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experimental basis of
Special Relativity?" has documented, with brief commentary, many of the tests that, over the years, have been used to validate various aspects of special relativity. Kaivola et al. (1985) and McGowan et al. (1993) are examples of experiments classified in this resource as time dilation experiments. These two also represent tests of TDE. These experiments compared the frequency of two lasers, one locked to the frequency of a neon atom transition in a fast beam, the other locked to the same transition in thermal neon. The 1993 version of the experiment verified time dilation, and hence TDE, to an accuracy of 2.3Ă10.
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receiver starts to move along the positive x-axis. The tuning fork continues until it reaches A, at which point it stops emitting waves: a wavepacket has therefore been generated, and all the waves in the wavepacket are received by the receiver with the last wave reaching it at B. The proper time for the duration of the packet in the tuning fork's frame of reference is the length of OA while the proper time for the duration of the wavepacket in the receiver's frame of reference is the length of OB. If
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observation of two counter-rotating particle beams. In other repetitions, the energies of gamma rays emitted by a rapidly moving particle beam have been measured at opposite angles relative to the direction of the particle beam. Since these experiments do not actually measure the wavelength of the particle beam at right angles to the beam, some authors have preferred to refer to the effect they are measuring as the "quadratic
Doppler shift" rather than TDE.
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4861.06 Angstrom line. On the left, longitudinal
Doppler shift results in broadening the emission line to such an extent that the TDE cannot be observed. The middle figures illustrate that even if one narrows one's view to the exact center of the beam, very small deviations of the beam from an exact right angle introduce shifts comparable to the predicted effect.
4103:
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2926:, the difficulty of analyzing a relativistic scenario often depends on the choice of reference frame. Attempting to analyze the scenario in the frame of the receiver involves much tedious algebra. It is much easier, almost trivial, to establish the lack of Doppler shift between emitter and absorber in the laboratory frame.
4368:, which enables the production of exceedingly narrow resonance lines for nuclear gamma ray emission and absorption. Mössbauer effect experiments have proven themselves easily capable of detecting TDE using emitter-absorber relative velocities on the order of 2Ă10 cm/s. These experiments include ones performed by Hay
4339:
used a concave mirror that allowed them to simultaneously observe a nearly longitudinal direct beam (blue) and its reflected image (red). Spectroscopically, three lines would be observed: An undisplaced emission line, and blueshifted and redshifted lines. The average of the redshifted and blueshifted
2917:
This conclusion generated some controversy. A certain persistent critic of relativity maintained that, although the experiment was consistent with general relativity, it refuted special relativity, his point being that since the emitter and absorber were in uniform relative motion, special relativity
2667:
Fig. 2b. It is much easier if, instead, we analyze the scenario from the frame of the source. An observer situated at the source knows, from the problem statement, that the receiver is at its closest point to him. That means that the receiver has no longitudinal component of motion to complicate
4384:
The transverse
Doppler effect and the kinematic time dilation of special relativity are closely related. All validations of TDE represent validations of kinematic time dilation, and most validations of kinematic time dilation have also represented validations of TDE. An online resource, "What is the
4138:
In both cases, the monochromatic stars ahead of and behind the observer are
Doppler-shifted towards invisible wavelengths. If, however, the observer had eyes that could see into the ultraviolet and infrared, he would see the stars ahead of him as brighter and more closely clustered together than the
2826:
the source as being at its closest point, the receiver observes a redshift, there obviously must exist a point where blueshift changes to a redshift. In Fig. 2, the signal travels perpendicularly to the receiver path and is blueshifted. In Fig. 3, the signal travels perpendicularly to the
2663:
Fig. 2a. If we analyze the scenario in the frame of the receiver, we find that the analysis is more complicated than it should be. The apparent position of a celestial object is displaced from its true position (or geometric position) because of the object's motion during the time it takes its
4356:
technology has made possible the production of particle beams of considerably higher energy than was available to Ives and
Stilwell. This has enabled the design of tests of the transverse Doppler effect directly along the lines of how Einstein originally envisioned them, i.e. by directly viewing a
4134:
In the non-relativistic case, the light ahead of the observer is blueshifted to a wavelength of 300 nm in the medium ultraviolet, while light behind the observer is redshifted to 5200 nm in the intermediate infrared. Because of the aberration of light, objects formerly at right angles to
4130:
In the relativistic case, the light ahead of the observer is blueshifted to a wavelength of 137 nm in the far ultraviolet, while light behind the observer is redshifted to 2400 nm in the short wavelength infrared. Because of the relativistic aberration of light, objects formerly at right
4162:
The
Doppler effect (with arbitrary direction) also modifies the perceived source intensity: this can be expressed concisely by the fact that source strength divided by the cube of the frequency is a Lorentz invariant This implies that the total radiant intensity (summing over all frequencies) is
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This scenario is equivalent to the receiver looking at a direct right angle to the path of the source. The analysis of this scenario is best conducted from the frame of the receiver. Figure 3 shows the receiver being illuminated by light from when the source was closest to the receiver, even
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In his seminal paper of 1905 introducing special relativity, Einstein had already published an expression for the
Doppler shift perceived by an observer moving at an arbitrary angle with respect to an infinitely distant source of light. Einstein's 1907 derivation of the TDE represented a trivial
4147:
distribution. It is not necessarily true that stars ahead of the observer would show a bluer color. This is because the whole spectral energy distribution is shifted. At the same time that visible light is blueshifted into invisible ultraviolet wavelengths, infrared light is blueshifted into the
4409:
Consider the spacetime diagram in Fig. 10. Worldlines for a tuning fork (the source) and a receiver are both illustrated on this diagram. The tuning fork and receiver start at O, at which point the tuning fork starts to vibrate, emitting waves and moving along the negative x-axis while the
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First-year physics textbooks almost invariably analyze
Doppler shift for sound in terms of Newtonian kinematics, while analyzing Doppler shift for light and electromagnetic phenomena in terms of relativistic kinematics. This gives the false impression that acoustic phenomena require a different
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Fig. 9 shows the results of attempting to measure the 4861 Angstrom line emitted by a beam of canal rays (a mixture of H1+, H2+, and H3+ ions) as they recombine with electrons stripped from the dilute hydrogen gas used to fill the Canal ray tube. Here, the predicted result of the TDE is a
2902:
Suppose source and receiver are located on opposite ends of a spinning rotor, as illustrated in Fig. 6. Kinematic arguments (special relativity) and arguments based on noting that there is no difference in potential between source and receiver in the pseudogravitational field of the rotor
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As seen in Fig. 4, null frequency shift occurs for a pulse that travels the shortest distance from source to receiver. When viewed in the frame where source and receiver have the same speed, this pulse is emitted perpendicularly to the source's path and is received perpendicularly to the
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Fig. 5 illustrates two variants of this scenario. Both variants can be analyzed using simple time dilation arguments. Figure 5a is essentially equivalent to the scenario described in Figure 2b, and the receiver observes light from the source as being blueshifted by a factor of
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Various of the subsequent repetitions of the Ives and
Stilwell experiment have adopted other strategies for measuring the mean of blueshifted and redshifted particle beam emissions. In some recent repetitions of the experiment, modern accelerator technology has been used to arrange for the
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Whether a scientific report describes TDE as being a redshift or blueshift depends on the particulars of the experimental arrangement being related. For example, Einstein's original description of the TDE in 1907 described an experimenter looking at the center (nearest point) of a beam of
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The only seeming complication is that the orbiting objects are in accelerated motion. An accelerated particle does not have an inertial frame in which it is always at rest. However, an inertial frame can always be found which is momentarily comoving with the particle. This frame, the
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light to reach an observer. The source would be time-dilated relative to the receiver, but the redshift implied by this time dilation would be offset by a blueshift due to the longitudinal component of the relative motion between the receiver and the apparent position of the source.
2635:" (a beam of positive ions that is created by certain types of gas-discharge tubes). According to special relativity, the moving ions' emitted frequency would be reduced by the Lorentz factor, so that the received frequency would be reduced (redshifted) by the same factor.
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can be extended in a straightforward fashion to calculate the Doppler shift for the case where the inertial motions of the source and receiver are at any specified angle. Fig. 7 presents the scenario from the frame of the receiver, with the source moving at speed
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though the source has moved on. Because the source's clock is time dilated as measured in the frame of the receiver, and because there is no longitudinal component of its motion, the light from the source, emitted from this closest point, is redshifted with frequency
5906:{\displaystyle {\frac {f_{r}}{f_{s}}}={\frac {1-{\frac {|\mathbf {v_{r}} |}{\mathbf {|C|} }}\cos(\theta _{\mathbf {C,v_{r}} })}{1-{\frac {|\mathbf {v_{s}} |}{\mathbf {|C|} }}\cos(\theta _{\mathbf {C,v_{s}} })}}{\sqrt {\frac {1-(v_{s}/c)^{2}}{1-(v_{r}/c)^{2}}}}}
4361:α line emitted by hydrogen atoms moving at speeds ranging from 2.53Ă10 cm/s to 9.28Ă10 cm/s, finding the coefficient of the second order term in the relativistic approximation to be 0.52±0.03, in excellent agreement with the theoretical value of 1/2.
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In this scenario, the point of closest approach is frame-independent and represents the moment where there is no change in distance versus time. Figure 2 demonstrates that the ease of analyzing this scenario depends on the frame in which it is analyzed.
4122:. Assume that the observer is uniformly surrounded in all directions by yellow stars emitting monochromatic light of 570 nm. The arrows in each diagram represent the observer's velocity vector relative to its surroundings, with a magnitude of 0.89
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Since the transverse Doppler effect is one of the main novel predictions of the special theory of relativity, the detection and precise quantification of this effect has been an important goal of experiments attempting to validate special relativity.
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The traditional analysis of the Doppler effect for sound represents a low speed approximation to the exact, relativistic analysis. The fully relativistic analysis for sound is, in fact, equally applicable to both sound and electromagnetic phenomena.
