|u|^{2}}}\right).}
10364:
7554:
843:
from the invariance of the spacetime interval on the spacetime manifold as consequences of the postulates of special relativity, not to specific application or
5150:
2068:
1147:
future-directed timelike vectors whose first component is positive (tip of vector located in causal future (also called the absolute future) in the figure) and
8062:
801:. It is principally this view of spacetime that is current nowadays, although the older view involving imaginary time has also influenced special relativity.
21507:
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20446:
372:
21659:
11578:. The pushforward is the smooth manifold version of this. It acts between tangent spaces, and is in coordinates represented by the Jacobian matrix of the
9502:. Minkowski space is a suitable basis for special relativity, a good description of physical systems over finite distances in systems without significant
7086:
Technically, a non-degenerate bilinear form provides a map between a vector space and its dual; in this context, the map is between the tangent spaces of
5500:{\displaystyle \eta =\left({\begin{array}{r}1&0&0&0\\0&-1&0&0\\0&0&-1&0\\0&0&0&-1\end{array}}\right)\!,}
14803:
One computes according to the standard rules for computing differentials (though one is really computing the rigorously defined exterior derivatives),
11592:
10898:
4275:
1476:: An important property is that the scalar product of two similarly directed time-like vectors is always positive. This can be seen from the reversed
9535:
to any point is a 4-dimensional Minkowski space. Thus, the structure of Minkowski space is still essential in the description of general relativity.
5064:
This quantity is not consistently named in the literature. The interval is sometimes referred to as the square root of the interval as defined here.
44:(1864–1909) found that the theory of special relativity could be best understood as a four-dimensional space, since known as the Minkowski spacetime.
11299:. (This is in accord with what was detailed about the pullback of the inclusion map. In the general case here, one cannot proceed as simply because
8567:
7150:
are the components of a vector in the cotangent space (a linear functional). Due to the identification of vectors in tangent spaces with vectors in
619:
10376:
8391:{\displaystyle dx^{\mu }\left({\frac {\partial }{\partial x^{\nu }}}\right)={\frac {\partial x^{\mu }}{\partial x^{\nu }}}=\delta _{\nu }^{\mu }.}
20619:
13568:{\displaystyle {\begin{aligned}\tau &={\frac {R(1-\lambda )}{\lambda }},\\\mathbf {x} &={\frac {\mathbf {u} }{\lambda }},\end{aligned}}}
5881:
3326:{\displaystyle {\begin{aligned}x'^{\mu }&={\Lambda ^{\mu }}_{\nu }x^{\nu },\\v'^{\mu }&={\Lambda ^{\mu }}_{\nu }v^{\nu }.\end{aligned}}}
22564:
13798:
2404:, which is necessary for spacetime to be modeled as a vector space. This addition is not required, and more complex treatments analogous to an
11259:{\displaystyle F^{*}(\alpha )\left(X_{1},\,X_{2},\,\ldots ,\,X_{k}\right)=\alpha \left(F_{*}X_{1},\,F_{*}X_{2},\,\ldots ,\,F_{*}X_{k}\right),}
816:) of certain vectors, and the Minkowski norm squared is referred to (somewhat cryptically, perhaps this is a translation dependent) as "sum".
21260:
7782:{\displaystyle \eta ^{-1}:M^{*}\times M^{*}\rightarrow \mathbf {R} ,\eta ^{-1}(\alpha ,\beta )=\eta (\eta ^{-1}(\alpha ),\eta ^{-1}(\beta ))}
3816:
3643:
1190:
are called timelike, spacelike, or null if the associated vectors are timelike, spacelike, or null at each point where the field is defined.
192:, in Minkowski spacetime, all frames of reference will agree on the total interval in spacetime between events. Minkowski space differs from
17:
6679:
1143:
Once a direction of time is chosen, timelike and null vectors can be further decomposed into various classes. For timelike vectors, one has
20501:
This corresponds to the time coordinate either increasing or decreasing when the proper time for any particle increases. An application of
13596:
9200:
A Lorentzian manifold is a generalization of Minkowski space in two ways. The total number of spacetime dimensions is not restricted to be
7819:
can be told apart by the argument each is evaluated on. This can then be used to raise indices. If a coordinate basis is used, the metric
6272:{\displaystyle \eta (v,w)=\eta _{\mu \nu }v^{\mu }w^{\nu }=v^{0}w_{0}+v^{1}w_{1}+v^{2}w_{2}+v^{3}w_{3}=v^{\mu }w_{\mu }=v_{\mu }w^{\mu },}
2480:. By the same principle, the tangent space at a point in flat spacetime can be thought of as a subspace of spacetime, which happens to be
22267:
5057:{\displaystyle c^{2}\left(t_{1}-t_{2}\right)^{2}-\left(x_{1}-x_{2}\right)^{2}-\left(y_{1}-y_{2}\right)^{2}-\left(z_{1}-z_{2}\right)^{2}.}
2497:, Proposition 3.13.) These identifications are routinely done in mathematics. They can be expressed formally in Cartesian coordinates as
4566:
3483:
nature of the metric and the true nature of Lorentz boosts, which are not rotations. It also needlessly complicates the use of tools of
22502:
20836:
9696:
7994:{\displaystyle \eta _{\mu \nu }dx^{\mu }\otimes dx^{\nu }=\eta _{\mu \nu }dx^{\mu }\odot dx^{\nu }=\eta _{\mu \nu }dx^{\mu }dx^{\nu }.}
7009:
as the same thing, mathematically, they are elements of different spaces, and act on the space of standard bases from different sides.
752:
concluded that time and space should be treated equally, and so arose his concept of events taking place in a unified four-dimensional
427:
22291:
22037:
10800:
8553:{\displaystyle \left.dx^{\mu }\right|_{p}\left(\left.{\frac {\partial }{\partial x^{\nu }}}\right|_{p}\right)=\delta _{\nu }^{\mu }.}
7265:) and proceed algebraically in a purely formal fashion. The time-proven robustness of the formalism itself, sometimes referred to as
3447:(or raising and lowering indices) to be described below. The inner product is instead affected by a straightforward extension of the
877:
At a time when Minkowski was giving the geometrical interpretation of special relativity by extending the Euclidean three-space to a
10500:
Behavior of tensors under inclusion, pullback of covariant tensors under general maps and pushforward of vectors under general maps
7586:
5282:
18709:{\displaystyle \left(\sigma ^{-1}\right)^{*}h_{R}^{1(n)}={\frac {4R^{2}\left}{\left(R^{2}-|u|^{2}\right)^{2}}}\equiv h_{R}^{2(n)}.}
15738:{\displaystyle \left(\sigma ^{-1}\right)^{*}h_{R}^{1(n)}={\frac {4R^{2}\left}{\left(R^{2}-|u|^{2}\right)^{2}}}\equiv h_{R}^{2(n)}.}
10093:
3384:
In general, but with several exceptions, mathematicians and general relativists prefer spacelike vectors to yield a positive sign,
402:
5070:
717:
3929:
14270:{\displaystyle h_{R}^{1(n)}=\eta |_{\mathbf {H} _{R}^{1(n)}}=\left(dx^{1}\right)^{2}+\cdots +\left(dx^{n}\right)^{2}-d\tau ^{2}}
13472:{\displaystyle {\begin{aligned}U&=(0,\mathbf {u} )\\P&=(\tau (\mathbf {u} ),\mathbf {x} (\mathbf {u} )).\end{aligned}},}
6480:
5794:
22073:
22027:
4062:. The positive-definite condition has been replaced by the weaker condition of non-degeneracy. The bilinear form is said to be
6872:
6814:
22211:
21618:
21590:
21567:
21378:
21355:
21245:
21226:
21207:
21158:
21082:
20956:
Y. Friedman, A Physically Meaningful Relativistic Description of the Spin State of an Electron, Symmetry 2021, 13(10), 1853;
20924:.). Where Lee refers to positive definiteness to show the injectivity of the map, one needs instead appeal to non-degeneracy.
20863:
20815:
20656:
7280:
20821:
12519:
11450:
21695:
4209:
5566:
5510:
21640:
20610:
12370:
9566:(positive curvature). The reason is the indefiniteness of the Minkowski metric. Minkowski space is, in particular, not a
8972:{\displaystyle g(p)_{\mu \nu }\left.dx^{\mu }\right|_{p}\left.dx^{\nu }\right|_{p}(a,b)=g(p)_{\mu \nu }a^{\mu }b^{\nu },}
6461:{\displaystyle \eta (v,v)=\eta _{\mu \nu }v^{\mu }v^{\nu }=v^{0}v_{0}+v^{1}v_{1}+v^{2}v_{2}+v^{3}v_{3}=v^{\mu }v_{\mu }.}
521:
11555:
9670:
be isometrically embedded in spaces of one more dimension when the embedding space is endowed with the Minkowski metric
3356:. They offer various degrees of sophistication (and rigor) depending on which part of the material one chooses to read.
881:
four-space that included time, Einstein was already aware that this is not valid, because it excludes the phenomenon of
21948:
21540:
20960:
5067:
The invariance of the interval under coordinate transformations between inertial frames follows from the invariance of
3530:
367:
592:
To understand this concept, one should consider the coordinates of an event in spacetime represented as a four-vector
22319:
21953:
21448:
21188:
3467:. This works in the flat spacetime of special relativity, but not in the curved spacetime of general relativity, see
727:
published in 1908 called "The Fundamental Equations for Electromagnetic Processes in Moving Bodies". He reformulated
452:
10451:. The present purpose is to describe this and similar operations as a preparation for the actual demonstration that
1640:
is another consequence of the convexity of either light cone. For two distinct similarly directed time-like vectors
21275:
11562:
of partial derivatives of the component functions. The differential is the best linear approximation of a function
10448:
10151:
7164:. In the same fashion, the inverse of the map from tangent to cotangent spaces, explicitly given by the inverse of
4850:
22273:
10271:. The 1 in the upper index refers to an enumeration of the different model spaces of hyperbolic geometry, and the
4074:
pseudo-Riemannian metric in Cartesian coordinates. As such, it is a nondegenerate symmetric bilinear form, a type
475:
21986:
13788:{\displaystyle -{\frac {R^{2}(1-\lambda )^{2}}{\lambda ^{2}}}+{\frac {|\mathbf {u} |^{2}}{\lambda ^{2}}}=-R^{2},}
9384:
13256:{\displaystyle {\begin{aligned}R&=\lambda (\tau +R),\\\mathbf {u} &=\lambda \mathbf {x} .\end{aligned}}}
6930:
22089:
21105:
8183:
4504:
4490:{\displaystyle \eta (au+v,\,w)=a\eta (u,\,w)+\eta (v,\,w),\quad \forall u,\,v\in M,\;\forall a\in \mathbb {R} }
1150:
past-directed timelike vectors whose first component is negative (causal past (also called the absolute past)).
804:
In the English translation of Minkowski's paper, the Minkowski metric, as defined below, is referred to as the
31:
8034:
7852:
A full-blown version of the Minkowski metric in coordinates as a tensor field on spacetime has the appearance
7426:
22554:
22539:
22520:
22262:
22167:
22097:
21326:
20711:
10454:
10181:
6545:
Any two such bases are related in some sense by a Lorentz transformation, either by a change-of-basis matrix
4812:
3397:
1477:
793:
associated with each point, and events not on the light cone are classified by their relation to the apex as
22230:
6976:
6548:
22549:
22084:
20807:
The Geometry of Minkowski Spacetime: An Introduction to the Mathematics of the Special Theory of Relativity
11589:
from the dual of the range tangent space to the dual of the domain tangent space, i.e. it is a linear map,
8999:
7247:
7111:
7018:
21259:
21240:. Springer Graduate Texts in Mathematics. Vol. 176. New York · Berlin · Heidelberg: Springer Verlag.
10340:
2408:
can remove the extra structure. However, this is not the introductory convention and is not covered here.
22559:
22125:
21859:
21844:
21512:
Robb A A: Optical Geometry of Motion; a New View of the Theory of Relativity Cambridge 1911, (Heffers).
7474:
988:
809:
154:
11072:{\displaystyle T^{k}V=\underbrace {V^{*}\otimes V^{*}\otimes \cdots \otimes V^{*}} _{k{\text{ times}}}.}
10024:
8059:
vectors are, in this formalism, given in terms of a basis of differential operators of the first order,
3894:
3699:
22569:
22412:
21854:
21807:
21653:
21610:
11267:
9179:
3628:
3595:
227:. When time is appended as a fourth dimension, the further transformations of translations in time and
10083:
9613:
9584:
6653:{\displaystyle \Lambda _{\rho }^{\mu }\eta _{\mu \nu }\Lambda _{\sigma }^{\nu }=\eta _{\rho \sigma }.}
22442:
22068:
21749:
21180:
11586:
9642:
3634:
Then mathematically, the metric is a bilinear form on an abstract four-dimensional real vector space
1464:{\displaystyle \eta (u_{1},u_{2})=u_{1}\cdot u_{2}=c^{2}t_{1}t_{2}-x_{1}x_{2}-y_{1}y_{2}-z_{1}z_{2}.}
16369:{\displaystyle {\frac {\partial }{\partial \tau }}{\frac {2R^{2}u^{1}}{R^{2}-|u|^{2}}}d\tau ^{2}=0.}
22467:
22017:
21777:
21688:
21266:
Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse
11765:
4819:
4763:
723:
This idea, which was mentioned only briefly by Poincaré, was elaborated by Minkowski in a paper in
377:
243:
220:
9449:. These generalizations are used in theories where spacetime is assumed to have more or less than
5849:
3778:
3745:
22196:
22118:
21738:
21729:
20537:
and curved space at infinitesimally small distance scales is foundational to the definition of a
8773:{\displaystyle \eta _{\mu \nu }dx^{\mu }\otimes dx^{\nu }(a,b)=\eta _{\mu \nu }a^{\mu }b^{\nu },}
8033:, i.e. the type that expects two contravariant vectors as arguments. On the right-hand side, the
7211:
7068:
5596:
A standard or orthonormal basis for Minkowski space is a set of four mutually orthogonal vectors
2423:
886:
823:, and he uses it to define concepts and demonstrate properties of Lorentz transformations (e.g.,
514:
271:
20975:
Jackson, J.D., Classical Electrodynamics, 3rd ed.; John Wiley \& Sons: Hoboken, NJ, US, 1998
9249:
of the space. The imaginary part, on the other hand, may consist of four pseudovectors, such as
7792:
3742:
may be equivalently viewed as an element of this space. By making a choice of orthonormal basis
1140:
of constant velocity associated with it, represented by a straight line in a Minkowski diagram.
22422:
22285:
22191:
22149:
21823:
21787:
20559:
9511:
9469:
9246:
8124:
5340:
4793:
4070:
is the metric tensor of Minkowski space. It is a pseudo-Euclidean metric, or more generally, a
4037:
4023:
4011:
3584:
3364:
The metric signature refers to which sign the Minkowski inner product yields when given space (
3113:
878:
832:
585:
580:
540:
342:
224:
193:
20663:
18739:
11686:
10307:
10236:
5784:{\displaystyle -\eta (e_{0},e_{0})=\eta (e_{1},e_{1})=\eta (e_{2},e_{2})=\eta (e_{3},e_{3})=1}
4834:-dimensional flat spacetime with the remaining ambiguity only being the signature convention.
22427:
22032:
21971:
21393:
20436:
9760:
9524:
9482:
can be formulated as a submanifold of generalized Minkowski space as can the model spaces of
9195:
5276:
3484:
437:
337:
21513:
20853:
20833:
4006:
An interesting example of non-inertial coordinates for (part of) Minkowski spacetime is the
3373:
1628:{\displaystyle \left\|u\right\|={\sqrt {\eta (u,u)}}={\sqrt {c^{2}t^{2}-x^{2}-y^{2}-z^{2}}}}
851:
Minkowski spacetime still provides a springboard as curved spacetime is locally Lorentzian.
831:) and to provide geometrical interpretation to the generalization of Newtonian mechanics to
789:. Points in this space correspond to events in spacetime. In this space, there is a defined
22332:
22258:
22102:
21704:
21520:
21474:
21422:
20920:. One point in Lee's proof of the existence of this map needs modification (Lee deals with
18722:
This last equation shows that the metric on the ball is identical to the Riemannian metric
10157:
9673:
7559:
7515:
7320:
7235:
5575:
has been used in a derivation, go back to the earliest point where it was used, substitute
4027:
3444:
3335:
This definition is equivalent to the definition given above under a canonical isomorphism.
2396:
It is assumed below that spacetime is endowed with a coordinate system corresponding to an
613:
21066:
13340:{\displaystyle \sigma (\tau ,\mathbf {x} )=\mathbf {u} ={\frac {R\mathbf {x} }{R+\tau }}.}
10313:
10280:
10059:
9570:
and not a Riemannian manifold with a Riemannian metric. However, Minkowski space contains
8:
22407:
22402:
22392:
22224:
22173:
21991:
21976:
21802:
21733:
21681:
21578:
20470:
10222:
9559:
9499:
9483:
9191:
9071:
8652:. (This can be taken as a definition, but may also be proved in a more general setting.)
8004:
3571:
3521:
3423:
choose a signature at all, but instead, opt to coordinatize spacetime such that the time
890:
753:
22397:
21478:
10346:
7118:
on the vector space, the same holds for the Minkowski inner product of Minkowski space.
4189:, are denoted in italics, and not, as is common in the Euclidean setting, with boldface
2446:. The tangent space at each event is a vector space of the same dimension as spacetime,
22472:
22372:
22296:
22154:
22135:
22129:
22080:
22022:
21931:
21849:
21767:
21744:
21712:
21552:
21529:
21498:
21434:
21410:
21345:
21140:
21117:
21049:
9507:
9266:
is introduced, which also changes sign with a change of orientation. Thus, elements of
8816:
7539:
7512:. Note it does not matter which argument is partially evaluated due to the symmetry of
7022:
4854:
4843:
3606:
3574:. Its metric tensor is in coordinates with the same symmetric matrix at every point of
3517:
3109:
2133:
927:
Subdivision of Minkowski spacetime with respect to an event in four disjoint sets: the
828:
507:
493:
457:
314:
309:
262:
208:
185:
181:
177:
150:
146:
121:
20694:
s:Translation:The Fundamental Equations for Electromagnetic Processes in Moving Bodies
3388:, while particle physicists tend to prefer timelike vectors to yield a positive sign,
1165:
future-directed null vectors whose first component is positive (upper light cone), and
854:
Minkowski, aware of the fundamental restatement of the theory which he had made, said
22544:
22482:
22314:
22306:
21907:
21864:
21614:
21604:
21586:
21563:
21536:
21502:
21444:
21374:
21351:
21333:
21261:"Die Grundgleichungen für die elektromagnetischen Vorgänge in bewegten Körpern"
21255:
21241:
21222:
21203:
21184:
21154:
21144:
21088:
21078:
21041:
20921:
20859:
20811:
20723:
20652:
20517:
10849:
10226:
10175:
9544:
8003:
Explanation: The coordinate differentials are 1-form fields. They are defined as the
7115:
7103:
5975:
4823:
4801:
820:
728:
432:
158:
41:
22352:
21550:
Shaw, R. (1982). "§ 6.6 Minkowski space, § 6.7,8 Canonical forms pp 221–242".
21494:
21458:
21264:[The Fundamental Equations for Electromagnetic Processes in Moving Bodies],
21053:
7110:). Just as an authentic inner product on a vector space with one argument fixed, by
4853:, together with homogeneity of spacetime and isotropy of space, it follows that the
836:
546:
267:
261:
depending on the context. The Minkowski inner product is defined so as to yield the
232:
166:
22387:
22377:
22324:
22301:
21879:
21490:
21482:
21402:
21388:
21289:
21109:
21070:
21033:
10230:
9250:
9208:
or more) and a Lorentzian manifold need not be flat, i.e. it allows for curvature.
9111:
7536:
this is enough to conclude the partial evaluation map is a linear isomorphism from
7266:
4007:
3440:
2435:
956:
918:
447:
139:
63:
22447:
12740:{\displaystyle U=\left(0,u^{1}(P),\ldots ,u^{n}(P)\right)\equiv (0,\mathbf {u} ).}
12631:{\displaystyle \left\{\left(\tau ,x^{1},\ldots ,x^{n}\right)\in M:\tau =0\right\}}
7156:
itself, this is mostly ignored, and vectors with lower indices are referred to as
22462:
22437:
22362:
22357:
22240:
22201:
22163:
22107:
21981:
21600:
21440:
21418:
21370:
21341:
21172:
21136:
20964:
20840:
20805:
20731:
20646:
20614:
20451:
12509:{\displaystyle P=\left(\tau ,x^{1},\ldots ,x^{n}\right)\in \mathbf {H} _{R}^{n},}
11559:
9479:
9254:
7317:, the lowered version of a vector can be thought of as the partial evaluation of
7257:
One may, of course, ignore geometrical views altogether (as is the style in e.g.
7231:
7091:
7071:
4681:
3583:
Introducing more terminology (but not more structure), Minkowski space is thus a
3393:
3108:
are any two events, and the second basis vector identification is referred to as
870:
848:
724:
239:
changing the scale applied to the frame in motion and shifts the phase of light.
216:
212:
200:
173:
162:
22245:
7243:, offers the same geometrical view of these objects (but mentions no piercing).
6664:, a linear map on the abstract vector space satisfying, for any pair of vectors
5588:, and retrace forward to the desired formula with the desired metric signature.
3148:
may be defined, here specialized to Cartesian coordinates in Lorentz frames, as
1168:
past-directed null vectors whose first component is negative (lower light cone).
22487:
22159:
22143:
22139:
22042:
22012:
21869:
21772:
21559:
14580:
The pulled back metric can be obtained by straightforward methods of calculus;
11693:
In order to exhibit the metric, it is necessary to pull it back via a suitable
10087:
9506:. However, in order to take gravity into account, physicists use the theory of
9332:
8022:
5144:
4808:
4041:
3696:. The space of bilinear maps forms a vector space which can be identified with
3435:(which may seem like an extra burden in an introductory course), and one needs
3130:. This definition of tangent vectors is not the only possible one, as ordinary
2397:
1212:
576:
566:
480:
204:
21647:
21074:
8667:, both expanded in terms of the basis coordinate vector fields, the result is
5557:
with which this section started by assuming its existence, is now identified.
1739:{\displaystyle \eta (u_{1},u_{2})>\left\|u_{1}\right\|\left\|u_{2}\right\|}
37:
22533:
22477:
22457:
22452:
22367:
22235:
22063:
22007:
21839:
21792:
21664:
21430:
21092:
21045:
20607:
20521:
13171:{\displaystyle {\overrightarrow {SU}}=\lambda (\tau ){\overrightarrow {SP}}.}
10444:
9532:
9454:
9279:
9242:
9238:
8998:
is still a metric tensor but now depending on spacetime and is a solution of
7079:
4647:
4265:
4033:
3580:, and its arguments can, per above, be taken as vectors in spacetime itself.
3432:
2427:
2139:
The proof uses the algebraic definition with the reversed Cauchy inequality:
709:
246:
236:
228:
189:
10436:{\displaystyle \iota :\mathbf {H} _{R}^{1(n)}\rightarrow \mathbf {M} ^{n+1}}
7414:{\displaystyle \eta (\cdot ,-):M\rightarrow M^{*};v\mapsto \eta (v,\cdot ).}
4732:. For a geometric interpretation of orthogonality in the special case, when
22497:
22417:
22382:
21912:
21874:
21531:
Relativistic Mechanics - Special Relativity and Classical Particle Dynamics
21293:
20726:(1972) "Einstein's Path from Special to General Relativity", pages 5–19 of
9581:
be isometrically embedded in Euclidean space with one more dimension, i.e.
9567:
9516:
9258:
9234:
8225:
8042:
7027:
2419:
2405:
1187:
786:
442:
21311:
9514:. When this geometry is used as a model of physical space, it is known as
5389:
3340:
For some purposes, it is desirable to identify tangent vectors at a point
22281:
22250:
21797:
21660:
The Geometry of Special Relativity: The Minkowski Space – Time Light Cone
21150:
9787:
9571:
9503:
8800:, and the relation may as well be interpreted as the Minkowski metric at
7583:. This then allows the definition of the inverse partial evaluation map,
5268:{\displaystyle u\cdot v=c^{2}t_{1}t_{2}-x_{1}x_{2}-y_{1}y_{2}-z_{1}z_{2}}
4774:
4672:
The most important feature of the inner product and norm squared is that
3448:
2416:
2125:{\displaystyle \left\|u+w\right\|\geq \left\|u\right\|+\left\|w\right\|,}
1176:
1123:
882:
824:
422:
196:
insofar as it treats time differently than the three spatial dimensions.
21302:
The Fundamental Equations for Electromagnetic Processes in Moving Bodies
21121:
20858:(illustrated, herdruk ed.). Cambridge University Press. p. 7.
13368:
must be calculated. Use the same considerations as before, but now with
11681:
9523:
Even in curved space, Minkowski space is still a good description in an
8108:{\displaystyle \left.{\frac {\partial }{\partial x^{\mu }}}\right|_{p},}
6477:
There are many possible choices of standard basis obeying the condition
6057:
In terms of components, the Minkowski inner product between two vectors
3538:, which is a nondegenerate symmetric bilinear form on the tangent space
22492:
22058:
21902:
21897:
21668:
21486:
21414:
21337:
21280:
21168:
21037:
21024:
Corry, L. (1997). "Hermann Minkowski and the postulate of relativity".
20785:, See Lee's discussion on geometric tangent vectors early in chapter 3.
20534:
20520:, which provides a positive definite symmetric bilinear form, i. e. an
20441:
9169:
8446:
7230:
One quantum mechanical analogy explored in the literature is that of a
7215:
7075:
5367:. While possibly confusing, it is common practice to denote with just
4714:
3878:{\displaystyle \mathbf {R} ^{1,3}:=(\mathbf {R} ^{4},\eta _{\mu \nu })}
1127:
928:
790:
21579:"Minkowski, Mathematicians, and the Mathematical Theory of Relativity"
20957:
9498:, the three spatial components of Minkowski spacetime always obey the
2458:
808:. The Minkowski inner product below appears unnamed when referring to
21889:
21113:
4818:
More terminology (but not more structure): The Minkowski metric is a
1137:
550:
332:
298:
290:
135:
117:
21462:
21406:
21129:
20492:
Translate the coordinate system so that the event is the new origin.
