4261:
187:) in the COM frame (where, by definition, the momentum of the system is zero). However, since the invariant mass of any system is also the same quantity in all inertial frames, it is a quantity often calculated from the total energy in the COM frame, then used to calculate system energies and momenta in other frames where the momenta are not zero, and the system total energy will necessarily be a different quantity than in the COM frame. As with energy and momentum, the invariant mass of a system cannot be destroyed or changed, and it is thus conserved, so long as the system is closed to all influences. (The technical term is
778:(such as a photon, or a theoretical graviton) moves at the speed of light in every frame of reference. In this case there is no transformation that will bring the particle to rest. The total energy of such particles becomes smaller and smaller in frames which move faster and faster in the same direction. As such, they have no rest mass, because they can never be measured in a frame where they are at rest. This property of having no rest mass is what causes these particles to be termed "massless". However, even massless particles have a relativistic mass, which varies with their observed energy in various frames of reference.
22:
1009:
6375:
2543:, is identical to it. This invariant mass for a system remains the same quantity in any inertial frame, although the system total energy and total momentum are functions of the particular inertial frame which is chosen, and will vary in such a way between inertial frames as to keep the invariant mass the same for all observers. Invariant mass thus functions for systems of particles in the same capacity as "rest mass" does for single particles.
2752:
because for bound systems these merely act to change the inertial frame of the system or the observer. Though such actions may change the total energy or momentum of the bound system, these two changes cancel, so that there is no change in the system's invariant mass. This is just the same result as with single particles: their calculated rest mass also remains constant no matter how fast they move, or how fast an observer sees them move.
714:, relativistic mass is used to explain electron orbital contraction in heavy elements. The notion of mass as a property of an object from Newtonian mechanics does not bear a precise relationship to the concept in relativity. Relativistic mass is not referenced in nuclear and particle physics, and a survey of introductory textbooks in 2005 showed that only 5 of 24 texts used the concept, although it is still prevalent in popularizations.
2580:
2149:
912:
3449:
2783:
which (as a system) preserve the invariant mass of the particle which produced them. Also a box of moving non-interacting particles (e.g., photons, or an ideal gas) will have a larger invariant mass than the sum of the rest masses of the particles which compose it. This is because the total energy of all particles and fields in a system must be summed, and this quantity, as seen in the
2756:
on (do work on) only one part of such an unbound system, this is equivalent to allowing energy into or out of the system, and the condition of "closure" to massâenergy (total isolation) is violated. In this case, conservation of invariant mass of the system also will no longer hold. Such a loss of rest mass in systems when energy is removed, according to
3231:
2755:
On the other hand, for systems which are unbound, the "closure" of the system may be enforced by an idealized surface, inasmuch as no massâenergy can be allowed into or out of the test-volume over time, if conservation of system invariant mass is to hold during that time. If a force is allowed to act
2676:
Total energy is an additive conserved quantity (for single observers) in systems and in reactions between particles, but rest mass (in the sense of being a sum of particle rest masses) may not be conserved through an event in which rest masses of particles are converted to other types of energy, such
130:
The term "relativistic mass" tends not to be used in particle and nuclear physics and is often avoided by writers on special relativity, in favor of referring to the body's relativistic energy. In contrast, "invariant mass" is usually preferred over rest energy. The measurable inertia and the warping
4230:
The concept of "relativistic mass" is subject to misunderstanding. That's why we don't use it. First, it applies the name mass â belonging to the magnitude of a 4-vector â to a very different concept, the time component of a 4-vector. Second, it makes increase of energy of an object with velocity or
4222:
and how the public's interpretation of the equation has largely informed how it is taught in higher education. He instead supposes that the difference between rest and relativistic mass should be explicitly taught, so that students know why mass should be thought of as invariant "in most discussions
3741:
only applies to a slower-than-light particle (a particle with a nonzero rest mass). Tolman remarked on this relation that "We have, moreover, of course the experimental verification of the expression in the case of moving electrons ... We shall hence have no hesitation in accepting the expression as
2805:
allowed to escape (the system is closed and isolated), the energy will continue to contribute to the system rest mass, and the system mass will not change. Only if the energy is released to the environment will the mass be lost; this is because the associated mass has been allowed out of the system,
2734:
All conservation laws in special relativity (for energy, mass, and momentum) require isolated systems, meaning systems that are totally isolated, with no massâenergy allowed in or out, over time. If a system is isolated, then both total energy and total momentum in the system are conserved over time
717:
If a stationary box contains many particles, its weight increases in its rest frame the faster the particles are moving. Any energy in the box (including the kinetic energy of the particles) adds to the mass, so that the relative motion of the particles contributes to the mass of the box. But if the
4126:
For many years it was conventional to enter the discussion of dynamics through derivation of the relativistic mass, that is the massâvelocity relation, and this is probably still the dominant mode in textbooks. More recently, however, it has been increasingly recognized that relativistic mass is a
2782:
Again, in special relativity, the rest mass of a system is not required to be equal to the sum of the rest masses of the parts (a situation which would be analogous to gross mass-conservation in chemistry). For example, a massive particle can decay into photons which individually have no mass, but
770:
of the object is at rest (another way of stating this is that it is the frame in which the momenta of the system's parts add to zero). For compound objects (made of many smaller objects, some of which may be moving) and sets of unbound objects (some of which may also be moving), only the center of
735:
Relativistic mass and rest mass are both traditional concepts in physics, but the relativistic mass corresponds to the total energy. The relativistic mass is the mass of the system as it would be measured on a scale, but in some cases (such as the box above) this fact remains true only because the
709:
concept of the theory, it has been argued that this is wrong as the fundamentals of the theory relate to spaceâtime. There is disagreement over whether the concept is pedagogically useful. It explains simply and quantitatively why a body subject to a constant acceleration cannot reach the speed of
753:
The invariant mass is proportional to the value of the total energy in one reference frame, the frame where the object as a whole is at rest (as defined below in terms of center of mass). This is why the invariant mass is the same as the rest mass for single particles. However, the invariant mass
749:
and weighed, or the scale were somehow sent after it, it would not be moving with respect to the scale, and again the relativistic and rest masses would be the same for the single electron (and would be smaller). In general, relativistic and rest masses are equal only in systems which have no net
2796:
In special relativity, mass is not "converted" to energy, for all types of energy still retain their associated mass. Neither energy nor invariant mass can be destroyed in special relativity, and each is separately conserved over time in closed systems. Thus, a system's invariant mass may change
2751:
Conservation of invariant mass also requires the system to be enclosed so that no heat and radiation (and thus invariant mass) can escape. As in the example above, a physically enclosed or bound system does not need to be completely isolated from external forces for its mass to remain constant,
744:
is moving in circles with a relativistic velocity, the mass of the cyclotron+electron system is increased by the relativistic mass of the electron, not by the electron's rest mass. But the same is also true of any closed system, such as an electron-and-box, if the electron bounces at high speed
177:
The concept of invariant mass does not require bound systems of particles, however. As such, it may also be applied to systems of unbound particles in high-speed relative motion. Because of this, it is often employed in particle physics for systems which consist of widely separated high-energy
165:
frame (COM frame), as when any closed system (for example a bottle of hot gas) is weighed, which requires that the measurement be taken in the center of momentum frame where the system has no net momentum. Under such circumstances the invariant mass is equal to the relativistic mass (discussed
722:
is moving), there remains the question of whether the kinetic energy of the overall motion should be included in the mass of the system. The invariant mass is calculated excluding the kinetic energy of the system as a whole (calculated using the single velocity of the box, which is to say the
2708:
also invariant. For example: A closed container of gas (closed to energy as well) has a system "rest mass" in the sense that it can be weighed on a resting scale, even while it contains moving components. This mass is the invariant mass, which is equal to the total relativistic energy of the
2677:
as kinetic energy. Finding the sum of individual particle rest masses would require multiple observers, one for each particle rest inertial frame, and these observers ignore individual particle kinetic energy. Conservation laws require a single observer and a single inertial frame.
4121:
The concept of relativistic mass is widely used in popular science writing and in high school and undergraduate textbooks. Authors such as Okun and A. B. Arons have argued against this as archaic and confusing, and not in accord with modern relativistic theory. Arons wrote:
2692:
automatically means that relativistic mass is conserved for any given observer and inertial frame. However, this quantity, like the total energy of a particle, is not invariant. This means that, even though it is conserved for any observer during a reaction, its absolute
2550:(i.e., one closed to both mass and energy) is also independent of observer or inertial frame, and is a constant, conserved quantity for isolated systems and single observers, even during chemical and nuclear reactions. The concept of invariant mass is widely used in
131:
of spacetime by a body in a given frame of reference is determined by its relativistic mass, not merely its invariant mass. For example, photons have zero rest mass but contribute to the inertia (and weight in a gravitational field) of any system containing them.
3444:{\displaystyle {\begin{aligned}f_{\text{x}}&=m\gamma ^{3}a_{\text{x}}&=m_{\text{L}}a_{\text{x}},\\f_{\text{y}}&=m\gamma a_{\text{y}}&=m_{\text{T}}a_{\text{y}},\\f_{\text{z}}&=m\gamma a_{\text{z}}&=m_{\text{T}}a_{\text{z}}.\end{aligned}}}
4211:
C. Alder takes a similarly dismissive stance on mass in relativity. Writing on said subject matter, he says that "its introduction into the theory of special relativity was much in the way of a historical accident", noting towards the widespread knowledge of
160:
of single particles. The more general invariant mass (calculated with a more complicated formula) loosely corresponds to the "rest mass" of a "system". Thus, invariant mass is a natural unit of mass used for systems which are being viewed from their
1906:
2680:
In general, for isolated systems and single observers, relativistic mass is conserved (each observer sees it constant over time), but is not invariant (that is, different observers see different values). Invariant mass, however, is both conserved
178:
particles. If such systems were derived from a single particle, then the calculation of the invariant mass of such systems, which is a never-changing quantity, will provide the rest mass of the parent particle (because it is conserved over time).
2721:
has a velocity). In this case, its total relativistic mass and energy increase. However, in such a situation, although the container's total relativistic energy and total momentum increase, these energy and momentum increases subtract out in the
2716:
The container may even be subjected to a force which gives it an overall velocity, or else (equivalently) it may be viewed from an inertial frame in which it has an overall velocity (that is, technically, a frame in which its
2492:
351:
3140:(although Abraham used more complicated expressions than Lorentz's relativistic ones). So, according to Lorentz's theory no body can reach the speed of light because the mass becomes infinitely large at this velocity.
1637:
4231:
momentum appear to be connected with some change in internal structure of the object. In reality, the increase of energy with velocity originates not in the object but in the geometric properties of spacetime itself.
3857:
506:
2245:
The rest mass of a composite system is not the sum of the rest masses of the parts, unless all the parts are at rest. The total mass of a composite system includes the kinetic energy and field energy in the system.