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3533:
In his 1905 paper on special relativity, Einstein obtained a somewhat different looking equation for the Doppler shift equation. After changing the variable names in Einstein's equation to be consistent with those used here, his equation reads
2668:
the analysis. (i.e. dr/dt = 0 where r is the distance between receiver and source) Since the receiver's clocks are time-dilated relative to the source, the light that the receiver receives is blue-shifted by a factor of gamma. In other words,
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1701:
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1294:
2879:, enables application of special relativity to the analysis of accelerated particles. If an inertial observer looks at an accelerating clock, only the clock's instantaneous speed is important when computing time dilation.
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Relativistic Doppler shift for the longitudinal case, with source and receiver moving directly towards or away from each other, is often derived as if it were the classical phenomenon, but modified by the addition of a
4328:" at right angles to the beam. Attempts to measure TDE following this scheme proved to be impractical, since the maximum speed of a particle beam available at the time was only a few thousandths of the speed of light.
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lines would be compared with the wavelength of the undisplaced emission line. The difference that Ives and Stilwell measured corresponded, within experimental limits, to the effect predicted by special relativity.
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Given that, in the case where the inertially moving source and receiver are geometrically at their nearest approach to each other, the receiver observes a blueshift, whereas in the case where the receiver
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The leading ratio has the form of the classical Doppler effect, while the square root term represents the relativistic correction. If we consider the angles relative to the frame of the source, then
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As a matter of fact, however, Champeney and Moon's experiment said nothing either pro or con about special relativity. Because of the symmetry of the setup, it turns out that virtually
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visible range. Precisely what changes in the colors one sees depends on the physiology of the human eye and on the spectral characteristics of the light sources being observed.
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testing exactly this scenario, and found that the Mössbauer absorption process was unaffected by rotation. They concluded that their findings supported special relativity.
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was spun in a rapid circular path around a central Mössbauer emitter. As explained below, this experimental arrangement resulted in KĂŒndig's measurement of a blueshift.
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3010:
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In the literature, most reports of transverse Doppler shift analyze the effect in terms of the receiver pointed at direct right angles to the path of the source, thus
5986:
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5472:
Analysis of the spacetime diagram in Fig. 10 gave a general formula for source and receiver moving directly along their line of sight, i.e. in collinear motion.
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Hay, H. J.; Schiffer, J. P.; Cranshaw, T. E.; Egelstaff, P. A. (1960). "Measurement of the Red Shift in an Accelerated System Using the Mössbauer Effect in Fe".
3033:
2922:, that it is simply not true that a Doppler shift must always be observed between two frames in uniform relative motion. Furthermore, as demonstrated in section
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Rather than being equidistant from the center, suppose the emitter and absorber were at differing distances from the rotor's center. For an emitter at radius
6409:
Olin, A.; Alexander, T. K.; HĂ€usser, O.; McDonald, A. B.; Ewan, G. T. (1973). "Measurement of the Relativistic Doppler Effect Using 8.6-MeV Capture Îł Rays".
5018:
since otherwise their passage through the medium will set up shock waves, invalidating the calculation. Some routine algebra gives the ratio of frequencies:
2595:, which dictates that the result can not depend on which object is considered to be the one at rest. In contrast, the classic nonrelativistic Doppler effect
2843:
Figure 5. Transverse Doppler effect for two scenarios: (a) receiver moving in a circle around the source; (b) source moving in a circle around the receiver.
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2886:
objects are in accelerated motion requires a somewhat more sophisticated analysis. Not understanding this point has led to confusion and misunderstanding.
433:
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2654:
Figure 2. Source and receiver are at their points of closest approach. (a) Analysis in the frame of the receiver. (b) Analysis in the frame of the source.
1215:
7072:. Course of Theoretical Physics: Volume 2. Trans. Morton Hamermesh (Fourth revised English ed.). Elsevier Butterworth-Heinemann. pp. 116â117.
1879:{\displaystyle \lambda _{s}+vt_{r,s}=ct_{r,s}\Longleftrightarrow \lambda _{s}=ct_{r,s}(1-v/c)\Longleftrightarrow t_{r,s}={\frac {1}{f_{s}(1-\beta )}},}
500:
and do not involve the medium of propagation as a reference point. They describe the total difference in observed frequencies and possess the required
2627:
the emitter as being at its closest approach. The transverse Doppler effect is one of the main novel predictions of the special theory of relativity.
6952:
2384:
5479:
Figure 11. A source and receiver are moving in different directions and speeds in a frame where the speed of sound is independent of direction.
3202:
is the angular velocity of the rotor. The source and emitter do not have to be 180° apart, but can be at any angle with respect to the center.
4016:
3544:
3937:
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6642:
McGowan, Roger W.; Giltner, David M.; Sternberg, Scott J.; Lee, Siu Au (1993). "New measurement of the relativistic Doppler shift in neon".
8009:
1957:
8244:
3313:
The equation below can be interpreted as the classical Doppler shift for a stationary and moving source modified by the Lorentz factor
8033:
7779:
2607:
Suppose that a source and a receiver are both approaching each other in uniform inertial motion along paths that do not collide. The
2047:
Thus far, the equations have been identical to those of the classical Doppler effect with a stationary source and a moving receiver.
1411:
Following this approach towards deriving the relativistic longitudinal Doppler effect, assume the receiver and the source are moving
7124:
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6857:
2316:
426:
6105:"On the Possibility of a New Test of the Relativity Principle (Ăber die Möglichkeit einer neuen PrĂŒfung des RelativitĂ€tsprinzips)"
6075:, i.e., power per unit solid angle and per unit frequency, expressed in watts per steradian per hertz; for spectral intensity in
5028:
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5463:
of the source and receiver merge into a single relative speed independent of any reference to a fixed medium. Indeed, we obtain
27:
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7769:
2933:
conceivable theory of the Doppler shift between frames in uniform inertial motion must yield a null result in this experiment.
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7953:
7104:
6876:
6707:
2587:
Identical expressions for relativistic Doppler shift are obtained when performing the analysis in the reference frame of the
4846:
4756:
7437:
6231:
Champeney, D. C.; Moon, P. B. (1961). "Absence of Doppler Shift for Gamma Ray Source and Detector on Same Circular Orbit".
4177:
419:
5955:, Einstein's 1905 formula for the Doppler effect. If we consider the angles relative to the frame of the receiver, then
2831:
receiver's path. The pulse is emitted slightly before the point of closest approach, and it is received slightly after.
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measured in the frame of the receiver. The radial component of the source's motion along the line of sight is equal to
47:
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3214:
Figure 7. Doppler shift with source moving at an arbitrary angle with respect to the line between source and receiver.
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2903:(general relativity) both lead to the conclusion that there should be no Doppler shift between source and receiver.
2868:. Figure 5b is essentially equivalent to the scenario described in Figure 3, and the light is redshifted.
2754:
1408:
term. This is the approach employed in first-year physics or mechanics textbooks such as those by Feynman or Morin.
971:
855:
7271:
Saburo Nasu (2013). "General Introduction to Mössbauer Spectroscopy". In Yoshida, Yutaka; Langouche, Guido (eds.).
7173:
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2918:
demanded that a Doppler shift be observed. The fallacy with this critic's argument was, as demonstrated in section
67:
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8015:
6557:
Champeney, D. C.; Isaak, G. R.; Khan, A. M. (1965). "A time dilatation experiment based on the Mössbauer effect".
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2111:
614:
388:
7728:
2577:{\displaystyle {\frac {\lambda _{r}}{\lambda _{s}}}={\frac {f_{s}}{f_{r}}}={\sqrt {\frac {1+\beta }{1-\beta }}},}
2196:
is the time in the frame in which the source is at rest. The receiver will measure the received frequency to be
777:{\displaystyle {\frac {\lambda _{r}}{\lambda _{s}}}={\frac {f_{s}}{f_{r}}}={\sqrt {\frac {1+\beta }{1-\beta }}}}
7831:
8262:
8004:
7909:
7839:
6444:
Mandelberg, Hirsch I.; Witten, Louis (1962). "Experimental Verification of the Relativistic Doppler Effect".
4488:
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3435:
2817:
Figure 4. Null frequency shift occurs for a pulse that travels the shortest distance from source to receiver.
244:
7972:
4280:
a black body spectrum seen through a Doppler shift (with arbitrary direction) is still a black body spectrum
3172:{\displaystyle {\frac {f'}{f}}=\left({\frac {c^{2}-R^{2}\omega ^{2}}{c^{2}-R'^{2}\omega ^{2}}}\right)^{1/2}}
1482:
1160:{\displaystyle {\frac {f'}{f}}=\left({\frac {c^{2}-R^{2}\omega ^{2}}{c^{2}-R'^{2}\omega ^{2}}}\right)^{1/2}}
7826:
6971:
Sama, Nicholas (1969). "Some Comments on a Relativistic Frequency-Shift Experiment of Champeney and Moon".
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2057:
5475:
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Figure 9. Why it is difficult to measure the transverse Doppler effect accurately using a transverse beam.
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5275:{\displaystyle ={\frac {1-v_{r}/c_{s}}{1+v_{s}/c_{s}}}{\sqrt {\frac {1-(v_{s}/c)^{2}}{1-(v_{r}/c)^{2}}}}}
406:
149:
124:
72:
6479:
Hasselkamp, D.; Mondry, E.; Sharmann, A. (1979). "Direct observation of the transversal Doppler-shift".
4106:
Figure 8. Comparison of the relativistic Doppler effect (top) with the non-relativistic effect (bottom).
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at the time of reception. The analysis is performed in a coordinate system in which the signal's speed
4110:
Fig. 8 helps us understand, in a rough qualitative sense, how the relativistic Doppler effect and
129:
119:
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5486:
2619:
that occurs when the emitter and receiver are at their points of closest approach; or (b) the nominal
2460:
of the source relative to the receiver. (This terminology is particularly prevalent in the subject of
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7491:
4139:
stars behind, but the stars would be far brighter and far more concentrated in the relativistic case.
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dependent on whether it is the source or the receiver that is stationary with respect to the medium.