18754:
The pullback can be computed in a different fashion. By definition,
8231:. A vector field is an assignment of a tangent vector to each point
7234:(scaled by a factor of Planck's reduced constant) associated with a
2049:
From this, the positive property of the scalar product can be seen.
785:
of space and time in the coordinate form in a four-dimensional real
235:. Minkowski's model follows special relativity, where motion causes
231:
are added, and the group of all these transformations is called the
169:, and others said it "was grown on experimental physical grounds".
22432:
21782:
21436:
Road to Reality : A Complete Guide to the Laws of the Universe
20538:
9528:
9458:
9336:
8049:
a tensor. For elaboration on the differences and similarities, see
4186:
3570:. Minkowski space is thus a comparatively simple special case of a
3427:(but not time itself!) is imaginary. This removes the need for the
125:
21102:
Minkowski's Space–Time: from visual thinking to the absolute world
15495:
and substitutes the results into the right hand side. This yields
9577:
Model spaces of hyperbolic geometry of low dimension, say 2 or 3,
4632:{\displaystyle \eta (u,\,v)=0,\;\forall v\in M\ \Rightarrow \ u=0}
4358:. The Minkowski inner product satisfies the following properties.
21656:
visualizing Minkowski space in the context of special relativity.
10881:(one taking only contravariant vectors as arguments) under a map
9750:{\displaystyle \mathbf {H} _{R}^{1(n)}\subset \mathbf {M} ^{n+1}}
9427:-dimensional Minkowski space is a vector space of real dimension
397:
49:
21519:
Robb A A: Geometry of Time and Space, 1936 Cambridge Univ Press
11663:{\displaystyle F^{*}\colon T_{F(p)}^{*}N\rightarrow T_{p}^{*}M.}
10969:{\displaystyle F^{*}\colon T_{F(p)}^{k}N\rightarrow T_{p}^{k}M,}
9110:
is future-directed null or future-directed timelike. It gives a
7195:. These maps between a vector space and its dual can be denoted
4347:{\displaystyle u\cdot u=\eta (u,u)\equiv \|u\|^{2}\equiv u^{2},}
2388:
The result now follows by taking the square root on both sides.
22112:
21635:
9574:
endowed with a Riemannian metric yielding hyperbolic geometry.
9563:
9257:, which change their direction with a change of orientation. A
7251:
7220:
4268:, but it describes a different geometry. It is also called the
3498:
3116:
operator on the set of smooth functions. This is promoted to a
2467:
20710:, pp. 75–88 Various English translations on Wikisource: "
11362:
is by definition the same as pushing forward the vectors from
8625:{\displaystyle \alpha \otimes \beta (a,b)=\alpha (a)\beta (b)}
3368:
to be specific, defined further down) and time basis vectors (
698:{\displaystyle x^{2}+y^{2}+z^{2}+(ict)^{2}={\text{constant}}.}
21673:
21179:. Course of Theoretical Physics. Vol. 2 (4th ed.).
21104:. Historical Studies in the Physical Sciences. Vol. 10.
11689:; it projects to the brown geodesic on the green hyperboloid.
8796:
of the vector fields. The above equation holds at each point
7254:, which geometrical description can as well be found in MTW.
7106:(so the dimension of the cotangent space at an event is also
1172:
Together with spacelike vectors, there are 6 classes in all.
923:
128:
81:
11735:, a parametrization is just the inverse of a coordinate map
10225:. It is one of the model spaces of Riemannian geometry, the
5371:. The matrix is read off from the explicit bilinear form as
4800:
consisting of mutually orthogonal unit vectors is called an
2462:
A pictorial representation of the tangent space at a point,
997:
is the constant representing the universal speed limit, and
22093:
14589:
9433:
on which there is a constant Minkowski metric of signature
8879:
8850:
8491:
8458:
8068:
3996:{\displaystyle \eta _{\mu \nu }={\text{diag}}(-1,+1,+1,+1)}
2970:
2941:
2882:
2841:
2800:
2759:
2708:
2667:
2626:
2585:
132:
93:
72:
21391:(1956). "The Imbedding Problem for Riemannian Manifolds".
7239:
the arrow pierces the planes. The mathematical reference,
6538:{\displaystyle \eta (e_{\mu },e_{\nu })=\eta _{\mu \nu }.}
5939:{\displaystyle \eta (e_{\mu },e_{\nu })=\eta _{\mu \nu }.}
4674:
these are quantities unaffected by Lorentz transformations
105:
21514:
http://www.archive.org/details/opticalgeometryoOOrobbrich
21238:
Riemannian Manifolds – An Introduction to Curvature
21202:. Springer Graduate Texts in Mathematics. Vol. 218.
13868:{\displaystyle \lambda ={\frac {R^{2}-|u|^{2}}{2R^{2}}}.}
8176:, i.e. an assignment of a cotangent vector to each point
7337:, that is, there is an associated partial evaluation map
5948:
Relative to a standard basis, the components of a vector
5560:
For definiteness and shorter presentation, the signature
4642:
The first two conditions imply bilinearity. The defining
4132:. Due to the above-mentioned canonical identification of
69:
21521:
http://www.archive.org/details/geometryoftimean032218mbp
21100:
Galison, P. L. (1979). R McCormach; et al. (eds.).
20810:(illustrated ed.). Courier Corporation. p. 8.
9024:
6920:{\displaystyle e_{\mu }'=e_{\nu }\Lambda _{\mu }^{\nu }}
6862:{\displaystyle e_{\mu }'=e_{\nu }\Lambda _{\mu }^{\nu }}
7310:{\displaystyle \eta :M\times M\rightarrow \mathbf {R} }
7272:
3673:{\displaystyle \eta :V\times V\rightarrow \mathbf {R} }
2470:. This vector space can be thought of as a subspace of
731:
as a symmetrical set of equations in the four variables
21310:
11676:
8655:
Thus when the metric tensor is fed two vectors fields
7127:
are the components of a vector in tangent space, then
6734:{\displaystyle \eta (\Lambda u,\Lambda v)=\eta (u,v).}
5878:
These conditions can be written compactly in the form
5852:
3690:, and signature is a coordinate-invariant property of
3497:
Mathematically associated with the bilinear form is a
3041:
1488:
20651:(illustrated ed.). CRC Press. pp. 184–185.
20516:
For comparison and motivation of terminology, take a
20447:
Introduction to the mathematics of general relativity
20134:
19497:
19070:
18760:
18472:
18199:
17817:
17001:
16604:
16384:
16267:
15762:
15501:
14809:
14586:
14283:
14113:
13889:
13801:
13673:
13660:{\displaystyle -\tau ^{2}+|\mathbf {x} |^{2}=-R^{2},}
13599:
13485:
13374:
13277:
13186:
13119:
12909:
12755:
12644:
12554:
12522:
12431:
12373:
12167:
11825:
11757:. This is illustrated in the figure to the right for
11595:
11453:
11087:
10986:
10901:
10803:
10536:
10457:
10379:
10349:
10316:
10283:
10239:
10184:
10160:
10096:
10062:
10027:
9795:
9763:
9699:
9676:
9645:
9616:
9587:
9562:(negative curvature) and the geometry modeled by the
8825:
8673:
8570:
8455:
8286:
8186:
8155:
fixed. They provide a basis for the tangent space at
8065:
7858:
7795:
7641:
7589:
7562:
7542:
7518:
7477:
7429:
7343:
7323:
7283:
6979:
6933:
6875:
6817:
6682:
6587:
6551:
6483:
6285:
6073:
5884:
5797:
5634:
5513:
5377:
5343:
5285:
5153:
5073:
4871:
4569:
4507:
4370:
4278:
4212:
3932:
3926:
are not just vector spaces but have added structure.
3897:
3819:
3781:
3748:
3702:
3646:
3202:
2938:
2503:
2145:
2071:
1752:
1664:
1522:
1297:
869:
Though Minkowski took an important step for physics,
622:
549:
showed how, by taking time to be an imaginary fourth
108:
102:
90:
87:
84:
78:
13113:
By construction of stereographic projection one has
7834:
7223:(though the latter is usually reserved for covector
2932:
with basis vectors in the tangent spaces defined by
21332:
20906:
20728:
General Relativity: Papers in Honour of J. L. Synge
11753:. The parametrization to be used is the inverse of
11403:Further unwinding the definitions, the pushforward
9070:is future-directed timelike. This relation has the
8050:
7839:The present purpose is to show semi-rigorously how
3885:. The notation is meant to emphasize the fact that
3468:
3353:
2430:at each point in spacetime, here simply called the
1158:the zero vector, whose components in any basis are
99:
75:
66:
21551:
21528:
20415:
20118:
19476:
19054:
18708:
18456:
18185:
17797:
16985:
16590:
16368:
16253:
15737:
15487:
14793:
14570:
14269:
14091:
13867:
13787:
13659:
13567:
13471:
13339:
13255:
13170:
13103:
12895:
12739:
12630:
12540:
12508:
12415:
12359:
12106:
11662:
11502:
11258:
11071:
10968:
10840:
10762:
10486:
10435:
10358:
10331:
10298:
10263:
10213:
10166:
10142:
10074:
10048:
10013:
9778:
9749:
9682:
9658:
9631:
9602:
9275:are independent of the choice of the orientation.
8971:
8772:
8624:
8552:
8390:
8216:
8107:
7993:
7811:
7781:
7627:
7575:
7548:
7524:
7504:
7463:
7413:
7329:
7309:
7012:
6997:
6965:
6919:
6861:
6733:
6652:
6569:
6537:
6460:
6271:
5938:
5868:
5838:
5783:
5549:
5499:
5355:
5329:
5267:
5135:
5056:
4631:
4551:
4489:
4346:
4246:
3995:
3918:
3877:
3805:
3767:
3728:
3672:
3392:. Authors covering several areas of physics, e.g.
3325:
3138:Definitions of tangent vectors as ordinary vectors
3086:
2924:
2378:
2124:
2039:
1738:
1627:
1463:
1193:
897:For further historical information see references
697:
21128:Giulini D The rich structure of Minkowski space,
18742:, another standard model of hyperbolic geometry.
9372:is the four-velocity of the particle, satisfying
8119:is an event. This operator applied to a function
7635:which allows the inverse metric to be defined as
5493:
22531:
21135:
20803:
20743:See Schutz's proof p 148, also Naber p. 48
8416:provide a basis for the cotangent space at each
7170:in matrix representation, can be used to define
6010:while the other three components are called the
3416:
3162:Lorentz frame related by Lorentz transformation
1100:. This can be expressed in terms of the sign of
893:, and the heavy mathematical apparatus entailed.
20936:, The tangent-cotangent isomorphism p. 282
20851:
20566:), but the embedding dimension is much higher,
12516:then it is geometrically clear that the vector
10841:{\displaystyle \iota ^{*}\alpha =\alpha |_{S},}
10233:. It is a space of constant negative curvature
9211:
5279:. This bilinear form can in turn be written as
2052:
1126:at an event of Minkowski space constitutes the
612:. A Lorentz transformation is represented by a
21308:
21254:
21167:
20891:
20707:
20689:
18748:Alternative calculation using the pushforward
11325:The pushforward of vectors under general maps:
9510:, which is formulated in the mathematics of a
9408:
7628:{\displaystyle \eta ^{-1}:M^{*}\rightarrow M,}
7262:
5330:{\displaystyle u\cdot v=u^{\textsf {T}}\,\,v,}
5143:provided the transformations are linear. This
2132:where the equality holds when the vectors are
534:
21689:
21457:
20674:
13271:and one obtains for stereographic projection
10143:{\displaystyle h_{R}^{1(n)}=\iota ^{*}\eta ,}
7174:. The components of this inverse are denoted
4260:, previously and also henceforth, called the
3562:. In coordinates, it may be represented by a
2057:For two similarly directed time-like vectors
1481:times either of different or the same signs.
515:
21467:Rendiconti del Circolo Matematico di Palermo
21325:Various English translations on Wikisource:
20620:Random House Webster's Unabridged Dictionary
20053:
20036:
19902:
19885:
19739:
19722:
5136:{\displaystyle c^{2}t^{2}-x^{2}-y^{2}-z^{2}}
4319:
4312:
3762:
3749:
30:For the use in algebraic number theory, see
21465:[On the Dynamics of the Electron],
20692:, pp. 53–111 *Wikisource translation:
20677:, pp. 129–176 Wikisource translation:
20648:Independent Axioms for Minkowski Space–Time
10856:to accept as input vectors tangent to some
9216:Complexified Minkowski space is defined as
8017:provide a basis for the cotangent space at
4654:required to be positive definite, that is,
4199:-vector part (to be introduced below) of a
4195:. The latter is generally reserved for the
4017:
4010:. Another useful set of coordinates is the
3621:to emphasize the chosen signature, or just
2476:itself. Then vectors in it would be called
759:
172:Minkowski space is closely associated with
21696:
21682:
21583:The Expanding Worlds of General Relativity
21581:. In Goenner, Hubert; et al. (eds.).
21221:. Springer Graduate Texts in Mathematics.
20703:
20701:
14102:
9339:or any half spin particle is described by
9233:. Its real part is the Minkowski space of
9019:and can no longer be moved around freely.
6966:{\displaystyle e_{\mu }'=\Lambda e_{\mu }}
5839:{\displaystyle \eta (e_{\mu },e_{\nu })=0}
4598:
4472:
4354:the resulting quantity will be called the
4254:yields an inner product-like structure on
3479:). MTW also argues that it hides the true
522:
508:
157:in which they are recorded. Mathematician
116:) is the main mathematical description of
20167:
20047:
19896:
19733:
19012:
18895:
18822:
18466:Now add this contribution to finally get
14500:
14392:
11227:
11220:
11196:
11146:
11139:
11125:
10741:
10734:
10720:
10670:
10663:
10639:
10589:
10582:
10568:
9383:is the 4D spin vector, which is also the
9245:, which are independent of the choice of
8632:for general one-forms on a tangent space
7848:A formal approach to the Minkowski metric
6973:. While it might be tempting to think of
5862:
5320:
5310:
5304:
4582:
4552:{\displaystyle \eta (u,\,v)=\eta (v,\,u)}
4542:
4520:
4483:
4459:
4439:
4417:
4392:
4237:
3605:. Elements of Minkowski space are called
2554:
2540:
2526:
2391:
242:Spacetime is equipped with an indefinite
21599:
21554:Linear Algebra and Group Representations
11680:
11328:Heuristically, pulling back a tensor to
9639:respectively, with the Euclidean metric
8400:Since this relation holds at each point
8013:. These quantities evaluated at a point
7464:{\displaystyle \eta (v,\cdot )\in M^{*}}
7258:
7026:
2457:
922:
545:In his second relativity paper in 1905,
36:
21429:
21099:
21063:The Mathematics of Minkowski Space-Time
20698:
20585:for a Riemannian manifold of dimension
11386:feeding them to the tensor residing at
10870:Pullback of tensors under general maps:
10487:{\displaystyle \mathbf {H} _{R}^{1(n)}}
10214:{\displaystyle \mathbf {H} _{R}^{1(n)}}
7102:, the tangent and cotangent spaces are
4680:classical groups definable this way in
3513:matrix representing the bilinear form.
3505:at each point in spacetime, called the
1130:of that event. Given a timelike vector
1032:is classified according to the sign of
898:
14:
22532:
21585:. Boston: Birkhäuser. pp. 45–86.
21576:
21060:
20644:
13180:This leads to the system of equations
12541:{\displaystyle {\overrightarrow {PS}}}
11503:{\displaystyle F_{*}(X)f=X(f\circ F),}
10510:For inclusion maps from a submanifold
10086:of the generalized Lorentz group. The
9015:be tangent vectors at spacetime point
8266:, the coordinate function itself, and
7114:, may be expressed as the action of a
6998:{\displaystyle \Lambda _{\nu }^{\mu }}
6570:{\displaystyle \Lambda _{\nu }^{\mu }}
5147:can be used to define a bilinear form
4851:second postulate of special relativity
4846:which is also called Minkowski metric.
4646:between a pseudo-inner product and an
4032:The Minkowski inner product is not an
2411:For an overview, Minkowski space is a
1154:Null vectors fall into three classes:
906:
27:Spacetime used in theory of relativity
22565:Exact solutions in general relativity
21677:
21364:
21273:
21023:
20902:
20900:
11697:. A parametrization of a submanifold
11266:where the subscript star denotes the
10447:and the superscript star denotes the
9025:Chronological and causality relations
4272:. If the two arguments are the same,
4247:{\displaystyle u\cdot v=\eta (u,\,v)}
1001:is time, the four-dimensional vector
902:
885:. He was still far from the study of
403:Newton's law of universal gravitation
21549:
21526:
21387:
20878:
20752:Schutz p. 148, Naber p. 49
20563:
20423:and the same conclusion is reached.
10507:Behavior of tensors under inclusion:
7273:Coordinate free raising and lowering
7207:(eta-sharp) by the musical analogy.
5550:{\displaystyle u\cdot v=\eta (u,v),}
4857:between two arbitrary events called
4177:As a notational convention, vectors
3172:in a frame related to some frame by
991:coordinates in 3-dimensional space,
952:
22268:Tolman–Oppenheimer–Volkoff equation
22221:Friedmann–Lemaître–Robertson–Walker
21433:(2005). "18 Minkowskian geometry".
21367:The Geometry of Minkowski Spacetime
21235:
21216:
21197:
21065:. Frontiers in Mathematics. Basel:
21008:
20996:
20984:
20958:https://doi.org/10.3390/sym13101853
20945:
20933:
20917:
20782:
20770:
20632:
20524:proper at each point on a manifold.
12416:{\displaystyle S=(-R,0,\ldots ,0).}
11755:hyperbolic stereographic projection
11677:Hyperbolic stereographic projection
9114:of spacetime and so can be written
7505:{\displaystyle u\mapsto \eta (v,u)}
7240:
7078:intersected by a vector equals the
6743:Then if two different bases exist,
4837:
3609:. Minkowski space is often denoted
3359:
3354:Misner, Thorne & Wheeler (1973)
2494:
2490:
1489:Norm and reversed Cauchy inequality
1116:
912:
24:
20897:
19538:
19534:
19462:
19458:
19431:
19427:
19400:
19396:
19380:
19371:
19340:
19336:
19320:
19305:
19280:
19276:
19258:
19250:
19218:
19214:
19196:
19181:
19149:
19145:
17969:
17965:
17854:
17850:
16274:
16270:
16042:
16038:
15773:
15769:
15139:
15135:
15037:
15033:
14929:
14925:
11764:. It is instructive to compare to
11585:The corresponding pullback is the
10049:{\displaystyle \mathbf {M} ^{n+1}}
9468:. In string theory, there appears
9185:
8804:applied to two tangent vectors at
8499:
8495:
8351:
8336:
8310:
8306:
8076:
8072:
8051:Misner, Thorne & Wheeler (1973
7789:where the two different usages of
6981:
6950:
6903:
6845:
6698:
6689:
6617:
6589:
6553:
4692:(the Lorentz group) to the matrix
4599:
4473:
4450:
3919:{\displaystyle \mathbf {R} ^{1,3}}
3729:{\displaystyle M^{*}\otimes M^{*}}
3469:Misner, Thorne & Wheeler (1973
3290:
3232:
2978:
2974:
2453:
819:Minkowski's principal tool is the
368:Introduction to general relativity
25:
22581:
22038:Hamilton–Jacobi–Einstein equation
21629:
21130:https://arxiv.org/abs/0802.4345v1
20907:Misner, Thorne & Wheeler 1973
20469:This makes spacetime distance an
16995:Summing this formula one obtains
13265:The first of these is solved for
11719:whose range is an open subset of
7835:Formalism of the Minkowski metric
5591:
4078:tensor. It accepts two arguments
4058:need not be positive for nonzero
3813:can be identified with the space
1206:
453:Mathematics of general relativity
373:Mathematics of general relativity
22516:
22515:
21634:
21463:"Sur la dynamique de l'électron"
21309:Minkowski, Hermann (1908–1909),
21219:Introduction to Smooth manifolds
21200:Introduction to Smooth manifolds
21061:Catoni, F.; et al. (2008).
20049:
20040:
19898:
19889:
19735:
19726:
18947:
18384:
18373:
18264:
18253:
17728:
17717:
17408:
17397:
17350:
17339:
17328:
17317:
17080:
17030:
16912:
16901:
16821:
16810:
16628:
16525:
16514:
16402:
15750:Detailed outline of computation
15368:
15256:
14831:
14770:
14733:
14683:
14631:
14518:
14402:
14394:
14382:
14362:
14317:
14302:
14153:
14039:
13927:
13907:
13737:
13622:
13549:
13535:
13449:
13441:
13430:
13399:
13316:
13302:
13291:
13242:
13227:
13084:
13058:
13035:
12989:
12969:
12946:
12727:
12488:
12242:
12170:
12050:
11934:
11914:
11872:
11854:
11843:
11685:Red circular arc is geodesic in
11447:between manifolds is defined by
10494:actually is a hyperbolic space.
10460:
10417:
10388:
10368:
10187:
10030:
9883:
9798:
9731:
9702:
9632:{\displaystyle \mathbf {R} ^{4}}
9619:
9603:{\displaystyle \mathbf {R} ^{3}}
9590:
8029:) yields a tensor field of type
7825:is indeed the matrix inverse to
7685:
7303:
3900:
3846:
3822:
3666:
3120:of tangent vectors in manifolds
3011:
2945:
2897:
2856:
2815:
2774:
2723:
2682:
2641:
2600:
2422:equipped with a non-degenerate,
839:as symmetry group of spacetime)
488:
487:
474:
297:
194:four-dimensional Euclidean space
62:
21508:On the Dynamics of the Electron
21002:
20990:
20978:
20969:
20950:
20939:
20927:
20911:
20884:
20872:
20845:
20824:from the original on 2022-12-26
20797:
20788:
20776:
20764:
20755:
20746:
20737:
20679:On the Dynamics of the Electron
20544:
20527:
20510:
20495:
20486:
19798:
14344:
11352:feeding it vectors residing at
10021:in generalized Minkowski space
9659:{\displaystyle {\overline {g}}}
8135:in the direction of increasing
7013:Raising and lowering of indices
4449:
4362:Linearity in the first argument
3568:depending on spacetime position
1493:The norm of a time-like vector
1194:Properties of time-like vectors
21845:Mass–energy equivalence (E=mc)
21703:
21177:The Classical Theory of Fields
21106:Johns Hopkins University Press
20717:
20683:
20668:
20638:
20626:
20601:
20476:
20463:
20407:
20389:
20384:
20378:
20339:
20330:
20299:
20290:
20261:
20251:
20086:
20077:
19999:
19990:
19964:
19955:
19822:
19813:
19772:
19763:
19680:
19671:
19609:
19600:
18966:
18960:
18940:
18851:
18845:
18826:
18813:
18808:
18802:
18698:
18692:
18653:
18644:
18520:
18514:
18428:
18419:
18389:
18369:
18303:
18294:
18163:
18154:
18042:
18033:
18007:
17998:
17934:
17925:
17899:
17890:
17765:
17756:
17732:
17713:
17661:
17652:
17531:
17522:
17452:
17443:
17413:
17393:
17383:
17374:
17354:
17335:
17332:
17313:
17298:
17289:
17154:
17145:
17084:
17076:
17034:
17026:
16957:
16948:
16789:
16780:
16694:
16685:
16632:
16624:
16562:
16553:
16529:
16510:
16456:
16447:
16406:
16398:
16334:
16325:
16208:
16199:
16109:
16100:
15993:
15984:
15907:
15898:
15840:
15831:
15727:
15721:
15682:
15673:
15549:
15543:
15454:
15445:
15419:
15410:
15372:
15364:
15328:
15319:
15260:
15252:
15199:
15190:
15104:
15095:
14996:
14987:
14903:
14894:
14835:
14827:
14774:
14766:
14737:
14729:
14687:
14679:
14650:
14644:
14547:
14538:
14484:
14475:
14449:
14440:
14409:
14406:
14398:
14386:
14378:
14372:
14366:
14358:
14336:
14330:
14312:
14172:
14166:
14146:
14133:
14127:
14068:
14059:
14006:
13997:
13971:
13962:
13931:
13917:
13911:
13903:
13834:
13825:
13743:
13732:
13703:
13690:
13628:
13617:
13518:
13506:
13456:
13453:
13445:
13434:
13426:
13420:
13403:
13389:
13295:
13281:
13216:
13204:
13147:
13141:
13088:
13068:
13062:
13045:
13039:
13025:
12993:
12979:
12973:
12956:
12950:
12936:
12859:
12791:
12731:
12717:
12706:
12700:
12678:
12672:
12407:
12380:
12079:
12070:
12017:
12008:
11982:
11973:
11938:
11924:
11918:
11910:
11847:
11833:
11636:
11623:
11617:
11494:
11482:
11470:
11464:
11433:of a vector field under a map
11104:
11098:
10942:
10929:
10923:
10825:
10479:
10473:
10412:
10407:
10401:
10326:
10320:
10293:
10287:
10206:
10200:
10116:
10110:
9817:
9811:
9721:
9715:
8934:
8927:
8918:
8906:
8836:
8829:
8728:
8716:
8619:
8613:
8607:
8601:
8592:
8580:
8239:can be expanded at each point
8199:
8193:
7776:
7773:
7767:
7748:
7742:
7726:
7717:
7705:
7681:
7616:
7499:
7487:
7481:
7445:
7433:
7405:
7393:
7387:
7368:
7359:
7347:
7299:
6725:
6713:
6704:
6686:
6513:
6487:
6301:
6289:
6089:
6077:
6014:. The spatial components of a
5914:
5888:
5827:
5801:
5772:
5746:
5737:
5711:
5702:
5676:
5667:
5641:
5541:
5529:
5317:
5311:
4614:
4586:
4573:
4546:
4533:
4524:
4511:
4443:
4430:
4421:
4408:
4396:
4374:
4306:
4294:
4241:
4228:
3990:
3954:
3872:
3841:
3800:
3788:
3662:
3492:
3417:Kleppner & Kolenkow (1978)
3380:The choice of metric signature
3124:necessarily being embedded in
3023:
2751:
2577:
2354:
2348:
2340:
2334:
2314:
2308:
2299:
2293:
2288:
2282:
2265:
2259:
2237:
2231:
2191:
2185:
2166:
2152:
2115:
2109:
2101:
2095:
2087:
2073:
1732:
1719:
1714:
1701:
1694:
1668:
1638:The reversed Cauchy inequality
1554:
1542:
1530:
1524:
1327:
1301:
675:
662:
161:developed it from the work of
32:Minkowski space (number field)
13:
1:
21016:
19486:One has from the formula for
8053:, Box 3.2 and section 13.2.)