394:. However, for given single frames of reference and for isolated systems, the relativistic mass is also a conserved quantity. The relativistic mass is also the proportionality factor between velocity and momentum,
1715:
385:
and the relativistic mass equals the rest mass. In other frames, the relativistic mass (of a body or system of bodies) includes a contribution from the "net" kinetic energy of the body (the kinetic energy of the
4205:
1787:
2747:
particle rest mass (a special case of the invariant mass) no matter how they move (what inertial frame they choose), but different observers see different total energies and momenta for the same particle.
2900:
1976:
1388:
2388:
3628:
3458:
In special relativity, an object that has nonzero rest mass cannot travel at the speed of light. As the object approaches the speed of light, the object's energy and momentum increase without bound.
435:
3897:
1145:
5413:
3236:
1799:
3521:
2801:
because invariant mass is allowed to escape, perhaps as light or heat. Thus, when reactions (whether chemical or nuclear) release energy in the form of heat and light, if the heat and light is
1794:
3967:
3056:
3578:
703:
3739:
1483:
745:
inside the box. It is only the lack of total momentum in the system (the system momenta sum to zero) which allows the kinetic energy of the electron to be "weighed". If the electron is
3461:
In the first years after 1905, following
Lorentz and Einstein, the terms longitudinal and transverse mass were still in use. However, those expressions were replaced by the concept of
2954:
1435:
4088:
2713:. Just as is the case for single particles, the calculated "rest mass" of such a container of gas does not change when it is in motion, although its "relativistic mass" does change.
4892:
4865:
3696:
3173:
by an unfortunate force definition, which was later corrected), and in another paper in 1906. However, he later abandoned velocity dependent mass concepts (see quote at the end of
2990:
253:
1525:
897:
650:
598:
2063:
843:
3765:
is simply equal to 1, and the relativistic mass is reduced to the rest mass as one can see in the next two equations below. As the velocity increases toward the speed of light
1218:
3924:
2825:
Concepts that were similar to what nowadays is called "relativistic mass", were already developed before the advent of special relativity. For example, it was recognized by
3171:
3133:
3103:
2105:
2523:
2317:
2280:
2406:
1303:
4010:
548:
3787:
3763:
3226:
6078:
282:
1247:
528:
383:
3198:
1542:
277:
3796:
445:
150:
in special relativity usually refers to the rest mass of the object, which is the
Newtonian mass as measured by an observer moving along with the object. The
3146:
also initially used the concepts of longitudinal and transverse mass in his 1905 electrodynamics paper (equivalent to those of
Lorentz, but with a different
4249:. The connection of velocity to hyperbolic geometry enables the 3-velocity-dependent relativistic mass to be related to the 4-velocity Minkowski formalism.
1644:
4134:
1720:
4127:
troublesome and dubious concept. ... The sound and rigorous approach to relativistic dynamics is through direct development of that expression for
1314:
3074:
is the speed of light). Only when the force is perpendicular to the velocity, Lorentz's mass is equal to what is now called "relativistic mass".
5323:
2913:. He defined mass as the ratio of force to acceleration, not as the ratio of momentum to velocity, so he needed to distinguish between the mass
2322:
3583:
3180:
The precise relativistic expression (which is equivalent to
Lorentz's) relating force and acceleration for a particle with non-zero rest mass
397:
4464:
1068:
2905:
Then, it was pointed out by
Thomson and Searle that this electromagnetic mass also increases with velocity. This was further elaborated by
2726:
definition, so that the moving container's invariant mass will be calculated as the same value as if it were measured at rest, on a scale.
6125:
2644:
2597:
2213:
2166:
976:
929:
5501:
3476:
1392:
To find the form of the momentum and energy as a function of velocity, it can be noted that the four-velocity, which is proportional to
390:
of the body), and is larger the faster the body moves. Thus, unlike the invariant mass, the relativistic mass depends on the observer's
6360:
2739:
will vary, according to different observers in different inertial frames. The invariant mass of the system is also conserved, but does
2616:
2536:
where momenta sum to zero, again the system mass (called the invariant mass) corresponds to the total system energy or, in units where
2185:
948:
6149:
5895:
5522:
4274:
2282:
of the system, a vector quantity, can also be computed by adding together the momenta of all its components. Given the total energy
711:
3526:
4090:
of a moving body for which no clear definition can be given. It is better to introduce no other mass concept than the ârest massâ
2623:
2192:
955:
2840:
1911:
5931:
5885:
5087:"Note on the Derivation from the Principle of Relativity of the Fifth Fundamental Equation of the MaxwellâLorentz Theory"
4791:
2403:, for systems of particles (whether bound or unbound) the total system invariant mass is given equivalently by the following:
6069:
5511:
5390:
5195:
4770:
4744:
4706:
3862:
2774:
is the change in rest mass, reflect changes of mass associated with movement of energy, not "conversion" of mass to energy.
2630:
2199:
962:
736:
system on average must be at rest to be weighed (it must have zero net momentum, which is to say, the measurement is in its
5553:
795:
215:
4442:
Ketterle, W. and
Jamison, A. O. (2020). "An atomic physics perspective on the kilogramâs new definition", "Physics Today"
2529:) is used. This results in a scalar number, which is subtracted from the scalar value of the square of the total energy.
2253:
of a composite system can be determined by adding together the sum of the energies of its components. The total momentum
1488:
856:
5002:
4952:
4353:
2012:
805:
181:
It is often convenient in calculation that the invariant mass of a system is the total energy of the system (divided by
5806:
4020:. Einstein never derived an equation for "relativistic mass", and in later years he expressed his dislike of the idea:
2612:
2181:
944:
2829:
in 1881 that a charged body is harder to set in motion than an uncharged body, which was worked out in more detail by
1168:
6177:
5811:
5218:
4859:
4730:
3929:
2663:
2232:
995:
65:
43:
2743:
change with different observers. This is also the familiar situation with single particles: all observers calculate
36:
2995:
6131:
4260:
658:
5844:
3701:
2834:
1440:
5947:
2916:
2601:
2170:
1437:, is the only four-vector associated with the particle's motion, so that if there is a conserved four-momentum
1395:
1269:
933:
771:
mass of the system is required to be at rest, for the object's relativistic mass to be equal to its rest mass.
4226:
Many contemporary authors such as Taylor and
Wheeler avoid using the concept of relativistic mass altogether:
6378:
6120:
6025:
5955:
5003:"Ăber eine Methode zur Bestimmung des VerhĂ€ltnisses der transversalen und longitudinalen Masse des Elektrons"
4027:
1251:
A photon's momentum is a function of its energy, but it is not proportional to the velocity, which is always
100:
6088:
5488:
4245:
where relativistic mass plays an analogous role to that of
Newtonian mass in the barycentric coordinates of
3972:
Even though
Einstein initially used the expressions "longitudinal" and "transverse" mass in two papers (see
3653:
5942:
4790:
2959:
1062:
1901:{\displaystyle {\begin{aligned}E_{0}&=mc^{2},\\E&=\gamma mc^{2},\\p&=mv\gamma ,\end{aligned}}}
1223:
723:
velocity of the box's center of mass), while the relativistic mass is calculated including invariant mass
6397:
5983:
5717:
5702:
4516:
4409:
4306:
603:
557:
210:
191:
meaning that an idealized boundary is drawn around the system, and no mass/energy is allowed across it.)
112:
2637:
2206:
969:
6270:
5712:
135:
5354:
5046:
4551:
6300:
5926:
5607:
5274:
4298:
3902:
1485:, it must be proportional to this vector. This allows expressing the ratio of energy to momentum as
6325:
5875:
5635:
5546:
4736:
4242:
2784:
2710:
2533:
759:
30:
3149:
3111:
3081:
2777:
758:
is at rest for systems of many particles. This special frame where this occurs is also called the
6054:
5976:
5596:
5587:
5432:
4887:"Electromagnetic phenomena in a system moving with any velocity smaller than that of light"
4858:
4497:
2590:
2159:
2070:
922:
5482:
5232:
4235:
While spacetime has the unbounded geometry of
Minkowski space, the velocity-space is bounded by
2499:
2293:
2256:
6280:
6143:
6049:
6007:
5681:
5645:
5269:
4831:
4797:
4641:
3790:
2689:
439:
47:
4825:
3979:
727:
the kinetic energy of the system which is calculated from the velocity of the center of mass.
533:
6285:
5890:
5829:
4777:
4712:
3772:
3748:
3211:
2697:
will change with the frame of the observer, and for different observers in different frames.
5181:
6190:
6116:
5960:
5562:
5451:
5338:
5261:
5019:
4969:
4926:
4650:
4605:
4525:
4481:
4418:
4370:
4315:
2910:
2820:
5519:
4594:"Why is mercury liquid? Or, why do relativistic effects not get into chemistry textbooks?"
513:
8:
6265:
6260:
6250:
6082:
6031:
5849:
5834:
5660:
5591:
5539:
4912:
4792:"On the Electric and Magnetic Effects produced by the Motion of Electrified Bodies"
3969:
is generally not a constant (see the section above on transverse and longitudinal mass).
3698:
holds for all particles, including those moving at the speed of light, while the formula
1992:
362:
123:
6255:
5456:
5342:
5265:
5023:
4973:
4930:
4654:
4609:
4529:
4485:
4422:
4374:
4319:
4098:
it is better to mention the expression for the momentum and energy of a body in motion.
2837:(1897). So the electrostatic energy behaves as having some sort of electromagnetic mass
1308:
The rest mass is only proportional to the total energy in the rest frame of the object.
6330:
6230:
6154:
6012:
5993:
5987:
5938:
5880:
5789:
5707:
5625:
5602:
5570:
5465:
5441:
5067:
5010:
4960:
4917:
4666:
4574:
4361:
4331:
4246:
3183:
737:
391:
356:
262:
162:
96:
86:
5086:
2685:
invariant (all single observers see the same value, which does not change over time).
6340:
6172:
6164:
5765:
5722:
5507:
5396:
5386:
5214:
5201:
5191:
5118:
4766:
4740:
4702:
4670:
4621:
4389:
4335:
4327:
3470:
2525:
are first summed as vectors, and then the square of their resulting total magnitude (
853:
when the rest mass is constant. The four-dimensional form of Newton's second law is:
846:
775:
6210:
4578:
2709:
container (including the kinetic energy of the gas) only when it is measured in the
750:
momentum and the system center of mass is at rest; otherwise they may be different.
652:, which some interpret as the relativistic mass of the emitted photon since it also
6245:
6235:
6182:
6159:
5737:
5346:
5279:
5163:
5131:
5099:
5059:
5027:
4988:
4977:
4934:
4840:
4806:
4658:
4613:
4566:
4533:
4489:
4426:
4378:
4323:
3466:
2830:
2551:
653:
104:
6305:
5045:
4639:
E. Eriksen; K. VĂžyenli (1976), "The classical and relativistic concepts of mass",
2487:{\displaystyle m^{2}=\left(\sum E\right)^{2}-\left\|\sum {\vec {p}}\ \right\|^{2}}
2111:, a length which is associated with rest mass or invariant mass in systems. Where
6320:
6295:
6220:
6215:
6098:
6059:
6021:
5965:
5839:
5775:
5526:
5187:
4760:
4696:
4593:
4104:
3143:
3136:
2906:
2547:
1981:
188:
6103:
5119:"Non-Newtonian Mechanics :â The Direction of Force and Acceleration."