534:
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7759:
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7197:"Brightness and color of rapidly moving objects: The visual appearance of a large sphere revisited"
6374:
Ives, H. E.; Stilwell, G. R. (1938). "An experimental study of the rate of a moving atomic clock".
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1436:
as measured by an observer on the receiver or the source (The sign convention adopted here is that
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Figure 6. Source and receiver are placed on opposite ends of a rotor, equidistant from the center.
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KĂŒndig, Walter (1963). "Measurement of the Transverse Doppler Effect in an Accelerated System".
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4364:
Other direct tests of the TDE on rotating platforms were made possible by the discovery of the
4324:
Einstein (1907) had initially suggested that the TDE might be measured by observing a beam of "
520:
516:
453:
Figure 1. A source of light waves moving to the right, relative to observers, with velocity 0.7
87:
82:
7111:
3802:{\displaystyle \cos \theta _{r}={\frac {\cos \theta _{s}-\beta }{1-\beta \cos \theta _{s}}}\,}
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A four-vector approach to deriving these results may be found in Landau and Lifshitz (2005).
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of light, caused by the relative motion of the source and the observer (as in the classical
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7335:
Moriconi, M (1 November 2006). "Special theory of relativity through the Doppler effect".
1954:, the frequency of at which wavefronts impinge on the receiver in the source's frame, is:
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457:. The frequency is higher for observers on the right, and lower for observers on the left.
8:
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Figure 10. The relativistic Doppler shift formula is applicable to both sound and light.
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523:(where space itself stretches). This article concerns itself only with Doppler shifts.
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Fig. 11 illustrates a scenario in two dimensions. The source moves with velocity
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7096:
Einstein's Physics: Atoms, Quanta, and Relativity - Derived, Explained, and Appraised
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3830:, demonstrating the consistency of these alternate equations for the Doppler shift.
3422:{\displaystyle f_{r}={\frac {f_{s}}{\gamma \left(1+\beta \cos \theta _{r}\right)}}.}
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1289:{\displaystyle f_{r}={\frac {f_{s}}{\gamma \left(1+\beta \cos \theta _{r}\right)}}}
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This result provides one of the pieces of evidence that serves to distinguish the
4143:
Real stars are not monochromatic, but emit a range of wavelengths approximating a
8204:
8179:
8104:
8099:
7982:
7943:
7905:
7849:
7723:
7659:
7245:
7094:
7065:
6793:
6772:
6697:
4357:
particle beam at a 90° angle. For example, Hasselkamp et al. (1979) observed the
4286:
4167:
321:
316:
291:
268:
178:
7987:
2923:
790:
8335:
8323:
8229:
7901:
7885:
7881:
7784:
7754:
7611:
7514:
6926:
6813:
6663:
6620:
6003:
4115:
2160:
1921:
is the speed of the receiver in terms of the speed of light. The corresponding
1614:
489:
478:
393:
361:
222:
77:
6699:
Relativistic Electron Mirrors: from High Intensity LaserâNanofoil Interactions
6543:
6148:
4547:
represents the speed of signal propagation (i.e. the speed of sound) to event
2645:
8345:
8219:
8199:
8194:
8109:
7977:
7805:
7749:
7581:
7534:
7039:
6801:
6217:
6175:
6130:
6028:
3923:
In electromagnetic waves both the electric and the magnetic field amplitudes
2051:
1405:
493:
227:
109:
6430:
8299:
8239:
8159:
8124:
7654:
7616:
6776:
6671:
6628:
6465:
6395:
6360:
6341:
5994:, the alternative form of the Doppler shift equation discussed previously.
3652:
with respect to the source rest frame rather than the receiver rest frame.
2461:
2446:{\displaystyle {\frac {f_{s}}{f_{r}}}={\sqrt {\frac {1+\beta }{1-\beta }}}}
846:
356:
4316:
8023:
7992:
7539:
5359:, the above equation reduces to the classical Doppler formula for sound.
4303:
IvesâStilwell-, Mössbauer rotor-, and Spectroscopy tests of time dilation
2722:
341:
217:
7349:
5598:
The ratio between the proper frequencies for the source and receiver is
4087:{\displaystyle B_{r}=\gamma \left(1-\beta \cos \theta _{s}\right)B_{s}.}
3615:{\displaystyle f_{r}=\gamma \left(1-\beta \cos \theta _{s}\right)f_{s}.}
8234:
7800:
7644:
7639:
7403:. Physics and Astronomy Education Group, Hildesheim University, Germany
7061:
6556:
6500:
5512:(at the time of emission). It emits a signal which travels at velocity
4325:
4282:
with a temperature multiplied by the same Doppler factor as frequency.
4144:
4005:{\displaystyle E_{r}=\gamma \left(1-\beta \cos \theta _{s}\right)E_{s}}
2735:
Figure 3. Transverse Doppler shift for the scenario where the receiver
2632:
2472:
1557:
1379:{\displaystyle f_{r}=\gamma \left(1-\beta \cos \theta _{s}\right)f_{s}}
488:
The relativistic Doppler effect is different from the non-relativistic
470:
371:
351:
193:
188:
8330:
7172:. Math Department, University of California, Riverside. Archived from
6992:
7631:
6292:
6267:
6023:
3625:
The differences stem from the fact that Einstein evaluated the angle
2612:
1589:
1476:
512:
474:
466:
169:
7223:
4163:
multiplied by the fourth power of the Doppler factor for frequency.
2882:
The converse, however, is not true. The analysis of scenarios where
8174:
7524:
7396:
6018:
2805:
the source as being at its closest point and observing a redshift.
2620:
2591:
with a moving source. This matches up with the expectations of the
962:
906:
508:
346:
6831:
4388:
4347:
2890:
Source and receiver both in circular motion around a common center
1479:
arrives at the receiver. The next wavefront is then at a distance
2834:
2638:
On the other hand, KĂŒndig (1963) described an experiment where a
2050:
However, due to relativistic effects, clocks on the receiver are
7312:. Department of Mathematics, University of California, Riverside
6865:
Introduction to Classical Mechanics: With Problems and Solutions
1694:, which is the period of light waves impinging on the receiver,
449:
7854:
7391:
Computer program demonstrating the relativistic Doppler effect.
7397:"Space Time Travel: Visualization of the theory of relativity"
6182:
English translation: âOn the Electrodynamics of Moving Bodiesâ
3917:, the expression for relativistic longitudinal Doppler shift.
2307:{\displaystyle ={\frac {1-\beta }{\sqrt {1-\beta ^{2}}}}f_{s}}
7415:
7388:
7252:. Astronomy Department, University of California, Los Angeles
3219:
2368:{\displaystyle ={\sqrt {\frac {1-\beta }{1+\beta }}}\,f_{s}.}
679:
571:
the receiver and the source are moving away from each other,
7099:(illustrated, reprinted ed.). OUP Oxford. p. 164.
6408:
4174:
as having a spectral intensity in frequency proportional to
4102:
7835:
5469:, the formula for relativistic longitudinal Doppler shift.
5102:{\displaystyle {\frac {f_{r}}{f_{s}}}={\frac {|OA|}{|OB|}}}
2924:
Source and receiver are at their points of closest approach
2646:
Source and receiver are at their points of closest approach
1398:
8294:
6641:
1641:, but at the same time the receiver moves away with speed
2163:. In order to know which time is dilated, we recall that
6927:
Misner, C. W., Thorne, K. S., and Wheeler, J. A (1973).
6478:
6315:
Johnson, Montgomery H.; Teller, Edward (February 1982).
6058:
consequence of his earlier published general expression.
4751: (speeds of source and receiver)
4631:{\displaystyle c_{s}={\frac {x_{B}-x_{A}}{t_{B}-t_{A}}}}
7306:"What is the experimental basis of Special Relativity?"
7166:"What would a relativistic interstellar traveller see?"
6594:"Measurement of the Relativistic Doppler Shift in Neon"
4925:{\displaystyle |OB|={\sqrt {t_{B}^{2}-(x_{B}/c)^{2}}}}
4835:{\displaystyle |OA|={\sqrt {t_{A}^{2}-(x_{A}/c)^{2}}}}
4311:
4131:
angles to the observer appear shifted forwards by 63°.
2114:
617:
8271:
5961:
5922:
5611:
5569:
5540:
5534:
towards the receiver, which is traveling at velocity
5518:
5489:
5442:
5415:
5395:
5368:
5345:
5318:
5291:
5114:
5031:
4994:
4967:
4940:
4849:
4759:
4703:
4646:
4560:
4491:
4436:
4416:
4264:
4244:
4231:{\displaystyle {\frac {f^{3}}{e^{\frac {hf}{kT}}-1}}}
4180:
4019:
3940:
3878:
3839:
3723:
3685:
3658:
3631:
3547:
3485:
3438:
3348:
3319:
3276:
3249:
3229:
3188:
3048:
3018:
2990:
2967:
2942:
2854:
2757:
2682:
2488:
2387:
2319:
2257:
2209:
2169:
2060:
1960:
1927:
1892:
1704:
1667:
1647:
1626:
1598:
1566:
1534:
1485:
1442:
1421:
1314:
1218:
1206:
1177:
1036:
974:
918:
858:
802:
691:
597:
577:
537:
481:), when taking into account effects described by the
6792:
1302:
6732:
Journal of the Royal Astronomical Society of Canada
6373:
4335:Rather than attempt direct measurement of the TDE,
3205:
7237:
6592:Kaivola, Matti; Riis, Erling; Lee, Siu Au (1985).