7031:Linear functionals (1-forms)
5869:{\textstyle \mu \neq \nu \,.}
4830:Lorentz metric, reserved for
4650:proper is that the former is
4152:itself, it accepts arguments
3190:way in which the coordinates
3134:-tuples can be used as well.
864:Hermann Minkowski, 1908, 1909
207:(maps preserving the regular
18:Null vector (Minkowski space)
21606:The Quantum Theory of Fields
21535:. New York: W. A. Benjamin.
21146:An Introduction to Mechanics
20730:, L. O'Raifeartaigh editor,
20554:an isometric embedding into
13593:lying in the hyperboloid is
10873:The pullback of a covariant
9651:
9489:
9212:Complexified Minkowski space
8642:and general tangent vectors
8007:of the coordinate functions
7248:electromagnetic field tensor
7112:Riesz representation theorem
7019:Raising and lowering indices
3806:{\displaystyle M:=(V,\eta )}
3768:{\displaystyle \{e_{\mu }\}}
3529:is likewise equipped with a
3142:A tangent vector at a point
2053:Reversed triangle inequality
1474:Positivity of scalar product
7:
21860:Relativistic Doppler effect
21181:Butterworth–Heinemann
20430:
10976:where for any vector space
9538:
9409:Generalized Minkowski space
9385:Pauli–Lubanski pseudovector
4264:, similar to the Euclidean
2478:geometrical tangent vectors
712:in physical spacetime with
535:Complex Minkowski spacetime
398:Introduction to gravitation
155:inertial frame of reference
145:The model helps show how a
10:
22586:
22331:In computational physics:
21855:Relativity of simultaneity
21611:Cambridge University Press
21315:[Space and Time],
20892:Landau & Lifshitz 2002
20457:
12548:intersects the hyperplane
11729:has the same dimension as
9542:
9189:
9180:relativity of simultaneity
9000:Einstein's field equations
8815:This situation changes in
8243:in the basis given by the
8180:, by definition such that
7812:{\displaystyle \eta ^{-1}}
7263:Landau & Lifshitz 2002
7016:
6474:with the metric was used.
4841:
4826:, even more specifically,
4813:Sylvester's law of inertia
4021:
3629:pseudo-Riemannian manifold
951:. The terminology is from
916:
538:
277:
29:
22513:
22345:
22210:
22182:
22168:Lense–Thirring precession
22051:
22000:
21962:
21941:
21930:
21888:
21832:
21816:
21758:
21750:Doubly special relativity
21722:
21711:
21577:Walter, Scott A. (1999).
21317:Physikalische Zeitschrift
21274:Carus, Edward H. (1918).
21075:10.1007/978-3-7643-8614-6
20804:Gregory L. Naber (2003).
11771:Stereographic projection
11580:coordinate representation
10341:Poincaré half-space model
8217:{\displaystyle df(X)=Xf,}
7074:. The number of (1-form)
6022:may be identified with a
5356:{\displaystyle 4\times 4}
4688:is identical in the case
1478:Cauchy–Schwarz inequality
1215:of two time-like vectors
428:Derivations of relativity
22028:Post-Newtonian formalism
22018:Einstein field equations
21954:Mathematical formulation
21778:Hyperbolic orthogonality
21506:Wikisource translation:
21495:2027/uiug.30112063899089
21300:Wikisource translation:
20852:Sean M. Carroll (2019).
20645:Schutz, John W. (1977).
20595:
20533:This similarity between
16378:With this one may write
11766:stereographic projection
11558:, which is given by the
10373:In the definition above
10264:{\displaystyle -1/R^{2}}
10154:of the Minkowski metric
10084:surfaces of transitivity
9549:The meaning of the term
9470:conformal field theories
9053:chronologically precedes
8162:The exterior derivative
4842:Not to be confused with
4820:pseudo-Riemannian metric
4764:hyperbolic orthogonality
4270:relativistic dot product
4018:Pseudo-Euclidean metrics
3627:. It is an example of a
3176:transforms according to
760:Real Minkowski spacetime
378:Einstein field equations
21739:Galilean transformation
21730:Principle of relativity
21272:Published translation:
14103:Pulling back the metric
12114:where, for simplicity,
10518:and a covariant tensor
10056:of spacetime dimension
9779:{\displaystyle ct>0}
9757:to be the upper sheet (
9461:are two examples where
9140:simultaneous hyperplane
8564:. Furthermore, one has
8278:coordinate vector field
8037:(denoted by the symbol
8025:(denoted by the symbol
5363:matrix associated with
4822:, more specifically, a
4698:to be displayed below.
4262:Minkowski inner product
4120:, the tangent space at
4066:. The Minkowski metric
3475:) (who, by the way use
3471:, Box 2.1, Farewell to
3196:transform. Explicitly,
2493:, Proposition 3.8.) or
2432:Minkowski inner product
2424:symmetric bilinear form
887:curvilinear coordinates
859:an independent reality.
708:coordinate space, is a
586:Lorentz transformations
581:Lorentz transformations
343:Lorentz transformations
259:Minkowski inner product
21824:Lorentz transformation
21294:10.5840/monist19182826
20855:Spacetime and Geometry
20560:Nash embedding theorem
20417:
20217:
20120:
19858:
19478:
19056:
18710:
18458:
18187:
18084:
17847:
17799:
16987:
16592:
16370:
16255:
15739:
15489:
14795:
14572:
14271:
14093:
13869:
13789:
13661:
13569:
13473:
13341:
13257:
13172:
13105:
12897:
12741:
12638:once in point denoted
12632:
12542:
12510:
12417:
12361:
12108:
11690:
11664:
11504:
11260:
11073:
10970:
10842:
10764:
10488:
10437:
10360:
10333:
10300:
10265:
10215:
10174:under inclusion, is a
10168:
10144:
10076:
10050:
10015:
9780:
9751:
9684:
9660:
9633:
9604:
9512:non-Euclidean geometry
9476:spacetime dimensions.
9331:. A relativistic pure
9138:is timelike. Then the
9074:and so can be written
8973:
8774:
8626:
8554:
8392:
8218:
8125:directional derivative
8109:
7995:
7813:
7783:
7629:
7577:
7550:
7526:
7506:
7465:
7415:
7331:
7311:
7277:Given a bilinear form
7083:
6999:
6967:
6921:
6869:can be represented as
6863:
6735:
6654:
6571:
6539:
6462:
6273:
5940:
5870:
5840:
5785:
5551:
5507:and the bilinear form
5501:
5357:
5331:
5269:
5137:
5058:
4633:
4553:
4491:
4356:Minkowski norm squared
4348:
4248:
4024:Pseudo-Euclidean space
4012:light-cone coordinates
3997:
3920:
3879:
3807:
3769:
3730:
3674:
3585:pseudo-Euclidean space
3327:
3114:directional derivative
3088:
2926:
2485:
2392:Mathematical structure
2380:
2126:
2041:
1740:
1629:
1465:
960:
895:
867:
833:relativistic mechanics
699:
541:Four-dimensional space
255:Minkowski norm squared
153:is independent of the
45:
22292:Weyl−Lewis−Papapetrou
22033:Raychaudhuri equation
21972:Equivalence principle
21643:at Wikimedia Commons
21394:Annals of Mathematics
21365:Naber, G. L. (1992).
21312:"Raum und Zeit"
21026:Arch. Hist. Exact Sci
20794:Giulini 2008 pp. 5, 6
20507:flips this direction.
20437:Hyperbolic quaternion
20418:
20197:
20121:
19838:
19479:
19057:
18711:
18459:
18188:
18064:
17827:
17800:
16988:
16593:
16371:
16256:
15740:
15490:
14796:
14573:
14272:
14094:
13870:
13790:
13662:
13570:
13474:
13342:
13258:
13173:
13106:
12898:
12742:
12633:
12543:
12511:
12418:
12362:
12109:
11684:
11665:
11505:
11261:
11074:
10971:
10852:) the restriction of
10848:meaning (with slight
10843:
10793:are vector fields on
10765:
10489:
10438:
10361:
10334:
10301:
10277:for its dimension. A
10266:
10216:
10169:
10167:{\displaystyle \eta }
10145:
10090:on this submanifold,
10077:
10051:
10016:
9781:
9752:
9685:
9683:{\displaystyle \eta }
9661:
9634:
9605:
9196:Super Minkowski space
8974:
8775:
8627:
8555:
8393:
8256:. Applying this with
8219:
8110:
7996:
7814:
7784:
7630:
7578:
7576:{\displaystyle M^{*}}
7551:
7527:
7525:{\displaystyle \eta }
7507:
7471:is then the dual map
7466:
7416:
7332:
7330:{\displaystyle \eta }
7312:
7162:contravariant vectors
7030:
7000:
6968:
6922:
6864:
6736:
6655:
6572:
6540:
6463:
6274:
5941:
5871:
5841:
5786:
5552:
5502:
5358:
5332:
5277:polarization identity
5270:
5138:
5059:
4762:(or vice versa), see
4634:
4554:
4492:
4349:
4249:
3998:
3921:
3880:
3808:
3770:
3731:
3675:
3587:with total dimension
3485:differential geometry
3445:contravariant vectors
3328:
3166:such that the vector
3089:
2927:
2461:
2381:
2127:
2042:
1741:
1630:
1466:
926:
875:
856:
700:
438:Differential geometry
338:Equivalence principle
215:. It is generated by
40:
22555:Lorentzian manifolds
22540:Equations of physics
22333:Numerical relativity
22174:pulsar timing arrays
21527:Sard, R. D. (1970).
20132:
19495:
19068:
18758:
18470:
18197:
17815:
16999:
16602:
16382:
16265:
15760:
15499:
14807:
14584:
14281:
14111:
13887:
13799:
13671:
13597:
13587:. The condition for
13483:
13372:
13275:
13184:
13117:
12907:
12753:
12642:
12552:
12520:
12429:
12371:
12165:
11823:
11593:
11451:
11085:
10984:
10899:
10801:
10534:
10455:
10377:
10347:
10332:{\displaystyle 3(n)}
10314:
10299:{\displaystyle 2(2)}
10281:
10237:
10182:
10158:
10094:
10075:{\displaystyle n+1.}
10060:
10025:
9793:
9761:
9697:
9674:
9643:
9614:
9585:
9525:infinitesimal region
9182:in Minkowski space.
8823:
8671:
8568:
8453:
8284:
8184:
8063:
7856:
7793:
7639:
7587:
7560:
7540:
7516:
7475:
7427:
7341:
7321:
7281:
7236:momentum four-vector
6977:
6931:
6873:
6815:
6680:
6585:
6549:
6481:
6472:lowering of an index
6283:
6071:
5882:
5850:
5795:
5632:
5511:
5375:
5341:
5283:
5151:
5071:
4869:
4684:. There, the matrix
4567:
4505:
4368:
4276:
4210:
4028:Lorentzian manifolds
3930:
3895:
3817:
3779:
3746:
3700:
3644:
3346:displacement vectors
3200:
2936:
2501:
2143:
2069:
2065:, the inequality is
1750:
1662:
1520:
1295:
873:saw its limitation:
620:
416:Relevant mathematics
22550:Minkowski spacetime
22225:Friedmann equations
22119:Hulse–Taylor binary
22081:Gravitational waves
21977:Riemannian geometry
21803:Proper acceleration
21788:Maxwell's equations
21734:Galilean relativity
21479:1906RCMP...21..129P
21236:Lee, J. M. (1997).
21217:Lee, J. M. (2012).
21198:Lee, J. M. (2003).
21108:. pp. 85–121.
20708:Minkowski 1908–1909
20690:Minkowski 1907–1908
20664:Extract of page 184
20388:
20185:
20160:
18970:
18855:
18812:
18740:Poincaré ball model
18702:
18524:
15731:
15553:
14654:
14340:
14176:
14137:
12502:
12184:
12158:Detailed derivation
12148:are coordinates on
12136:are coordinates on
11687:Poincaré disk model
11653:
11632:
11536:the pushforward of
10959:
10938:
10483:
10411:
10339:corresponds to the
10308:Poincaré disk model
10306:corresponds to the
10223:Riemannian manifold
10210:
10178:. With this metric
10120:
10082:This is one of the
9821:
9725:
9560:hyperbolic geometry
9500:Pythagorean Theorem
9484:hyperbolic geometry
9282:-like structure on
9192:Lorentzian manifold
9178:varies, there is a
9072:transitive property
8794:component functions
8546:
8384:
8005:exterior derivative
7423:The lowered vector
7252:differential 2-form
7172:raising of an index
6994:
6946:
6916:
6888:
6858:
6830:
6630:
6602:
6566:
3572:Lorentzian manifold
3522:Lorentzian manifold
3516:For comparison, in
3398:Landau and Lifshitz
2400:. This provides an
2029:
2011:
1993:
1975:
1940:
1922:
1904:
1886:
1658:this inequality is
891:Riemannian geometry
754:spacetime continuum
718:hyperbolic rotation
285:Part of a series on
270:(as opposed to the
58:Minkowski spacetime
22560:Special relativity
22274:Reissner–Nordström
22192:Brans–Dicke theory
22023:Linearized gravity
21850:Length contraction
21768:Frame of reference
21745:Special relativity
21641:Minkowski diagrams
21487:10.1007/BF03013466
21334:Misner, Charles W.
21256:Minkowski, Hermann
21038:10.1007/BF00518231
20963:2023-08-13 at the
20922:Riemannian metrics
20839:2022-12-26 at the
20761:Schutz p. 148
20675:Poincaré 1905–1906
20613:2019-06-22 at the
20413:
20365:
20168:
20143:
20116:
20114:
19474:
19052:
18945:
18832:
18789:
18706:
18679:
18501:
18454:
18183:
17795:
17793:
16983:
16588:
16366:
16251:
16249:
15735:
15708:
15530:
15485:
15483:
14791:
14629:
14568:
14315:
14267:
14151:
14114:
14089:
13865:
13785:
13657:
13565:
13563:
13469:
13464:
13349:Next, the inverse
13337:
13253:
13251:
13168:
13101:
13096:
12893:
12891:
12737:
12628:
12538:
12506:
12486:
12413:
12357:
12168:
12159:
12104:
12102:
11691:
11660:
11639:
11609:
11500:
11256:
11069:
11065:
11053:
10966:
10945:
10915:
10838:
10760:
10484:
10458:
10433:
10386:
10359:{\displaystyle n.}
10356:
10329:
10296:
10261:
10211:
10185:
10164:
10140:
10097:
10072:
10046:
10011:
9796:
9776:
9747:
9700:
9680:
9656:
9629:
9600:
9508:general relativity
8969:
8817:general relativity
8770:
8622:
8550:
8532:
8388:
8370:
8214:
8105:
7991:
7809:
7779:
7625:
7573:
7546:
7522:
7502:
7461:
7411:
7327:
7307:
7180:. It happens that
7104:dual vector spaces
7084:
7023:tensor contraction
6995:
6980:
6963:
6934:
6917:
6902:
6876:
6859:
6844:
6818:
6731:
6650:
6616:
6588:
6581:matrix satisfying
6567:
6552:
6535:
6458:
6269:
6012:spatial components
6004:timelike component
5936:
5866:
5836:
5781:
5547:
5497:
5487:
5353:
5327:
5265:
5133:
5054:
4855:spacetime interval
4844:Minkowski distance
4629:
4549:
4487:
4344:
4244:
4036:, since it is not
3993:
3916:
3875:
3803:
3765:
3726:
3670:
3518:general relativity
3439:be concerned with
3431:introduction of a
3323:
3321:
3110:parallel transport
3084:
3070:
2922:
2920:
2486:
2376:
2374:
2134:linearly dependent
2122:
2037:
2015:
1997:
1979:
1961:
1926:
1908:
1890:
1872:
1746:or algebraically,
1736:
1625:
1461:
1201:similarly directed
961:
829:length contraction
695:
481:Physics portal
458:Spacetime topology
433:Spacetime diagrams
361:General relativity
333:Spacetime manifold
326:Spacetime concepts
315:General relativity
310:Special relativity
263:spacetime interval
209:Euclidean distance
186:length contraction
182:general relativity
178:special relativity
147:spacetime interval
120:in the absence of
46:
22570:Hermann Minkowski
22527:
22526:
22341:
22340:
22320:Ozsváth–Schücking
21926:
21925:
21908:Minkowski diagram
21865:Thomas precession
21808:Relativistic mass
21639:Media related to
21620:978-0-521-55001-7
21592:978-0-8176-4060-6
21569:978-0-12-639201-2
21380:978-0-387-97848-2
21357:978-0-7167-0344-0
21350:, W. H. Freeman,
21247:978-0-387-98322-6
21228:978-1-4419-9981-8
21209:978-0-387-95448-6
21160:978-0-07-035048-9
21084:978-3-7643-8613-9
21067:Birkhäuser Verlag
20865:978-1-108-48839-6
20834:Extract of page 8
20817:978-0-486-43235-9
20724:Cornelius Lanczos
20658:978-0-582-31760-4
20558:according to the
20518:Riemannian metric
20428:
20427:
20360:
20107:
20010:
19807:
19793:
19691:
19620:
19552:
19469:
19445:
19407:
19377:
19354:
19311:
19287:
19272:
19232:
19210:
19163:
18720:
18719:
18674:
18449:
18324:
18178:
18053:
17976:
17945:
17868:
17786:
17682:
17473:
16978:
16583:
16345:
16281:
16229:
16120:
16056:
16014:
15851:
15787:
15703:
15465:
15339:
15231:
15228:
15210:
15146:
15115:
15051:
15007:
14943:
14914:
14558:
14495:
14079:
14017:
13860:
13764:
13723:
13556:
13525:
13332:
13163:
13133:
13016:
12927:
12875:
12847:
12807:
12779:
12536:
12157:
12090:
12028:
11888:
11674:
11673:
11582:of the function.
11514:is a function on
11081:It is defined by
11062:
11005:
11003:
10850:abuse of notation
10227:hyperboloid model
10176:Riemannian metric
9654:
9545:Hyperboloid model
9093:causally precedes
8513:
8365:
8324:
8235:. In coordinates
8090:
8035:symmetric product
7549:{\displaystyle M}
7158:covariant vectors
7116:linear functional
5978:is used to write
5976:Einstein notation
5306:
4824:Lorentzian metric
4802:orthonormal basis
4619:
4613:
4038:positive-definite
3952:
3441:covariant vectors
3079:
3007:
2992:
2756:
2582:
2572:
2035:
1623:
1557:
943:(also called the
935:(also called the
821:Minkowski diagram
729:Maxwell equations
690:
532:
531:
391:Classical gravity
266:distance) is the
199:In 3-dimensional
159:Hermann Minkowski
42:Hermann Minkowski
16:(Redirected from
22577:
22519:
22518:
22302:van Stockum dust
22074:Two-body problem
21992:Mach's principle
21939:
21938:
21880:Terrell rotation
21720:
21719:
21698:
21691:
21684:
21675:
21674:
21650:
21638:
21623:
21596:
21573:
21557:
21546:
21534:
21505:
21454:
21426:
21384:
21360:
21342:Wheeler, John A.
21324:
21314:
21297:
21288:(288): 288–302.