1305:
Note that the formula is true only for particles or systems with zero momentum.
1008:
6345:
6017:
6001:
5997:
5900:
5870:
5727:
5630:
5252:
Eugene Hecht (19 August 2009). "Einstein Never Approved of Relativistic Mass".
4860:"Simplified Theory of Electrical and Optical Phenomena in Moving Systems"
4266:
3632:
Tolman in 1912 further elaborated on this concept, and stated: "the expression
3059:
2778:
The system invariant mass vs. the individual rest masses of parts of the system
2718:
2569:
2526:
787:
767:
763:
755:
719:
387:
171:
152:
91:
5167:
5135:
5103:
4844:
4810:
346:{\displaystyle m_{\text{rel}}={\frac {m}{\sqrt {1-{\dfrac {v^{2}}{c^{2}}}}}}.}
6402:
6391:
6335:
6315:
6310:
6225:
6093:
5921:
5865:
5697:
5650:
5400:
5031:
4982:
4938:
4625:
4472:
4383:
2826:
2700:
By contrast, the rest mass and invariant masses of systems and particles are
2393:
799:
791:
2554:, because the invariant mass of a particle's decay products is equal to its
6355:
6275:
6240:
5770:
5732:
5416:, Abraham A. Ungar, Commun. Math. Anal. Volume 10, Number 1 (2011), 30â56.
5380:
4824:
1632:{\displaystyle E^{2}=\left(mc^{2}\right)^{2}+E^{2}{\frac {v^{2}}{c^{2}}},}
111:
is dependent on the velocity of the observer. According to the concept of
6139:
6108:
5655:
5477:
3852:{\displaystyle \mathbf {f} ={\frac {d(m_{\text{rel}}\mathbf {v} )}{dt}},}
3075:
501:{\displaystyle \mathbf {f} ={\frac {d(m_{\text{rel}}\mathbf {v} )}{dt}}.}
117:
5470:
5071:
4570:
6350:
5916:
5760:
5755:
5497:
5150:
4886:
4683:
4662:
2559:
2108:
850:
5283:
4617:
4537:
3769:, the denominator of the right side approaches zero, and consequently
2558:. This is used to make measurements of the mass of particles like the
5747:
5063:
4493:
2563:
2555:
1710:{\displaystyle E={\frac {mc^{2}}{\sqrt {1-{\dfrac {v^{2}}{c^{2}}}}}}}
741:
5350:
4430:
2579:
2148:
911:
710:
light, and why the mass of a system emitting a photon decreases. In
203:
is the sum total quantity of energy in a body or system (divided by
6290:
5640:
5043:
4354:"Ist die TrÀgheit eines Körpers von seinem Energieinhalt abhÀngig?"
4200:{\displaystyle p={m_{0}v \over {\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}}
4108:
2000:
1995:, all the relativistic equations are simplified and the quantities
5446:
5085:
1782:{\displaystyle p={\frac {mv}{\sqrt {1-{\dfrac {v^{2}}{c^{2}}}}}}.}
5117:
4893:
Proceedings of the Royal Netherlands Academy of Arts and Sciences
4866:
Proceedings of the Royal Netherlands Academy of Arts and Sciences
5205:
5047:"The Principle of Relativity, and Non-Newtonian Mechanics"
5970:
5464:
Oas, Gary (2005), "On the Abuse and Use of Relativistic Mass",
4514:
L. B. Okun (2009), "Mass versus relativistic and rest masses",
3068:
is the relative velocity between the ether and the object, and
2902:, which can increase the normal mechanical mass of the bodies.
1996:
551:
255:
is the relativistic mass. For a particle of non-zero rest mass
5382:
Hyperbolic Triangle Centers: The Special Relativistic Approach
2067:
The equation is often written this way because the difference
5531:
5210:
3473:
in 1909. They defined the total energy and mass of a body as
2895:{\textstyle m_{\text{em}}={\frac {4}{3}}E_{\text{em}}/c^{2}}
2735:
for any observer in any single inertial frame, though their
2729:
2570:
Conservation versus invariance of mass in special relativity
2125:, this equation again expresses the massâenergy equivalence
1971:{\textstyle \gamma ={1}/{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}.}
1383:{\displaystyle E={\sqrt {\left(mc^{2}\right)^{2}+(pc)^{2}}}}
5951:
5151:"Non-Newtonian Mechanics. The Mass of a Moving Body."
2809:
2004:
166:
below), which is the total energy of the system divided by
82:
3650:
In 1934, Tolman argued that the relativistic mass formula
4728:
2383:{\displaystyle m={\frac {\sqrt {E^{2}-(pc)^{2}}}{c^{2}}}}
1151:
is the rest mass, or the invariant mass for systems, and
5052:
Proceedings of the American Academy of Arts and Sciences
4826:"On the Steady Motion of an Electrified Ellipsoid"
4682:
Oas, "On the Abuse and Use of Relativistic Mass," 2005,
4407:
T. R. Sandin (1991), "In defense of relativistic mass",
3623:{\displaystyle {\frac {m_{\text{rel}}}{m_{0}}}=\gamma .}
1311:
When the object is moving, the total energy is given by
901:
430:{\displaystyle \mathbf {p} =m_{\text{rel}}\mathbf {v} .}
5485:
by Philip Gibbs et al., 2002, retrieved August 10, 2006
3892:{\displaystyle \mathbf {f} =m_{\text{rel}}\mathbf {a} }
3742:
correct in general for the mass of a moving particle."
705:. Although some authors present relativistic mass as a
4030:
2998:
2843:
2806:
where it contributes to the mass of the surroundings.
1914:
1140:{\displaystyle E^{2}-(pc)^{2}=\left(mc^{2}\right)^{2}}
5491:
by Matt Austern et al., 1998, retrieved June 27, 2007
5430:
Silagadze, Z. K. (2008), "Relativity without tears",
4137:
4131:
that ensures conservation of momentum in all frames:
3982:
3973:
3932:
3905:
3865:
3799:
3775:
3751:
3704:
3656:
3586:
3529:
3479:
3234:
3214:
3186:
3152:
3114:
3084:
2962:
2919:
2502:
2409:
2325:
2296:
2259:
2073:
2015:
1797:
1748:
1723:
1679:
1647:
1545:
1491:
1443:
1398:
1317:
1272:
1226:
1171:
1071:
859:
808:
661:
606:
560:
536:
516:
448:
400:
365:
312:
285:
265:
218:
4638:
4256:
4024:
It is not good to introduce the concept of the mass
2688:
The relativistic mass corresponds to the energy, so
5149:
5044:Lewis, Gilbert N. & Tolman, Richard C. (1909),
4885:
4765:(2nd ed.). John Wiley & Sons. p. 70.
2814:
2604:. Unsourced material may be challenged and removed.
2173:. Unsourced material may be challenged and removed.
1158:The equation is also valid for photons, which have
936:. Unsourced material may be challenged and removed.
4199:
4082:
4004:
3961:
3918:
3891:
3851:
3781:
3757:
3733:
3690:
3622:
3572:
3516:{\displaystyle m_{\text{rel}}={\frac {E}{c^{2}}},}
3515:
3443:
3220:
3192:
3165:
3127:
3097:
3050:
2984:
2948:
2894:
2517:
2486:
2382:
2311:
2274:
2099:
2057:
1970:
1900:
1781:
1709:
1631:
1519:
1477:
1429:
1382:
1297:
1241:
1212:
1139:
891:
837:
697:
644:
592:
542:
522:
500:
429:
377:
345:
271:
247:
3647:) is best suited for the mass of a moving body."
2956:parallel to the direction of motion and the mass
6389:
5506:, Courier Dover Publications, pp. 177â178,
5503:Concepts of Mass in Classical and Modern Physics
5414:When Relativistic Mass Meets Hyperbolic Geometry
5230:
4116:
2992:perpendicular to the direction of motion (where
2138:
4822:
3962:{\displaystyle {d(m_{\text{rel}}\mathbf {v} )}}
3051:{\textstyle \gamma =1/{\sqrt {1-v^{2}/c^{2}}}}
5547:
4883:
4856:
4788:
4552:"Relativistic effects on chemical properties"
3573:{\displaystyle m_{0}={\frac {E_{0}}{c^{2}}},}
730:
5251:
4406:
698:{\displaystyle p=m_{\text{rel}}c=h/\lambda }
5324:"Does mass really depend on velocity, dad?"
5179:
5000:
4950:
4436:
4351:
3734:{\displaystyle m_{\text{rel}}=\gamma m_{0}}
3465:, an expression which was first defined by
754:also represents the measured mass when the
121:, while relativistic mass is equivalent to
5554:
5540:
5496:
5299:
4910:
4543:
4513:
4462:
1478:{\displaystyle \left(E,{\vec {p}}c\right)}
740:frame). For example, if an electron in a
5469:
5455:
5445:
5429:
5273:
5183:Relativity, Thermodynamics, and Cosmology
5147:
5115:
5083:
4981:
4758:
4724:
4722:
4720:
4585:
4382:
4347:
4345:
4275:Tests of relativistic energy and momentum
2949:{\displaystyle m_{\text{L}}=\gamma ^{3}m}
2730:Closed (meaning totally isolated) systems
2664:Learn how and when to remove this message
2233:Learn how and when to remove this message
2107:is the relativistic length of the energy
1430:{\displaystyle \left(c,{\vec {v}}\right)}
996:Learn how and when to remove this message
66:Learn how and when to remove this message
5302:A Guide to Introductory Physics Teaching
5295:
5293:
4402:
4400:
4398:
4083:{\textstyle M=m/{\sqrt {1-v^{2}/c^{2}}}}
2810:History of the relativistic mass concept
1007:
794:(the four-dimensional generalization of
29:This article includes a list of general
4694:
4507:
4292:
4290:
6390:
4913:"Prinzipien der Dynamik des Elektrons"
4717:
4632:
4591:
4549:
4458:
4456:
4454:
4452:
4342:
4207:rather than through relativistic mass.
3691:{\displaystyle m_{\text{rel}}=E/c^{2}}
3453:
3174:
78:Meanings of mass in special relativity
5535:
5378:
5321:
5290:
4395:
4296:
2985:{\displaystyle m_{\text{T}}=\gamma m}
1012:Dependency between the rest mass and
902:Relativistic energyâmomentum equation
510:When a body emits light of frequency
248:{\displaystyle E=m_{\text{rel}}c^{2}}
5173:
4953:"Zur Elektrodynamik bewegter Körper"
4729:E. F. Taylor; J. A. Wheeler (1992),
4684:http://arxiv.org/abs/physics/0504110
4287:
3745:When the relative velocity is zero,
2602:adding citations to reliable sources
2573:
2171:adding citations to reliable sources
2142:
1791:these expressions can be written as
1260:For an object at rest, the momentum
934:adding citations to reliable sources
905:
600:, the mass of the body decreases by
279:relative to the observer, one finds
194:
15:
6126:TolmanâOppenheimerâVolkoff equation
6079:FriedmannâLemaĂźtreâRobertsonâWalker
5463:
4759:Mandl, Franz; Shaw, Graham (2013).