5980:
5941:
5905:
5587:
5555:
5526:
5504:
5455:
5428:
5401:
5381:
5351:
5331:
5304:
5274:
5101:
5010:
4980:
4953:
4924:
4834:
4743:
4689:
4630:
4532:
4477:
4422:
4289:from alternative theories proposed to explain the
4270:
4250:
4230:
4086:
4004:
3897:
3858:
3812:Substituting the relativistic aberration equation
3801:
3698:
3671:
3644:
3614:
3522:
3464:
3421:
3328:
3303:
3262:
3235:
3194:
3171:
3027:
3004:
2984:on the rotor, the ratio of the emitter frequency,
2973:
2953:
2919:
2860:
2790:
2711:
2623:predicted by special relativity when the receiver
2576:
2445:
2367:
2306:
2244:
2188:
2151:
2100:
2038:{\displaystyle f_{r,s}=1/t_{r,s}=f_{s}(1-\beta ).}
2037:
1946:
1913:
1878:
1686:
1653:
1633:
1605:
1580:
1548:
1520:
1449:
1428:
1378:
1288:
1194:
1159:
1007:
947:
891:
831:
776:
654:
603:
583:
563:
519:(due to light exiting a gravitational field); and
6268:"Group Motions in Space-time and Doppler Effects"
5772:
5764:
5684:
5676:
3523:{\displaystyle f_{r}={\frac {f_{s}}{\gamma }}.\,}
8343:
6443:
4278:the frequency), we can draw the conclusion that
3931:transform in a similar manner as the frequency:
2808:
7012:. Delhi, India: Alekh Prakashan. pp. 60â61
6079:, the cube should be replaced by a fifth power.
4389:Relativistic Doppler effect for sound and light
4348:Direct measurement of transverse Doppler effect
531:In the following table, it is assumed that for
7060:
6702:(illustrated ed.). Springer. p. 22.
6591:
6195:
6142:
6140:
6098:
6096:
4151:
3473:
2835:One object in circular motion around the other
2791:{\displaystyle f_{r}={\frac {f_{s}}{\gamma }}}
1008:{\displaystyle f_{r}={\frac {f_{s}}{\gamma }}}
892:{\displaystyle f_{r}={\frac {f_{s}}{\gamma }}}
7431:
6771:. London: Logos Press Limited. pp. 6â9.
6695:
6314:
6230:
4690:{\displaystyle v_{s}=-{\frac {x_{A}}{t_{A}}}}
4379:
2152:{\textstyle \gamma =1/{\sqrt {1-\beta ^{2}}}}
655:{\textstyle \gamma =1/{\sqrt {1-\beta ^{2}}}}
427:
7122:
7092:
6951:: CS1 maint: multiple names: authors list (
4744:{\displaystyle v_{r}={\frac {x_{B}}{t_{B}}}}
4296:
4135:the observer appear shifted forwards by 42°.
2602:
7270:
6137:
6093:
5409:, it can be shown that the absolute speeds
3710:. The relativistic aberration equation is:
526:
7438:
7424:
7303:
7116:
7033:
7031:
7029:
7027:
6899:
6897:
6895:
4158:Black-body radiation § Doppler effect
1461:if the receiver and the source are moving
434:
420:
7348:
7188:
7157:
6810:The Feynman Lectures on Physics: Volume 1
6788:
6786:
6446:Journal of the Optical Society of America
6376:Journal of the Optical Society of America
6350:
6340:
6317:"Intensity changes in the Doppler effect"
6291:
6224:
6174:
3798:
3519:
2739:the source as being at its closest point.
2351:
1910:
1630:
1602:
1577:
1545:
1517:
1446:
1425:
7334:
6367:
6191:
6189:
6146:
6102:
5474:
4392:
4315:
4101:
3220:Relativistic longitudinal Doppler effect
3209:
2893:
2838:
2812:
2730:
2727:the source as being at its closest point
2649:
1592:of the waves that the source emits, and
1399:Relativistic longitudinal Doppler effect
448:
7297:
7024:
7005:
6999:
6966:
6964:
6962:
6892:
6851:
6849:
6847:
6845:
6843:
6841:
4533:{\displaystyle f_{r}={\frac {n}{|OB|}}}
4478:{\displaystyle f_{s}={\frac {n}{|OA|}}}
3465:{\displaystyle \theta _{r}=90^{\circ }}
1696:as observed in the frame of the source.
8344:
7054:
6920:
6783:
6762:
6760:
1521:{\displaystyle \lambda _{s}=c/f_{s}\,}
1415:from each other with a relative speed
7419:
7394:
7275:Mössbauer Spectroscopy: Tutorial Book
7194:
7163:
7123:Savage, C. M.; Searle, A. C. (1999).
6858:"Chapter 11: Relativity (Kinematics)"
6855:
6265:
6259:
6186:
4697:
4402:analysis than light and radio waves.
2101:{\displaystyle t_{r}=t_{r,s}/\gamma }
507:Astronomers know of three sources of
6970:
6959:
6838:
6766:
6725:
6149:"Zur Elektrodynamik bewegter Körper"
3304:{\displaystyle v\cos {\theta _{r}}.}
2875:momentarily comoving reference frame
2245:{\displaystyle f_{r}=f_{r,s}\gamma }
8010:TolmanâOppenheimerâVolkoff equation
7963:FriedmannâLemaĂźtreâRobertsonâWalker
7040:"Doppler Shift for Sound and Light"
6806:"Relativistic Effects in Radiation"
6757:
6559:Proceedings of the Physical Society
5362:If the speed of signal propagation
4312:Ives and Stilwell-type measurements
2611:(TDE) may refer to (a) the nominal
13:
7328:
7243:
6086:
6067:Here, "source strength" refers to
2712:{\displaystyle f_{r}=\gamma f_{s}}
2054:relative to clocks at the source:
948:{\displaystyle f_{r}=\gamma f_{s}}
832:{\displaystyle f_{r}=\gamma f_{s}}
14:
8368:
7780:HamiltonâJacobiâEinstein equation
7389:Warp Special Relativity Simulator
7382:
7246:"Errors in Tired Light Cosmology"
7037:
6903:
6728:"The Relativistic Doppler Effect"
4114:differ from the non-relativistic
8329:
8317:
8305:
8293:
8281:
8258:
8257:
7304:Roberts, Tom; Schleif, Siegmar.
7125:"Visualizing Special Relativity"
7009:Origin and Concept of Relativity
5805:
5801:
5797:
5794:
5768:
5750:
5746:
5717:
5713:
5709:
5706:
5680:
5662:
5658:
5576:
5556:{\displaystyle \mathbf {v_{r}} }
5547:
5543:
5520:
5505:{\displaystyle \mathbf {v_{s}} }
5496:
5492:
4097:
3206:Motion in an arbitrary direction
591:being the relative velocity and
401:
400:
387:
26:
7310:The Original Usenet Physics FAQ
7264:
7250:Ned Wright's Cosmology Tutorial
7086:
6635:
6585:
6550:
6515:
6472:
6437:
6402:
6308:
6061:
2827:source path and is redshifted.
1620:The wavefront moves with speed
564:{\displaystyle \beta =v/c>0}
7587:Massâenergy equivalence (E=mc)
7445:
7070:The Classical Theory of Fields
6719:
6689:
6051:
5890:
5868:
5851:
5829:
5813:
5785:
5757:
5740:
5725:
5697:
5669:
5652:
5588:{\displaystyle |\mathbf {C} |}
5581:
5571:
5259:
5237:
5220:
5198:
5092:
5081:
5074:
5063:
4911:
4889:
4862:
4851:
4821:
4799:
4772:
4761:
4523:
4512:
4468:
4457:
4258:is the source temperature and
2029:
2017:
1867:
1855:
1817:
1814:
1794:
1759:
1549:{\displaystyle \lambda _{s}\,}
1528:away from the receiver (where
1:
6682:
5595:is independent of direction.
3898:{\displaystyle \theta _{s}=0}
3859:{\displaystyle \theta _{r}=0}
3218:The analysis used in section
2920:Point of null frequency shift
2809:Point of null frequency shift
1393:
1303:Motion in arbitrary direction
1207:Motion in arbitrary direction
492:as the equations include the
5988:and the equation reduces to
5949:and the equation reduces to
5602:
5527:{\displaystyle \mathbf {C} }
4988:are assumed to be less than
3012:and the absorber frequency,
1914:{\displaystyle \beta =v/c\,}
1468:Consider the problem in the
483:special theory of relativity
7:
7602:Relativistic Doppler effect
7337:European Journal of Physics
7244:Wright, Edward L. ("Ned").
6973:American Journal of Physics
6034:Gravitational time dilation
6009:Differential Doppler effect
5997:
5990:
5951:
5465:
5022:
4376:(1965), and KĂŒndig (1963).
4152:Doppler effect on intensity
3913:
3907:
3868:
3826:
3820:
3814:
3714:
3699:{\displaystyle \theta _{r}}
3672:{\displaystyle \theta _{s}}
3645:{\displaystyle \theta _{s}}
3538:
3339:
3263:{\displaystyle \theta _{r}}
3039:
2961:and the absorber at radius
2748:
2673:
2479:
2200:
463:relativistic Doppler effect
135:Relativistic Doppler effect
73:Inertial frame of reference
10:
8373:
8073:In computational physics:
7597:Relativity of simultaneity
7395:Kraus, Ute; Zahn, Corvin.
7367:10.1088/0143-0807/27/6/015
6869:Cambridge University Press
6812:. Reading, Massachusetts:
6664:10.1103/PhysRevLett.70.251
6621:10.1103/PhysRevLett.54.255
6579:10.1088/0370-1328/85/3/317
6321:Proc. Natl. Acad. Sci. USA
6253:10.1088/0370-1328/77/2/318
4551:. We can therefore write:
4380:Time dilation measurements
4300:
4155:
2912:Mössbauer rotor experiment
1209:measured in receiver frame
847:Transverse Doppler effect,
793:geometric closest approach
791:Transverse Doppler effect,
130:Relativity of simultaneity
8255:
8087:
7952:
7924:
7910:LenseâThirring precession
7793:
7742:
7704:
7683:
7672:
7630:
7574:
7558:
7500:
7492:Doubly special relativity
7464:
7453:
6544:10.1103/PhysRevLett.4.165
6147:Einstein, Albert (1905).
6103:Einstein, Albert (1907).
4430:waves were emitted, then
4297:Experimental verification
3706:because of the effect of
3474:transverse Doppler effect
2609:transverse Doppler effect
2603:Transverse Doppler effect
1025:in circular motion around
907:TDE, receiver in circular
680:Relativistic longitudinal
7770:Post-Newtonian formalism
7760:Einstein field equations
7696:Mathematical formulation
7520:Hyperbolic orthogonality
6931:. Freeman. p. 163.