21276:"Space and Time"
21269:
21263:
21251:
21232:
21213:
21194:
21164:
21125:
21114:10.2307/27757388
21096:
21057:
21011:
21006:
21000:
20994:
20988:
20982:
20976:
20973:
20967:
20954:
20948:
20943:
20937:
20931:
20925:
20915:
20909:
20904:
20895:
20888:
20882:
20876:
20870:
20869:
20849:
20843:
20832:
20830:
20829:
20801:
20795:
20792:
20786:
20780:
20774:
20768:
20762:
20759:
20753:
20750:
20744:
20741:
20735:
20721:
20715:
20705:
20696:
20687:
20681:
20672:
20666:
20662:
20642:
20636:
20630:
20624:
20605:
20590:
20588:
20584:
20557:
20548:
20542:
20531:
20525:
20514:
20508:
20506:
20499:
20493:
20490:
20484:
20480:
20474:
20467:
20422:
20420:
20419:
20414:
20387:
20373:
20361:
20359:
20358:
20353:
20349:
20348:
20347:
20342:
20333:
20325:
20324:
20309:
20308:
20307:
20302:
20293:
20288:
20287:
20274:
20269:
20268:
20247:
20246:
20241:
20237:
20236:
20235:
20216:
20211:
20193:
20189:
20184:
20176:
20159:
20151:
20125:
20123:
20122:
20117:
20115:
20108:
20106:
20105:
20100:
20096:
20095:
20094:
20089:
20080:
20072:
20071:
20056:
20052:
20043:
20035:
20034:
20021:
20016:
20012:
20011:
20009:
20008:
20007:
20002:
19993:
19985:
19984:
19974:
19973:
19972:
19967:
19958:
19950:
19949:
19939:
19909:
19905:
19901:
19892:
19878:
19877:
19868:
19867:
19857:
19852:
19831:
19830:
19825:
19816:
19808:
19805:
19794:
19792:
19791:
19786:
19782:
19781:
19780:
19775:
19766:
19758:
19757:
19742:
19738:
19729:
19721:
19720:
19711:
19710:
19697:
19692:
19690:
19689:
19688:
19683:
19674:
19666:
19665:
19655:
19654:
19653:
19644:
19643:
19630:
19625:
19621:
19619:
19618:
19617:
19612:
19603:
19595:
19594:
19584:
19583:
19582:
19573:
19572:
19559:
19553:
19551:
19550:
19549:
19533:
19531:
19530:
19514:
19513:
19491:
19483:
19481:
19480:
19475:
19470:
19468:
19457:
19446:
19444:
19443:
19442:
19426:
19424:
19423:
19408:
19406:
19395:
19393:
19392:
19391:
19378:
19370:
19368:
19367:
19355:
19353:
19352:
19351:
19335:
19333:
19332:
19331:
19318:
19317:
19312:
19304:
19301:
19300:
19288:
19286:
19275:
19273:
19271:
19270:
19269:
19256:
19248:
19246:
19245:
19233:
19231:
19230:
19229:
19213:
19211:
19209:
19208:
19207:
19194:
19193:
19192:
19179:
19177:
19176:
19164:
19162:
19161:
19160:
19144:
19142:
19141:
19132:
19131:
19126:
19122:
19121:
19098:
19097:
19092:
19088:
19087:
19064:In coordinates,
19061:
19059:
19058:
19053:
19048:
19044:
19040:
19039:
19034:
19030:
19029:
19005:
19004:
18999:
18995:
18994:
18972:
18971:
18969:
18955:
18950:
18943:
18931:
18927:
18923:
18922:
18917:
18913:
18912:
18888:
18887:
18882:
18878:
18877:
18854:
18840:
18811:
18797:
18788:
18787:
18782:
18778:
18777:
18745:
18744:
18737:
18736:
18735:
18715:
18713:
18712:
18707:
18701:
18687:
18675:
18673:
18672:
18667:
18663:
18662:
18661:
18656:
18647:
18639:
18638:
18623:
18622:
18618:
18617:
18616:
18611:
18607:
18606:
18605:
18578:
18577:
18572:
18568:
18567:
18566:
18543:
18542:
18529:
18523:
18509:
18500:
18499:
18494:
18490:
18489:
18463:
18461:
18460:
18455:
18450:
18448:
18447:
18442:
18438:
18437:
18436:
18431:
18422:
18414:
18413:
18398:
18397:
18396:
18387:
18376:
18368:
18367:
18354:
18352:
18351:
18336:
18335:
18330:
18326:
18325:
18323:
18322:
18317:
18313:
18312:
18311:
18306:
18297:
18289:
18288:
18273:
18272:
18268:
18267:
18256:
18246:
18245:
18232:
18215:
18214:
18192:
18190:
18189:
18184:
18179:
18177:
18173:
18172:
18171:
18166:
18157:
18149:
18148:
18134:
18133:
18132:
18120:
18119:
18110:
18109:
18096:
18094:
18093:
18083:
18078:
18054:
18052:
18051:
18050:
18045:
18036:
18028:
18027:
18017:
18016:
18015:
18010:
18001:
17993:
17992:
17982:
17977:
17975:
17964:
17959:
17958:
17946:
17944:
17943:
17942:
17937:
17928:
17920:
17919:
17909:
17908:
17907:
17902:
17893:
17885:
17884:
17874:
17869:
17867:
17866:
17865:
17849:
17846:
17841:
17810:
17804:
17802:
17801:
17796:
17794:
17787:
17785:
17784:
17779:
17775:
17774:
17773:
17768:
17759:
17751:
17750:
17735:
17731:
17720:
17712:
17711:
17698:
17696:
17695:
17683:
17681:
17680:
17675:
17671:
17670:
17669:
17664:
17655:
17647:
17646:
17631:
17630:
17626:
17625:
17624:
17619:
17615:
17614:
17613:
17586:
17585:
17580:
17576:
17575:
17574:
17551:
17550:
17545:
17541:
17540:
17539:
17534:
17525:
17517:
17516:
17501:
17500:
17487:
17483:
17474:
17472:
17471:
17466:
17462:
17461:
17460:
17455:
17446:
17438:
17437:
17422:
17421:
17420:
17411:
17400:
17392:
17391:
17386:
17377:
17372:
17371:
17353:
17342:
17331:
17320:
17312:
17308:
17307:
17306:
17301:
17292:
17284:
17283:
17269:
17268:
17253:
17249:
17248:
17247:
17242:
17238:
17237:
17236:
17209:
17208:
17203:
17199:
17198:
17197:
17174:
17173:
17168:
17164:
17163:
17162:
17157:
17148:
17140:
17139:
17124:
17123:
17110:
17106:
17097:
17096:
17091:
17087:
17083:
17075:
17074:
17047:
17046:
17041:
17037:
17033:
17025:
17024:
17005:
16992:
16990:
16989:
16984:
16979:
16977:
16976:
16971:
16967:
16966:
16965:
16960:
16951:
16943:
16942:
16927:
16926:
16925:
16920:
16916:
16915:
16904:
16893:
16892:
16887:
16883:
16882:
16868:
16867:
16852:
16851:
16839:
16838:
16829:
16825:
16824:
16813:
16803:
16799:
16798:
16797:
16792:
16783:
16775:
16774:
16760:
16759:
16744:
16743:
16738:
16734:
16733:
16732:
16714:
16713:
16708:
16704:
16703:
16702:
16697:
16688:
16680:
16679:
16664:
16663:
16650:
16645:
16644:
16639:
16635:
16631:
16623:
16622:
16597:
16595:
16594:
16589:
16584:
16582:
16581:
16576:
16572:
16571:
16570:
16565:
16556:
16548:
16547:
16532:
16528:
16517:
16509:
16508:
16499:
16498:
16483:
16482:
16470:
16466:
16465:
16464:
16459:
16450:
16442:
16441:
16427:
16426:
16413:
16405:
16397:
16396:
16375:
16373:
16372:
16367:
16359:
16358:
16346:
16344:
16343:
16342:
16337:
16328:
16320:
16319:
16309:
16308:
16307:
16298:
16297:
16284:
16282:
16280:
16269:
16260:
16258:
16257:
16252:
16250:
16243:
16242:
16230:
16228:
16227:
16222:
16218:
16217:
16216:
16211:
16202:
16194:
16193:
16178:
16177:
16176:
16167:
16166:
16157:
16156:
16143:
16134:
16133:
16121:
16119:
16118:
16117:
16112:
16103:
16095:
16094:
16084:
16083:
16082:
16073:
16072:
16059:
16057:
16055:
16054:
16053:
16037:
16028:
16027:
16015:
16013:
16012:
16007:
16003:
16002:
16001:
15996:
15987:
15979:
15978:
15963:
15962:
15961:
15956:
15952:
15951:
15937:
15936:
15921:
15917:
15916:
15915:
15910:
15901:
15893:
15892:
15874:
15865:
15864:
15852:
15850:
15849:
15848:
15843:
15834:
15826:
15825:
15815:
15814:
15813:
15804:
15803:
15790:
15788:
15786:
15785:
15784:
15768:
15747:
15746:
15744:
15742:
15741:
15736:
15730:
15716:
15704:
15702:
15701:
15696:
15692:
15691:
15690:
15685:
15676:
15668:
15667:
15652:
15651:
15647:
15646:
15645:
15640:
15636:
15635:
15634:
15607:
15606:
15601:
15597:
15596:
15595:
15572:
15571:
15558:
15552:
15538:
15529:
15528:
15523:
15519:
15518:
15494:
15492:
15491:
15486:
15484:
15471:
15467:
15466:
15464:
15463:
15462:
15457:
15448:
15440:
15439:
15429:
15428:
15427:
15422:
15413:
15405:
15404:
15394:
15371:
15344:
15340:
15338:
15337:
15336:
15331:
15322:
15314:
15313:
15303:
15302:
15301:
15292:
15291:
15278:
15259:
15251:
15250:
15229:
15226:
15224:
15211:
15209:
15208:
15207:
15202:
15193:
15185:
15184:
15174:
15173:
15172:
15163:
15162:
15149:
15147:
15145:
15134:
15129:
15128:
15116:
15114:
15113:
15112:
15107:
15098:
15090:
15089:
15079:
15078:
15077:
15068:
15067:
15054:
15052:
15050:
15049:
15048:
15032:
15021:
15020:
15008:
15006:
15005:
15004:
14999:
14990:
14982:
14981:
14971:
14970:
14969:
14960:
14959:
14946:
14944:
14942:
14941:
14940:
14924:
14919:
14915:
14913:
14912:
14911:
14906:
14897:
14889:
14888:
14878:
14877:
14876:
14867:
14866:
14853:
14834:
14826:
14825:
14800:
14798:
14797:
14792:
14787:
14786:
14781:
14777:
14773:
14750:
14749:
14744:
14740:
14736:
14728:
14727:
14700:
14699:
14694:
14690:
14686:
14678:
14677:
14656:
14655:
14653:
14639:
14634:
14627:
14623:
14619:
14618:
14613:
14609:
14608:
14577:
14575:
14574:
14569:
14564:
14560:
14559:
14557:
14556:
14555:
14550:
14541:
14533:
14532:
14522:
14521:
14516:
14515:
14502:
14496:
14494:
14493:
14492:
14487:
14478:
14470:
14469:
14459:
14458:
14457:
14452:
14443:
14435:
14434:
14424:
14405:
14397:
14385:
14365:
14357:
14356:
14339:
14325:
14320:
14311:
14310:
14305:
14296:
14295:
14276:
14274:
14273:
14268:
14266:
14265:
14250:
14249:
14244:
14240:
14239:
14238:
14211:
14210:
14205:
14201:
14200:
14199:
14178:
14177:
14175:
14161:
14156:
14149:
14136:
14122:
14098:
14096:
14095:
14090:
14085:
14081:
14080:
14078:
14077:
14076:
14071:
14062:
14054:
14053:
14043:
14042:
14037:
14036:
14023:
14018:
14016:
14015:
14014:
14009:
14000:
13992:
13991:
13981:
13980:
13979:
13974:
13965:
13957:
13956:
13946:
13930:
13910:
13902:
13901:
13882:
13874:
13872:
13871:
13866:
13861:
13859:
13858:
13857:
13844:
13843:
13842:
13837:
13828:
13820:
13819:
13809:
13794:
13792:
13791:
13786:
13781:
13780:
13765:
13763:
13762:
13753:
13752:
13751:
13746:
13740:
13735:
13729:
13724:
13722:
13721:
13712:
13711:
13710:
13689:
13688:
13678:
13666:
13664:
13663:
13658:
13653:
13652:
13637:
13636:
13631:
13625:
13620:
13612:
13611:
13592:
13586:
13580:
13574:
13572:
13571:
13566:
13564:
13557:
13552:
13547:
13538:
13526:
13521:
13501:
13478:
13476:
13475:
13470:
13465:
13452:
13444:
13433:
13402:
13367:
13346:
13344:
13343:
13338:
13333:
13331:
13320:
13319:
13310:
13305:
13294:
13270:
13262:
13260:
13259:
13254:
13252:
13245:
13230:
13177:
13175:
13174:
13169:
13164:
13159:
13151:
13134:
13129:
13121:
13110:
13108:
13107:
13102:
13097:
13087:
13061:
13038:
13017:
13012:
13004:
12992:
12972:
12949:
12928:
12923:
12915:
12902:
12900:
12899:
12894:
12892:
12876:
12871:
12863:
12848:
12843:
12835:
12808:
12803:
12795:
12780:
12775:
12767:
12746:
12744:
12743:
12738:
12730:
12713:
12709:
12699:
12698:
12671:
12670:
12637:
12635:
12634:
12629:
12627:
12623:
12604:
12600:
12599:
12598:
12580:
12579:
12547:
12545:
12544:
12539:
12537:
12532:
12524:
12515:
12513:
12512:
12507:
12501:
12496:
12491:
12482:
12478:
12477:
12476:
12458:
12457:
12422:
12420:
12419:
12414:
12366:
12364:
12363:
12358:
12356:
12352:
12339:
12338:
12323:
12322:
12317:
12313:
12312:
12289:
12288:
12283:
12279:
12278:
12261:
12260:
12245:
12237:
12233:
12232:
12231:
12213:
12212:
12183:
12178:
12173:
12153:
12147:
12141:
12135:
12123:
12113:
12111:
12110:
12105:
12103:
12096:
12092:
12091:
12089:
12088:
12087:
12082:
12073:
12065:
12064:
12054:
12053:
12048:
12047:
12034:
12029:
12027:
12026:
12025:
12020:
12011:
12003:
12002:
11992:
11991:
11990:
11985:
11976:
11968:
11967:
11957:
11937:
11917:
11909:
11908:
11889:
11887:
11876:
11875:
11866:
11857:
11846:
11818:
11817:
11816:
11795:and its inverse
11794:
11789:
11788:
11763:
11752:
11734:
11728:
11724:
11718:
11704:
11700:
11669:
11667:
11666:
11661:
11652:
11647:
11631:
11626:
11605:
11604:
11577:
11571:
11565:
11553:
11539:
11535:
11517:
11513:
11509:
11507:
11506:
11501:
11463:
11462:
11446:
11432:
11399:
11385:
11371:
11361:
11351:
11337:
11320:
11298:
11284:
11273:
11265:
11263:
11262:
11257:
11252:
11248:
11247:
11246:
11237:
11236:
11216:
11215:
11206:
11205:
11192:
11191:
11182:
11181:
11161:
11157:
11156:
11155:
11135:
11134:
11121:
11120:
11097:
11096:
11078:
11076:
11075:
11070:
11064:
11063:
11060:
11054:
11049:
11048:
11047:
11029:
11028:
11016:
11015:
10996:
10995:
10979:
10975:
10973:
10972:
10967:
10958:
10953:
10937:
10932:
10911:
10910:
10895:is a linear map
10894:
10880:
10876:
10865:
10855:
10847:
10845:
10844:
10839:
10834:
10833:
10828:
10813:
10812:
10796:
10792:
10769:
10767:
10766:
10761:
10756:
10752:
10751:
10750:
10730:
10729:
10716:
10715:
10695:
10691:
10690:
10689:
10680:
10679:
10659:
10658:
10649:
10648:
10635:
10634:
10625:
10624:
10604:
10600:
10599:
10598:
10578:
10577:
10564:
10563:
10546:
10545:
10529:
10525:
10521:
10517:
10513:
10497:
10496:
10493:
10491:
10490:
10485:
10482:
10468:
10463:
10442:
10440:
10439:
10434:
10432:
10431:
10420:
10410:
10396:
10391:
10365:
10363:
10362:
10357:
10338:
10336:
10335:
10330:
10305:
10303:
10302:
10297:
10276:
10270:
10268:
10267:
10262:
10260:
10259:
10250:
10231:hyperbolic space
10220:
10218:
10217:
10212:
10209:
10195:
10190:
10173:
10171:
10170:
10165:
10149:
10147:
10146:
10141:
10133:
10132:
10119:
10105:
10081:
10079:
10078:
10073:
10055:
10053:
10052:
10047:
10045:
10044:
10033:
10020:
10018:
10017:
10012:
10010:
10006:
9990:
9989:
9977:
9976:
9971:
9967:
9966:
9943:
9942:
9937:
9933:
9932:
9915:
9914:
9905:
9904:
9892:
9891:
9886:
9877:
9873:
9872:
9871:
9853:
9852:
9820:
9806:
9801:
9785:
9783:
9782:
9777:
9756:
9754:
9753:
9748:
9746:
9745:
9734:
9724:
9710:
9705:
9689:
9687:
9686:
9681:
9665:
9663:
9662:
9657:
9655:
9647:
9638:
9636:
9635:
9630:
9628:
9627:
9622:
9609:
9607:
9606:
9601:
9599:
9598:
9593:
9475:
9467:
9452:
9448:
9440:
9432:
9426:
9420:
9404:
9393:
9382:
9378:
9371:
9365:
9351:
9330:
9313:
9290:
9274:
9265:
9251:angular velocity
9232:
9207:
9203:
9177:
9167:
9147:
9137:
9123:
9112:partial ordering
9109:
9099:
9091:
9083:
9069:
9059:
9051:
9042:
9018:
9011:
8997:
8993:
8978:
8976:
8975:
8970:
8965:
8964:
8955:
8954:
8945:
8944:
8905:
8904:
8899:
8895:
8894:
8893:
8876:
8875:
8870:
8866:
8865:
8864:
8847:
8846:
8819:. There one has
8807:
8803:
8799:
8791:
8785:
8779:
8777:
8776:
8771:
8766:
8765:
8756:
8755:
8746:
8745:
8715:
8714:
8699:
8698:
8686:
8685:
8666:
8660:
8651:
8641:
8631:
8629:
8628:
8623:
8563:
8559:
8557:
8556:
8551:
8545:
8540:
8528:
8524:
8523:
8518:
8514:
8512:
8511:
8510:
8494:
8484:
8483:
8478:
8474:
8473:
8472:
8444:
8431:
8419:
8415:
8403:
8397:
8395:
8394:
8389:
8383:
8378:
8366:
8364:
8363:
8362:
8349:
8348:
8347:
8334:
8329:
8325:
8323:
8322:
8321:
8305:
8299:
8298:
8275:
8265:
8255:
8242:
8238:
8234:
8230:
8223:
8221:
8220:
8215:
8179:
8171:
8167:
8158:
8154:
8140:
8134:
8130:
8122:
8118:
8114:
8112:
8111:
8106:
8101:
8100:
8095:
8091:
8089:
8088:
8087:
8071:
8040:
8032:
8028:
8020:
8016:
8012:
8000:
7998:
7997:
7992:
7987:
7986:
7974:
7973:
7961:
7960:
7945:
7944:
7929:
7928:
7916:
7915:
7900:
7899:
7884:
7883:
7871:
7870:
7830:
7824:
7818:
7816:
7815:
7810:
7808:
7807:
7788:
7786:
7785:
7780:
7766:
7765:
7741:
7740:
7704:
7703:
7688:
7680:
7679:
7667:
7666:
7654:
7653:
7634:
7632:
7631:
7626:
7615:
7614:
7602:
7601:
7582:
7580:
7579:
7574:
7572:
7571:
7555:
7553:
7552:
7547:
7531:
7529:
7528:
7523:
7511:
7509:
7508:
7503:
7470:
7468:
7467:
7462:
7460:
7459:
7420:
7418:
7417:
7412:
7380:
7379:
7336:
7334:
7333:
7328:
7316:
7314:
7313:
7308:
7306:
7267:index gymnastics
7206:
7200:
7194:
7179:
7169:
7155:
7149:
7126:
7109:
7101:
7098:. At a point in
7097:
7092:cotangent spaces
7089:
7066:
7060:
7054:
7048:
7042:
7036:
7008:
7004:
7002:
7001:
6996:
6993:
6988:
6972:
6970:
6969:
6964:
6962:
6961:
6942:
6926:
6924:
6923:
6918:
6915:
6910:
6901:
6900:
6884:
6868:
6866:
6865:
6860:
6857:
6852:
6843:
6842:
6826:
6810:
6774:
6740:
6738:
6737:
6732:
6675:
6669:
6663:
6659:
6657:
6656:
6651:
6646:
6645:
6629:
6624:
6615:
6614:
6601:
6596:
6580:
6576:
6574:
6573:
6568:
6565:
6560:
6544:
6542:
6541:
6536:
6531:
6530:
6512:
6511:
6499:
6498:
6467:
6465:
6464:
6459:
6454:
6453:
6444:
6443:
6431:
6430:
6421:
6420:
6408:
6407:
6398:
6397:
6385:
6384:
6375:
6374:
6362:
6361:
6352:
6351:
6339:
6338:
6329:
6328:
6319:
6318:
6278:
6276:
6275:
6270:
6265:
6264:
6255:
6254:
6242:
6241:
6232:
6231:
6219:
6218:
6209:
6208:
6196:
6195:
6186:
6185:
6173:
6172:
6163:
6162:
6150:
6149:
6140:
6139:
6127:
6126:
6117:
6116:
6107:
6106:
6064:
6060:
6053:
6025:
6021:
6017:
6009:
6001:
5996:. The component
5995:
5973:
5953:
5945:
5943:
5942:
5937:
5932:
5931:
5913:
5912:
5900:
5899:
5875:
5873:
5872:
5867:
5845:
5843:
5842:
5837:
5826:
5825:
5813:
5812:
5790:
5788:
5787:
5782:
5771:
5770:
5758:
5757:
5736:
5735:
5723:
5722:
5701:
5700:
5688:
5687:
5666:
5665:
5653:
5652:
5627:
5587:
5580:
5574:
5563:
5556:
5554:
5553:
5548:
5506:
5504:
5503:
5498:
5492:
5488:
5370:
5366:
5362:
5360:
5359:
5354:
5336:
5334:
5333:
5328:
5309:
5308:
5307:
5274:
5272:
5271:
5266:
5264:
5263:
5254:
5253:
5241:
5240:
5231:
5230:
5218:
5217:
5208:
5207:
5195:
5194:
5185:
5184:
5175:
5174:
5142:
5140:
5139:
5134:
5132:
5131:
5119:
5118:
5106:
5105:
5093:
5092:
5083:
5082:
5063:
5061:
5060:
5055:
5050:
5049:
5044:
5040:
5039:
5038:
5026:
5025:
5007:
5006:
5001:
4997:
4996:
4995:
4983:
4982:
4964:
4963:
4958:
4954:
4953:
4952:
4940:
4939:
4921:
4920:
4915:
4911:
4910:
4909:
4897:
4896:
4881:
4880:
4864:
4860:
4838:Minkowski metric
4833:
4799:
4791:
4772:
4761:
4746:
4731:
4712:
4706:
4697:
4691:
4687:
4668:
4638:
4636:
4635:
4630:
4617:
4611:
4558:
4556:
4555:
4550:
4496:
4494:
4493:
4488:
4486:
4353:
4351:
4350:
4345:
4340:
4339:
4327:
4326:
4259:
4253:
4251:
4250:
4245:
4206:The definition
4202:
4198:
4194:
4184:
4180:
4173:
4169:
4165:
4161:
4151:
4145:
4131:
4125:
4119:
4097:
4077:
4069:
4061:
4057:
4008:Born coordinates
4002:
4000:
3999:
3994:
3953:
3950:
3945:
3944:
3925:
3923:
3922:
3917:
3915:
3914:
3903:
3890:
3884:
3882:
3881:
3876:
3871:
3870:
3855:
3854:
3849:
3837:
3836:
3825:
3812:
3810:
3809:
3804:
3774:
3772:
3771:
3766:
3761:
3760:
3741:
3735:
3733:
3732:
3727:
3725:
3724:
3712:
3711:
3695:
3689:
3685:
3679:
3677:
3676:
3671:
3669:
3639:
3626:
3620:
3614:
3604:
3600:
3593:
3579:
3565:
3561:
3555:
3551:
3537:
3528:
3512:
3507:Minkowski metric
3504:
3478:
3466:
3456:
3414:
3407:
3403:
3391:
3387:
3376:in Relativity.
3360:Metric signature
3351:
3343:
3332:
3330:
3329:
3324:
3322:
3315:
3314:
3305:
3304:
3299:
3298:
3297:
3279:
3278:
3277:
3257:
3256:
3247:
3246:
3241:
3240:
3239:
3221:
3220:
3219:
3195:
3185:
3175:
3171:
3165:
3157:
3151:
3147:
3129:
3107:
3101:
3093:
3091:
3090:
3085:
3080:
3077:
3075:
3071:
3032:
3031:
3026:
3020:
3019:
3014:
3008:
3005:
3003:
3002:
2997:
2993:
2991:
2990:
2989:
2973:
2964:
2963:
2958:
2954:
2953:
2948:
2931:
2929:
2928:
2923:
2921:
2917:
2916:
2911:
2907:
2906:
2905:
2900:
2894:
2893:
2876:
2875:
2870:
2866:
2865:
2864:
2859:
2853:
2852:
2835:
2834:
2829:
2825:
2824:
2823:
2818:
2812:
2811:
2794:
2793:
2788:
2784:
2783:
2782:
2777:
2771:
2770:
2754:
2747:
2743:
2742:
2737:
2733:
2732:
2731:
2726:
2720:
2719:
2702:
2701:
2696:
2692:
2691:
2690:
2685:
2679:
2678:
2661:
2660:
2655:
2651:
2650:
2649:
2644:
2638:
2637:
2620:
2619:
2614:
2610:
2609:
2608:
2603:
2597:
2596:
2580:
2570:
2569:
2565:
2564:
2563:
2550:
2549:
2536:
2535:
2522:
2521:
2475:
2465:
2449:
2445:
2441:
2436:metric signature
2414:
2385:
2383:
2382:
2377:
2375:
2368:
2367:
2362:
2358:
2357:
2343:
2323:
2322:
2317:
2302:
2291:
2274:
2273:
2268:
2250:
2246:
2245:
2240:
2225:
2221:
2200:
2199:
2194:
2175:
2174:
2169:
2165:
2131:
2129:
2128:
2123:
2118:
2104:
2090:
2086:
2064:
2060:
2046:
2044:
2043:
2038:
2036:
2034:
2030:
2028:
2023:
2010:
2005:
1992:
1987:
1974:
1969:
1960:
1959:
1945:
1941:
1939:
1934:
1921:
1916:
1903:
1898:
1885:
1880:
1871:
1870:
1856:
1851:
1850:
1841:
1840:
1828:
1827:
1818:
1817:
1805:
1804:
1795:
1794:
1782:
1781:
1772:
1771:
1762:
1761:
1745:
1743:
1742:
1737:
1735:
1731:
1730:
1717:
1713:
1712:
1693:
1692:
1680:
1679:
1657:
1648:
1634:
1632:
1631:
1626:
1624:
1622:
1621:
1609:
1608:
1596:
1595:
1583:
1582:
1573:
1572:
1563:
1558:
1538:
1533:
1515:
1470:
1468:
1467:
1462:
1457:
1456:
1447:
1446:
1434:
1433:
1424:
1423:
1411:
1410:
1401:
1400:
1388:
1387:
1378:
1377:
1368:
1367:
1355:
1354:
1342:
1341:
1326:
1325:
1313:
1312:
1290:
1252:
1161:
1135:
1114:
1099:
1078:
1061:
1044:
1031:
1000:
996:
986:
980:
974:
968:
957:Causal structure
919:Causal structure
913:Causal structure
865:
812:(which he calls
784:
750:
704:
702:
701:
696:
691:
688:
683:
682:
658:
657:
645:
644:
632:
631:
611:
574:
564:
558:
524:
517:
510:
496:
491:
490:
483:
479:
478:
448:Curved spacetime
301:
282:
281:
251:Minkowski metric
149:between any two
140:four-dimensional
115:
114:
111:
110:
107:
104:
101:
96:
95:
92:
89:
86:
83:
80:
77:
74:
71:
68:
21:
22585:
22584:
22580:
22579:
22578:
22576:
22575:
22574:
22530:
22529:
22528:
22523:
22509:
22337:
22241:BKL singularity
22231:Lemaître–Tolman
22206:
22202:Quantum gravity
22184:
22178:
22164:geodetic effect
22138:(together with
22108:LISA Pathfinder
22047:
21996:
21982:Penrose diagram
21964:
21958:
21933:
21922:
21918:Minkowski space
21884:
21828:
21812:
21760:
21754:
21714:
21707:
21702:
21665:Minkowski space
21648:
21632:
21627:
21621:
21609:, vol. 1,
21593:
21570:
21543:
21459:Poincaré, Henri
21451:
21441:Alfred A. Knopf
21407:10.2307/1969989
21381:
21371:Springer-Verlag
21358:
21338:Thorne, Kip. S.
21248:
21229:
21210:
21191:
21161:
21141:Kolenkow, R. J.