4752:
4449:
2546:Note that the invariant mass of an
2496:Where, again, the particle momenta
1520:{\displaystyle pc=E{\frac {v}{c}},}
892:{\displaystyle F^{\mu }=mA^{\mu }.}
645:{\displaystyle E/c^{2}=h/\lambda c}
593:{\displaystyle E=h\nu =hc/\lambda }
13:
5322:Adler, Carl (September 30, 1986).
5077:
2793:, is the system's invariant mass.
2532:For such a system, in the special
2319:, the invariant mass is given by:
2058:{\displaystyle m^{2}=E^{2}-p^{2}.}
838:{\displaystyle p^{\mu }=mv^{\mu }}
115:, invariant mass is equivalent to
35:it lacks sufficient corresponding
14:
6414:
5896:HamiltonâJacobiâEinstein equation
5483:"Does mass change with velocity?"
5423:
2909:(1899, 1904) in the framework of
2007:have the same natural dimension:
1053:The relativistic expressions for
781:
6374:
6373:
4259:
4016:as what would now be called the
3951:
3885:
3867:
3828:
3801:
2815:Transverse and longitudinal mass
2578:
2147:
1527:resulting in a relation between
1213:{\displaystyle E^{2}-(pc)^{2}=0}
910:
477:
450:
420:
402:
20:
5489:"What is the mass of a photon?"
5407:
5372:
5368:– via HUIT Sites Hosting.
5315:
5245:
5224:
5141:
5109:
5037:
4994:
4944:
4904:
4877:
4850:
4816:
4782:
4688:
4676:
2835:George Frederick Charles Searle
2589:needs additional citations for
2158:needs additional citations for
921:needs additional citations for
5703:Massâenergy equivalence (E=mc)
5561:
3955:
3937:
3919:{\displaystyle m_{\text{rel}}}
3832:
3814:
3208:and associated Lorentz factor
2509:
2474:
2464:
2451:
2358:
2348:
2303:
2266:
1461:
1416:
1369:
1359:
1195:
1185:
1095:
1085:
712:relativistic quantum chemistry
481:
463:
134:The concept is generalized in
95:(also called rest mass) is an
1:
5309:Teaching Introductory Physics
4735:(second ed.), New York:
4598:Journal of Chemical Education
4559:Accounts of Chemical Research
4280:
4117:Popular science and textbooks
4103:Albert Einstein in letter to
2290:of the total momentum vector
2139:The mass of composite systems
5233:"What is relativistic mass?"
4695:McGlinn, William D. (2004),
3166:{\displaystyle m_{\text{T}}}
3128:{\displaystyle m_{\text{T}}}
3098:{\displaystyle m_{\text{L}}}
2613:"Mass in special relativity"
2182:"Mass in special relativity"
945:"Mass in special relativity"
141:
127:(also called total energy).
7:
5718:Relativistic Doppler effect
5520:Mass as a Variable Quantity
5331:American Journal of Physics
5209:Reissued (1987), New York:
4550:Pitzer, Kenneth S. (1979).
4517:American Journal of Physics
4410:American Journal of Physics
4307:European Journal of Physics
4252:
4107:, 19 June 1948 (quote from
3789:approaches infinity. While
2770:is the energy removed, and
2286:and the length (magnitude)
2100:{\displaystyle E^{2}-p^{2}}
10:
6419:
6189:In computational physics:
5713:Relativity of simultaneity
5379:Ungar, Abraham A. (2010).
4698:Introduction to relativity
4328:10.1088/0143-0807/26/2/002
3793:remains valid in the form
2818:
2518:{\displaystyle {\vec {p}}}
2312:{\displaystyle {\vec {p}}}
2275:{\displaystyle {\vec {p}}}
731:Relativistic vs. rest mass
718:box itself is moving (its
442:remains valid in the form
136:mass in general relativity
99:which is the same for all
6371:
6203:
6068:
6040:
6026:LenseâThirring precession
5909:
5858:
5820:
5799:
5788:
5746:
5690:
5674:
5616:
5608:Doubly special relativity
5580:
5569:
5168:10.1080/14786440308637231
5136:10.1080/14786440908637142
5104:10.1080/14786440308637034
4845:10.1080/14786449708621072
4811:10.1080/14786448108627008
4701:, JHU Press, p. 43,
4094:. Instead of introducing
3976:), in his first paper on
1298:{\displaystyle E=mc^{2}.}
845:and is also the ratio of
209:). Thus, the mass in the
5886:Post-Newtonian formalism
5876:Einstein field equations
5812:Mathematical formulation
5636:Hyperbolic orthogonality
5231:Philip Gibbs; Jim Carr.
5032:10.1002/andp.19063261310
4983:10.1002/andp.19053221004
4939:10.1002/andp.19023150105
4823:G. F. C. Searle (1897),
4737:W.H. Freeman and Company
4592:Norrby, Lars J. (1991).
4384:10.1002/andp.19053231314
4241:and has the geometry of
4005:{\displaystyle E=mc^{2}}
3204:direction with velocity
2785:center of momentum frame
2711:center of momentum frame
2534:center of momentum frame
1063:energyâmomentum relation
762:, and is defined as the
760:center of momentum frame
543:{\displaystyle \lambda }
156:is another name for the
5597:Galilean transformation
5588:Principle of relativity
5433:Acta Physica Polonica B
5385:. Dordrecht: Springer.
3782:{\displaystyle \gamma }
3758:{\displaystyle \gamma }
3221:{\displaystyle \gamma }
113:massâenergy equivalence
50:more precise citations.
5682:Lorentz transformation
5156:Philosophical Magazine
5124:Philosophical Magazine
5092:Philosophical Magazine
4884:H. A. Lorentz (1904),
4857:H. A. Lorentz (1899),
4832:Philosophical Magazine
4798:Philosophical Magazine
4789:J. J. Thomson (1881),
4642:Foundations of Physics
4233:
4209:
4201:
4114:
4084:
4006:
3963:
3920:
3893:
3853:
3783:
3759:
3735:
3692:
3624:
3574:
3523:and of a body at rest
3517:
3445:
3222:
3194:
3167:
3129:
3099:
3052:
2986:
2950:
2896:
2690:conservation of energy
2519:
2488:
2384:
2313:
2276:
2101:
2059:
1972:
1902:
1783:
1711:
1633:
1521:
1479:
1431:
1384:
1299:
1243:
1214:
1141:
1061:obey the relativistic
1050:
1016:, given in 4-momentum
893:
839:
699:
646:
594:
544:
524:
502:
431:
379:
347:
273:
249:
85:" has two meanings in
6150:WeylâLewisâPapapetrou
5891:Raychaudhuri equation
5830:Equivalence principle
4465:"The Concept of Mass"
4228:
4202:
4124:
4085:
4022:
4007:
3964:
3921:
3899:is not valid because
3894:
3854:
3784:
3760:
3736:
3693:
3625:
3575:
3518:
3446:
3223:
3195:
3168:
3130:
3100:
3053:
2987:
2951:
2897:
2819:Further information:
2520:
2489:
2385:
2314:
2277:
2102:
2060:
1973:
1903:
1784:
1712:
1634:
1522:
1480:
1432:
1385:
1300:
1244:
1215:
1155:is the total energy.
1142:
1011:
894:
840:
700:
647:
595:
545:
525:
503:
432:
380:
348:
274:
250:
6191:Numerical relativity
6032:pulsar timing arrays
5180:R.C. Tolman (1934),
5001:A. Einstein (1906),
4951:A. Einstein (1905),
4762:Quantum Field Theory
4739:, pp. 248â249,
4352:A. Einstein (1905),
4135:
4028:
3980:
3930:
3903:
3863:
3797:
3773:
3749:
3702:
3654:
3584:
3527:
3477:
3232:
3212:
3184:
3150:
3112:
3082:
2996:
2960:
2917:
2911:Lorentz ether theory
2841:
2821:Electromagnetic mass
2598:improve this article
2500:
2407:
2323:
2294:
2257:
2167:improve this article
2109:momentum four-vector
2071:
2013:
1912:
1795:
1721:
1645:
1543:
1489:
1441:
1396:
1315:
1270:
1242:{\displaystyle E=pc}
1224:
1169:
1069:
930:improve this article
857:
806:
659:
604:
558:
534:
523:{\displaystyle \nu }
514:
446:
398:
363:
283:
263:
216:
6083:Friedmann equations
5977:HulseâTaylor binary
5939:Gravitational waves
5835:Riemannian geometry
5661:Proper acceleration
5646:Maxwell's equations
5592:Galilean relativity
5457:2008AcPPB..39..811S
5343:1987AmJPh..55..739A
5311:, 2001, p. 308
5300:A.B. Arons (1990),
5266:2009PhTea..47..336H
5254:The Physics Teacher
5024:1906AnP...326..583E
4989:English translation
4974:1905AnP...322..891E
4931:1902AnP...315..105A
4911:M. Abraham (1903),
4655:1976FoPh....6..115E
4610:1991JChEd..68..110N
4571:10.1021/ar50140a001
4530:2009AmJPh..77..430O
4486:1989PhT....42f..31O
4463:L. B. Okun (1989),
4423:1991AmJPh..59.1032S
4390:English translation
4375:1905AnP...323..639E
4320:2005EJPh...26..225R
4243:hyperbolic geometry
4111:(1989), p. 42)
3791:Newton's second law
1993:natural unit system
1266:is zero, therefore
1034:coordinates, where
440:Newton's second law
378:{\displaystyle v=0}
124:relativistic energy
6398:Special relativity
6132:ReissnerâNordström
6050:BransâDicke theory
5881:Linearized gravity
5708:Length contraction
5626:Frame of reference
5603:Special relativity
5525:2015-07-16 at the
5478:Usenet Physics FAQ
5148:R. Tolman (1912),
5116:R. Tolman (1911),
5084:R. Tolman (1911),
5011:Annalen der Physik
4961:Annalen der Physik
4918:Annalen der Physik
4778:Extract of page 70
4713:Extract of page 43
4663:10.1007/BF00708670
4362:Annalen der Physik
4247:Euclidean geometry
4197:
4080:
4012:(1905) he treated
4002:
3959:
3916:
3889:
3849:
3779:
3755:
3731:
3688:
3620:
3570:
3513:
3441:
3439:
3218:
3190:
3163:
3125:
3095:
3048:
2982:
2946:
2892:
2515:
2484:
2380:
2309:
2272:
2097:
2055:
1968:
1898:
1896:
1779:
1771:
1707:
1702:
1629:
1517:
1475:
1427:
1380:
1295:
1239:
1210:
1137:
1051:
889:
835:
796:classical momentum
738:center of momentum
695:
642:
590:
540:
520:
498:
427:
392:frame of reference
375:
357:center of momentum
343:
335:
269:
259:moving at a speed
245:
163:center of momentum
97:invariant quantity
87:special relativity
6385:
6384:
6199:
6198:
6178:OzsvĂĄthâSchĂŒcking
5784:
5783:
5766:Minkowski diagram
5723:Thomas precession
5666:Relativistic mass
5513:978-0-486-29998-3
5392:978-90-481-8636-5
5284:10.1119/1.3204111
5197:978-0-486-65383-9
4772:978-1-118-71665-6
4746:978-0-7167-2327-1
4732:Spacetime Physics
4708:978-0-8018-7047-7
4618:10.1021/ed068p110
4538:10.1119/1.3056168
4417:(11): 1032â1036,
4297:Roche, J (2005).