6481:Zeitschrift fĂŒr Physik A
6218:10.1103/PhysRev.129.2371
6176:10.1002/andp.19053221004
6131:10.1002/andp.19073280613
6044:
5339:are small compared with
4337:Ives and Stilwell (1938)
4166:As a consequence, since
3329:{\displaystyle \gamma :}
1305:measured in source frame
1023:TDE, source and receiver
611:the speed of light, and
527:Summary of major results
150:Bell's spaceship paradox
7481:Galilean transformation
7472:Principle of relativity
7006:Keswani, G. H. (1965).
6524:Physical Review Letters
6431:10.1103/PhysRevD.8.1633
5981:{\displaystyle v_{r}=0}
5942:{\displaystyle v_{s}=0}
4112:relativistic aberration
3708:relativistic aberration
3195:{\displaystyle \omega }
2906:In 1961, Champeney and
2861:{\displaystyle \gamma }
2593:principle of relativity
2189:{\displaystyle t_{r,s}}
1947:{\displaystyle f_{r,s}}
1687:{\displaystyle t_{r,s}}
1581:{\displaystyle f_{s}\,}
963:TDE, source in circular
849:visual closest approach
517:gravitational redshifts
264:Galilean transformation
125:Massâenergy equivalence
38:Principle of relativity
7566:Lorentz transformation
6696:Daniel Kiefer (2014).
6466:10.1364/JOSA.52.000529
6396:10.1364/JOSA.28.000215
6342:10.1073/pnas.79.4.1340
5982:
5943:
5907:
5589:
5557:
5528:
5506:
5480:
5457:
5430:
5403:
5383:
5353:
5333:
5306:
5276:
5103:
5012:
5011:{\displaystyle c_{s},}
4982:
4955:
4926:
4836:
4745:
4691:
4632:
4534:
4479:
4424:
4398:
4321:
4272:
4252:
4232:
4107:
4088:
4006:
3899:
3860:
3803:
3700:
3673:
3646:
3616:
3524:
3466:
3423:
3330:
3305:
3264:
3237:
3215:
3196:
3173:
3029:
3006:
2975:
2955:
2899:
2862:
2844:
2818:
2792:
2740:
2713:
2655:
2578:
2447:
2369:
2308:
2246:
2190:
2153:
2102:
2039:
1948:
1915:
1880:
1688:
1655:
1635:
1607:
1582:
1550:
1522:
1451:
1430:
1380:
1290:
1196:
1161:
1009:
965:motion around receiver
949:
893:
833:
778:
656:
605:
585:
565:
521:cosmological expansion
458:
88:Lorentz transformation
8034:WeylâLewisâPapapetrou
7775:Raychaudhuri equation
7714:Equivalence principle
7279:. Springer. pp.
7138:(141). Archived from
7093:Ta-Pei Cheng (2013).
6856:Morin, David (2008).
6266:Synge, J. L. (1963).
5983:
5944:
5908:
5590:
5558:
5529:
5507:
5478:
5458:
5456:{\displaystyle v_{r}}
5431:
5429:{\displaystyle v_{s}}
5404:
5384:
5382:{\displaystyle c_{s}}
5354:
5334:
5332:{\displaystyle v_{s}}
5307:
5305:{\displaystyle v_{r}}
5277:
5104:
5013:
4983:
4981:{\displaystyle v_{r}}
4956:
4954:{\displaystyle v_{s}}
4927:
4837:
4746:
4692:
4633:
4543:the inverse slope of
4535:
4480:
4425:
4396:
4319:
4291:cosmological redshift
4273:
4253:
4233:
4156:Further information:
4118:and non-relativistic
4105:
4089:
4007:
3900:
3861:
3804:
3701:
3674:
3647:
3617:
3525:
3467:
3424:
3331:
3306:
3265:
3238:
3213:
3197:
3174:
3030:
3007:
2976:
2956:
2897:
2863:
2842:
2816:
2793:
2734:
2714:
2653:
2579:
2448:
2370:
2309:
2247:
2191:
2154:
2103:
2040:
1949:
1916:
1881:
1689:
1656:
1636:
1608:
1583:
1551:
1523:
1452:
1431:
1381:
1291:
1197:
1162:
1010:
950:
894:
834:
779:
657:
606:
586:
566:
452:
446:Scientific phenomenon
68:Einstein's postulates
8075:Numerical relativity
7916:pulsar timing arrays
6906:"The Doppler Effect"
6871:. pp. 539â543.
6767:Gill, T. P. (1965).
6014:Relativistic beaming
5959:
5920:
5609:
5567:
5538:
5516:
5487:
5440:
5413:
5393:
5366:
5343:
5316:
5289:
5112:
5029:
4992:
4965:
4938:
4847:
4757:
4701:
4644:
4558:
4489:
4434:
4414:
4354:particle accelerator
4262:
4242:
4178:
4172:black-body radiation
4017:
3938:
3876:
3837:
3721:
3683:
3656:
3629:
3545:
3483:
3436:
3346:
3317:
3274:
3247:
3227:
3186:
3046:
3016:
2988:
2965:
2940:
2852:
2755:
2680:
2486:
2466:relativistic beaming
2385:
2317:
2255:
2207:
2167:
2112:
2058:
1958:
1925:
1890:
1702:
1665:
1645:
1624:
1596:
1564:
1532:
1483:
1440:
1419:
1312:
1216:
1195:{\displaystyle R=R'}
1175:
1034:
972:
916:
909:motion around source
856:
800:
689:
615:
595:
575:
535:
43:Theory of relativity
7967:Friedmann equations
7861:HulseâTaylor binary
7823:Gravitational waves
7719:Riemannian geometry
7545:Proper acceleration
7530:Maxwell's equations
7476:Galilean relativity
7401:SpacetimeTravel.org
7359:2006EJPh...27.1409M
7216:2000AmJPh..68...56K
7164:Brandeker, Alexis.
7112:Extract of page 164
6985:1969AmJPh..37..832S
6816:. pp. 34â7 f.
6798:Leighton, Robert B.
6794:Feynman, Richard P.
6744:1968JRASC..62..105S
6656:1993PhRvL..70..251M
6613:1985PhRvL..54..255K
6571:1965PPS....85..583C
6536:1960PhRvL...4..165H
6493:1979ZPhyA.289..151H
6458:1962JOSA...52..529M
6423:1973PhRvD...8.1633O
6388:1938JOSA...28..215I
6333:1982PNAS...79.1340J
6284:1963Natur.198..679S
6245:1961PPS....77..350C
6210:1963PhRv..129.2371K
6167:1905AnP...322..891E
6123:1907AnP...328..197E
4885:
4795:
4120:aberration of light
3005:{\displaystyle f',}
1634:{\displaystyle c\,}
1606:{\displaystyle c\,}
1450:{\displaystyle v\,}
1429:{\displaystyle v\,}
259:Galilean relativity
179:Minkowski spacetime
83:Maxwell's equations
8352:Special relativity
8016:ReissnerâNordström
7934:BransâDicke theory
7765:Linearized gravity
7592:Length contraction
7510:Frame of reference
7487:Special relativity
7195:Kraus, U. (2000).
6769:The Doppler Effect
6715:Extract of page 22
6501:10.1007/BF01435932
6154:Annalen der Physik
6110:Annalen der Physik
6069:spectral intensity
6039:Special relativity
5978:
5939:
5903:
5585:
5553:
5524:
5502:
5481:
5453:
5426:
5399:
5379:
5349:
5329:
5302:
5272:
5099:
5008:
4978:
4951:
4922:
4871:
4832:
4781:
4741:
4687:
4628:
4530:
4475:
4420:
4399:
4372:(1960), Champeney
4322:
4268:
4248:
4228:
4108:
4084:
4002:
3895:
3856:
3799:
3696:
3669:
3642:
3612:
3520:
3472:, one obtains the
3462:
3419:
3326:
3301:
3260:
3233:
3216:
3192:
3169:
3028:{\displaystyle f,}
3025:
3002:
2971:
2954:{\displaystyle R'}
2951:
2900:
2858:
2845:
2819:
2788:
2741:
2709:
2656:
2640:Mössbauer absorber
2617:special relativity
2574:
2471:The corresponding
2443:
2365:
2304:
2242:
2186:
2149:
2098:
2035:
1944:
1911:
1876:
1684:
1651:
1631:
1603:
1578:
1546:
1518:
1447:
1426:
1376:
1286:
1192:
1157:
1005:
945:
889:
829:
774:
652:
601:
581:
561:
515:: Doppler shifts;
498:special relativity
459:
394:Physics portal
115:Length contraction
20:Special relativity
8269:
8268:
8083:
8082:
8062:OzsvĂĄthâSchĂŒcking
7668:
7667:
7650:Minkowski diagram
7607:Thomas precession
7550:Relativistic mass
7106:978-0-19-966991-2
6993:10.1119/1.1975859
6878:978-1-139-46837-4
6804:(February 1977).
6726:Sher, D. (1968).
6709:978-3-319-07752-9
5901:
5900:
5817:
5777:
5689:
5634:
5402:{\displaystyle c}
5352:{\displaystyle c}
5270:
5269:
5186:
5097:
5054:
4920:
4830:
4739:
4685:
4626:
4528:
4473:
4423:{\displaystyle n}
4271:{\displaystyle f}
4251:{\displaystyle T}
4226:
4216:
3796:
3514:
3432:In the case when
3414:
3236:{\displaystyle v}
3149:
3062:
2974:{\displaystyle R}
2786:
2569:
2568:
2538:
2511:
2441:
2440:
2410:
2349:
2348:
2292:
2291:
2147:
1871:
1654:{\displaystyle v}
1391:
1390:
1284:
1137:
1050:
1003:
887:
772:
771:
741:
714:
650:
604:{\displaystyle c}
584:{\displaystyle v}
465:is the change in
444:
443:
184:Spacetime diagram
155:Ehrenfest paradox
145:Relativistic disk
140:Thomas precession
120:Relativistic mass
8364:
8334:
8333:
8322:
8321:
8310:
8309:
8298:
8297:
8286:
8285:
8284:
8277:
8261:
8260:
8044:van Stockum dust
7816:Two-body problem
7734:Mach's principle
7681:
7680:
7622:Terrell rotation
7462:
7461:
7440:
7433:
7426:
7417:
7416:
7412:
7410:
7408:
7378:
7352:
7343:(6): 1409â1423.