21085:
21019:
21014:
21007:
21003:
20995:
20991:
20983:
20979:
20974:
20970:
20965:Wayback Machine
20955:
20951:
20944:
20940:
20932:
20928:
20916:
20912:
20905:
20898:
20889:
20885:
20877:
20873:
20866:
20850:
20846:
20841:Wayback Machine
20827:
20825:
20818:
20802:
20798:
20793:
20789:
20781:
20777:
20769:
20765:
20760:
20756:
20751:
20747:
20742:
20738:
20732:Clarendon Press
20722:
20718:
20706:
20699:
20688:
20684:
20673:
20669:
20659:
20643:
20639:
20631:
20627:
20615:Wayback Machine
20606:
20602:
20598:
20593:
20586:
20567:
20555:
20549:
20545:
20532:
20528:
20515:
20511:
20502:
20500:
20496:
20491:
20487:
20481:
20477:
20468:
20464:
20460:
20452:Minkowski plane
20433:
20374:
20369:
20354:
20343:
20338:
20337:
20329:
20320:
20316:
20315:
20311:
20310:
20303:
20298:
20297:
20289:
20283:
20279:
20275:
20273:
20264:
20260:
20242:
20231:
20227:
20223:
20219:
20218:
20212:
20201:
20177:
20172:
20152:
20147:
20142:
20138:
20133:
20130:
20129:
20113:
20112:
20101:
20090:
20085:
20084:
20076:
20067:
20063:
20062:
20058:
20057:
20048:
20039:
20030:
20026:
20022:
20020:
20003:
19998:
19997:
19989:
19980:
19976:
19975:
19968:
19963:
19962:
19954:
19945:
19941:
19940:
19938:
19934:
19930:
19920:
19911:
19910:
19897:
19888:
19873:
19869:
19863:
19859:
19853:
19842:
19826:
19821:
19820:
19812:
19804:
19803:
19799:
19787:
19776:
19771:
19770:
19762:
19753:
19749:
19748:
19744:
19743:
19734:
19725:
19716:
19712:
19706:
19702:
19698:
19696:
19684:
19679:
19678:
19670:
19661:
19657:
19656:
19649:
19645:
19639:
19635:
19631:
19629:
19613:
19608:
19607:
19599:
19590:
19586:
19585:
19578:
19574:
19568:
19564:
19560:
19558:
19554:
19545:
19541:
19537:
19532:
19526:
19522:
19515:
19509:
19505:
19498:
19496:
19493:
19492:
19487:
19461:
19456:
19438:
19434:
19430:
19425:
19419:
19415:
19399:
19394:
19387:
19383:
19379:
19369:
19363:
19359:
19347:
19343:
19339:
19334:
19327:
19323:
19319:
19313:
19303:
19302:
19296:
19292:
19279:
19274:
19265:
19261:
19257:
19249:
19247:
19241:
19237:
19225:
19221:
19217:
19212:
19203:
19199:
19195:
19188:
19184:
19180:
19178:
19172:
19168:
19156:
19152:
19148:
19143:
19137:
19133:
19127:
19114:
19110:
19106:
19105:
19093:
19080:
19076:
19072:
19071:
19069:
19066:
19065:
19035:
19022:
19018:
19014:
19013:
19000:
18987:
18983:
18979:
18978:
18977:
18973:
18956:
18951:
18946:
18944:
18939:
18938:
18918:
18905:
18901:
18897:
18896:
18883:
18870:
18866:
18862:
18861:
18860:
18856:
18841:
18836:
18798:
18793:
18783:
18770:
18766:
18762:
18761:
18759:
18756:
18755:
18734:
18729:
18728:
18727:
18723:
18688:
18683:
18668:
18657:
18652:
18651:
18643:
18634:
18630:
18629:
18625:
18624:
18612:
18601:
18597:
18593:
18589:
18588:
18573:
18562:
18558:
18554:
18550:
18549:
18548:
18544:
18538:
18534:
18530:
18528:
18510:
18505:
18495:
18482:
18478:
18474:
18473:
18471:
18468:
18467:
18443:
18432:
18427:
18426:
18418:
18409:
18405:
18404:
18400:
18399:
18392:
18388:
18383:
18372:
18363:
18359:
18355:
18353:
18347:
18343:
18331:
18318:
18307:
18302:
18301:
18293:
18284:
18280:
18279:
18275:
18274:
18263:
18252:
18251:
18247:
18241:
18237:
18233:
18231:
18227:
18223:
18222:
18210:
18206:
18198:
18195:
18194:
18167:
18162:
18161:
18153:
18144:
18140:
18139:
18135:
18128:
18124:
18115:
18111:
18105:
18101:
18097:
18095:
18089:
18085:
18079:
18068:
18046:
18041:
18040:
18032:
18023:
18019:
18018:
18011:
18006:
18005:
17997:
17988:
17984:
17983:
17981:
17968:
17963:
17954:
17950:
17938:
17933:
17932:
17924:
17915:
17911:
17910:
17903:
17898:
17897:
17889:
17880:
17876:
17875:
17873:
17861:
17857:
17853:
17848:
17842:
17831:
17816:
17813:
17812:
17808:
17807:Similarly, for
17792:
17791:
17780:
17769:
17764:
17763:
17755:
17746:
17742:
17741:
17737:
17736:
17727:
17716:
17707:
17703:
17699:
17697:
17691:
17687:
17676:
17665:
17660:
17659:
17651:
17642:
17638:
17637:
17633:
17632:
17620:
17609:
17605:
17601:
17597:
17596:
17581:
17570:
17566:
17562:
17558:
17557:
17556:
17552:
17546:
17535:
17530:
17529:
17521:
17512:
17508:
17507:
17503:
17502:
17496:
17492:
17488:
17486:
17484:
17482:
17476:
17475:
17467:
17456:
17451:
17450:
17442:
17433:
17429:
17428:
17424:
17423:
17416:
17412:
17407:
17396:
17387:
17382:
17381:
17373:
17367:
17363:
17349:
17338:
17327:
17316:
17302:
17297:
17296:
17288:
17279:
17275:
17274:
17270:
17264:
17260:
17243:
17232:
17228:
17224:
17220:
17219:
17204:
17193:
17189:
17185:
17181:
17180:
17179:
17175:
17169:
17158:
17153:
17152:
17144:
17135:
17131:
17130:
17126:
17125:
17119:
17115:
17111:
17109:
17107:
17105:
17099:
17098:
17092:
17079:
17070:
17066:
17062:
17058:
17057:
17042:
17029:
17020:
17016:
17012:
17008:
17007:
17002:
17000:
16997:
16996:
16972:
16961:
16956:
16955:
16947:
16938:
16934:
16933:
16929:
16928:
16921:
16911:
16900:
16899:
16895:
16894:
16888:
16878:
16874:
16870:
16869:
16863:
16859:
16847:
16843:
16834:
16830:
16820:
16809:
16808:
16804:
16793:
16788:
16787:
16779:
16770:
16766:
16765:
16761:
16755:
16751:
16739:
16728:
16724:
16720:
16716:
16715:
16709:
16698:
16693:
16692:
16684:
16675:
16671:
16670:
16666:
16665:
16659:
16655:
16651:
16649:
16640:
16627:
16618:
16614:
16610:
16606:
16605:
16603:
16600:
16599:
16577:
16566:
16561:
16560:
16552:
16543:
16539:
16538:
16534:
16533:
16524:
16513:
16504:
16500:
16494:
16490:
16478:
16474:
16460:
16455:
16454:
16446:
16437:
16433:
16432:
16428:
16422:
16418:
16414:
16412:
16401:
16392:
16388:
16383:
16380:
16379:
16354:
16350:
16338:
16333:
16332:
16324:
16315:
16311:
16310:
16303:
16299:
16293:
16289:
16285:
16283:
16273:
16268:
16266:
16263:
16262:
16248:
16247:
16238:
16234:
16223:
16212:
16207:
16206:
16198:
16189:
16185:
16184:
16180:
16179:
16172:
16168:
16162:
16158:
16152:
16148:
16144:
16142:
16135:
16129:
16125:
16113:
16108:
16107:
16099:
16090:
16086:
16085:
16078:
16074:
16068:
16064:
16060:
16058:
16049:
16045:
16041:
16036:
16033:
16032:
16023:
16019:
16008:
15997:
15992:
15991:
15983:
15974:
15970:
15969:
15965:
15964:
15957:
15947:
15943:
15939:
15938:
15932:
15928:
15911:
15906:
15905:
15897:
15888:
15884:
15883:
15879:
15875:
15873:
15866:
15860:
15856:
15844:
15839:
15838:
15830:
15821:
15817:
15816:
15809:
15805:
15799:
15795:
15791:
15789:
15780:
15776:
15772:
15767:
15763:
15761:
15758:
15757:
15717:
15712:
15697:
15686:
15681:
15680:
15672:
15663:
15659:
15658:
15654:
15653:
15641:
15630:
15626:
15622:
15618:
15617:
15602:
15591:
15587:
15583:
15579:
15578:
15577:
15573:
15567:
15563:
15559:
15557:
15539:
15534:
15524:
15511:
15507:
15503:
15502:
15500:
15497:
15496:
15482:
15481:
15458:
15453:
15452:
15444:
15435:
15431:
15430:
15423:
15418:
15417:
15409:
15400:
15396:
15395:
15393:
15389:
15385:
15375:
15367:
15355:
15354:
15332:
15327:
15326:
15318:
15309:
15305:
15304:
15297:
15293:
15287:
15283:
15279:
15277:
15273:
15263:
15255:
15246:
15242:
15236:
15235:
15222:
15221:
15203:
15198:
15197:
15189:
15180:
15176:
15175:
15168:
15164:
15158:
15154:
15150:
15148:
15138:
15133:
15124:
15120:
15108:
15103:
15102:
15094:
15085:
15081:
15080:
15073:
15069:
15063:
15059:
15055:
15053:
15044:
15040:
15036:
15031:
15016:
15012:
15000:
14995:
14994:
14986:
14977:
14973:
14972:
14965:
14961:
14955:
14951:
14947:
14945:
14936:
14932:
14928:
14923:
14907:
14902:
14901:
14893:
14884:
14880:
14879:
14872:
14868:
14862:
14858:
14854:
14852:
14848:
14838:
14830:
14821:
14817:
14810:
14808:
14805:
14804:
14782:
14769:
14759:
14755:
14754:
14745:
14732:
14723:
14719:
14715:
14711:
14710:
14695:
14682:
14673:
14669:
14665:
14661:
14660:
14640:
14635:
14630:
14628:
14614:
14601:
14597:
14593:
14592:
14591:
14588:
14587:
14585:
14582:
14581:
14551:
14546:
14545:
14537:
14528:
14524:
14523:
14517:
14511:
14507:
14503:
14501:
14488:
14483:
14482:
14474:
14465:
14461:
14460:
14453:
14448:
14447:
14439:
14430:
14426:
14425:
14423:
14419:
14415:
14401:
14393:
14381:
14361:
14349:
14345:
14326:
14321:
14316:
14306:
14301:
14300:
14288:
14284:
14282:
14279:
14278:
14261:
14257:
14245:
14234:
14230:
14226:
14222:
14221:
14206:
14195:
14191:
14187:
14183:
14182:
14162:
14157:
14152:
14150:
14145:
14144:
14123:
14118:
14112:
14109:
14108:
14105:
14100:
14072:
14067:
14066:
14058:
14049:
14045:
14044:
14038:
14032:
14028:
14024:
14022:
14010:
14005:
14004:
13996:
13987:
13983:
13982:
13975:
13970:
13969:
13961:
13952:
13948:
13947:
13945:
13941:
13937:
13926:
13906:
13894:
13890:
13888:
13885:
13884:
13878:
13853:
13849:
13845:
13838:
13833:
13832:
13824:
13815:
13811:
13810:
13808:
13800:
13797:
13796:
13776:
13772:
13758:
13754:
13747:
13742:
13741:
13736:
13731:
13730:
13728:
13717:
13713:
13706:
13702:
13684:
13680:
13679:
13677:
13672:
13669:
13668:
13648:
13644:
13632:
13627:
13626:
13621:
13616:
13607:
13603:
13598:
13595:
13594:
13588:
13582:
13576:
13562:
13561:
13548:
13546:
13539:
13534:
13531:
13530:
13502:
13500:
13493:
13486:
13484:
13481:
13480:
13463:
13462:
13448:
13440:
13429:
13413:
13407:
13406:
13398:
13382:
13375:
13373:
13370:
13369:
13350:
13321:
13315:
13311:
13309:
13301:
13290:
13276:
13273:
13272:
13266:
13250:
13249:
13241:
13231:
13226:
13223:
13222:
13194:
13187:
13185:
13182:
13181:
13152:
13150:
13122:
13120:
13118:
13115:
13114:
13095:
13094:
13083:
13057:
13034:
13018:
13005:
13003:
13000:
12999:
12988:
12968:
12945:
12929:
12916:
12914:
12910:
12908:
12905:
12904:
12890:
12889:
12864:
12862:
12849:
12836:
12834:
12825:
12824:
12796:
12794:
12781:
12768:
12766:
12756:
12754:
12751:
12750:
12726:
12694:
12690:
12666:
12662:
12655:
12651:
12643:
12640:
12639:
12594:
12590:
12575:
12571:
12564:
12560:
12559:
12555:
12553:
12550:
12549:
12525:
12523:
12521:
12518:
12517:
12497:
12492:
12487:
12472:
12468:
12453:
12449:
12442:
12438:
12430:
12427:
12426:
12372:
12369:
12368:
12334:
12330:
12318:
12308:
12304:
12300:
12299:
12284:
12274:
12270:
12266:
12265:
12256:
12252:
12241:
12227:
12223:
12208:
12204:
12197:
12193:
12192:
12188:
12179:
12174:
12169:
12166:
12163:
12162:
12149:
12143:
12137:
12125:
12115:
12101:
12100:
12083:
12078:
12077:
12069:
12060:
12056:
12055:
12049:
12043:
12039:
12035:
12033:
12021:
12016:
12015:
12007:
11998:
11994:
11993:
11986:
11981:
11980:
11972:
11963:
11959:
11958:
11956:
11952:
11948:
11941:
11933:
11913:
11901:
11897:
11894:
11893:
11877:
11871:
11867:
11865:
11858:
11853:
11842:
11826:
11824:
11821:
11820:
11815:
11810:
11809:
11808:
11796:
11787:
11782:
11781:
11780:
11772:
11758:
11736:
11730:
11726:
11720:
11706:
11702:
11698:
11695:parametrization
11679:
11648:
11643:
11627:
11613:
11600:
11596:
11594:
11591:
11590:
11573:
11567:
11563:
11560:Jacobian matrix
11554:, the ordinary
11541:
11537:
11519:
11515:
11511:
11458:
11454:
11452:
11449:
11448:
11434:
11431:
11417:
11408:
11404:
11387:
11373:
11363:
11353:
11339:
11329:
11327:
11319:
11312:
11306:
11300:
11294:
11286:
11285:are vectors in
11275:
11271:
11242:
11238:
11232:
11228:
11211:
11207:
11201:
11197:
11187:
11183:
11177:
11173:
11172:
11168:
11151:
11147:
11130:
11126:
11116:
11112:
11111:
11107:
11092:
11088:
11086:
11083:
11082:
11059:
11055:
11043:
11039:
11024:
11020:
11011:
11007:
11006:
11004:
10991:
10987:
10985:
10982:
10981:
10977:
10954:
10949:
10933:
10919:
10906:
10902:
10900:
10897:
10896:
10882:
10878:
10874:
10872:
10857:
10853:
10829:
10824:
10823:
10808:
10804:
10802:
10799:
10798:
10794:
10791:
10784:
10777:
10771:
10746:
10742:
10725:
10721:
10711:
10707:
10706:
10702:
10685:
10681:
10675:
10671:
10654:
10650:
10644:
10640:
10630:
10626:
10620:
10616:
10615:
10611:
10594:
10590:
10573:
10569:
10559:
10555:
10554:
10550:
10541:
10537:
10535:
10532:
10531:
10527:
10523:
10519:
10515:
10511:
10509:
10469:
10464:
10459:
10456:
10453:
10452:
10421:
10416:
10415:
10397:
10392:
10387:
10378:
10375:
10374:
10371:
10348:
10345:
10344:
10315:
10312:
10311:
10282:
10279:
10278:
10272:
10255:
10251:
10246:
10238:
10235:
10234:
10196:
10191:
10186:
10183:
10180:
10179:
10159:
10156:
10155:
10128:
10124:
10106:
10101:
10095:
10092:
10091:
10061:
10058:
10057:
10034:
10029:
10028:
10026:
10023:
10022:
9985:
9981:
9972:
9962:
9958:
9954:
9953:
9938:
9928:
9924:
9920:
9919:
9910:
9906:
9900:
9896:
9887:
9882:
9881:
9867:
9863:
9848:
9844:
9834:
9830:
9829:
9825:
9807:
9802:
9797:
9794:
9791:
9790:
9762:
9759:
9758:
9735:
9730:
9729:
9711:
9706:
9701:
9698:
9695:
9694:
9675:
9672:
9671:
9646:
9644:
9641:
9640:
9623:
9618:
9617:
9615:
9612:
9611:
9594:
9589:
9588:
9586:
9583:
9582:
9547:
9541:
9492:
9480:de Sitter space
9473:
9462:
9450:
9442:
9434:
9428:
9422:
9415:
9411:
9395:
9388:
9380:
9373:
9367:
9353:
9350:
9340:
9329:
9315:
9292:
9289:
9283:
9273:
9267:
9261:
9255:magnetic moment
9223:
9217:
9214:
9205:
9201:
9198:
9190:Main articles:
9188:
9186:Generalizations
9173:
9149:
9143:
9129:
9115:
9101:
9095:
9087:
9075:
9061:
9055:
9047:
9030:
9027:
9022:
9021:
9016:
9003:
8995:
8980:
8960:
8956:
8950:
8946:
8937:
8933:
8900:
8889:
8885:
8881:
8878:
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8824:
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8805:
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8797:
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8761:
8757:
8751:
8747:
8738:
8734:
8710:
8706:
8694:
8690:
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8674:
8672:
8669:
8668:
8662:
8656:
8643:
8633:
8569:
8566:
8565:
8561:
8541:
8536:
8519:
8506:
8502:
8498:
8493:
8490:
8489:
8485:
8479:
8468:
8464:
8460:
8457:
8456:
8454:
8451:
8450:
8449:to each other,
8443:
8433:
8430:
8421:
8417:
8414:
8405:
8401:
8379:
8374:
8358:
8354:
8350:
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8317:
8313:
8309:
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8300:
8294:
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8267:
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8254:
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8240:
8236:
8232:
8228:
8185:
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8177:
8169:
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8142:
8136:
8132:
8128:
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7476:
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7428:
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7375:
7371:
7342:
7339:
7338:
7322:
7319:
7318:
7302:
7282:
7279:
7278:
7275:
7259:Weinberg (2002)
7232:de Broglie wave
7202:
7201:(eta-flat) and
7196:
7193:
7181:
7175:
7165:
7151:
7148:
7136:
7128:
7122:
7107:
7099:
7095:
7087:
7072:Euclidean space
7062:
7056:
7050:
7044:
7038:
7032:
7025:
7017:Main articles:
7015:
7006:
6989:
6984:
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4797:
4778:
4770:
4748:
4733:
4718:
4713:are said to be
4708:
4702:
4693:
4689:
4685:
4682:classical group
4655:
4568:
4565:
4564:
4506:
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4369:
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4099:
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4022:Main articles:
4020:
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3452:
3409:
3405:
3401:
3394:Steven Weinberg
3389:
3385:
3381:
3374:sign convention
3362:
3349:
3341:
3338:
3337:
3320:
3319:
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3293:
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3211:
3207:
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3167:
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3153:
3152:column vectors
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2517:
2513:
2512:
2508:
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2499:
2498:
2471:
2463:
2456:
2454:Tangent vectors
2447:
2443:
2439:
2412:
2394:
2373:
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2195:
2184:
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2024:
2019:
2006:
2001:
1988:
1983:
1970:
1965:
1955:
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1930:
1917:
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1260:
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1216:
1209:
1196:
1159:
1131:
1122:The set of all
1101:
1088:
1067:
1050:
1033:
1002:
998:
992:
982:
976:
970:
964:
937:absolute future
921:
915:
879:quasi-Euclidean
871:Albert Einstein
866:
863:
766:
762:
732:
687:
678:
674:
653:
649:
640:
636:
627:
623:
621:
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593:
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499:
486:
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417:
409:
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392:
384:
383:
382:
362:
354:
353:
352:
348:Minkowski space
327:
319:
280:
213:Euclidean group
201:Euclidean space
163:Hendrik Lorentz
98:
65:
61:
54:Minkowski space
35:
28:
23:
22:
15:
12:
11:
5:
22583:
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22428:Choquet-Bruhat
22425:
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22395:
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22335:
22328:
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22317:
22310:
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22299:
22294:
22289:
22280:Axisymmetric:
22277:
22276:
22271:
22265:
22254:
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22248:
22243:
22238:
22233:
22228:
22219:Cosmological:
22216:
22214:
22208:
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22205:
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22199:
22194:
22188:
22186:
22180:
22179:
22177:
22176:
22171:
22160:frame-dragging
22157:
22152:
22147:
22144:Einstein rings
22140:Einstein cross
22133:
22122:
22121:
22116:
22110:
22105:
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22087:
22077:
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22049:
22048:
22046:
22045:
22043:Ernst equation
22040:
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22030:
22025:
22020:
22015:
22013:BSSN formalism
22010:
22004:
22002:
21998:
21997:
21995:
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21984:
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21910:
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21877:
21872:
21870:Ladder paradox
21867:
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21790:
21785:
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21775:
21773:Speed of light
21770:
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21747:
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21649:Animation clip
21631:
21630:External links
21628:
21626:
21625:
21619:
21597:
21591:
21574:
21568:
21560:Academic Press
21547:
21542:978-0805384918
21541:
21524:
21517:
21510:
21455:
21449:
21431:Penrose, Roger
21427:
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21379:
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21356:
21330:
21327:Space and Time
21306:
21305:
21304:
21298:
21252:
21246:
21233:
21227:
21214:
21208:
21195:
21189:
21173:Lifshitz, E.M.
21165:
21159:
21133:
21126:
21097:
21083:
21058:
21032:(4): 273–314.