4195:
4193:
4191:
4078:
3947:
3913:
3881:
3859:the derived form
3844:
3824:
3712:
3664:
3609:
3596:
3565:
3508:
3487:
3471:Richard C. Tolman
3463:relativistic mass
3454:Relativistic mass
3431:
3421:
3406:
3383:
3366:
3356:
3341:
3318:
3301:
3291:
3276:
3246:
3193:{\displaystyle m}
3160:
3122:
3106:longitudinal mass
3092:
3046:
2970:
2927:
2874:
2865:
2851:
2787:, and divided by
2674:
2673:
2666:
2648:
2512:
2472:
2467:
2392:In the system of
2378:
2367:
2306:
2269:
2249:The total energy
2243:
2242:
2235:
2217:
1963:
1961:
1908:where the factor
1774:
1773:
1770:
1705:
1704:
1701:
1641:This results in
1624:
1512:
1464:
1419:
1378:
1006:
1005:
998:
980:
847:four-acceleration
776:massless particle
675:
493:
473:
416:
338:
337:
334:
293:
272:{\displaystyle v}
232:
201:relativistic mass
195:Relativistic mass
109:relativistic mass
76:
75:
68:
6410:
6377:
6376:
6160:van Stockum dust
5932:Two-body problem
5850:Mach's principle
5797:
5796:
5738:Terrell rotation
5578:
5577:
5556:
5549:
5542:
5533:
5532:
5516:
5474:
5473:
5460:
5459:
5449:
5417:
5411:
5405:
5404:
5376:
5370:
5369:
5367:
5365:
5359:
5353:. Archived from
5328:
5319:
5313:
5312:
5305:
5297:
5288:
5287:
5277:
5249:
5243:
5242:
5240:
5239:
5228:
5222:
5208:
5177:
5171:
5170:
5162:(135): 375â380,
5153:
5145:
5139:
5138:
5130:(129): 458â463,
5121:
5113:
5107:
5106:
5098:(123): 296â301,
5089:
5081:
5075:
5074:
5064:10.2307/20022495
5049:
5041:
5035:
5034:
5007:
4998:
4992:
4986:
4985:
4957:
4948:
4942:
4941:
4908:
4902:
4901:
4889:
4881:
4875:
4874:
4862:
4854:
4848:
4847:
4839:(269): 329â341,
4828:
4820:
4814:
4813:
4794:
4786:
4780:
4776:
4756:
4750:
4749:
4726:
4715:
4711:
4692:
4686:
4680:
4674:
4673:
4636:
4630:
4629:
4589:
4583:
4582:
4556:
4547:
4541:
4540:
4511:
4505:
4504:
4502:
4496:, archived from
4494:10.1063/1.881171
4469:
4460:
4447:
4440:
4434:
4433:
4404:
4393:
4387:
4386:
4358:
4349:
4340:
4339:
4303:
4294:
4269:
4264:
4263:
4240:
4221:
4206:
4204:
4203:
4198:
4196:
4194:
4192:
4190:
4189:
4180:
4179:
4170:
4162:
4160:
4156:
4155:
4145:
4112:
4089:
4087:
4086:
4081:
4079:
4077:
4076:
4067:
4062:
4061:
4046:
4044:
4015:
4011:
4009:
4008:
4003:
4001:
4000:
3974:previous section
3968:
3966:
3965:
3960:
3958:
3954:
3949:
3948:
3945:
3925:
3923:
3922:
3917:
3915:
3914:
3911:
3898:
3896:
3895:
3890:
3888:
3883:
3882:
3879:
3870:
3858:
3856:
3855:
3850:
3845:
3843:
3835:
3831:
3826:
3825:
3822:
3809:
3804:
3788:
3786:
3785:
3780:
3764:
3762:
3761:
3756:
3740:
3738:
3737:
3732:
3730:
3729:
3714:
3713:
3710:
3697:
3695:
3694:
3689:
3687:
3686:
3677:
3666:
3665:
3662:
3629:
3627:
3626:
3621:
3610:
3608:
3607:
3598:
3597:
3594:
3588:
3579:
3577:
3576:
3571:
3566:
3564:
3563:
3554:
3553:
3544:
3539:
3538:
3522:
3520:
3519:
3514:
3509:
3507:
3506:
3494:
3489:
3488:
3485:
3467:Gilbert N. Lewis
3450:
3448:
3447:
3442:
3440:
3433:
3432:
3429:
3423:
3422:
3419:
3408:
3407:
3404:
3385:
3384:
3381:
3368:
3367:
3364:
3358:
3357:
3354:
3343:
3342:
3339:
3320:
3319:
3316:
3303:
3302:
3299:
3293:
3292:
3289:
3278:
3277:
3274:
3268:
3267:
3248:
3247:
3244:
3227:
3225:
3224:
3219:
3199:
3197:
3196:
3191:
3172:
3170:
3169:
3164:
3162:
3161:
3158:
3134:
3132:
3131:
3126:
3124:
3123:
3120:
3104:
3102:
3101:
3096:
3094:
3093:
3090:
3073:
3067:
3057:
3055:
3054:
3049:
3047:
3045:
3044:
3035:
3030:
3029:
3014:
3012:
2991:
2989:
2988:
2983:
2972:
2971:
2968:
2955:
2953:
2952:
2947:
2942:
2941:
2929:
2928:
2925:
2901:
2899:
2898:
2893:
2891:
2890:
2881:
2876:
2875:
2872:
2866:
2858:
2853:
2852:
2849:
2831:Oliver Heaviside
2792:
2773:
2769:
2765:
2669:
2662:
2658:
2655:
2649:
2647:
2606:
2582:
2574:
2552:particle physics
2542:
2524:
2522:
2521:
2516:
2514:
2513:
2505:
2493:
2491:
2490:
2485:
2483:
2482:
2477:
2473:
2470:
2469:
2468:
2460:
2445:
2444:
2439:
2435:
2419:
2418:
2402:
2389:
2387:
2386:
2381:
2379:
2377:
2376:
2366:
2365:
2344:
2343:
2334:
2333:
2318:
2316:
2315:
2310:
2308:
2307:
2299:
2289:
2285:
2281:
2279:
2278:
2273:
2271:
2270:
2262:
2252:
2238:
2231:
2227:
2224:
2218:
2216:
2175:
2151:
2143:
2134:
2124:
2117:
2106:
2104:
2103:
2098:
2096:
2095:
2083:
2082:
2064:
2062:
2061:
2056:
2051:
2050:
2038:
2037:
2025:
2024:
1990:
1980:When working in
1977:
1975:
1974:
1969:
1964:
1962:
1960:
1959:
1950:
1949:
1940:
1932:
1930:
1925:
1907:
1905:
1904:
1899:
1897:
1864:
1863:
1831:
1830:
1811:
1810:
1788:
1786:
1785:
1780:
1775:
1772:
1769:
1768:
1759:
1758:
1749:
1740:
1739:
1731:
1716:
1714:
1713:
1708:
1706:
1703:
1700:
1699:
1690:
1689:
1680:
1671:
1670:
1669:
1668:
1655:
1638:
1636:
1635:
1630:
1625:
1623:
1622:
1613:
1612:
1603:
1601:
1600:
1588:
1587:
1582:
1578:
1577:
1576:
1555:
1554:
1538:
1532:
1526:
1524:
1523:
1518:
1513:
1505:
1484:
1482:
1481:
1476:
1474:
1470:
1466:
1465:
1457:
1436:
1434:
1433:
1428:
1426:
1422:
1421:
1420:
1412:
1389:
1387:
1386:
1381:
1379:
1377:
1376:
1355:
1354:
1349:
1345:
1344:
1343:
1325:
1304:
1302:
1301:
1296:
1291:
1290:
1265:
1256:
1248:
1246:
1245:
1240:
1219:
1217:
1216:
1211:
1203:
1202:
1181:
1180:
1164:
1154:
1146:
1144:
1143:
1138:
1136:
1135:
1130:
1126:
1125:
1124:
1103:
1102:
1081:
1080:
1060:
1056:
1049:
1033:
1001:
994:
990:
987:
981:
979:
938:
914:
906:
898:
896:
895:
890:
885:
884:
869:
868:
844:
842:
841:
836:
834:
833:
818:
817:
790:is the ratio of
704:
702:
701:
696:
691:
677:
676:
673:
651:
649:
648:
643:
635:
624:
623:
614:
599:
597:
596:
591:
586:
549:
547:
546:
541:
529:
527:
526:
521:
507:
505:
504:
499:
494:
492:
484:
480:
475:
474:
471:
458:
453:
436:
434:
433:
428:
423:
418:
417:
414:
405:
384:
382:
381:
376:
352:
350:
349:
344:
339:
336:
333:
332:
323:
322:
313:
304:
300:
295:
294:
291:
278:
276:
275:
270:
258:
254:
252:
251:
246:
244:
243:
234:
233:
230:
208:
186:
105:reference frames
71:
64:
60:
57:
51:
46:this article by
37:inline citations
24:
23:
16:
6418:
6417:
6413:
6412:
6411:
6409:
6408:
6407:
6388:
6387:
6386:
6381:
6367:
6195:
6099:BKL singularity
6089:LemaĂźtreâTolman
6064:
6060:Quantum gravity
6042:
6036:
6022:geodetic effect
5996:(together with
5966:LISA Pathfinder
5905:
5854:
5840:Penrose diagram
5822:
5816:
5791:
5780:
5776:Minkowski space
5742:
5686:
5670:
5618:
5612:
5572:
5565:
5560:
5527:Wayback Machine
5514:
5471:physics/0504110
5426:
5421:
5420:
5412:
5408:
5393:
5377:
5373:
5363:
5361:
5357:
5351:10.1119/1.15314
5326:
5320:
5316:
5307:
5298:
5291:
5275:10.1.1.205.5072
5250:
5246:
5237:
5235:
5229:
5225:
5198:
5188:Clarendon Press
5178:
5174:
5146:
5142:
5114:
5110:
5082:
5078:
5058:(25): 709â726,
5042:
5038:
5018:(13): 583â586,
5005:
4999:
4995:
4968:(10): 891â921,
4955:
4949:
4945:
4909:
4905:
4882:
4878:
4855:
4851:
4821:
4817:
4805:(68): 229â249,
4787:
4783:
4773:
4757:
4753:
4747:
4727:
4718:
4709:
4693:
4689:
4681:
4677:
4637:
4633:
4590:
4586:
4554:
4548:
4544:
4512:
4508:
4500:
4467:
4461:
4450:
4441:
4437:
4431:10.1119/1.16642
4405:
4396:
4369:(13): 639â643,
4356:
4350:
4343:
4301:
4299:"What is mass?"