7322:
7321:
7319:
7317:
7301:
7295:
7294:
7278:
7268:
7262:
7261:
7259:
7257:
7241:
7235:
7234:
7232:
7230:
7201:
7192:
7186:
7185:
7183:
7181:
7161:
7155:
7154:
7152:
7150:
7144:
7129:
7120:
7114:
7110:
7090:
7084:
7083:
7058:
7052:
7051:
7049:
7047:
7038:Brown, Kevin S.
7035:
7022:
7021:
7019:
7017:
7003:
6997:
6996:
6968:
6957:
6956:
6950:
6942:
6924:
6918:
6917:
6915:
6913:
6904:Brown, Kevin S.
6901:
6890:
6889:
6888:on 4 April 2018.
6887:
6881:. Archived from
6862:
6853:
6836:
6835:
6790:
6781:
6780:
6764:
6755:
6754:
6752:
6750:
6723:
6717:
6713:
6693:
6676:
6675:
6639:
6633:
6632:
6598:
6589:
6583:
6582:
6554:
6548:
6547:
6519:
6513:
6512:
6476:
6470:
6469:
6441:
6435:
6434:
6417:(6): 1633â1639.
6406:
6400:
6399:
6371:
6365:
6364:
6354:
6344:
6312:
6306:
6305:
6295:
6293:10.1038/198679a0
6263:
6257:
6256:
6228:
6222:
6221:
6204:(6): 2371â2375.
6193:
6184:
6180:
6178:
6144:
6135:
6134:
6100:
6080:
6065:
6059:
6055:
5987:
5985:
5984:
5979:
5971:
5970:
5948:
5946:
5945:
5940:
5932:
5931:
5912:
5910:
5909:
5904:
5902:
5899:
5898:
5897:
5885:
5880:
5879:
5860:
5859:
5858:
5846:
5841:
5840:
5821:
5820:
5818:
5816:
5812:
5811:
5810:
5809:
5808:
5778:
5776:
5775:
5767:
5761:
5760:
5755:
5754:
5753:
5743:
5737:
5728:
5724:
5723:
5722:
5721:
5720:
5690:
5688:
5687:
5679:
5673:
5672:
5667:
5666:
5665:
5655:
5649:
5640:
5635:
5633:
5632:
5623:
5622:
5613:
5604:
5594:
5592:
5591:
5586:
5584:
5579:
5574:
5562:
5560:
5559:
5554:
5552:
5551:
5550:
5533:
5531:
5530:
5525:
5523:
5511:
5509:
5508:
5503:
5501:
5500:
5499:
5462:
5460:
5459:
5454:
5452:
5451:
5435:
5433:
5432:
5427:
5425:
5424:
5408:
5406:
5405:
5400:
5388:
5386:
5385:
5380:
5378:
5377:
5358:
5356:
5355:
5350:
5338:
5336:
5335:
5330:
5328:
5327:
5311:
5309:
5308:
5303:
5301:
5300:
5281:
5279:
5278:
5273:
5271:
5268:
5267:
5266:
5254:
5249:
5248:
5229:
5228:
5227:
5215:
5210:
5209:
5190:
5189:
5187:
5185:
5184:
5183:
5174:
5169:
5168:
5152:
5151:
5150:
5141:
5136:
5135:
5119:
5108:
5106:
5105:
5100:
5098:
5096:
5095:
5084:
5078:
5077:
5066:
5060:
5055:
5053:
5052:
5043:
5042:
5033:
5024:
5017:
5015:
5014:
5009:
5004:
5003:
4987:
4985:
4984:
4979:
4977:
4976:
4960:
4958:
4957:
4952:
4950:
4949:
4931:
4929:
4928:
4923:
4921:
4919:
4918:
4906:
4901:
4900:
4884:
4879:
4870:
4865:
4854:
4841:
4839:
4838:
4833:
4831:
4829:
4828:
4816:
4811:
4810:
4794:
4789:
4780:
4775:
4764:
4750:
4748:
4747:
4742:
4740:
4738:
4737:
4728:
4727:
4718:
4713:
4712:
4696:
4694:
4693:
4688:
4686:
4684:
4683:
4674:
4673:
4664:
4656:
4655:
4638:(speed of sound)
4637:
4635:
4634:
4629:
4627:
4625:
4624:
4623:
4611:
4610:
4600:
4599:
4598:
4586:
4585:
4575:
4570:
4569:
4539:
4537:
4536:
4531:
4529:
4527:
4526:
4515:
4506:
4501:
4500:
4484:
4482:
4481:
4476:
4474:
4472:
4471:
4460:
4451:
4446:
4445:
4429:
4427:
4426:
4421:
4366:Mössbauer effect
4277:
4275:
4274:
4269:
4257:
4255:
4254:
4249:
4237:
4235:
4234:
4229:
4227:
4225:
4218:
4217:
4215:
4207:
4199:
4192:
4191:
4182:
4093:
4091:
4090:
4085:
4080:
4079:
4070:
4066:
4065:
4064:
4029:
4028:
4011:
4009:
4008:
4003:
4001:
4000:
3991:
3987:
3986:
3985:
3950:
3949:
3904:
3902:
3901:
3896:
3888:
3887:
3865:
3863:
3862:
3857:
3849:
3848:
3808:
3806:
3805:
3800:
3797:
3795:
3794:
3793:
3768:
3761:
3760:
3744:
3739:
3738:
3716:
3705:
3703:
3702:
3697:
3695:
3694:
3679:is not equal to
3678:
3676:
3675:
3670:
3668:
3667:
3651:
3649:
3648:
3643:
3641:
3640:
3621:
3619:
3618:
3613:
3608:
3607:
3598:
3594:
3593:
3592:
3557:
3556:
3540:
3529:
3527:
3526:
3521:
3515:
3510:
3509:
3500:
3495:
3494:
3471:
3469:
3468:
3463:
3461:
3460:
3448:
3447:
3428:
3426:
3425:
3420:
3415:
3413:
3412:
3408:
3407:
3406:
3373:
3372:
3363:
3358:
3357:
3341:
3335:
3333:
3332:
3327:
3310:
3308:
3307:
3302:
3297:
3296:
3295:
3269:
3267:
3266:
3261:
3259:
3258:
3242:
3240:
3239:
3234:
3201:
3199:
3198:
3193:
3178:
3176:
3175:
3170:
3168:
3167:
3163:
3154:
3150:
3148:
3147:
3146:
3137:
3136:
3135:
3119:
3118:
3108:
3107:
3106:
3097:
3096:
3084:
3083:
3073:
3063:
3058:
3050:
3041:
3034:
3032:
3031:
3026:
3011:
3009:
3008:
3003:
2998:
2980:
2978:
2977:
2972:
2960:
2958:
2957:
2952:
2950:
2867:
2865:
2864:
2859:
2797:
2795:
2794:
2789:
2787:
2782:
2781:
2772:
2767:
2766:
2750:
2718:
2716:
2715:
2710:
2708:
2707:
2692:
2691:
2675:
2583:
2581:
2580:
2575:
2570:
2567:
2556:
2545:
2544:
2539:
2537:
2536:
2527:
2526:
2517:
2512:
2510:
2509:
2500:
2499:
2490:
2481:
2452:
2450:
2449:
2444:
2442:
2439:
2428:
2417:
2416:
2411:
2409:
2408:
2399:
2398:
2389:
2374:
2372:
2371:
2366:
2361:
2360:
2350:
2347:
2336:
2325:
2324:
2313:
2311:
2310:
2305:
2303:
2302:
2293:
2290:
2289:
2274:
2273:
2262:
2251:
2249:
2248:
2243:
2238:
2237:
2219:
2218:
2202:
2195:
2193:
2192:
2187:
2185:
2184:
2158:
2156:
2155:
2150:
2148:
2146:
2145:
2130:
2128:
2107:
2105:
2104:
2099:
2094:
2089:
2088:
2070:
2069:
2044:
2042:
2041:
2036:
2016:
2015:
2003:
2002:
1987:
1976:
1975:
1953:
1951:
1950:
1945:
1943:
1942:
1920:
1918:
1917:
1912:
1906:
1885:
1883:
1882:
1877:
1872:
1870:
1854:
1853:
1840:
1835:
1834:
1810:
1793:
1792:
1771:
1770:
1758:
1757:
1736:
1735:
1714:
1713:
1693:
1691:
1690:
1685:
1683:
1682:
1660:
1658:
1657:
1652:
1640:
1638:
1637:
1632:
1612:
1610:
1609:
1604:
1587:
1585:
1584:
1579:
1576:
1575:
1555:
1553:
1552:
1547:
1544:
1543:
1527:
1525:
1524:
1519:
1516:
1515:
1506:
1495:
1494:
1456:
1454:
1453:
1448:
1435:
1433:
1432:
1427:
1385:
1383:
1382:
1377:
1375:
1374:
1365:
1361:
1360:
1359:
1324:
1323:
1295:
1293:
1292:
1287:
1285:
1283:
1282:
1278:
1277:
1276:
1243:
1242:
1233:
1228:
1227:
1201:
1199:
1198:
1193:
1191:
1169:No Doppler shift
1166:
1164:
1163:
1158:
1156:
1155:
1151:
1142:
1138:
1136:
1135:
1134:
1125:
1124:
1123:
1107:
1106:
1096:
1095:
1094:
1085:
1084:
1072:
1071:
1061:
1051:
1046:
1038:
1014:
1012:
1011:
1006:
1004:
999:
998:
989:
984:
983:
954:
952:
951:
946:
944:
943:
928:
927:
898:
896:
895:
890:
888:
883:
882:
873:
868:
867:
838:
836:
835:
830:
828:
827:
812:
811:
783:
781:
780:
775:
773:
770:
759:
748:
747:
742:
740:
739:
730:
729:
720:
715:
713:
712:
703:
702:
693:
665:
664:
661:
659:
658:
653:
651:
649:
648:
633:
631:
610:
608:
607:
602:
590:
588:
587:
582:
570:
568:
567:
562:
551:
502:Lorentz symmetry
436:
429:
422:
409:
404:
403:
396:
392:
391:
284:
211:
172:
103:
61:
30:
16:
15:
8372:
8371:
8367:
8366:
8365:
8363:
8362:
8361:
8357:Doppler effects
8342:
8341:
8340:
8328:
8316:
8304:
8292:
8282:
8280:
8272:
8270:
8265:
8251:
8079:
7983:BKL singularity
7973:LemaĂźtreâTolman
7948:
7944:Quantum gravity
7926:
7920:
7906:geodetic effect
7880:(together with
7850:LISA Pathfinder
7789:
7738:
7724:Penrose diagram
7706:
7700:
7675:
7664:
7660:Minkowski space
7626:
7570:
7554:
7502:
7496:
7456:
7449:
7444:
7406:
7404:
7385:
7350:physics/0605204
7331:
7329:Further reading
7326:
7325:
7315:
7313:
7302:
7298:
7291:
7269:
7265:
7255:
7253:
7242:
7238:
7228:
7226:
7224:10.1119/1.19373
7199:
7193:
7189:
7179:
7177:
7162:
7158:
7148:
7146:
7142:
7127:
7121:
7117:
7107:
7091:
7087:
7080:
7059:
7055:
7045:
7043:
7036:
7025:
7015:
7013:
7004:
7000:
6969:
6960:
6944:
6943:
6939:
6925:
6921:
6911:
6909:
6902:
6893:
6885:
6879:
6860:
6854:
6839:
6824:
6791:
6784:
6765:
6758:
6748:
6746:
6724:
6720:
6710:
6694:
6690:
6685:
6680:
6679:
6644:Phys. Rev. Lett
6640:
6636:
6601:Phys. Rev. Lett
6596:
6590:
6586:
6555:
6551:
6520:
6516:
6477:
6473:
6442:
6438:
6407:
6403:
6372:
6368:
6313:
6309:
6264:
6260:
6233:Proc. Phys. Soc
6229:
6225:
6198:Physical Review
6194:
6187:
6161:(10): 891â921.