21020:
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20712:Space and Time
20697:
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20204:
20200:
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19571:
19567:
19563:
19557:
19548:
19544:
19540:
19536:
19529:
19525:
19521:
19518:
19516:
19512:
19508:
19504:
19501:
19500:
19473:
19467:
19464:
19460:
19455:
19452:
19449:
19441:
19437:
19433:
19429:
19422:
19418:
19414:
19411:
19405:
19402:
19398:
19390:
19386:
19382:
19376:
19373:
19366:
19362:
19358:
19350:
19346:
19342:
19338:
19330:
19326:
19322:
19316:
19310:
19307:
19299:
19295:
19291:
19285:
19282:
19278:
19268:
19264:
19260:
19255:
19252:
19244:
19240:
19236:
19228:
19224:
19220:
19216:
19206:
19202:
19198:
19191:
19187:
19183:
19175:
19171:
19167:
19159:
19155:
19151:
19147:
19140:
19136:
19130:
19125:
19120:
19117:
19113:
19109:
19104:
19101:
19096:
19091:
19086:
19083:
19079:
19075:
19051:
19047:
19043:
19038:
19033:
19028:
19025:
19021:
19017:
19011:
19008:
19003:
18998:
18993:
18990:
18986:
18982:
18976:
18968:
18965:
18962:
18959:
18954:
18949:
18942:
18937:
18934:
18930:
18926:
18921:
18916:
18911:
18908:
18904:
18900:
18894:
18891:
18886:
18881:
18876:
18873:
18869:
18865:
18859:
18853:
18850:
18847:
18844:
18839:
18835:
18831:
18828:
18825:
18821:
18818:
18815:
18810:
18807:
18804:
18801:
18796:
18792:
18786:
18781:
18776:
18773:
18769:
18765:
18750:
18749:
18730:
18718:
18717:
18705:
18700:
18697:
18694:
18691:
18686:
18682:
18678:
18671:
18666:
18660:
18655:
18650:
18646:
18642:
18637:
18633:
18628:
18621:
18615:
18610:
18604:
18600:
18596:
18592:
18587:
18584:
18581:
18576:
18571:
18565:
18561:
18557:
18553:
18547:
18541:
18537:
18533:
18527:
18522:
18519:
18516:
18513:
18508:
18504:
18498:
18493:
18488:
18485:
18481:
18477:
18453:
18446:
18441:
18435:
18430:
18425:
18421:
18417:
18412:
18408:
18403:
18395:
18391:
18386:
18382:
18379:
18375:
18371:
18366:
18362:
18358:
18350:
18346:
18342:
18339:
18334:
18329:
18321:
18316:
18310:
18305:
18300:
18296:
18292:
18287:
18283:
18278:
18271:
18266:
18262:
18259:
18255:
18250:
18244:
18240:
18236:
18230:
18226:
18221:
18218:
18213:
18209:
18205:
18202:
18182:
18176:
18170:
18165:
18160:
18156:
18152:
18147:
18143:
18138:
18131:
18127:
18123:
18118:
18114:
18108:
18104:
18100:
18092:
18088:
18082:
18077:
18074:
18071:
18067:
18063:
18060:
18057:
18049:
18044:
18039:
18035:
18031:
18026:
18022:
18014:
18009:
18004:
18000:
17996:
17991:
17987:
17980:
17974:
17971:
17967:
17962:
17957:
17953:
17949:
17941:
17936:
17931:
17927:
17923:
17918:
17914:
17906:
17901:
17896:
17892:
17888:
17883:
17879:
17872:
17864:
17860:
17856:
17852:
17845:
17840:
17837:
17834:
17830:
17826:
17823:
17820:
17790:
17783:
17778:
17772:
17767:
17762:
17758:
17754:
17749:
17745:
17740:
17734:
17730:
17726:
17723:
17719:
17715:
17710:
17706:
17702:
17694:
17690:
17686:
17679:
17674:
17668:
17663:
17658:
17654:
17650:
17645:
17641:
17636:
17629:
17623:
17618:
17612:
17608:
17604:
17600:
17595:
17592:
17589:
17584:
17579:
17573:
17569:
17565:
17561:
17555:
17549:
17544:
17538:
17533:
17528:
17524:
17520:
17515:
17511:
17506:
17499:
17495:
17491:
17485:
17481:
17478:
17477:
17470:
17465:
17459:
17454:
17449:
17445:
17441:
17436:
17432:
17427:
17419:
17415:
17410:
17406:
17403:
17399:
17395:
17390:
17385:
17380:
17376:
17370:
17366:
17362:
17359:
17356:
17352:
17348:
17345:
17341:
17337:
17334:
17330:
17326:
17323:
17319:
17315:
17311:
17305:
17300:
17295:
17291:
17287:
17282:
17278:
17273:
17267:
17263:
17259:
17256:
17252:
17246:
17241:
17235:
17231:
17227:
17223:
17218:
17215:
17212:
17207:
17202:
17196:
17192:
17188:
17184:
17178:
17172:
17167:
17161:
17156:
17151:
17147:
17143:
17138:
17134:
17129:
17122:
17118:
17114:
17108:
17104:
17101:
17100:
17095:
17090:
17086:
17082:
17078:
17073:
17069:
17065:
17061:
17056:
17053:
17050:
17045:
17040:
17036:
17032:
17028:
17023:
17019:
17015:
17011:
17006:
17004:
16982:
16975:
16970:
16964:
16959:
16954:
16950:
16946:
16941:
16937:
16932:
16924:
16919:
16914:
16910:
16907:
16903:
16898:
16891:
16886:
16881:
16877:
16873:
16866:
16862:
16858:
16855:
16850:
16846:
16842:
16837:
16833:
16828:
16823:
16819:
16816:
16812:
16807:
16802:
16796:
16791:
16786:
16782:
16778:
16773:
16769:
16764:
16758:
16754:
16750:
16747:
16742:
16737:
16731:
16727:
16723:
16719:
16712:
16707:
16701:
16696:
16691:
16687:
16683:
16678:
16674:
16669:
16662:
16658:
16654:
16648:
16643:
16638:
16634:
16630:
16626:
16621:
16617:
16613:
16609:
16587:
16580:
16575:
16569:
16564:
16559:
16555:
16551:
16546:
16542:
16537:
16531:
16527:
16523:
16520:
16516:
16512:
16507:
16503:
16497:
16493:
16489:
16486:
16481:
16477:
16473:
16469:
16463:
16458:
16453:
16449:
16445:
16440:
16436:
16431:
16425:
16421:
16417:
16411:
16408:
16404:
16400:
16395:
16391:
16387:
16365:
16362:
16357:
16353:
16349:
16341:
16336:
16331:
16327:
16323:
16318:
16314:
16306:
16302:
16296:
16292:
16288:
16279:
16276:
16272:
16246:
16241:
16237:
16233:
16226:
16221:
16215:
16210:
16205:
16201:
16197:
16192:
16188:
16183:
16175:
16171:
16165:
16161:
16155:
16151:
16147:
16141:
16138:
16136:
16132:
16128:
16124:
16116:
16111:
16106:
16102:
16098:
16093:
16089:
16081:
16077:
16071:
16067:
16063:
16052:
16048:
16044:
16040:
16035:
16034:
16031:
16026:
16022:
16018:
16011:
16006:
16000:
15995:
15990:
15986:
15982:
15977:
15973:
15968:
15960:
15955:
15950:
15946:
15942:
15935:
15931:
15927:
15924:
15920:
15914:
15909:
15904:
15900:
15896:
15891:
15887:
15882:
15878:
15872:
15869:
15867:
15863:
15859:
15855:
15847:
15842:
15837:
15833:
15829:
15824:
15820:
15812:
15808:
15802:
15798:
15794:
15783:
15779:
15775:
15771:
15766:
15765:
15752:
15751:
15734:
15729:
15726:
15723:
15720:
15715:
15711:
15707:
15700:
15695:
15689:
15684:
15679:
15675:
15671:
15666:
15662:
15657:
15650:
15644:
15639:
15633:
15629:
15625:
15621:
15616:
15613:
15610:
15605:
15600:
15594:
15590:
15586:
15582:
15576:
15570:
15566:
15562:
15556:
15551:
15548:
15545:
15542:
15537:
15533:
15527:
15522:
15517:
15514:
15510:
15506:
15480:
15477:
15474:
15470:
15461:
15456:
15451:
15447:
15443:
15438:
15434:
15426:
15421:
15416:
15412:
15408:
15403:
15399:
15392:
15388:
15384:
15381:
15378:
15376:
15374:
15370:
15366:
15363:
15360:
15357:
15356:
15353:
15350:
15347:
15343:
15335:
15330:
15325:
15321:
15317:
15312:
15308:
15300:
15296:
15290:
15286:
15282:
15276:
15272:
15269:
15266:
15264:
15262:
15258:
15254:
15249:
15245:
15241:
15238:
15237:
15234:
15225:
15223:
15220:
15217:
15214:
15206:
15201:
15196:
15192:
15188:
15183:
15179:
15171:
15167:
15161:
15157:
15153:
15144:
15141:
15137:
15132:
15127:
15123:
15119:
15111:
15106:
15101:
15097:
15093:
15088:
15084:
15076:
15072:
15066:
15062:
15058:
15047:
15043:
15039:
15035:
15030:
15027:
15024:
15019:
15015:
15011:
15003:
14998:
14993:
14989:
14985:
14980:
14976:
14968:
14964:
14958:
14954:
14950:
14939:
14935:
14931:
14927:
14922:
14918:
14910:
14905:
14900:
14896:
14892:
14887:
14883:
14875:
14871:
14865:
14861:
14857:
14851:
14847:
14844:
14841:
14839:
14837:
14833:
14829:
14824:
14820:
14816:
14813:
14812:
14790:
14785:
14780:
14776:
14772:
14768:
14765:
14762:
14758:
14753:
14748:
14743:
14739:
14735:
14731:
14726:
14722:
14718:
14714:
14709:
14706:
14703:
14698:
14693:
14689:
14685:
14681:
14676:
14672:
14668:
14664:
14659:
14652:
14649:
14646:
14643:
14638:
14633:
14626:
14622:
14617:
14612:
14607:
14604:
14600:
14596:
14590:
14567:
14563:
14554:
14549:
14544:
14540:
14536:
14531:
14527:
14520:
14514:
14510:
14506:
14499:
14491:
14486:
14481:
14477:
14473:
14468:
14464:
14456:
14451:
14446:
14442:
14438:
14433:
14429:
14422:
14418:
14414:
14411:
14408:
14404:
14400:
14396:
14391:
14388:
14384:
14380:
14377:
14374:
14371:
14368:
14364:
14360:
14355:
14352:
14348:
14343:
14338:
14335:
14332:
14329:
14324:
14319:
14314:
14309:
14304:
14299:
14294:
14291:
14287:
14264:
14260:
14256:
14253:
14248:
14243:
14237:
14233:
14229:
14225:
14220:
14217:
14214:
14209:
14204:
14198:
14194:
14190:
14186:
14181:
14174:
14171:
14168:
14165:
14160:
14155:
14148:
14143:
14140:
14135:
14132:
14129:
14126:
14121:
14117:
14104:
14101:
14088:
14084:
14075:
14070:
14065:
14061:
14057:
14052:
14048:
14041:
14035:
14031:
14027:
14021:
14013:
14008:
14003:
13999:
13995:
13990:
13986:
13978:
13973:
13968:
13964:
13960:
13955:
13951:
13944:
13940:
13936:
13933:
13929:
13925:
13922:
13919:
13916:
13913:
13909:
13905:
13900:
13897:
13893:
13883:, one obtains
13864:
13856:
13852:
13848:
13841:
13836:
13831:
13827:
13823:
13818:
13814:
13807:
13804:
13784:
13779:
13775:
13771:
13768:
13761:
13757:
13750:
13745:
13739:
13734:
13727:
13720:
13716:
13709:
13705:
13701:
13698:
13695:
13692:
13687:
13683:
13676:
13656:
13651:
13647:
13643:
13640:
13635:
13630:
13624:
13619:
13615:
13610:
13606:
13602:
13560:
13555:
13551:
13545:
13542:
13540:
13537:
13533:
13532:
13529:
13524:
13520:
13517:
13514:
13511:
13508:
13505:
13499:
13496:
13494:
13492:
13489:
13488:
13468:
13461:
13458:
13455:
13451:
13447:
13443:
13439:
13436:
13432:
13428:
13425:
13422:
13419:
13416:
13414:
13412:
13409:
13408:
13405:
13401:
13397:
13394:
13391:
13388:
13385:
13383:
13381:
13378:
13377:
13336:
13330:
13327:
13324:
13318:
13314:
13308:
13304:
13300:
13297:
13293:
13289:
13286:
13283:
13280:
13248:
13244:
13240:
13237:
13234:
13232:
13229:
13225:
13224:
13221:
13218:
13215:
13212:
13209:
13206:
13203:
13200:
13197:
13195:
13193:
13190:
13189:
13167:
13162:
13158:
13155:
13149:
13146:
13143:
13140:
13137:
13132:
13128:
13125:
13100:
13093:
13090:
13086:
13082:
13079:
13076:
13073:
13070:
13067:
13064:
13060:
13056:
13053:
13050:
13047:
13044:
13041:
13037:
13033:
13030:
13027:
13024:
13021:
13019:
13015:
13011:
13008:
13002:
13001:
12998:
12995:
12991:
12987:
12984:
12981:
12978:
12975:
12971:
12967:
12964:
12961:
12958:
12955:
12952:
12948:
12944:
12941:
12938:
12935:
12932:
12930:
12926:
12922:
12919:
12913:
12912:
12888:
12885:
12882:
12879:
12874:
12870:
12867:
12861:
12858:
12855:
12852:
12850:
12846:
12842:
12839:
12833:
12830:
12827:
12826:
12823:
12820:
12817:
12814:
12811:
12806:
12802:
12799:
12793:
12790:
12787:
12784:
12782:
12778:
12774:
12771:
12765:
12762:
12759:
12758:
12736:
12733:
12729:
12725:
12722:
12719:
12716:
12712:
12708:
12705:
12702:
12697:
12693:
12689:
12686:
12683:
12680:
12677:
12674:
12669:
12665:
12661:
12658:
12654:
12650:
12647:
12626:
12622:
12619:
12616:
12613:
12610:
12607:
12603:
12597:
12593:
12589:
12586:
12583:
12578:
12574:
12570:
12567:
12563:
12558:
12535:
12531:
12528:
12505:
12500:
12495:
12490:
12485:
12481:
12475:
12471:
12467:
12464:
12461:
12456:
12452:
12448:
12445:
12441:
12437:
12434:
12412:
12409:
12406:
12403:
12400:
12397:
12394:
12391:
12388:
12385:
12382:
12379:
12376:
12355:
12351:
12348:
12345:
12342:
12337:
12333:
12329:
12326:
12321:
12316:
12311:
12307:
12303:
12298:
12295:
12292:
12287:
12282:
12277:
12273:
12269:
12264:
12259:
12255:
12251:
12248:
12244:
12240:
12236:
12230:
12226:
12222:
12219:
12216:
12211:
12207:
12203:
12200:
12196:
12191:
12187:
12182:
12177:
12172:
12156:
12099:
12095:
12086:
12081:
12076:
12072:
12068:
12063:
12059:
12052:
12046:
12042:
12038:
12032:
12024:
12019:
12014:
12010:
12006:
12001:
11997:
11989:
11984:
11979:
11975:
11971:
11966:
11962:
11955:
11951:
11947:
11944:
11942:
11940:
11936:
11932:
11929:
11926:
11923:
11920:
11916:
11912:
11907:
11904:
11900:
11896:
11895:
11892:
11886:
11883:
11880:
11874:
11870:
11864:
11861:
11859:
11856:
11852:
11849:
11845:
11841:
11838:
11835:
11832:
11829:
11828:
11811:
11783:
11701:of a manifold
11678:
11675:
11672:
11671:
11659:
11656:
11651:
11646:
11642:
11638:
11635:
11630:
11625:
11622:
11619:
11616:
11612:
11608:
11603:
11599:
11499:
11496:
11493:
11490:
11487:
11484:
11481:
11478:
11475:
11472:
11469:
11466:
11461:
11457:
11422:
11413:
11406:
11317:
11310:
11304:
11290:
11255:
11251:
11245:
11241:
11235:
11231:
11226:
11223:
11219:
11214:
11210:
11204:
11200:
11195:
11190:
11186:
11180:
11176:
11171:
11167:
11164:
11160:
11154:
11150:
11145:
11142:
11138:
11133:
11129:
11124:
11119:
11115:
11110:
11106:
11103:
11100:
11095:
11091:
11068:
11058:
11052:
11046:
11042:
11038:
11035:
11032:
11027:
11023:
11019:
11014:
11010:
11002:
10999:
10994:
10990:
10965:
10962:
10957:
10952:
10948:
10944:
10941:
10936:
10931:
10928:
10925:
10922:
10918:
10914:
10909:
10905:
10837:
10832:
10827:
10822:
10819:
10816:
10811:
10807:
10789:
10782:
10775:
10759:
10755:
10749:
10745:
10740:
10737:
10733:
10728:
10724:
10719:
10714:
10710:
10705:
10701:
10698:
10694:
10688:
10684:
10678:
10674:
10669:
10666:
10662:
10657:
10653:
10647:
10643:
10638:
10633:
10629:
10623:
10619:
10614:
10610:
10607:
10603:
10597:
10593:
10588:
10585:
10581:
10576:
10572:
10567:
10562:
10558:
10553:
10549:
10544:
10540:
10530:it holds that
10502:
10501:
10481:
10478:
10475:
10472:
10467:
10462:
10430:
10427:
10424:
10419:
10414:
10409:
10406:
10403:
10400:
10395:
10390:
10385:
10382:
10370:
10367:
10355:
10352:
10328:
10325:
10322:
10319:
10295:
10292:
10289:
10286:
10258:
10254:
10249:
10245:
10242:
10208:
10205:
10202:
10199:
10194:
10189:
10163:
10139:
10136:
10131:
10127:
10123:
10118:
10115:
10112:
10109:
10104:
10100:
10088:induced metric
10071:
10068:
10065:
10043:
10040:
10037:
10032:
10009:
10005:
10002:
9999:
9996:
9993:
9988:
9984:
9980:
9975:
9970:
9965:
9961:
9957:
9952:
9949:
9946:
9941:
9936:
9931:
9927:
9923:
9918:
9913:
9909:
9903:
9899:
9895:
9890:
9885:
9880:
9876:
9870:
9866:
9862:
9859:
9856:
9851:
9847:
9843:
9840:
9837:
9833:
9828:
9824:
9819:
9816:
9813:
9810:
9805:
9800:
9775:
9772:
9769:
9766:
9744:
9741:
9738:
9733:
9728:
9723:
9720:
9717:
9714:
9709:
9704:
9679:
9653:
9650:
9626:
9621:
9597:
9592:
9543:Main article:
9540:
9537:
9531:for which the
9496:flat spacetime
9491:
9488:
9410:
9407:
9348:
9327:
9291:is defined as
9287:
9271:
9237:, such as the
9221:
9213:
9210:
9187:
9184:
9126:
9125:
9085:
9026:
9023:
8968:
8963:
8959:
8953:
8949:
8943:
8940:
8936:
8932:
8929:
8926:
8923:
8920:
8917:
8914:
8911:
8908:
8903:
8898:
8892:
8888:
8884:
8880:
8874:
8869:
8863:
8859:
8855:
8851:
8845:
8842:
8838:
8834:
8831:
8828:
8769:
8764:
8760:
8754:
8750:
8744:
8741:
8737:
8733:
8730:
8727:
8724:
8721:
8718:
8713:
8709:
8705:
8702:
8697:
8693:
8689:
8684:
8681:
8677:
8621:
8618:
8615:
8612:
8609:
8606:
8603:
8600:
8597:
8594:
8591:
8588:
8585:
8582:
8579:
8576:
8573:
8549:
8544:
8539:
8535:
8531:
8527:
8522:
8517:
8509:
8505:
8501:
8497:
8492:
8488:
8482:
8477:
8471:
8467:
8463:
8459:
8439:
8426:
8420:and the bases
8410:
8387:
8382:
8377:
8373:
8369:
8361:
8357:
8353:
8346:
8342:
8338:
8332:
8328:
8320:
8316:
8312:
8308:
8303:
8297:
8293:
8289:
8280:, one obtains
8250:
8213:
8210:
8207:
8204:
8201:
8198:
8195:
8192:
8189:
8174:covector field
8168:of a function
8104:
8099:
8094:
8086:
8082:
8078:
8074:
8069:
8023:tensor product
7990:
7985:
7981:
7977:
7972:
7968:
7964:
7959:
7956:
7952:
7948:
7943:
7939:
7935:
7932:
7927:
7923:
7919:
7914:
7911:
7907:
7903:
7898:
7894:
7890:
7887:
7882:
7878:
7874:
7869:
7866:
7862:
7850:
7847:
7846:
7836:
7833:
7806:
7803:
7799:
7778:
7775:
7772:
7769:
7764:
7761:
7757:
7753:
7750:
7747:
7744:
7739:
7736:
7732:
7728:
7725:
7722:
7719:
7716:
7713:
7710:
7707:
7702:
7699:
7695:
7691:
7687:
7683:
7678:
7674:
7670:
7665:
7661:
7657:
7652:
7649:
7645:
7624:
7621:
7618:
7613:
7609:
7605:
7600:
7597:
7593:
7570:
7566:
7545:
7521:
7501:
7498:
7495:
7492:
7489:
7486:
7483:
7480:
7458:
7454:
7450:
7447:
7444:
7441:
7438:
7435:
7432:
7410:
7407:
7404:
7401:
7398:
7395:
7392:
7389:
7386:
7383:
7378:
7374:
7370:
7367:
7364:
7361:
7358:
7355:
7352:
7349:
7346:
7326:
7305:
7301:
7298:
7295:
7292:
7289:
7286:
7274:
7271:
7189:
7144:
7132:
7043:and their sum
7014:
7011:
6992:
6987:
6983:
6960:
6956:
6952:
6949:
6945:
6941:
6937:
6914:
6909:
6905:
6899:
6895:
6891:
6887:
6883:
6879:
6856:
6851:
6847:
6841:
6837:
6833:
6829:
6825:
6821:
6806:
6798:
6790:
6782:
6770:
6763:
6756:
6749:
6730:
6727:
6724:
6721:
6718:
6715:
6712:
6709:
6706:
6703:
6700:
6697:
6694:
6691:
6688:
6685:
6649:
6644:
6641:
6637:
6633:
6628:
6623:
6619:
6613:
6610:
6606:
6600:
6595:
6591:
6564:
6559:
6555:
6534:
6529:
6526:
6522:
6518:
6515:
6510:
6506:
6502:
6497:
6493:
6489:
6486:
6457:
6452:
6448:
6442:
6438:
6434:
6429:
6425:
6419:
6415:
6411:
6406:
6402:
6396:
6392:
6388:
6383:
6379:
6373:
6369:
6365:
6360:
6356:
6350:
6346:
6342:
6337:
6333:
6327:
6323:
6317:
6314:
6310:
6306:
6303:
6300:
6297:
6294:
6291:
6288:
6268:
6263:
6259:
6253:
6249:
6245:
6240:
6236:
6230:
6226:
6222:
6217:
6213:
6207:
6203:
6199:
6194:
6190:
6184:
6180:
6176:
6171:
6167:
6161:
6157:
6153:
6148:
6144:
6138:
6134:
6130:
6125:
6121:
6115:
6111:
6105:
6102:
6098:
6094:
6091:
6088:
6085:
6082:
6079:
6076:
6049:
6042:
6035:
6002:is called the
5990:
5935:
5930:
5927:
5923:
5919:
5916:
5911:
5907:
5903:
5898:
5894:
5890:
5887:
5865:
5861:
5858:
5855:
5835:
5832:
5829:
5824:
5820:
5816:
5811:
5807:
5803:
5800:
5791:and for which
5780:
5777:
5774:
5769:
5765:
5761:
5756:
5752:
5748:
5745:
5742:
5739:
5734:
5730:
5726:
5721:
5717:
5713:
5710:
5707:
5704:
5699:
5695:
5691:
5686:
5682:
5678:
5675:
5672:
5669:
5664:
5660:
5656:
5651:
5647:
5643:
5640:
5637:
5623:
5616:
5609:
5602:
5593:
5592:Standard basis
5590:
5546:
5543:
5540:
5537:
5534:
5531:
5528:
5525:
5522:
5519:
5516:
5496:
5491:
5484:
5481:
5478:
5476:
5473:
5471:
5468:
5466:
5463:
5462:
5459:
5456:
5454:
5451:
5448:
5446:
5443:
5441:
5438:
5437:
5434:
5431:
5429:
5426:
5424:
5421:
5418:
5416:
5413:
5412:
5409:
5406:
5404:
5401:
5399:
5396:
5394:
5391:
5390:
5387:
5383:
5380:
5352:
5349:
5346:
5326:
5323:
5319:
5316:
5313:
5301:
5297:
5294:
5291:
5288:
5262:
5258:
5252:
5248:
5244:
5239:
5235:
5229:
5225:
5221:
5216:
5212:
5206:
5202:
5198:
5193:
5189:
5183:
5179:
5173:
5169:
5165:
5162:
5159:
5156:
5145:quadratic form
5130:
5126:
5122:
5117:
5113:
5109:
5104:
5100:
5096:
5091:
5087:
5081:
5077:
5053:
5048:
5043:
5037:
5033:
5029:
5024:
5020:
5015:
5010:
5005:
5000:
4994:
4990:
4986:
4981:
4977:
4972:
4967:
4962:
4957:
4951:
4947:
4943:
4938:
4934:
4929:
4924:
4919:
4914:
4908:
4904:
4900:
4895:
4891:
4886:
4879:
4875:
4839:
4836:
4809:inertial frame
4640:
4639:
4628:
4625:
4622:
4616:
4610:
4607:
4604:
4601:
4597:
4594:
4591:
4588:
4585:
4581:
4578:
4575:
4572:
4562:
4561:Non-degeneracy
4559:
4548:
4545:
4541:
4538:
4535:
4532:
4529:
4526:
4523:
4519:
4516:
4513:
4510:
4500:
4497:
4485:
4481:
4478:
4475:
4471:
4468:
4465:
4462:
4458:
4455:
4452:
4448:
4445:
4442:
4438:
4435:
4432:
4429:
4426:
4423:
4420:
4416:
4413:
4410:
4407:
4404:
4401:
4398:
4395:
4391:
4388:
4385:
4382:
4379:
4376:
4373:
4363:
4343:
4338:
4334:
4330:
4325:
4321:
4317:
4314:
4311:
4308:
4305:
4302:
4299:
4296:
4293:
4290:
4287:
4284:
4281:
4243:
4240:
4236:
4233:
4230:
4227:
4224:
4221:
4218:
4215:
4137:
4103:
4092:
4083:
4042:quadratic form
4019:
4016:
3992:
3989:
3986:
3983:
3980:
3977:
3974:
3971:
3968:
3965:
3962:
3959:
3956:
3948:
3943:
3940:
3936:
3913:
3910:
3907:
3902:
3874:
3869:
3866:
3862:
3858:
3853:
3848:
3843:
3840:
3835:
3832:
3829:
3824:
3802:
3799:
3796:
3793:
3790:
3787:
3784:
3764:
3759:
3755:
3751:
3723:
3719:
3715:
3710:
3706:
3686:has signature
3668:
3664:
3661:
3658:
3655:
3652:
3649:
3552:at each point
3543:
3494:
3491:
3382:
3379:
3378:
3361:
3358:
3318:
3313:
3309:
3303:
3296:
3292:
3286:
3283:
3281:
3276:
3272:
3268:
3264:
3263:
3260:
3255:
3251:
3245:
3238:
3234:
3228:
3225:
3223:
3218:
3214:
3210:
3206:
3205:
3186:. This is the
3158:associated to
3140:
3137:
3136:
3083:
3074:
3067:
3064:
3063:
3060:
3057:
3056:
3053:
3050:
3049:
3046:
3043:
3042:
3039:
3035:
3030:
3025:
3018:
3013:
3006: or
3001:
2996:
2988:
2984:
2980:
2976:
2971:
2967:
2962:
2957:
2952:
2947:
2942:
2915:
2910:
2904:
2899:
2892:
2888:
2883:
2879:
2874:
2869:
2863:
2858:
2851:
2847:
2842:
2838:
2833:
2828:
2822:
2817:
2810:
2806:
2801:
2797:
2792:
2787:
2781:
2776:
2769:
2765:
2760:
2753:
2750:
2748:
2746:
2741:
2736:
2730:
2725:
2718:
2714:
2709:
2705:
2700:
2695:
2689:
2684:
2677:
2673:
2668:
2664:
2659:
2654:
2648:
2643:
2636:
2632:
2627:
2623:
2618:
2613:
2607:
2602:
2595:
2591:
2586:
2579:
2576:
2574:
2568:
2562:
2558:
2553:
2548:
2544:
2539:
2534:
2530:
2525:
2520:
2516:
2511:
2507:
2506:
2455:
2452:
2398:inertial frame
2393:
2390:
2371:
2366:
2361:
2356:
2353:
2350:
2346:
2342:
2339:
2336:
2331:
2326:
2321:
2316:
2313:
2310:
2305:
2301:
2298:
2295:
2290:
2287:
2284:
2280:
2277:
2272:
2267:
2264:
2261:
2256:
2253:
2251:
2249:
2244:
2239:
2236:
2233:
2228:
2224:
2220:
2217:
2214:
2210:
2206:
2203:
2198:
2193:
2190:
2187:
2182:
2179:
2177:
2173:
2168:
2164:
2161:
2158:
2154:
2149:
2148:
2121:
2117:
2114:
2111:
2107:
2103:
2100:
2097:
2093:
2089:
2085:
2082:
2079:
2075:
2054:
2051:
2033:
2027:
2022:
2018:
2014:
2009:
2004:
2000:
1996:
1991:
1986:
1982:
1978:
1973:
1968:
1964:
1958:
1954:
1949:
1944:
1938:
1933:
1929:
1925:
1920:
1915:
1911:
1907:
1902:
1897:
1893:
1889:
1884:
1879:
1875:
1869:
1865:
1860:
1854:
1849:
1845:
1839:
1835:
1831:
1826:
1822:
1816:
1812:
1808:
1803:
1799:
1793:
1789:
1785:
1780:
1776:
1770:
1766:
1760:
1756:
1734:
1729:
1725:
1721:
1716:
1711:
1707:
1703:
1699:
1696:
1691:
1687:
1683:
1678:
1674:
1670:
1667:
1654:
1645:
1620:
1616:
1612:
1607:
1603:
1599:
1594:
1590:
1586:
1581:
1577:
1571:
1567:
1561:
1556:
1553:
1550:
1547:
1544:
1541:
1536:
1532:
1529:
1526:
1516:is defined as
1490:
1487:
1460:
1455:
1451:
1445:
1441:
1437:
1432:
1428:
1422:
1418:
1414:
1409:
1405:
1399:
1395:
1391:
1386:
1382:
1376:
1372:
1366:
1362:
1358:
1353:
1349:
1345:
1340:
1336:
1332:
1329:
1324:
1320:
1316:
1311:
1307:
1303:
1300:
1286:
1279:
1272:
1265:
1258:
1248:
1241:
1234:
1227:
1220:
1213:scalar product
1208:
1207:Scalar product
1205:
1195:
1192:
1170:
1169:
1166:
1163:
1152:
1151:
1148:
1117:scalar product
1115:, also called
1045:. A vector is
917:Main article:
914:
911:
899:Galison (1979)
861:
837:Poincaré group
761:
758:
694:
686:
681:
677:
673:
670:
667:
664:
661:
656:
652:
648:
643:
639:
635:
630:
626:
577:imaginary unit
567:speed of light
547:Henri Poincaré
536:
533:
530:
529:
527:
526:
519:
512:
504:
501:
500:
498:
497:
484:
469:
466:
465:
461:
460:
455:
450:
445:
440:
435:
430:
425:
419:
418:
415:
414:
411:
410:
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294:
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279:
276:
272:Galilean group
268:Poincaré group
244:non-degenerate
233:Poincaré group
229:Lorentz boosts
205:isometry group
167:Henri Poincaré
124:. It combines
26:
9:
6:
4:
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2:
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22098:collaboration
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22064:Event horizon
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22008:ADM formalism
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21913:Biquaternions
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21840:Time dilation
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21793:Proper length
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21461:(1905–1906),
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21258:(1907–1908),
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14479:
14471:
14466:
14462:
14454:
14444:
14436:
14431:
14427:
14420:
14416:
14412:
14389:
14375:
14369:
14353:
14350:
14346:
14341:
14333:
14327:
14322:
14307:
14297:
14292:
14289:
14285:
14262:
14258:
14254:
14251:
14246:
14241:
14235:
14231:
14227:
14223:
14218:
14215:
14212:
14207:
14202:
14196:
14192:
14188:
14184:
14179:
14169:
14163:
14158:
14141:
14138:
14130:
14124:
14119:
14115:
14099:
14086:
14082:
14073:
14063:
14055:
14050:
14046:
14033:
14029:
14025:
14019:
14011:
14001:
13993:
13988:
13984:
13976:
13966:
13958:
13953:
13949:
13942:
13938:
13934:
13923:
13920:
13914:
13898:
13895:
13891:
13881:
13875:
13862:
13854:
13850:
13846:
13839:
13829:
13821:
13816:
13812:
13805:
13802:
13782:
13777:
13773:
13769:
13766:
13759:
13755:
13748:
13725:
13718:
13714:
13707:
13699:
13696:
13693:
13685:
13681:
13674:
13654:
13649:
13645:
13641:
13638:
13633:
13613:
13608:
13604:
13600:
13591:
13585:
13581:depending on
13579:
13575:but now with
13558:
13553:
13543:
13541:
13527:
13522:
13515:
13512:
13509:
13503:
13497:
13495:
13490:
13466:
13459:
13437:
13423:
13417:
13415:
13410:
13395:
13392:
13386:
13384:
13379:
13365:
13361:
13357:
13353:
13347:
13334:
13328:
13325:
13322:
13312:
13306:
13298:
13287:
13284:
13278:
13269:
13263:
13246:
13238:
13235:
13233:
13219:
13213:
13210:
13207:
13201:
13198:
13196:
13191:
13178:
13165:
13160:
13156:
13153:
13144:
13138:
13135:
13130:
13126:
13123:
13111:
13098:
13091:
13080:
13077:
13074:
13071:
13065:
13054:
13051:
13048:
13042:
13031:
13028:
13022:
13020:
13013:
13009:
13006:
12996:
12985:
12982:
12976:
12965:
12962:
12959:
12953:
12942:
12939:
12933:
12931:
12924:
12920:
12917:
12886:
12883:
12880:
12877:
12872:
12868:
12865:
12856:
12853:
12851:
12844:
12840:
12837:
12831:
12828:
12821:
12818:
12815:
12812:
12809:
12804:
12800:
12797:
12788:
12785:
12783:
12776:
12772:
12769:
12763:
12760:
12747:
12734:
12723:
12720:
12714:
12710:
12703:
12695:
12691:
12687:
12684:
12681:
12675:
12667:
12663:
12659:
12656:
12652:
12648:
12645:
12624:
12620:
12617:
12614:
12611:
12608:
12605:
12601:
12595:
12591:
12587:
12584:
12581:
12576:
12572:
12568:
12565:
12561:
12556:
12533:
12529:
12526:
12503:
12498:
12493:
12483:
12479:
12473:
12469:
12465:
12462:
12459:
12454:
12450:
12446:
12443:
12439:
12435:
12432:
12423:
12410:
12404:
12401:
12398:
12395:
12392:
12389:
12386:
12383:
12377:
12374:
12353:
12349:
12346:
12343:
12340:
12335:
12331:
12327:
12324:
12319:
12314:
12309:
12305:
12301:
12296:
12293:
12290:
12285:
12280:
12275:
12271:
12267:
12262:
12257:
12253:
12249:
12246:
12238:
12234:
12228:
12224:
12220:
12217:
12214:
12209:
12205:
12201:
12198:
12194:
12189:
12185:
12180:
12175:
12155:
12152:
12146:
12140:
12133:
12129:
12122:
12118:
12097:
12093:
12084:
12074:
12066:
12061:
12057:
12044:
12040:
12036:
12030:
12022:
12012:
12004:
11999:
11995:
11987:
11977:
11969:
11964:
11960:
11953:
11949:
11945:
11943:
11930:
11927:
11921:
11905:
11902:
11898:
11890:
11884:
11881:
11878:
11868:
11862:
11860:
11850:
11839:
11836:
11830:
11819:are given by
11814:
11807:
11803:
11799:
11793:
11786:
11779:
11775:
11769:
11768:for spheres.