4295:
4288:
4283:
4265:
4258:
4255:
4236:
4213:
4185:
4181:
4175:
4171:
4169:
4161:
4151:
4147:
4146:
4144:
4136:
4133:
4132:
4119:
4113:
4105:Lincoln Barnett
4102:
4072:
4068:
4063:
4057:
4053:
4045:
4040:
4029:
4026:
4025:
4013:
3996:
3992:
3981:
3978:
3977:
3950:
3944:
3940:
3933:
3931:
3928:
3927:
3910:
3906:
3904:
3901:
3900:
3884:
3878:
3874:
3866:
3864:
3861:
3860:
3836:
3827:
3821:
3817:
3810:
3808:
3800:
3798:
3795:
3794:
3774:
3771:
3770:
3750:
3747:
3746:
3725:
3721:
3709:
3705:
3703:
3700:
3699:
3682:
3678:
3673:
3661:
3657:
3655:
3652:
3651:
3638:
3603:
3599:
3593:
3589:
3587:
3585:
3582:
3581:
3580:with the ratio
3559:
3555:
3549:
3545:
3543:
3534:
3530:
3528:
3525:
3524:
3502:
3498:
3493:
3484:
3480:
3478:
3475:
3474:
3456:
3438:
3437:
3428:
3424:
3418:
3414:
3409:
3403:
3399:
3386:
3380:
3376:
3373:
3372:
3363:
3359:
3353:
3349:
3344:
3338:
3334:
3321:
3315:
3311:
3308:
3307:
3298:
3294:
3288:
3284:
3279:
3273:
3269:
3263:
3259:
3249:
3243:
3239:
3235:
3233:
3230:
3229:
3213:
3210:
3209:
3185:
3182:
3181:
3157:
3153:
3151:
3148:
3147:
3144:Albert Einstein
3137:transverse mass
3119:
3115:
3113:
3110:
3109:
3089:
3085:
3083:
3080:
3079:
3069:
3063:
3040:
3036:
3031:
3025:
3021:
3013:
3008:
2997:
2994:
2993:
2967:
2963:
2961:
2958:
2957:
2937:
2933:
2924:
2920:
2918:
2915:
2914:
2907:Hendrik Lorentz
2886:
2882:
2877:
2871:
2867:
2857:
2848:
2844:
2842:
2839:
2838:
2823:
2817:
2812:
2788:
2780:
2771:
2767:
2757:
2737:absolute values
2732:
2670:
2659:
2653:
2650:
2607:
2605:
2595:
2583:
2572:
2548:isolated system
2537:
2504:
2503:
2501:
2498:
2497:
2478:
2459:
2458:
2454:
2450:
2449:
2440:
2428:
2424:
2423:
2414:
2410:
2408:
2405:
2404:
2397:
2372:
2368:
2361:
2357:
2339:
2335:
2332:
2324:
2321:
2320:
2298:
2297:
2295:
2292:
2291:
2287:
2283:
2261:
2260:
2258:
2255:
2254:
2250:
2239:
2228:
2222:
2219:
2176:
2174:
2164:
2152:
2141:
2126:
2119:
2112:
2091:
2087:
2078:
2074:
2072:
2069:
2068:
2046:
2042:
2033:
2029:
2020:
2016:
2014:
2011:
2010:
1991:, known as the
1985:
1955:
1951:
1945:
1941:
1939:
1931:
1926:
1921:
1913:
1910:
1909:
1895:
1894:
1875:
1869:
1868:
1859:
1855:
1842:
1836:
1835:
1826:
1822:
1812:
1806:
1802:
1798:
1796:
1793:
1792:
1764:
1760:
1754:
1750:
1747:
1732:
1730:
1722:
1719:
1718:
1695:
1691:
1685:
1681:
1678:
1664:
1660:
1656:
1654:
1646:
1643:
1642:
1618:
1614:
1608:
1604:
1602:
1596:
1592:
1583:
1572:
1568:
1564:
1560:
1559:
1550:
1546:
1544:
1541:
1540:
1537:
1534:
1531:
1528:
1504:
1490:
1487:
1486:
1456:
1455:
1448:
1444:
1442:
1439:
1438:
1411:
1410:
1403:
1399:
1397:
1394:
1393:
1372:
1368:
1350:
1339:
1335:
1331:
1327:
1326:
1324:
1316:
1313:
1312:
1286:
1282:
1271:
1268:
1267:
1264:
1261:
1252:
1225:
1222:
1221:
1198:
1194:
1176:
1172:
1170:
1167:
1166:
1162:
1159:
1152:
1150:
1131:
1120:
1116:
1112:
1108:
1107:
1098:
1094:
1076:
1072:
1070:
1067:
1066:
1058:
1054:
1041:
1035:
1031:
1024:
1017:
1002:
991:
985:
982:
939:
937:
927:
915:
904:
880:
876:
864:
860:
858:
855:
854:
829:
825:
813:
809:
807:
804:
803:
784:
733:
687:
672:
668:
660:
657:
656:
631:
619:
615:
610:
605:
602:
601:
582:
559:
556:
555:
535:
532:
531:
530:and wavelength
515:
512:
511:
485:
476:
470:
466:
459:
457:
449:
447:
444:
443:
419:
413:
409:
401:
399:
396:
395:
364:
361:
360:
328:
324:
318:
314:
311:
299:
290:
286:
284:
281:
280:
264:
261:
260:
256:
239:
235:
229:
225:
217:
214:
213:
204:
197:
189:isolated system
182:
144:
79:
72:
61:
55:
52:
42:Please help to
41:
25:
21:
12:
11:
5:
6416:
6406:
6405:
6400:
6383:
6382:
6372:
6369:
6368:
6366:
6365:
6358:
6353:
6348:
6343:
6338:
6333:
6328:
6323:
6318:
6313:
6308:
6303:
6298:
6293:
6288:
6286:Choquet-Bruhat
6283:
6278:
6273:
6268:
6263:
6258:
6253:
6248:
6243:
6238:
6233:
6228:
6223:
6218:
6213:
6207:
6205:
6201:
6200:
6197:
6196:
6194:
6193:
6186:
6185:
6180:
6175:
6168:
6167:
6162:
6157:
6152:
6147:
6138:Axisymmetric:
6135:
6134:
6129:
6123:
6112:
6111:
6106:
6101:
6096:
6091:
6086:
6077:Cosmological:
6074:
6072:
6066:
6065:
6063:
6062:
6057:
6052:
6046:
6044:
6038:
6037:
6035:
6034:
6029:
6018:frame-dragging
6015:
6010:
6005:
6002:Einstein rings
5998:Einstein cross
5991:
5980:
5979:
5974:
5968:
5963:
5958:
5945:
5935:
5934:
5929:
5924:
5919:
5913:
5911:
5907:
5906:
5904:
5903:
5901:Ernst equation
5898:
5893:
5888:
5883:
5878:
5873:
5871:BSSN formalism
5868:
5862:
5860:
5856:
5855:
5853:
5852:
5847:
5842:
5837:
5832:
5826:
5824:
5818:
5817:
5815:
5814:
5809:
5803:
5801:
5794:
5786:
5785:
5782:
5781:
5779:
5778:
5773:
5768:
5763:
5758:
5752:
5750:
5744:
5743:
5741:
5740:
5735:
5730:
5728:Ladder paradox
5725:
5720:
5715:
5710:
5705:
5700:
5694:
5692:
5688:
5687:
5685:
5684:
5678:
5676:
5672:
5671:
5669:
5668:
5663:
5658:
5653:
5648:
5643:
5638:
5633:
5631:Speed of light
5628:
5622:
5620:
5614:
5613:
5611:
5610:
5605:
5600:
5594:
5584:
5582:
5575:
5567:
5566:
5559:
5558:
5551:
5544:
5536:
5530:
5529:
5517:
5512:
5494:
5493:
5492:
5486:
5475:
5461:
5440:(4): 811â885,
5425:
5424:External links
5422:
5419:
5418:
5406:
5391:
5371:
5360:on May 6, 2021
5337:(8): 739â743.
5314:
5289:
5260:(6): 336â341.
5244:
5223:
5196:
5172:
5140:
5108:
5076:
5036:
4993:
4943:
4925:(1): 105â179,
4903:
4876:
4849:
4815:
4781:
4771:
4751:
4745:
4716:
4707:
4687:
4675:
4649:(1): 115â124,
4631:
4604:(2): 110â113.
4584:
4565:(8): 271â276.