6145:
6138:
6101:
6094:
6089:
6087:Primary sources
6084:
6083:
6066:
6062:
6056:
6052:
6047:
6000:
5991:Equation 6
5966:
5962:
5960:
5957:
5956:
5952:Equation 7
5927:
5923:
5921:
5918:
5917:
5893:
5889:
5881:
5875:
5871:
5861:
5854:
5850:
5842:
5836:
5832:
5822:
5819:
5804:
5800:
5793:
5792:
5788:
5771:
5763:
5762:
5756:
5749:
5745:
5744:
5739:
5738:
5736:
5729:
5716:
5712:
5705:
5704:
5700:
5683:
5675:
5674:
5668:
5661:
5657:
5656:
5651:
5650:
5648:
5641:
5639:
5628:
5624:
5618:
5614:
5612:
5610:
5607:
5606:
5580:
5575:
5570:
5568:
5565:
5564:
5546:
5542:
5541:
5539:
5536:
5535:
5519:
5517:
5514:
5513:
5495:
5491:
5490:
5488:
5485:
5484:
5466:Equation 1
5447:
5443:
5441:
5438:
5437:
5420:
5416:
5414:
5411:
5410:
5394:
5391:
5390:
5373:
5369:
5367:
5364:
5363:
5344:
5341:
5340:
5323:
5319:
5317:
5314:
5313:
5296:
5292:
5290:
5287:
5286:
5262:
5258:
5250:
5244:
5240:
5230:
5223:
5219:
5211:
5205:
5201:
5191:
5188:
5179:
5175:
5170:
5164:
5160:
5153:
5146:
5142:
5137:
5131:
5127:
5120:
5118:
5113:
5110:
5109:
5091:
5080:
5079:
5073:
5062:
5061:
5059:
5048:
5044:
5038:
5034:
5032:
5030:
5027:
5026:
4999:
4995:
4993:
4990:
4989:
4972:
4968:
4966:
4963:
4962:
4945:
4941:
4939:
4936:
4935:
4914:
4910:
4902:
4896:
4892:
4880:
4875:
4869:
4861:
4850:
4848:
4845:
4844:
4824:
4820:
4812:
4806:
4802:
4790:
4785:
4779:
4771:
4760:
4758:
4755:
4754:
4733:
4729:
4723:
4719:
4717:
4708:
4704:
4702:
4699:
4698:
4679:
4675:
4669:
4665:
4663:
4651:
4647:
4645:
4642:
4641:
4619:
4615:
4606:
4602:
4601:
4594:
4590:
4581:
4577:
4576:
4574:
4565:
4561:
4559:
4556:
4555:
4522:
4511:
4510:
4505:
4496:
4492:
4490:
4487:
4486:
4467:
4456:
4455:
4450:
4441:
4437:
4435:
4432:
4431:
4415:
4412:
4411:
4391:
4382:
4350:
4314:
4305:
4299:
4287:Big Bang theory
4263:
4260:
4259:
4243:
4240:
4239:
4208:
4200:
4198:
4194:
4193:
4187:
4183:
4181:
4179:
4176:
4175:
4160:
4154:
4100:
4075:
4071:
4060:
4056:
4040:
4036:
4024:
4020:
4018:
4015:
4014:
3996:
3992:
3981:
3977:
3961:
3957:
3945:
3941:
3939:
3936:
3935:
3914:Equation 1
3908:Equation 7
3883:
3879:
3877:
3874:
3873:
3869:Equation 6
3844:
3840:
3838:
3835:
3834:
3827:Equation 7
3821:Equation 6
3815:Equation 8
3789:
3785:
3769:
3756:
3752:
3745:
3743:
3734:
3730:
3722:
3719:
3718:
3690:
3686:
3684:
3681:
3680:
3663:
3659:
3657:
3654:
3653:
3636:
3632:
3630:
3627:
3626:
3603:
3599:
3588:
3584:
3568:
3564:
3552:
3548:
3546:
3543:
3542:
3505:
3501:
3499:
3490:
3486:
3484:
3481:
3480:
3456:
3452:
3443:
3439:
3437:
3434:
3433:
3402:
3398:
3382:
3378:
3374:
3368:
3364:
3362:
3353:
3349:
3347:
3344:
3343:
3318:
3315:
3314:
3291:
3287:
3286:
3275:
3272:
3271:
3254:
3250:
3248:
3245:
3244:
3228:
3225:
3224:
3208:
3187:
3184:
3183:
3159:
3155:
3142:
3138:
3131:
3127:
3123:
3114:
3110:
3109:
3102:
3098:
3092:
3088:
3079:
3075:
3074:
3072:
3068:
3067:
3051:
3049:
3047:
3044:
3043:
3017:
3014:
3013:
2991:
2989:
2986:
2985:
2966:
2963:
2962:
2943:
2941:
2938:
2937:
2892:
2853:
2850:
2849:
2837:
2811:
2777:
2773:
2771:
2762:
2758:
2756:
2753:
2752:
2729:
2703:
2699:
2687:
2683:
2681:
2678:
2677:
2648:
2605:
2557:
2546:
2543:
2532:
2528:
2522:
2518:
2516:
2505:
2501:
2495:
2491:
2489:
2487:
2484:
2483:
2475:are related by
2429:
2418:
2415:
2404:
2400:
2394:
2390:
2388:
2386:
2383:
2382:
2356:
2352:
2337:
2326:
2323:
2318:
2315:
2314:
2298:
2294:
2285:
2281:
2263:
2261:
2256:
2253:
2252:
2227:
2223:
2214:
2210:
2208:
2205:
2204:
2174:
2170:
2168:
2165:
2164:
2141:
2137:
2129:
2124:
2113:
2110:
2109:
2090:
2078:
2074:
2065:
2061:
2059:
2056:
2055:
2011:
2007:
1992:
1988:
1983:
1965:
1961:
1959:
1956:
1955:
1932:
1928:
1926:
1923:
1922:
1902:
1891:
1888:
1887:
1849:
1845:
1844:
1839:
1824:
1820:
1806:
1782:
1778:
1766:
1762:
1747:
1743:
1725:
1721:
1709:
1705:
1703:
1700:
1699:
1672:
1668:
1666:
1663:
1662:
1646:
1643:
1642:
1625:
1622:
1621:
1597:
1594:
1593:
1571:
1567:
1565:
1562:
1561:
1539:
1535:
1533:
1530:
1529:
1511:
1507:
1502:
1490:
1486:
1484:
1481:
1480:
1472:of the source.