11767:
11761:
11756:
11751:
11747:
11743:
11739:
11733:
11723:
11717:
11713:
11709:
11696:
11688:
11683:
11670:
11657:
11654:
11649:
11644:
11640:
11633:
11628:
11620:
11614:
11610:
11606:
11601:
11597:
11588:
11583:
11581:
11576:
11570:
11561:
11557:
11552:
11548:
11544:
11534:
11530:
11526:
11522:
11497:
11491:
11488:
11485:
11479:
11476:
11473:
11467:
11459:
11455:
11445:
11441:
11437:
11429:
11425:
11421:
11416:
11412:
11401:
11398:
11394:
11390:
11384:
11380:
11376:
11370:
11366:
11360:
11356:
11350:
11346:
11342:
11336:
11332:
11326:
11322:
11321:in general.)
11316:
11309:
11303:
11297:
11293:
11289:
11282:
11278:
11269:
11253:
11249:
11243:
11239:
11233:
11229:
11224:
11221:
11217:
11212:
11208:
11202:
11198:
11193:
11188:
11184:
11178:
11174:
11169:
11165:
11162:
11158:
11152:
11148:
11143:
11140:
11136:
11131:
11127:
11122:
11117:
11113:
11108:
11101:
11093:
11089:
11079:
11066:
11056:
11050:
11044:
11040:
11036:
11033:
11030:
11025:
11021:
11017:
11012:
11008:
11000:
10997:
10992:
10988:
10963:
10960:
10955:
10950:
10946:
10939:
10934:
10926:
10920:
10916:
10912:
10907:
10903:
10893:
10889:
10885:
10871:
10867:
10864:
10860:
10851:
10835:
10830:
10820:
10817:
10814:
10809:
10805:
10788:
10781:
10774:
10757:
10753:
10747:
10743:
10738:
10735:
10731:
10726:
10722:
10717:
10712:
10708:
10703:
10699:
10696:
10692:
10686:
10682:
10676:
10672:
10667:
10664:
10660:
10655:
10651:
10645:
10641:
10636:
10631:
10627:
10621:
10617:
10612:
10608:
10605:
10601:
10595:
10591:
10586:
10583:
10579:
10574:
10570:
10565:
10560:
10556:
10551:
10547:
10542:
10538:
10508:
10504:
10503:
10499:
10498:
10495:
10476:
10470:
10465:
10450:
10446:
10445:inclusion map
10428:
10425:
10422:
10404:
10398:
10393:
10383:
10380:
10369:Preliminaries
10366:
10353:
10350:
10343:of dimension
10342:
10323:
10317:
10309:
10290:
10284:
10275:
10256:
10252:
10247:
10243:
10240:
10232:
10228:
10224:
10203:
10197:
10192:
10177:
10161:
10153:
10137:
10134:
10129:
10125:
10121:
10113:
10107:
10102:
10098:
10089:
10085:
10069:
10066:
10063:
10041:
10038:
10035:
10007:
10003:
10000:
9997:
9994:
9991:
9986:
9982:
9978:
9973:
9968:
9963:
9959:
9955:
9950:
9947:
9944:
9939:
9934:
9929:
9925:
9921:
9916:
9911:
9907:
9901:
9897:
9893:
9888:
9878:
9874:
9868:
9864:
9860:
9857:
9854:
9849:
9845:
9841:
9838:
9835:
9831:
9826:
9822:
9814:
9808:
9803:
9789:
9773:
9770:
9767:
9764:
9742:
9739:
9736:
9726:
9718:
9712:
9707:
9691:
9677:
9669:
9648:
9624:
9595:
9580:
9575:
9573:
9569:
9565:
9561:
9557:
9552:
9546:
9536:
9534:
9533:tangent space
9530:
9526:
9521:
9519:
9518:
9513:
9509:
9505:
9501:
9497:
9487:
9486:(see below).
9485:
9481:
9477:
9471:
9465:
9460:
9456:
9455:String theory
9446:
9438:
9431:
9425:
9418:
9406:
9402:
9398:
9391:
9386:
9376:
9370:
9364:
9360:
9356:
9347:
9343:
9338:
9334:
9326:
9322:
9318:
9311:
9307:
9303:
9299:
9295:
9286:
9281:
9280:inner product
9276:
9270:
9264:
9260:
9256:
9252:
9248:
9244:
9243:four-momentum
9240:
9239:four-velocity
9236:
9231:
9227:
9220:
9209:
9197:
9193:
9183:
9181:
9176:
9171:
9168:. Since this
9165:
9161:
9157:
9153:
9146:
9141:
9136:
9132:
9122:
9118:
9113:
9108:
9104:
9098:
9094:
9090:
9086:
9082:
9078:
9073:
9068:
9064:
9058:
9054:
9050:
9046:
9045:
9044:
9041:
9037:
9033:
9020:
9014:
9010:
9006:
9001:
8991:
8987:
8983:
8966:
8961:
8957:
8951:
8947:
8941:
8938:
8930:
8924:
8921:
8915:
8912:
8909:
8901:
8896:
8890:
8886:
8882:
8872:
8867:
8861:
8857:
8853:
8843:
8840:
8832:
8826:
8818:
8813:
8809:
8795:
8790:
8784:
8767:
8762:
8758:
8752:
8748:
8742:
8739:
8735:
8731:
8725:
8722:
8719:
8711:
8707:
8703:
8700:
8695:
8691:
8687:
8682:
8679:
8675:
8665:
8659:
8653:
8650:
8646:
8640:
8636:
8616:
8610:
8604:
8598:
8595:
8589:
8586:
8583:
8577:
8574:
8571:
8547:
8542:
8537:
8533:
8529:
8525:
8520:
8515:
8507:
8503:
8486:
8480:
8475:
8469:
8465:
8461:
8448:
8442:
8437:
8429:
8424:
8413:
8408:
8398:
8385:
8380:
8375:
8371:
8367:
8359:
8355:
8344:
8340:
8330:
8326:
8318:
8314:
8301:
8295:
8291:
8287:
8279:
8274:
8270:
8264:
8260:
8253:
8248:
8227:
8211:
8208:
8205:
8202:
8196:
8190:
8187:
8175:
8166:
8160:
8153:
8149:
8145:
8139:
8126:
8102:
8097:
8092:
8084:
8080:
8058:
8054:
8052:
8048:
8044:
8036:
8024:
8011:
8006:
8001:
7988:
7983:
7979:
7975:
7970:
7966:
7962:
7957:
7954:
7950:
7946:
7941:
7937:
7933:
7930:
7925:
7921:
7917:
7912:
7909:
7905:
7901:
7896:
7892:
7888:
7885:
7880:
7876:
7872:
7867:
7864:
7860:
7845:
7842:
7832:
7829:
7823:
7804:
7801:
7797:
7770:
7762:
7759:
7755:
7751:
7745:
7737:
7734:
7730:
7723:
7720:
7714:
7711:
7708:
7700:
7697:
7693:
7689:
7676:
7672:
7668:
7663:
7659:
7655:
7650:
7647:
7643:
7622:
7619:
7611:
7607:
7603:
7598:
7595:
7591:
7568:
7564:
7543:
7533:
7519:
7496:
7493:
7490:
7484:
7478:
7456:
7452:
7448:
7442:
7439:
7436:
7430:
7421:
7408:
7402:
7399:
7396:
7390:
7384:
7381:
7376:
7372:
7365:
7362:
7356:
7353:
7350:
7344:
7324:
7296:
7293:
7290:
7287:
7284:
7270:
7268:
7264:
7260:
7255:
7253:
7249:
7244:
7242:
7237:
7233:
7228:
7226:
7222:
7217:
7214:, which is a
7213:
7208:
7205:
7199:
7192:
7188:
7184:
7178:
7173:
7168:
7163:
7159:
7154:
7147:
7143:
7139:
7135:
7131:
7125:
7119:
7117:
7113:
7105:
7093:
7081:
7080:inner product
7077:
7073:
7070:
7065:
7059:
7053:
7047:
7041:
7035:
7029:
7024:
7020:
7010:
6990:
6985:
6958:
6954:
6947:
6943:
6939:
6935:
6912:
6907:
6897:
6893:
6889:
6885:
6881:
6877:
6854:
6849:
6839:
6835:
6831:
6827:
6823:
6819:
6804:
6796:
6788:
6780:
6769:
6762:
6755:
6748:
6741:
6728:
6722:
6719:
6716:
6710:
6707:
6701:
6695:
6692:
6683:
6674:
6668:
6647:
6642:
6639:
6635:
6631:
6626:
6621:
6611:
6608:
6604:
6598:
6593:
6562:
6557:
6532:
6527:
6524:
6520:
6516:
6508:
6504:
6500:
6495:
6491:
6484:
6475:
6473:
6468:
6455:
6450:
6446:
6440:
6436:
6432:
6427:
6423:
6417:
6413:
6409:
6404:
6400:
6394:
6390:
6386:
6381:
6377:
6371:
6367:
6363:
6358:
6354:
6348:
6344:
6340:
6335:
6331:
6325:
6321:
6315:
6312:
6308:
6304:
6298:
6295:
6292:
6286:
6266:
6261:
6257:
6251:
6247:
6243:
6238:
6234:
6228:
6224:
6220:
6215:
6211:
6205:
6201:
6197:
6192:
6188:
6182:
6178:
6174:
6169:
6165:
6159:
6155:
6151:
6146:
6142:
6136:
6132:
6128:
6123:
6119:
6113:
6109:
6103:
6100:
6096:
6092:
6086:
6083:
6080:
6074:
6066:
6055:
6048:
6041:
6034:
6030:
6013:
6005:
6000:
5993:
5989:
5986:
5982:
5977:
5971:
5967:
5963:
5959:
5952:
5946:
5933:
5928:
5925:
5921:
5917:
5909:
5905:
5901:
5896:
5892:
5885:
5876:
5863:
5859:
5856:
5853:
5833:
5830:
5822:
5818:
5814:
5809:
5805:
5798:
5778:
5775:
5767:
5763:
5759:
5754:
5750:
5743:
5740:
5732:
5728:
5724:
5719:
5715:
5708:
5705:
5697:
5693:
5689:
5684:
5680:
5673:
5670:
5662:
5658:
5654:
5649:
5645:
5638:
5635:
5622:
5615:
5608:
5601:
5589:
5586:
5579:
5573:
5568:
5558:
5544:
5538:
5535:
5532:
5526:
5523:
5520:
5517:
5514:
5494:
5489:
5482:
5479:
5474:
5469:
5464:
5457:
5452:
5449:
5444:
5439:
5432:
5427:
5422:
5419:
5414:
5407:
5402:
5397:
5392:
5385:
5381:
5378:
5350:
5347:
5344:
5324:
5321:
5314:
5299:
5295:
5292:
5289:
5286:
5278:
5260:
5256:
5250:
5246:
5242:
5237:
5233:
5227:
5223:
5219:
5214:
5210:
5204:
5200:
5196:
5191:
5187:
5181:
5177:
5171:
5167:
5163:
5160:
5157:
5154:
5146:
5128:
5124:
5120:
5115:
5111:
5107:
5102:
5098:
5094:
5089:
5085:
5079:
5075:
5065:
5051:
5046:
5041:
5035:
5031:
5027:
5022:
5018:
5013:
5008:
5003:
4998:
4992:
4988:
4984:
4979:
4975:
4970:
4965:
4960:
4955:
4949:
4945:
4941:
4936:
4932:
4927:
4922:
4917:
4912:
4906:
4902:
4898:
4893:
4889:
4884:
4877:
4873:
4856:
4852:
4845:
4835:
4829:
4825:
4821:
4816:
4814:
4810:
4805:
4803:
4795:
4789:
4785:
4781:
4776:
4767:
4765:
4759:
4755:
4751:
4744:
4740:
4736:
4729:
4725:
4721:
4716:
4711:
4705:
4699:
4696:
4683:
4679:
4675:
4670:
4666:
4662:
4658:
4653:
4649:
4648:inner product
4645:
4626:
4623:
4620:
4608:
4605:
4602:
4595:
4592:
4589:
4583:
4579:
4576:
4570:
4563:
4560:
4543:
4539:
4536:
4530:
4527:
4521:
4517:
4514:
4508:
4501:
4498:
4479:
4476:
4469:
4466:
4463:
4460:
4456:
4453:
4446:
4440:
4436:
4433:
4427:
4424:
4418:
4414:
4411:
4405:
4402:
4399:
4393:
4389:
4386:
4383:
4380:
4377:
4371:
4364:
4361:
4360:
4359:
4357:
4341:
4336:
4332:
4328:
4323:
4315:
4309:
4303:
4300:
4297:
4291:
4288:
4285:
4282:
4279:
4271:
4267:
4266:inner product
4263:
4258:
4238:
4234:
4231:
4225:
4222:
4219:
4216:
4213:
4204:
4193:
4188:
4175:
4160:
4156:
4150:
4144:
4140:
4136:
4130:
4124:
4118:
4114:
4110:
4106:
4102:
4098:, vectors in
4095:
4091:
4086:
4082:
4073:
4065:
4055:
4051:
4047:
4043:
4039:
4035:
4034:inner product
4029:
4025:
4015:
4013:
4009:
4004:
3987:
3984:
3981:
3978:
3975:
3972:
3969:
3966:
3963:
3960:
3957:
3946:
3941:
3938:
3934:
3911:
3908:
3905:
3889:
3867:
3864:
3860:
3856:
3851:
3838:
3833:
3830:
3827:
3797:
3794:
3791:
3785:
3782:
3757:
3753:
3740:
3721:
3717:
3713:
3708:
3704:
3694:
3684:
3659:
3656:
3653:
3650:
3647:
3638:
3632:
3630:
3625:
3619:
3613:
3608:
3597:
3591:
3586:
3581:
3578:
3573:
3569:
3560:
3550:
3546:
3542:
3536:
3532:
3531:metric tensor
3527:
3523:
3519:
3514:
3508:
3500:
3488:
3486:
3482:
3474:
3470:
3465:
3461:
3455:
3450:
3446:
3442:
3438:
3434:
3433:metric tensor
3430:
3426:
3422:
3418:
3412:
3399:
3395:
3377:
3375:
3371:
3367:
3357:
3355:
3347:
3336:
3333:
3316:
3311:
3307:
3301:
3294:
3284:
3282:
3274:
3270:
3266:
3258:
3253:
3249:
3243:
3236:
3226:
3224:
3216:
3212:
3208:
3194:
3189:
3184:
3180:
3170:
3161:
3156:
3146:
3135:
3133:
3128:
3123:
3119:
3115:
3111:
3106:
3100:
3094:
3081:
3072:
3065:
3058:
3051:
3044:
3037:
3033:
3028:
3016:
2999:
2994:
2986:
2982:
2965:
2960:
2955:
2950:
2913:
2908:
2902:
2890:
2886:
2877:
2872:
2867:
2861:
2849:
2845:
2836:
2831:
2826:
2820:
2808:
2804:
2795:
2790:
2785:
2779:
2767:
2763:
2749:
2739:
2734:
2728:
2716:
2712:
2703:
2698:
2693:
2687:
2675:
2671:
2662:
2657:
2652:
2646:
2634:
2630:
2621:
2616:
2611:
2605:
2593:
2589:
2575:
2566:
2560:
2556:
2551:
2546:
2542:
2537:
2532:
2528:
2523:
2518:
2514:
2509:
2496:
2492:
2484:of spacetime.
2483:
2479:
2474:
2469:
2460:
2451:
2437:
2433:
2429:
2428:tangent space
2425:
2421:
2418:
2415:-dimensional
2409:
2407:
2403:
2399:
2389:
2386:
2369:
2364:
2359:
2351:
2344:
2337:
2329:
2324:
2319:
2311:
2303:
2296:
2285:
2278:
2275:
2270:
2262:
2254:
2252:
2242:
2234:
2226:
2222:
2218:
2215:
2212:
2208:
2204:
2201:
2196:
2188:
2180:
2178:
2171:
2162:
2159:
2156:
2137:
2135:
2119:
2112:
2105:
2098:
2091:
2083:
2080:
2077:
2050:
2047:
2031:
2025:
2020:
2016:
2012:
2007:
2002:
1998:
1994:
1989:
1984:
1980:
1976:
1971:
1966:
1962:
1956:
1952:
1947:
1942:
1936:
1931:
1927:
1923:
1918:
1913:
1909:
1905:
1900:
1895:
1891:
1887:
1882:
1877:
1873:
1867:
1863:
1858:
1852:
1847:
1843:
1837:
1833:
1829:
1824:
1820:
1814:
1810:
1806:
1801:
1797:
1791:
1787:
1783:
1778:
1774:
1768:
1764:
1758:
1754:
1727:
1723:
1709:
1705:
1697:
1689:
1685:
1681:
1676:
1672:
1665:
1653:
1644:
1639:
1635:
1618:
1614:
1610:
1605:
1601:
1597:
1592:
1588:
1584:
1579:
1575:
1569:
1565:
1559:
1551:
1548:
1545:
1539:
1534:
1527:
1513:
1509:
1505:
1501:
1497:
1486:
1482:
1479:
1475:
1471:
1458:
1453:
1449:
1443:
1439:
1435:
1430:
1426:
1420:
1416:
1412:
1407:
1403:
1397:
1393:
1389:
1384:
1380:
1374:
1370:
1364:
1360:
1356:
1351:
1347:
1343:
1338:
1334:
1330:
1322:
1318:
1314:
1309:
1305:
1298:
1285:
1278:
1271:
1264:
1257:
1247:
1240:
1233:
1226:
1219:
1214:
1204:
1202:
1191:
1189:
1188:Vector fields
1185:
1183:
1178:
1173:
1167:
1164:
1157:
1156:
1155:
1149:
1146:
1145:
1144:
1141:
1139:
1136:, there is a
1134:
1129:
1125:
1120:
1118:
1112:
1108:
1104:
1098:
1094:
1091:
1086:
1082:
1077:
1073:
1070:
1065:
1060:
1056:
1053:
1048:
1043:
1039:
1036:
1029:
1025:
1021:
1017:
1013:
1009:
1005:
995:
990:
985:
979:
973:
969:is velocity,
967:
958:
954:
950:
946:
945:absolute past
942:
938:
934:
933:causal future
930:
925:
920:
910:
908:
907:Walter (1999)
904:
900:
894:
892:
888:
884:
880:
874:
872:
860:
855:
852:
850:
846:
842:
838:
834:
830:
826:
822:
817:
815:
811:
810:orthogonality
807:
802:
800:
796:
792:
788:
782:
778:
774:
770:
757:
755:
748:
744:
740:
736:
730:
726:
721:
719:
715:
711:
710:Lorentz boost
705:
692:
684:
679:
671:
668:
665:
659:
654:
650:
646:
641:
637:
633:
628:
624:
615:
609:
605:
601:
597:
590:
587:
582:
578:
573:
568:
563:
557:
552:
548:
542:
525:
520:
518:
513:
511:
506:
505:
503:
502:
495:
485:
482:
477:
471:
470:
468:
467:
459:
456:
454:
451:
449:
446:
444:
441:
439:
436:
434:
431:
429:
426:
424:
421:
420:
413:
412:
404:
401:
399:
396:
395:
388:
387:
379:
376:
374:
371:
369:
366:
365:
358:
357:
349:
346:
344:
341:
339:
336:
334:
331:
330:
323:
322:
316:
313:
311:
308:
307:
305:
304:
300:
296:
295:
292:
289:
288:
284:
283:
275:
273:
269:
264:
260:
256:
252:
249:, called the
248:
247:bilinear form
245:
240:
238:
237:time dilation
234:
230:
226:
222:
218:
214:
210:
206:
202:
197:
195:
191:
190:time dilation
187:
183:
179:
175:
170:
168:
164:
160:
156:
152:
148:
143:
141:
137:
134:
130:
127:
123:
119:
113:
59:
55:
51:
43:
39:
33:
19:
22503:
22197:Kaluza–Klein
21949:Introduction
21917:
21875:Twin paradox
21633:
21605:
21601:Weinberg, S.
21582:
21553:
21530:
21470:
21466:
21435:
21401:(1): 20–63.
21398:
21392:
21369:. New York:
21366:
21346:
21320:
21316:
21285:
21279:
21265:
21237:
21218:
21199:
21176:
21169:Landau, L.D.
21145:
21137:Kleppner, D.
21101:
21062:
21029:
21025:
21004:
20999:, p. 33
20992:
20987:, p. 66
20980:
20971:
20952:
20941:
20929:
20913:
20886:
20881:, p. 71
20874:
20854:
20847:
20826:. Retrieved
20806:
20799:
20790:
20778:
20773:, p. 15
20766:
20757:
20748:
20739:
20727:
20719:
20685:
20670:
20647:
20640:
20635:, p. 31
20628:
20618:
20603:
20580:
20576:
20572:
20568:
20551:
20546:
20529:
20512:
20503:
20497:
20488:
20478:
20465:
20127:
19488:
19485:
19063:
18753:
18731:
18724:
18721:
18465:
17806:
16994:
16377:
15755:
14802:
14579:
14277:and the map
14106:
13879:
13876:
13589:
13583:
13577:
13363:
13359:
13355:
13351:
13348:
13267:
13264:
13179:
13112:
12748:
12424:
12160:
12150:
12144:
12138:
12131:
12127:
12120:
12116:
11812:
11805:
11801:
11797:
11791:
11784:
11777:
11773:
11770:
11759:
11754:
11749:
11745:
11741:
11737:
11731:
11721:
11715:
11711:
11707:
11694:
11692:
11584:
11579:
11574:
11568:
11556:differential
11550:
11546:
11542:
11532:
11528:
11524:
11520:
11443:
11439:
11435:
11427:
11423:
11419:
11414:
11410:
11402:
11396:
11392:
11388:
11382:
11378:
11374:
11368:
11364:
11358:
11354:
11348:
11344:
11340:
11334:
11330:
11324:
11323:
11314:
11307:
11301:
11295:
11291:
11287:
11280:
11276:
11080:
10891:
10887:
10883:
10869:
10868:
10862:
10858:
10786:
10779:
10772:
10506:
10505:
10372:
10273:
9692:
9667:
9578:
9576:
9572:submanifolds
9568:metric space
9556:model spaces
9555:
9550:
9548:
9522:
9517:curved space
9515:
9495:
9493:
9478:
9463:
9453:dimensions.
9444:
9436:
9429:
9423:
9416:
9412:
9400:
9396:
9389:
9374:
9368:
9362:
9358:
9354:
9345:
9341:
9324:
9320:
9316:
9309:
9305:
9301:
9297:
9293:
9284:
9277:
9268:
9262:
9259:pseudoscalar
9235:four-vectors
9229:
9225:
9218:
9215:
9199:
9174:
9163:
9159:
9155:
9151:
9144:
9139:
9134:
9130:
9127:
9120:
9116:
9106:
9102:
9096:
9092:
9088:
9080:
9076:
9066:
9062:
9056:
9052:
9048:
9039:
9035:
9031:
9028:
9012:
9008:
9004:
9002:. Moreover,
8989:
8985:
8981:
8814:
8810:
8793:
8788:
8782:
8663:
8657:
8654:
8648:
8644:
8638:
8634:
8440:
8435:
8427:
8422:
8411:
8406:
8399:
8277:
8272:
8268:
8262:
8258:
8251:
8246:
8226:vector field
8173:
8164:
8161:
8151:
8147:
8143:
8137:
8056:
8055:
8046:
8043:line element
8009:
8002:
7851:
7840:
7838:
7827:
7821:
7534:
7422:
7276:
7256:
7245:
7229:
7224:
7209:
7203:
7197:
7190:
7186:
7182:
7176:
7171:
7166:
7161:
7157:
7152:
7145:
7141:
7137:
7133:
7129:
7123:
7120:
7085:
7063:
7057:
7051:
7049:and vectors
7045:
7039:
7033:
6802:
6794:
6786:
6778:
6767:
6760:
6753:
6746:
6742:
6672:
6666:
6476:
6471:
6469:
6067:
6065:is given by
6056:
6046:
6039:
6032:
6028:
6011:
6003:
5998:
5991:
5987:
5984:
5980:
5969:
5965:
5961:
5957:
5954:are written
5950:
5947:
5877:
5620:
5613:
5606:
5599:
5595:
5584:
5577:
5571:
5559:
5337:where is a
5066:
4848:
4827:
4817:
4807:For a given
4806:
4787:
4783:
4779:
4773:is called a
4768:
4757:
4753:
4749:
4742:
4738:
4734:
4727:
4723:
4719:
4709:
4703:
4701:Two vectors
4700:
4694:
4677:
4673:
4671:
4669:is allowed.