4542:
4524:(5): 430â431,
4506:
4448:
4435:
4394:
4341:
4285:
4284:
4282:
4279:
4278:
4277:
4271:
4270:
4267:Physics portal
4254:
4251:
4188:
4184:
4178:
4174:
4168:
4165:
4159:
4154:
4150:
4143:
4140:
4118:
4115:
4100:
4075:
4071:
4066:
4060:
4056:
4052:
4049:
4043:
4039:
4036:
4033:
3999:
3995:
3991:
3988:
3985:
3957:
3953:
3943:
3939:
3936:
3909:
3887:
3877:
3873:
3869:
3848:
3842:
3839:
3834:
3830:
3820:
3816:
3813:
3807:
3803:
3778:
3754:
3728:
3724:
3720:
3717:
3708:
3685:
3681:
3676:
3672:
3669:
3660:
3636:
3619:
3616:
3613:
3606:
3602:
3592:
3569:
3562:
3558:
3552:
3548:
3542:
3537:
3533:
3512:
3505:
3501:
3497:
3492:
3483:
3455:
3452:
3436:
3427:
3417:
3413:
3410:
3402:
3398:
3395:
3392:
3389:
3387:
3379:
3375:
3374:
3371:
3362:
3352:
3348:
3345:
3337:
3333:
3330:
3327:
3324:
3322:
3314:
3310:
3309:
3306:
3297:
3287:
3283:
3280:
3272:
3266:
3262:
3258:
3255:
3252:
3250:
3242:
3238:
3237:
3217:
3200:moving in the
3189:
3156:
3118:
3088:
3078:(1902) called
3060:Lorentz factor
3043:
3039:
3034:
3028:
3024:
3020:
3017:
3011:
3007:
3004:
3001:
2981:
2978:
2975:
2966:
2945:
2940:
2936:
2932:
2923:
2889:
2885:
2880:
2870:
2864:
2861:
2856:
2847:
2816:
2813:
2811:
2808:
2779:
2776:
2731:
2728:
2724:invariant mass
2719:center of mass
2707:
2703:
2672:
2671:
2586:
2584:
2577:
2571:
2568:
2527:Euclidean norm
2511:
2508:
2481:
2476:
2466:
2463:
2457:
2453:
2448:
2443:
2438:
2434:
2431:
2427:
2422:
2417:
2413:
2375:
2371:
2364:
2360:
2356:
2353:
2350:
2347:
2342:
2338:
2331:
2328:
2305:
2302:
2268:
2265:
2241:
2240:
2155:
2153:
2146:
2140:
2137:
2094:
2090:
2086:
2081:
2077:
2054:
2049:
2045:
2041:
2036:
2032:
2028:
2023:
2019:
1967:
1958:
1954:
1948:
1944:
1938:
1935:
1929:
1924:
1920:
1917:
1893:
1890:
1887:
1884:
1881:
1878:
1876:
1874:
1871:
1870:
1867:
1862:
1858:
1854:
1851:
1848:
1845:
1843:
1841:
1838:
1837:
1834:
1829:
1825:
1821:
1818:
1815:
1813:
1809:
1805:
1801:
1800:
1778:
1767:
1763:
1757:
1753:
1746:
1743:
1738:
1735:
1729:
1726:
1698:
1694:
1688:
1684:
1677:
1674:
1667:
1663:
1659:
1653:
1650:
1628:
1621:
1617:
1611:
1607:
1599:
1595:
1591:
1586:
1581:
1575:
1571:
1567:
1563:
1558:
1553:
1549:
1535:
1529:
1516:
1511:
1508:
1503:
1500:
1497:
1494:
1473:
1469:
1463:
1460:
1454:
1451:
1447:
1425:
1418:
1415:
1409:
1406:
1402:
1375:
1371:
1367:
1364:
1361:
1358:
1353:
1348:
1342:
1338:
1334:
1330:
1323:
1320:
1294:
1289:
1285:
1281:
1278:
1275:
1262:
1238:
1235:
1232:
1229:
1220:and therefore
1209:
1206:
1201:
1197:
1193:
1190:
1187:
1184:
1179:
1175:
1160:
1148:
1134:
1129:
1123:
1119:
1115:
1111:
1106:
1101:
1097:
1093:
1090:
1087:
1084:
1079:
1075:
1039:
1029:
1022:
1004:
1003:
918:
916:
909:
903:
900:
888:
883:
879:
875:
872:
867:
863:
832:
828:
824:
821:
816:
812:
788:invariant mass
783:
782:Invariant mass
780:
768:center of mass
764:inertial frame
756:center of mass
732:
729:
720:center of mass
694:
690:
686:
683:
680:
671:
667:
664:
641:
638:
634:
630:
627:
622:
618:
613:
609:
589:
585:
581:
578:
575:
572:
569:
566:
563:
539:
519:
497:
491:
488:
483:
479:
469:
465:
462:
456:
452:
426:
422:
412:
408:
404:
388:center of mass
374:
371:
368:
342:
331:
327:
321:
317:
310:
307:
303:
298:
289:
268:
242:
238:
228:
224:
221:
196:
193:
172:speed of light
153:invariant mass
143:
140:
92:invariant mass
77:
74:
73:
28:
26:
19:
9:
6:
4:
3:
2:
6415:
6404:
6401:
6399:
6396:
6395:
6393:
6380:
6370:
6364:
6363:
6359:
6357:
6354:
6352:
6349:
6347:
6344:
6342:
6339:
6337:
6334:
6332:
6329:
6327:
6324:
6322:
6319:
6317:
6314:
6312:
6309:
6307:
6304:
6302:
6299:
6297:
6294:
6292:
6289:
6287:
6284:
6282:
6279:
6277:
6274:
6272:
6271:Chandrasekhar
6269:
6267:
6264:
6262:
6259:
6257:
6254:
6252:
6249:
6247:
6244:
6242:
6239:
6237:
6234:
6232:
6231:Schwarzschild
6229:
6227:
6224:
6222:
6219:
6217:
6214:
6212:
6209:
6208:
6206:
6202:
6192:
6188:
6187:
6184:
6181:
6179:
6176:
6174:
6170:
6169:
6166:
6163:
6161:
6158:
6156:
6153:
6151:
6148:
6145:
6141:
6137:
6136:
6133:
6130:
6127:
6124:
6122:
6118:
6117:Schwarzschild
6114:
6113:
6110:
6107:
6105:
6102:
6100:
6097:
6095:
6092:
6090:
6087:
6084:
6080:
6076:
6075:
6073:
6071:
6067:
6061:
6058:
6056:
6053:
6051:
6048:
6047:
6045:
6039:
6033:
6030:
6027:
6023:
6019:
6016:
6014:
6013:Shapiro delay
6011:
6009:
6006:
6003:
5999:
5995:
5992:
5989:
5985:
5982:
5981:
5978:
5975:
5972:
5969:
5967:
5964:
5962:
5959:
5957:
5956:collaboration
5953:
5949:
5946:
5944:
5940:
5937:
5936:
5933:
5930:
5928:
5925:
5923:
5922:Event horizon
5920:
5918:
5915:
5914:
5912:
5908:
5902:
5899:
5897:
5894:
5892:
5889:
5887:
5884:
5882:
5879:
5877:
5874:
5872:
5869:
5867:
5866:ADM formalism
5864:
5863:
5861:
5857:
5851:
5848:
5846:
5843:
5841:
5838:
5836:
5833:
5831:
5828:
5827:
5825:
5819:
5813:
5810:
5808:
5805:
5804:
5802:
5798:
5795:
5793:
5787:
5777:
5774:
5772:
5771:Biquaternions
5769:
5767:
5764:
5762:
5759:
5757:
5754:
5753:
5751:
5749:
5745:
5739:
5736:
5734:
5731:
5729:
5726:
5724:
5721:
5719:
5716:
5714:
5711:
5709:
5706:
5704:
5701:
5699:
5698:Time dilation
5696:
5695:
5693:
5689:
5683:
5680:
5679:
5677:
5673:
5667:
5664:
5662:
5659:
5657:
5654:
5652:
5651:Proper length
5649:
5647:
5644:
5642:
5639:
5637:
5634:
5632:
5629:
5627:
5624:
5623:
5621:
5615:
5609:
5606:
5604:
5601:
5598:
5595:
5593:
5589:
5586:
5585:
5583:
5579:
5576:
5574:
5568:
5564:
5557:
5552:
5550:
5545:
5543:
5538:
5537:
5534:
5528:
5524:
5521:
5518:
5515:
5509:
5505:
5504:
5499:
5495:
5490:
5487:
5484:
5481:
5480:
5479:
5476:
5472:
5467:
5462:
5458:
5453:
5448:
5443:
5439:
5435:
5434:
5428:
5427:
5415:
5410:
5402:
5398:
5394:
5388:
5384:
5383:
5375:
5356:
5352:
5348:
5344:
5340:
5336:
5332:
5325:
5318:
5310:
5304:, p. 263
5303:
5296:
5294:
5285:
5281:
5276:
5271:
5267:
5263:
5259:
5255:
5248:
5234:
5227:
5220:
5219:0-486-65383-8
5216:
5212:
5207:
5203:
5199:
5193:
5189:
5185:
5184:
5176:
5169:
5165:
5161:
5157:
5152:
5144:
5137:
5133:
5129:
5125:
5120:
5112:
5105:
5101:
5097:
5093:
5088:
5080:
5073:
5069:
5065:
5061:
5057:
5053:
5048:
5040:
5033:
5029:
5025:
5021:
5017:
5014:(in German),
5013:
5012:
5004:
4997:
4990:
4984:
4979:
4975:
4971:
4967:
4964:(in German),
4963:
4962:
4954:
4947:
4940:
4936:
4932:
4928:
4924:
4920:
4919:
4914:
4907:
4899:
4895:
4894:
4888:
4880:
4872:
4868:
4867:
4861:
4853:
4846:
4842:
4838:
4834:
4833:
4827:
4819:
4812:
4808:
4804:
4800:
4799:
4793:
4785:
4779:
4774:
4768:
4764:
4763:
4755:
4748:
4742:
4738:
4734:
4733:
4725:
4723:
4721:
4714:
4710:
4704:
4700:
4699:
4691:
4685:
4679:
4672:
4668:
4664:
4660:
4656:
4652:
4648:
4644:
4643:
4635:
4627:
4623:
4619:
4615:
4611:
4607:
4603:
4599:
4595:
4588:
4580:
4576:
4572:
4568:
4564:
4560:
4553:
4546:
4539:
4535:
4531:
4527:
4523:
4519:
4518:
4510:
4503:on 2011-06-07
4499:
4495:
4491:
4487:
4483:
4479:
4475:
4474:
4473:Physics Today
4466:
4459:
4457:
4455:
4453:
4445:
4439:
4432:
4428:
4424:
4420:
4416:
4412:
4411:
4403:
4401:
4399:
4391:
4385:
4380:
4376:
4372:
4368:
4365:(in German),
4364:
4363:
4355:
4348:
4346:
4337:
4333:
4329:
4325:
4321:
4317:
4313:
4309:
4308:
4300:
4293:
4291:
4286:
4276:
4273:
4272:
4268:
4262:
4257:
4250:
4248:
4244:
4239:
4232:
4227:
4224:
4223:of inertia".