4664:
4660:
4656:
4651:
4643:
4641:
4355:
4269:
4261:
4256:
4205:
4191:
4176:
4158:
4154:
4148:
4142:
4138:
4134:
4128:
4122:
4116:
4112:
4108:
4104:
4100:
4093:
4089:
4084:
4080:
4071:
4063:
4053:
4049:
4045:
4031:
4005:
3887:
3738:
3692:
3688:(−, +, +, +)
3682:
3636:
3633:
3623:
3617:
3611:
3589:
3582:
3576:
3567:
3558:
3548:
3544:
3540:
3534:
3525:
3515:
3506:
3496:
3480:
3472:
3463:
3459:
3453:
3436:
3428:
3424:
3420:
3410:
3383:
3369:
3365:
3363:
3345:
3339:
3334:
3192:
3187:
3182:
3178:
3168:
3159:
3154:
3144:
3141:
3131:
3126:
3121:
3117:
3104:
3098:
3095:
2487:
2481:
2477:
2472:
2431:
2420:vector space
2410:
2406:affine space
2401:
2395:
2387:
2138:
2056:
2048:
1651:
1642:
1637:
1636:
1511:
1507:
1503:
1499:
1495:
1492:
1483:
1473:
1472:
1283:
1276:
1269:
1262:
1255:
1245:
1238:
1231:
1224:
1217:
1210:
1200:
1197:
1186:
1181:
1174:
1171:
1160:(0, 0, 0, 0)
1153:
1142:
1132:
1124:null vectors
1121:
1110:
1106:
1102:
1096:
1092:
1089:
1084:
1080:
1075:
1071:
1068:
1063:
1058:
1054:
1051:
1046:
1041:
1037:
1034:
1027:
1023:
1019:
1015:
1011:
1007:
1003:
993:
983:
977:
971:
965:
962:
948:
944:
940:
936:
932:
903:Corry (1997)
896:
876:
868:
857:
853:
844:
840:
818:
813:
806:line element
805:
803:
798:
794:
787:vector space
780:
776:
772:
768:
763:
746:
742:
738:
734:
722:
713:
706:
607:
603:
599:
595:
591:
571:
561:
555:
544:
443:Curved space
347:
258:
254:
250:
241:
225:translations
198:
176:theories of
171:
144:
57:
53:
47:
22286:Kerr–Newman
22257:Spherical:
22126:Other tests
22069:Singularity
22001:Formulation
21963:Fundamental
21817:Formulation
21798:Proper time
21759:Fundamental
21473:: 129–176,
21347:Gravitation
21151:McGraw-Hill
20894:, p. 4
20890:Minkowski,
20608:"Minkowski"
20564:Nash (1956)
20541:in general.
16598:from which
13795:leading to
11540:reduces to
11270:of the map
11268:pushforward
11061: times
9788:hyperboloid
9504:gravitation
9387:satisfying
9247:orientation
8276:, called a
7076:hyperplanes
4775:unit vector
4040:, i.e. the
3640:, that is,
3493:Terminology
3449:dot product
1177:orthonormal
955:, and from
953:Sard (1970)
941:causal past
883:gravitation
825:proper time
553:coordinate
423:Four-vector
221:reflections
122:gravitation
22534:Categories
22438:Zel'dovich
22346:Scientists
22325:Alcubierre
22132:of Mercury
22130:precession
22059:Black hole
21942:Background
21934:relativity
21903:World line
21898:Light cone
21723:Background
21715:relativity
21705:Relativity
21669:PhilPapers
21281:The Monist
21149:. London:
21017:References
20828:2022-12-26
20535:flat space
20442:Hyperspace
19806:here
13877:With this
9172:varies as
9170:hyperplane
8979:where now
8123:gives the
7241:Lee (2003)
7216:hyperplane
5974:where the
5628:such that
4715:orthogonal
4644:difference
4162:with both
4064:indefinite
3481:indefinite
3425:coordinate
3118:definition
1182:null basis
1128:light cone
929:light cone
845:derivation
791:light-cone
539:See also:
174:Einstein's
22408:Robertson
22393:Friedmann
22388:Eddington
22378:de Sitter
22212:Solutions
22090:detectors
22085:astronomy
22052:Phenomena
21987:Geodesics
21890:Spacetime
21833:Phenomena
21503:120211823
21175:(2002) .
21143:(1978) .
21093:1660-8046
21046:0003-9519
20879:Sard 1970
20471:invariant
20327:−
20258:τ
20249:−
20199:∑
20179:−
20174:∗
20170:σ
20154:−
20149:∗
20145:σ
20136:η
20074:−
20054:⟩
20037:⟨
19987:−
19918:τ
19903:⟩
19886:⟨
19880:≡
19840:∑
19760:−
19740:⟩
19723:⟨
19694:−
19668:−
19597:−
19539:∂
19535:∂
19466:τ
19463:∂
19459:∂
19454:τ
19432:∂
19428:∂
19404:τ
19401:∂
19397:∂
19381:∂
19375:τ
19372:∂
19341:∂
19337:∂
19321:∂
19306:∂
19284:τ
19281:∂
19277:∂
19259:∂
19254:τ
19251:∂
19219:∂
19215:∂
19197:∂
19182:∂
19150:∂
19146:∂
19129:∗
19116:−
19112:σ
19095:∗
19082:−
19078:σ
19037:∗
19024:−
19020:σ
19002:∗
18989:−
18985:σ
18936:η
18920:∗
18907:−
18903:σ
18885:∗
18872:−
18868:σ
18785:∗
18772:−
18768:σ
18677:≡
18641:−
18583:⋯
18497:∗
18484:−
18480:σ
18416:−
18378:⋅
18341:−
18291:−
18258:⋅
18220:−
18208:τ
18201:−
18193:yielding
18151:−
18066:∑
18059:τ
17973:τ
17970:∂
17966:∂
17855:∂
17851:∂
17829:∑
17822:τ
17811:one gets
17753:−
17722:⋅
17649:−
17591:⋯
17519:−
17440:−
17402:⋅
17344:⋅
17322:⋅
17286:−
17214:⋯
17142:−
17052:⋯
16945:−
16906:⋅
16815:⋅
16777:−
16682:−
16550:−
16519:⋅
16444:−
16352:τ
16322:−
16278:τ
16275:∂
16271:∂
16196:−
16097:−
16043:∂
16039:∂
15981:−
15895:−
15828:−
15774:∂
15770:∂
15706:≡
15670:−
15612:⋯
15526:∗
15513:−
15509:σ
15476:⋯
15442:−
15362:τ
15349:⋯
15316:−
15233:⋮
15216:τ
15187:−
15143:τ
15140:∂
15136:∂
15092:−
15038:∂
15034:∂
15026:⋯
14984:−
14930:∂
14926:∂
14891:−
14764:τ
14752:−
14705:⋯
14621:η
14616:∗
14603:−
14599:σ
14535:−
14472:−
14376:τ
14351:−
14347:σ
14313:→
14290:−
14286:σ
14259:τ
14252:−
14216:⋯
14142:η
14056:−
13994:−
13921:τ
13896:−
13892:σ
13822:−
13803:λ
13770:−
13756:λ
13715:λ
13700:λ
13697:−
13675:−
13642:−
13605:τ
13601:−
13554:λ
13523:λ
13516:λ
13513:−
13491:τ
13479:one gets
13424:τ
13329:τ
13285:τ
13279:σ
13239:λ
13208:τ
13202:λ
13161:→
13145:τ
13139:λ
13131:→
13072:τ
13049:−
13043:−
13029:τ
13014:→
12960:−
12954:−
12925:→
12884:−
12873:→
12860:⇒
12845:→
12816:−
12805:→
12792:⇒
12777:→
12715:≡
12685:…
12615:τ
12606:∈
12585:…
12566:τ
12534:→
12484:∈
12463:…
12444:τ
12399:…
12384:−
12344:τ
12328:−
12294:⋯
12254:τ
12250:−
12239:⊂
12218:…
12199:τ
12067:−
12005:−
11928:τ
11903:−
11899:σ
11885:τ
11837:τ
11831:σ
11705:is a map
11650:∗
11637:→
11629:∗
11607::
11602:∗
11489:∘
11460:∗
11234:∗
11222:…
11203:∗
11179:∗
11166:α
11141:…
11102:α
11094:∗
11051:⏟
11045:∗
11037:⊗
11034:⋯
11031:⊗
11026:∗
11018:⊗
11013:∗
10943:→
10913::
10908:∗
10821:α
10815:α
10810:∗
10806:ι
10736:…
10700:α
10677:∗
10673:ι
10665:…
10646:∗
10642:ι
10622:∗
10618:ι
10609:α
10584:…
10548:α
10543:∗
10539:ι
10522:of order
10413:→
10381:ι
10241:−
10162:η
10135:η
10130:∗
10126:ι
9951:−
9948:⋯
9945:−
9917:−
9879:∈
9858:…
9786:) of the
9727:⊂
9678:η
9652:¯
9490:Curvature
8962:ν
8952:μ
8942:ν
8939:μ
8891:ν
8862:μ
8844:ν
8841:μ
8763:ν
8753:μ
8743:ν
8740:μ
8736:η
8712:ν
8701:⊗
8696:μ
8683:ν
8680:μ
8676:η
8611:β
8599:α
8578:β
8575:⊗
8572:α
8543:μ
8538:ν
8534:δ
8508:ν
8500:∂
8496:∂
8470:μ
8381:μ
8376:ν
8372:δ
8360:ν
8352:∂
8345:μ
8337:∂
8319:ν
8311:∂
8307:∂
8296:μ
8224:for each
8085:μ
8077:∂
8073:∂
7984:ν
7971:μ
7958:ν
7955:μ
7951:η
7942:ν
7931:⊙
7926:μ
7913:ν
7910:μ
7906:η
7897:ν
7886:⊗
7881:μ
7868:ν
7865:μ
7861:η
7802:−
7798:η
7771:β
7760:−
7756:η
7746:α
7735:−
7731:η
7724:η
7715:β
7709:α
7698:−
7694:η
7682:→
7677:∗
7669:×
7664:∗
7648:−
7644:η
7617:→
7612:∗
7596:−
7592:η
7569:∗
7520:η
7485:η
7482:↦
7457:∗
7449:∈
7443:⋅
7431:η
7403:⋅
7391:η
7388:↦
7377:∗
7369:→
7357:−
7351:⋅
7345:η
7325:η
7300:→
7294:×
7285:η
6991:μ
6986:ν
6982:Λ
6959:μ
6951:Λ
6940:μ
6913:ν
6908:μ
6904:Λ
6898:ν
6882:μ
6855:ν
6850:μ
6846:Λ
6840:ν
6824:μ
6711:η
6699:Λ
6690:Λ
6684:η
6643:σ
6640:ρ
6636:η
6627:ν
6622:σ
6618:Λ
6612:ν
6609:μ
6605:η
6599:μ
6594:ρ
6590:Λ
6577:, a real
6563:μ
6558:ν
6554:Λ
6528:ν
6525:μ
6521:η
6509:ν
6496:μ
6485:η
6451:μ
6441:μ
6336:ν
6326:μ
6316:ν
6313:μ
6309:η
6287:η
6262:μ
6252:μ
6239:μ
6229:μ
6124:ν
6114:μ
6104:ν
6101:μ
6097:η
6075:η
5929:ν
5926:μ
5922:η
5910:ν
5897:μ
5886:η
5860:ν
5857:≠
5854:μ
5823:ν
5810:μ
5799:η
5744:η
5709:η
5674:η
5639:η
5636:−
5562:(− + + +)
5527:η
5518:⋅
5480:−
5450:−
5420:−
5379:η
5348:×
5315:η
5290:⋅
5243:−
5220:−
5197:−
5158:⋅
5121:−
5108:−
5095:−
5028:−
5009:−
4985:−
4966:−
4942:−
4923:−
4899:−
4849:From the
4769:A vector
4615:⇒
4606:∈
4600:∀
4571:η
4531:η
4509:η
4480:∈
4474:∀
4464:∈
4451:∀
4428:η
4406:η
4372:η
4329:≡
4320:‖
4313:‖
4310:≡
4292:η
4283:⋅
4226:η
4217:⋅
4203:-vector.
4187:4-vectors
4185:, called
3958:−
3942:ν
3939:μ
3935:η
3868:ν
3865:μ
3861:η
3798:η
3758:μ
3722:∗
3714:⊗
3709:∗
3663:→
3657:×
3648:η
3596:signature
3477:(− + + +)
3406:(+ − − −)
3402:(− + + +)
3390:(+ − − −)
3386:(− + + +)
3366:spacelike
3312:ν
3302:ν
3295:μ
3291:Λ
3275:μ
3254:ν
3244:ν
3237:μ
3233:Λ
3217:μ
2987:μ
2979:∂
2975:∂
2951:μ
2752:↔
2578:↔
2495:Lee (2012
2491:Lee (2003
2444:(− + + +)
2440:(+ − − −)
2255:≥
2092:≥
2013:−
1995:−
1977:−
1924:−
1906:−
1888:−
1830:−
1807:−
1784:−
1666:η
1611:−
1598:−
1585:−
1540:η
1436:−
1413:−
1390:−
1344:⋅
1299:η
1180:called a
1162:(origin),
1138:worldline
1085:lightlike
1064:spacelike
989:Cartesian
949:elsewhere
841:following
814:normality
795:spacelike
551:spacetime
291:Spacetime
217:rotations
211:) is the
136:manifolds
118:spacetime
22545:Geometry
22521:Category
22398:Lemaître
22363:Einstein
22353:Poincaré
22313:Others:
22297:Taub–NUT
22263:interior
22185:theories
22183:Advanced
22150:redshift
21965:concepts
21783:Rapidity
21761:concepts
21603:(2002),
21389:Nash, J.
21344:(1973),
21268:: 53–111
21122:27757388
21054:27016039
21009:Lee 1997
20997:Lee 1997
20985:Lee 1997
20961:Archived
20946:Lee 2003
20934:Lee 2003
20918:Lee 2003
20837:Archived
20822:Archived
20783:Lee 2003
20771:Lee 1997
20633:Lee 1997
20611:Archived
20539:manifold
20431:See also
20128:Lastly,
15756:One has
14107:One has
12749:One has
12367:and let
12142:and the
11587:dual map
10877:-tensor
10449:pullback
10310:, while
10152:pullback
9551:geometry
9539:Geometry
9529:manifold
9459:M-theory
9366:, where
9337:electron
9314:for any
9241:and the
9154: :
9128:Suppose
9043:. Here,
8994:, i.e.,
8792:are the
8560:at each
8045:. It is
7841:formally
7121:Thus if
7090:and the
6944:′
6886:′
6828:′
6026:-vector
6018:-vector
5275:via the
4667:) < 0
4499:Symmetry
4072:constant
3501:of type
3429:explicit
3370:timelike
3271:′
3213:′
2355:‖
2349:‖
2341:‖
2335:‖
2315:‖
2309:‖
2300:‖
2294:‖
2289:‖
2283:‖
2266:‖
2260:‖
2238:‖
2232:‖
2192:‖
2186:‖
2167:‖
2153:‖
2116:‖
2110:‖
2102:‖
2096:‖
2088:‖
2074:‖
1733:‖
1720:‖
1715:‖
1702:‖
1531:‖
1525:‖
1047:timelike
862:—
799:timelike
689:constant
559:, where
494:Category
142:model.
126:inertial
22463:Hawking
22458:Penrose
22443:Novikov
22423:Wheeler
22368:Hilbert
22358:Lorentz
22315:pp-wave
22136:lensing
21932:General
21713:Special
21654:YouTube
21475:Bibcode
21423:0075639
21415:1969989
21323:: 75–88
20458:Remarks
18738:in the
11518:. When
10443:is the
9693:Define
9439:− 1, 1)
8057:Tangent
4690:O(3, 1)
3566:matrix
2466:, on a
2438:either
2434:, with
2426:on the
947:), and
939:), the
575:is the
565:is the
278:History
138:into a
50:physics
22504:others
22493:Thorne
22483:Misner
22468:Taylor
22453:Geroch
22448:Ehlers
22418:Zwicky
22236:Kasner
21617:
21589:
21566:
21539:
21501:
21447:
21421:
21413:
21377:
21354:
21244:
21225:
21206:
21187:
21157:
21120:
21091:
21081:
21052:
21044:
20862:
20814:
20655:
20579:+ 1)(3
20550:There
20483:exist.
15230:
15227:
12124:. The
11510:where
11283:, …, X
11274:, and
10866:only.
10770:where
9579:cannot
9564:sphere
9466:> 4
9335:of an
9166:) = 0}
8780:where
8404:, the
8115:where
8031:(0, 2)
8021:. The
7225:fields
7221:1-form
7212:kernel
4790:) = ±1
4618:
4612:
4076:(0, 2)
3736:, and
3680:where
3607:events
3603:(1, 3)
3599:(3, 1)
3499:tensor
3096:Here,
2755:
2581:
2571:
2468:sphere
2402:origin
1079:, and
981:, and
963:Where
931:, the
725:German
614:matrix
492:
253:, the
203:, the
151:events
22498:Weiss
22478:Bondi
22473:Hulse
22403:Milne
22307:discs
22251:Milne
22246:Gödel
22103:Virgo
21499:S2CID
21411:JSTOR
21118:JSTOR
21050:S2CID
20596:Notes
20583:+ 11)
13358:) = (
11725:. If
11566:from
11338:from
10785:, …,
10514:into
10221:is a
9494:As a
9474:1 + 1
9472:with
9079:<
8271:= ∂/∂
8172:is a
8141:with
7250:is a
7067:, in
6579:4 × 4
6470:Here
5846:when
4794:basis
4760:) ≥ 0
4745:) ≤ 0
4730:) = 0
4146:with
3503:(0,2)
3419:, do
3344:with
3150:4 × 1
3078:, etc
1074:<
1057:>
1022:) = (
129:space
22433:Kerr
22383:Weyl
22282:Kerr
22142:and
22096:and
22094:LIGO
21615:ISBN
21587:ISBN
21564:ISBN
21537:ISBN
21445:ISBN
21375:ISBN
21352:ISBN
21242:ISBN
21223:ISBN
21204:ISBN
21185:ISBN
21155:ISBN
21089:ISSN
21079:ISBN
21042:ISSN
20860:ISBN
20812:ISBN
20653:ISBN
20575:/2)(
16261:and
12347:>
12161:Let
11395:) ∈
11381:) ∈
11347:) ∈
10150:the
10001:>
9771:>
9457:and
9447:− 1)
9443:(1,
9394:and
9392:= −1
9379:and
9333:spin
9278:The
9253:and
9194:and
9142:for
9029:Let
9013:must
8447:dual
8445:are
8432:and
7261:and
7246:The
7021:and
7005:and
6775:and
6279:and
6061:and
5581:for
5567:here
4865:is:
4861:and
4796:for
4792:. A
4747:and
4707:and
4166:and
4026:and
3951:diag
3891:and
3594:and
3520:, a
3443:and
3404:and
3396:and
3188:same
3160:each
3102:and
2417:real
2061:and
1853:>
1698:>
1649:and
1253:and
1211:The
1081:null
987:are
905:and
889:and
849:flat
827:and
714:real
569:and
223:and
188:and
180:and
133:time
131:and
56:(or
22488:Yau
22113:GEO
21667:at
21652:on
21491:hdl
21483:doi
21403:doi
21290:doi
21110:doi
21071:doi
21034:doi
20571:= (
13667:or
12903:or
12425:If
11762:= 2
11572:to
11549:→
11372:to
10526:on
10229:of
9668:can
9610:or
9558:in
9441:or
9419:≥ 2
9403:= 0
9377:= 1
9352:as
9148:is
9100:if
9060:if
8434:∂/∂
8245:∂/∂
8131:at
8127:of
8047:not
7556:to
7227:).
7219:or
7094:of
6927:or
6660:or
6031:= (
6006:of
4828:the
4777:if
4717:if
4678:all
4652:not
4181:in
4170:in
4126:in
3615:or
3601:or
3592:= 4
3564:4×4
3556:of
3511:4×4
3473:ict
3457:to
3451:in
3437:not
3421:not
3413:→ ∞
3348:at
3181:→ Λ
3122:not
2482:all
2442:or
1498:= (
1291:is
1261:= (
1223:= (
1175:An
1087:if
1083:or
1066:if
1049:if
1006:= (
797:or
747:ict
720:).
556:ict
274:).
257:or
60:) (
48:In
22536::
22162:/
22128::
22083::
21613:,
21562:.
21558:.
21497:,
21489:,
21481:,
21471:21
21469:,
21443:.
21439:.
21419:MR
21417:.
21409:.
21399:63
21397:.
21373:.
21340:;
21336:;
21321:10
21319:,
21286:28
21284:.
21278:.
21183:.
21171:;
21153:.
21139:;
21116:.
21087:.
21077:.
21069:.
21048:.
21040:.
21030:51
21028:.
20899:^
20820:.
20700:^
20617:.
20552:is
18357:16
17701:16
17361:16
17258:16
16857:16
16749:16
16364:0.
13362:,
12154:.
12130:,
12121:ct
12119:≡
11804:→
11800::
11790:→
11776::
11748:⊂
11744:→
11740::
11714:→
11710:⊂
11545::
11543:DF
11531:=
11527:,
11523:=
11442:→
11438::
11420:TN
11418:→
11411:TM
11409::
11400:.
11367:∈
11357:∈
11333:∈
11313:≠
11279:,
10980:,
10890:→
10886::
10861:∈
10778:,
10070:1.
9690:.
9520:.
9421:,
9405:.
9399:⋅
9363:is
9361:+
9357:=
9344:∈
9323:∈
9308:,
9300:=
9296:⋅
9230:iM
9228:⊕
9224:=
9162:,
9133:∈
9119:≤
9105:−
9065:−
9038:∈
9034:,
9007:,
8984:→
8808:.
8786:,
8661:,
8647:,
8637:,
8423:dx
8407:dx
8261:=
8165:df
8159:.
8150:≠
8146:,
7831:.
7532:.
7191:μν
7185:=
7140:=
7134:μν
7069:3d
7061:,
7055:,
7037:,
6811:,
6801:,
6793:,
6785:,
6766:,
6759:,
6752:,
6676:,
6670:,
6054:.
6045:,
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2352:w
2345:+
2338:u
2330:(
2325:=
2320:2
2312:w
2304:+
2297:w
2286:u
2279:2
2276:+
2271:2
2263:u
2243:2
2235:w
2227:+
2223:)
2219:w
2216:,
2213:u
2209:(
2205:2
2202:+
2197:2
2189:u
2181:=
2172:2
2163:w
2160:+
2157:u
2120:,
2113:w
2106:+
2099:u
2084:w
2081:+
2078:u
2063:w
2059:u
2032:)
2026:2
2021:2
2017:z
2008:2
2003:2
1999:y
1990:2
1985:2
1981:x
1972:2
1967:2
1963:t
1957:2
1953:c
1948:(
1943:)
1937:2
1932:1
1928:z
1919:2
1914:1
1910:y
1901:2
1896:1
1892:x
1883:2
1878:1
1874:t
1868:2
1864:c
1859:(
1848:2
1844:z
1838:1
1834:z
1825:2
1821:y
1815:1
1811:y
1802:2
1798:x
1792:1
1788:x
1779:2
1775:t
1769:1
1765:t
1759:2
1755:c
1728:2
1724:u
1710:1
1706:u
1695:)
1690:2
1686:u
1682:,
1677:1
1673:u
1669:(
1655:2
1652:u
1646:1
1643:u
1619:2
1615:z
1606:2
1602:y
1593:2
1589:x
1580:2
1576:t
1570:2
1566:c
1560:=
1555:)
1552:u
1549:,
1546:u
1543:(
1535:=
1528:u
1514:)
1512:z
1508:y
1504:x
1496:u
1459:.
1454:2
1450:z
1444:1
1440:z
1431:2
1427:y
1421:1
1417:y
1408:2
1404:x
1398:1
1394:x
1385:2
1381:t
1375:1
1371:t
1365:2
1361:c
1357:=
1352:2
1348:u
1339:1
1335:u
1331:=
1328:)
1323:2
1319:u
1315:,
1310:1
1306:u
1302:(
1289:)
1287:2
1284:z
1280:2
1277:y
1273:2
1270:x
1266:2
1263:t
1259:2
1256:u
1251:)
1249:1
1246:z
1242:1
1239:y
1235:1
1232:x
1228:1
1225:t
1221:1
1218:u
1133:v
1113:)
1111:v
1107:v
1105:(
1103:η
1097:r
1093:t
1090:c
1076:r
1072:t
1069:c
1059:r
1055:t
1052:c
1042:r
1038:t
1035:c
1030:)
1028:r
1020:z
1016:y
1012:x
1004:v
999:t
994:c
984:z
978:y
972:x
966:v
959:.
783:)
781:t
777:z
773:y
769:x
767:(
749:)
743:z
739:y
735:x
733:(
693:.
685:=
680:2
676:)
672:t
669:c
666:i
663:(
660:+
655:2
651:z
647:+
642:2
638:y
634:+
629:2
625:x
610:)
608:z
604:y
600:x
596:t
594:(
572:i
562:c
523:e
516:t
509:v
109:f
106:ɒ
103:k
100:ˈ
94:i
91:k
88:s
85:f
79:k
76:ˈ
73:ŋ
70:ɪ
67:m
64:/
34:.
20:)
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