4220:
4216:
4208:
4186:
4182:
4176:
4172:
4166:
4163:
4157:
4152:
4148:
4141:
4138:
4130:
4123:
4110:
4106:
4099:
4097:
4093:
4073:
4069:
4064:
4058:
4054:
4050:
4047:
4041:
4037:
4034:
4031:
4021:
4019:
3997:
3993:
3989:
3986:
3983:
3975:
3970:
3941:
3934:
3907:
3875:
3871:
3846:
3840:
3837:
3818:
3811:
3805:
3792:
3776:
3768:
3752:
3743:
3726:
3722:
3718:
3715:
3706:
3683:
3679:
3674:
3670:
3667:
3658:
3648:
3646:
3642:
3635:
3630:
3617:
3614:
3611:
3604:
3600:
3590:
3567:
3560:
3556:
3550:
3546:
3540:
3535:
3531:
3510:
3503:
3499:
3495:
3490:
3481:
3472:
3468:
3464:
3459:
3451:
3434:
3425:
3415:
3411:
3400:
3396:
3393:
3390:
3388:
3377:
3369:
3360:
3350:
3346:
3335:
3331:
3328:
3325:
3323:
3312:
3304:
3295:
3285:
3281:
3270:
3264:
3260:
3256:
3253:
3251:
3240:
3215:
3207:
3203:
3187:
3178:
3176:
3154:
3145:
3141:
3139:
3138:
3116:
3107:
3086:
3077:
3072:
3066:
3061:
3041:
3037:
3032:
3026:
3022:
3018:
3015:
3009:
3005:
3002:
2999:
2979:
2976:
2973:
2964:
2943:
2938:
2934:
2930:
2921:
2912:
2908:
2903:
2887:
2883:
2878:
2868:
2862:
2859:
2854:
2845:
2836:
2832:
2828:
2827:J. J. Thomson
2822:
2807:
2804:
2800:
2794:
2791:
2786:
2775:
2764:
2760:
2753:
2749:
2746:
2742:
2738:
2727:
2725:
2720:
2714:
2712:
2705:
2701:
2698:
2696:
2691:
2686:
2684:
2678:
2668:
2665:
2657:
2654:February 2016
2646:
2643:
2639:
2636:
2632:
2629:
2625:
2622:
2618:
2615: â
2614:
2610:
2609:Find sources:
2603:
2599:
2593:
2592:
2587:This article
2585:
2581:
2576:
2575:
2567:
2565:
2561:
2557:
2553:
2549:
2544:
2540:
2535:
2530:
2528:
2506:
2494:
2479:
2461:
2455:
2446:
2441:
2436:
2432:
2429:
2425:
2420:
2415:
2411:
2400:
2395:
2394:natural units
2390:
2373:
2369:
2362:
2354:
2351:
2345:
2340:
2336:
2329:
2326:
2300:
2263:
2247:
2237:
2234:
2226:
2223:February 2016
2215:
2212:
2208:
2205:
2201:
2198:
2194:
2191:
2187:
2184: â
2183:
2179:
2178:Find sources:
2172:
2168:
2162:
2161:
2156:This article
2154:
2150:
2145:
2144:
2136:
2133:
2129:
2122:
2115:
2110:
2092:
2088:
2084:
2079:
2075:
2065:
2052:
2047:
2043:
2039:
2034:
2030:
2026:
2021:
2017:
2008:
2006:
2002:
1998:
1994:
1988:
1983:
1978:
1965:
1956:
1952:
1946:
1942:
1936:
1933:
1927:
1922:
1918:
1915:
1891:
1888:
1885:
1882:
1879:
1877:
1872:
1865:
1860:
1856:
1852:
1849:
1846:
1844:
1839:
1832:
1827:
1823:
1819:
1816:
1814:
1807:
1803:
1789:
1776:
1765:
1761:
1755:
1751:
1744:
1741:
1736:
1733:
1727:
1724:
1696:
1692:
1686:
1682:
1675:
1672:
1665:
1661:
1657:
1651:
1648:
1639:
1626:
1619:
1615:
1609:
1605:
1597:
1593:
1589:
1584:
1579:
1573:
1569:
1565:
1561:
1556:
1551:
1547:
1514:
1509:
1506:
1501:
1498:
1495:
1492:
1471:
1467:
1458:
1452:
1449:
1445:
1423:
1413:
1407:
1404:
1400:
1390:
1373:
1365:
1362:
1356:
1351:
1346:
1340:
1336:
1332:
1328:
1321:
1318:
1309:
1306:
1292:
1287:
1283:
1279:
1276:
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1199:
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1177:
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1064:
1048:
1044:
1038:
1028:
1021:
1015:
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1000:
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989:
986:February 2016
978:
975:
971:
968:
964:
961:
957:
954:
950:
947: â
946:
942:
941:Find sources:
935:
931:
925:
924:
919:This article
917:
913:
908:
907:
899:
886:
881:
877:
873:
870:
865:
861:
852:
848:
830:
826:
822:
819:
814:
810:
801:
800:four-velocity
797:
793:
792:four-momentum
789:
779:
777:
772:
769:
766:in which the
765:
761:
757:
751:
748:
743:
739:
728:
726:
721:
715:
713:
708:
692:
688:
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665:
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587:
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489:
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125:
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98:
94:
93:
88:
84:
70:
67:
59:
49:
45:
39:
38:
32:
27:
18:
17:
6361:
6055:KaluzaâKlein
5807:Introduction
5733:Twin paradox
5665:
5502:
5437:
5431:
5409:
5381:
5374:
5364:December 12,
5362:. Retrieved
5355:the original
5334:
5330:
5317:
5308:
5301:
5257:
5253:
5247:
5236:. Retrieved
5226:
5182:
5175:
5159:
5155:
5143:
5127:
5123:
5111:
5095:
5091:
5079:
5055:
5051:
5039:
5015:
5009:
4996:
4965:
4959:
4946:
4922:
4916:
4906:
4897:
4891:
4879:
4870:
4864:
4852:
4836:
4830:
4818:
4802:
4796:
4784:
4761:
4754:
4731:
4697:
4690:
4678:
4646:
4640:
4634:
4601:
4597:
4587:
4562:
4558:
4545:
4521:
4515:
4509:
4498:the original
4480:(6): 31â36,
4477:
4471:
4443:
4438:
4414:
4408:
4366:
4360:
4311:
4305:
4237:
4234:
4229:
4225:
4218:
4214:
4210:
4128:
4125:
4120:
4095:
4091:
4023:
4017:
3971:
3766:
3744:
3649:
3644:
3640:
3633:
3631:
3462:
3460:
3457:
3205:
3201:
3179:
3175:next section
3142:
3135:
3105:
3070:
3064:
2904:
2824:
2802:
2798:
2795:
2789:
2781:
2762:
2758:
2754:
2750:
2744:
2740:
2736:
2733:
2723:
2715:
2699:
2694:
2687:
2682:
2679:
2675:
2660:
2651:
2641:
2634:
2627:
2620:
2608:
2596:Please help
2591:verification
2588:
2545:
2538:
2531:
2495:
2398:
2391:
2248:
2244:
2229:
2220:
2210:
2203:
2196:
2189:
2177:
2165:Please help
2160:verification
2157:
2131:
2127:
2120:
2113:
2066:
2009:
1986:
1979:
1790:
1640:
1391:
1310:
1307:
1259:
1253:
1250:
1157:
1052:
1046:
1042:
1036:
1026:
1019:
1013:
992:
983:
973:
966:
959:
952:
940:
928:Please help
923:verification
920:
785:
774:A so-called
773:
752:
746:
734:
724:
716:
706:
509:
438:
354:
205:
200:
198:
183:
180:
176:
167:
157:
151:
147:
145:
133:
129:
122:
116:
108:
107:, while the
90:
80:
62:
53:
34:
6144:KerrâNewman
6115:Spherical:
5984:Other tests
5927:Singularity
5859:Formulation
5821:Fundamental
5675:Formulation
5656:Proper time
5617:Fundamental
3076:Max Abraham
2833:(1889) and
707:fundamental
118:rest energy
56:August 2023
48:introducing
6392:Categories
6296:Zel'dovich
6204:Scientists
6183:Alcubierre
5990:of Mercury
5988:precession
5917:Black hole
5800:Background
5792:relativity
5761:World line
5756:Light cone
5581:Background
5573:relativity
5563:Relativity
5498:Max Jammer
5238:2011-09-27
5186:, Oxford:
4314:(2): 225.
4281:References
2704:conserved
2624:newspapers
2193:newspapers
1147:where the
956:newspapers
851:four-force
554:of energy
174:squared).
81:The word "
31:references
6266:Robertson
6251:Friedmann
6246:Eddington
6236:de Sitter
6070:Solutions
5948:detectors
5943:astronomy
5910:Phenomena
5845:Geodesics
5748:Spacetime
5691:Phenomena
5447:0708.0929
5401:663096629
5270:CiteSeerX
4900:: 809â831
4873:: 427â442
4671:120139174
4626:0021-9584
4336:122254861
4167:−
4109:L.B. Okun
4051:−
4018:rest mass
3777:γ
3753:γ
3719:γ
3615:γ
3397:γ
3332:γ
3261:γ
3216:γ
3019:−
3000:γ
2977:γ
2935:γ
2564:top quark
2556:rest mass
2510:→
2465:→
2456:∑
2447:−
2430:∑
2346:−
2304:→
2267:→
2085:−
2040:−
1937:−
1916:γ
1889:γ
1850:γ
1745:−
1676:−
1462:→
1417:→
1183:−
1083:−
882:μ
866:μ
831:μ
815:μ
742:cyclotron
693:λ
637:λ
588:λ
571:ν
538:λ
518:ν
309:−
158:rest mass
146:The term
142:Rest mass
101:observers
6379:Category
6256:LemaĂźtre
6221:Einstein
6211:Poincaré
6171:Others:
6155:TaubâNUT
6121:interior
6043:theories
6041:Advanced
6008:redshift
5823:concepts
5641:Rapidity
5619:concepts
5523:Archived
5500:(1997),
5306:Also in
5206:34032023
5072:20022495
4579:95601322
4253:See also
4129:momentum
4101:â
2745:the same
2475:‖
2452:‖
2001:momentum
654:fulfills
6321:Hawking
6316:Penrose
6301:Novikov
6281:Wheeler
6226:Hilbert
6216:Lorentz
6173:pp-wave
5994:lensing
5790:General
5571:Special
5452:Bibcode
5339:Bibcode
5262:Bibcode
5020:Bibcode
4970:Bibcode
4927:Bibcode
4651:Bibcode
4606:Bibcode
4526:Bibcode
4482:Bibcode
4446:, 32-38
4419:Bibcode
4371:Bibcode
4316:Bibcode
3058:is the
2638:scholar
2562:or the
2560:Z boson
2207:scholar
970:scholar
747:stopped
359:frame,
355:In the
211:formula
103:in all
44:improve
6362:others
6351:Thorne
6341:Misner
6326:Taylor
6311:Geroch
6306:Ehlers
6276:Zwicky
6094:Kasner
5510:
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5389:
5272:
5217:
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2209:
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2195:
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2180:
2116:> 0
2003:, and
1997:energy
1984:where
972:
965:
958:
951:
943:
552:photon
33:, but
6356:Weiss
6336:Bondi
6331:Hulse
6261:Milne
6165:discs
6109:Milne
6104:Gödel
5961:Virgo
5466:arXiv
5442:arXiv
5358:(PDF)
5327:(PDF)
5211:Dover
5068:JSTOR
5006:(PDF)
4956:(PDF)
4835:, 5,
4801:, 5,
4667:S2CID
4575:S2CID
4555:(PDF)
4501:(PDF)
4468:(PDF)
4357:(PDF)
4332:S2CID
4302:(PDF)
3639:(1 â
2695:value
2645:JSTOR
2631:books
2214:JSTOR
2200:books
1982:units
977:JSTOR
963:books
798:) to
550:as a
170:(the
6403:Mass
6291:Kerr
6241:Weyl
6140:Kerr
6000:and
5954:and
5952:LIGO
5508:ISBN
5397:OCLC
5387:ISBN
5366:2017
5215:ISBN
5202:LCCN
5192:ISBN
4767:ISBN
4741:ISBN
4703:ISBN
4622:ISSN
3469:and
3108:and
2799:only
2702:both
2617:news
2186:news
2118:and
2005:mass
1717:and
1533:and
1057:and
949:news
786:The
725:plus
199:The
148:mass
83:mass
6346:Yau
5971:GEO
5347:doi
5280:doi
5164:doi
5132:doi
5100:doi
5060:doi
5028:doi
4978:doi
4966:322
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4841:doi
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4659:doi
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4324:doi
3946:rel
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2803:not
2741:not
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2541:= 1
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2169:by
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674:rel
472:rel
415:rel
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6394::
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5986::
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1649:E
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926:.
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871:=
862:F
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685:h
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670:m
666:=
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640:c
633:/
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