1470:reference frame
1441:
1438:
1437:
1420:
1417:
1416:
1401:
1396:
1370:
1366:
1355:
1351:
1335:
1331:
1319:
1315:
1313:
1310:
1309:
1304:
1272:
1268:
1252:
1248:
1244:
1238:
1234:
1232:
1223:
1219:
1217:
1214:
1213:
1208:
1184:
1176:
1173:
1172:
1170:
1147:
1143:
1130:
1126:
1119:
1115:
1111:
1102:
1098:
1097:
1090:
1086:
1080:
1076:
1067:
1063:
1062:
1060:
1056:
1055:
1039:
1037:
1035:
1032:
1031:
1026:
1024:
994:
990:
988:
979:
975:
973:
970:
969:
964:
939:
935:
923:
919:
917:
914:
913:
908:
878:
874:
872:
863:
859:
857:
854:
853:
848:
823:
819:
807:
803:
801:
798:
797:
792:
760:
749:
746:
735:
731:
725:
721:
719:
708:
704:
698:
694:
692:
690:
687:
686:
681:
644:
640:
632:
627:
616:
613:
612:
596:
593:
592:
576:
573:
572:
547:
536:
533:
532:
529:
447:
440:
399:
386:
385:
378:
377:
376:
286:
285:
282:
274:
273:
269:Aether theories
254:
253:
252:
233:
232:
213:
212:
207:
199:
198:
174:
173:
168:
160:
159:
105:
104:
101:
93:
92:
63:
62:
59:
12:
11:
5:
8370:
8360:
8359:
8354:
8339:
8338:
8326:
8314:
8302:
8290:
8267:
8266:
8256:
8253:
8252:
8250:
8249:
8242:
8237:
8232:
8227:
8222:
8217:
8212:
8207:
8202:
8197:
8192:
8187:
8182:
8177:
8172:
8170:Choquet-Bruhat
8167:
8162:
8157:
8152:
8147:
8142:
8137:
8132:
8127:
8122:
8117:
8112:
8107:
8102:
8097:
8091:
8089:
8085:
8084:
8081:
8080:
8078:
8077:
8070:
8069:
8064:
8059:
8052:
8051:
8046:
8041:
8036:
8031:
8022:Axisymmetric:
8019:
8018:
8013:
8007:
7996:
7995:
7990:
7985:
7980:
7975:
7970:
7961:Cosmological:
7958:
7956:
7950:
7949:
7947:
7946:
7941:
7936:
7930:
7928:
7922:
7921:
7919:
7918:
7913:
7902:frame-dragging
7899:
7894:
7889:
7886:Einstein rings
7882:Einstein cross
7875:
7864:
7863:
7858:
7852:
7847:
7842:
7829:
7819:
7818:
7813:
7808:
7803:
7797:
7795:
7791:
7790:
7788:
7787:
7785:Ernst equation
7782:
7777:
7772:
7767:
7762:
7757:
7755:BSSN formalism
7752:
7746:
7744:
7740:
7739:
7737:
7736:
7731:
7726:
7721:
7716:
7710:
7708:
7702:
7701:
7699:
7698:
7693:
7687:
7685:
7678:
7670:
7669:
7666:
7665:
7663:
7662:
7657:
7652:
7647:
7642:
7636:
7634:
7628:
7627:
7625:
7624:
7619:
7614:
7612:Ladder paradox
7609:
7604:
7599:
7594:
7589:
7584:
7578:
7576:
7572:
7571:
7569:
7568:
7562:
7560:
7556:
7555:
7553:
7552:
7547:
7542:
7537:
7532:
7527:
7522:
7517:
7515:Speed of light
7512:
7506:
7504:
7498:
7497:
7495:
7494:
7489:
7484:
7478:
7468:
7466:
7459:
7451:
7450:
7443:
7442:
7435:
7428:
7420:
7414:
7413:
7392:
7384:
7383:External links
7381:
7380:
7379:
7330:
7327:
7324:
7323:
7296:
7290:978-3642322198
7289:
7263:
7236:
7187:
7156:
7115:
7105:
7085:
7078:
7066:Lifshitz, E.M.
7053:
7023:
6998:
6979:(8): 832â833.
6958:
6938:978-0716703440
6937:
6919:
6891:
6877:
6837:
6822:
6814:Addison-Wesley
6802:Sands, Matthew
6782:
6756:
6718:
6708:
6687:
6686:
6684:
6681:
6678:
6677:
6650:(3): 251â254.
6634:
6607:(4): 255â258.
6584:
6565:(3): 583â593.
6549:
6530:(4): 165â166.
6514:
6487:(2): 151â155.
6471:
6452:(5): 529â535.
6436:
6401:
6366:
6307:
6258:
6239:(2): 350â352.
6223:
6185:
6136:
6117:(6): 197â198.
6091:
6090:
6088:
6085:
6082:
6081:
6060:
6049:
6048:
6046:
6043:
6042:
6041:
6036:
6031:
6026:
6021:
6016:
6011:
6006:
6004:Doppler effect
5999:
5996:
5977:
5974:
5969:
5965:
5938:
5935:
5930:
5926:
5914:
5913:
5896:
5892:
5888:
5884:
5878:
5874:
5870:
5867:
5864:
5857:
5853:
5849:
5845:
5839:
5835:
5831:
5828:
5825:
5815:
5807:
5803:
5799:
5796:
5791:
5787:
5784:
5781:
5774:
5770:
5766:
5759:
5752:
5748:
5742:
5735:
5732:
5727:
5719:
5715:
5711:
5708:
5703:
5699:
5696:
5693:
5686:
5682:
5678:
5671:
5664:
5660:
5654:
5647:
5644:
5638:
5631:
5627:
5621:
5617:
5583:
5578:
5573:
5549:
5545:
5522:
5498:
5494:
5450:
5446:
5423:
5419:
5398:
5376:
5372:
5348:
5326:
5322:
5299:
5295:
5283:
5282:
5265:
5261:
5257:
5253:
5247:
5243:
5239:
5236:
5233:
5226:
5222:
5218:
5214:
5208:
5204:
5200:
5197:
5194:
5182:
5178:
5173:
5167:
5163:
5159:
5156:
5149:
5145:
5140:
5134:
5130:
5126:
5123:
5117:
5094:
5090:
5087:
5083:
5076:
5072:
5069:
5065:
5058:
5051:
5047:
5041:
5037:
5007:
5002:
4998:
4975:
4971:
4948:
4944:
4933:
4932:
4917:
4913:
4909:
4905:
4899:
4895:
4891:
4888:
4883:
4878:
4874:
4868:
4864:
4860:
4857:
4853:
4842:
4827:
4823:
4819:
4815:
4809:
4805:
4801:
4798:
4793:
4788:
4784:
4778:
4774:
4770:
4767:
4763:
4752:
4736:
4732:
4726:
4722:
4716:
4711:
4707:
4682:
4678:
4672:
4668:
4662:
4659:
4654:
4650:
4639:
4622:
4618:
4614:
4609:
4605:
4597:
4593:
4589:
4584:
4580:
4573:
4568:
4564:
4525:
4521:
4518:
4514:
4509:
4504:
4499:
4495:
4470:
4466:
4463:
4459:
4454:
4449:
4444:
4440:
4419:
4390:
4387:
4381:
4378:
4352:The advent of
4349:
4346:
4313:
4310:
4301:Main article:
4298:
4295:
4267:
4247:
4224:
4221:
4214:
4211:
4206:
4203:
4197:
4190:
4186:
4170:describes the
4153:
4150:
4141:
4140:
4136:
4132:
4116:Doppler effect
4099:
4096:
4095:
4094:
4083:
4078:
4074:
4069:
4063:
4059:
4055:
4052:
4049:
4046:
4043:
4039:
4035:
4032:
4027:
4023:
4012:
3999:
3995:
3990:
3984:
3980:
3976:
3973:
3970:
3967:
3964:
3960:
3956:
3953:
3948:
3944:
3894:
3891:
3886:
3882:
3855:
3852:
3847:
3843:
3810:
3809:
3792:
3788:
3784:
3781:
3778:
3775:
3772:
3767:
3764:
3759:
3755:
3751:
3748:
3742:
3737:
3733:
3729:
3726:
3693:
3689:
3666:
3662:
3639:
3635:
3623:
3622:
3611:
3606:
3602:
3597:
3591:
3587:
3583:
3580:
3577:
3574:
3571:
3567:
3563:
3560:
3555:
3551:
3531:
3530:
3518:
3513:
3508:
3504:
3498:
3493:
3489:
3459:
3455:
3451:
3446:
3442:
3430:
3429:
3418:
3411:
3405:
3401:
3397:
3394:
3391:
3388:
3385:
3381:
3377:
3371:
3367:
3361:
3356:
3352:
3325:
3322:
3300:
3294:
3290:
3285:
3282:
3279:
3257:
3253:
3232:
3207:
3204:
3191:
3180:
3179:
3166:
3162:
3158:
3153:
3145:
3141:
3134:
3130:
3126:
3122:
3117:
3113:
3105:
3101:
3095:
3091:
3087:
3082:
3078:
3071:
3066:
3061:
3057:
3054:
3024:
3021:
3001:
2997:
2994:
2970:
2949:
2946:
2891:
2888:
2857:
2836:
2833:
2810:
2807:
2799:
2798:
2785:
2780:
2776:
2770:
2765:
2761:
2728:
2721:
2720:
2719:
2706:
2702:
2698:
2695:
2690:
2686:
2670:
2669:
2665:
2647:
2644:
2604:
2601:
2585:
2584:
2573:
2566:
2563:
2560:
2555:
2552:
2549:
2542:
2535:
2531:
2525:
2521:
2515:
2508:
2504:
2498:
2494:
2458:Doppler factor
2456:is called the
2454:
2453:
2438:
2435:
2432:
2427:
2424:
2421:
2414:
2407:
2403:
2397:
2393:
2376:
2375:
2364:
2359:
2355:
2346:
2343:
2340:
2335:
2332:
2329:
2322:
2301:
2297:
2288:
2284:
2280:
2277:
2272:
2269:
2266:
2260:
2241:
2236:
2233:
2230:
2226:
2222:
2217:
2213:
2183:
2180:
2177:
2173:
2161:Lorentz factor
2144:
2140:
2136:
2133:
2127:
2123:
2120:
2117:
2097:
2093:
2087:
2084:
2081:
2077:
2073:
2068:
2064:
2034:
2031:
2028:
2025:
2022:
2019:
2014:
2010:
2006:
2001:
1998:
1995:
1991:
1986:
1982:
1979:
1974:
1971:
1968:
1964:
1941:
1938:
1935:
1931:
1909:
1905:
1901:
1898:
1895:
1875:
1869:
1866:
1863:
1860:
1857:
1852:
1848:
1843:
1838:
1833:
1830:
1827:
1823:
1819:
1816:
1813:
1809:
1805:
1802:
1799:
1796:
1791:
1788:
1785:
1781:
1777:
1774:
1769:
1765:
1761:
1756:
1753:
1750:
1746:
1742:
1739:
1734:
1731:
1728:
1724:
1720:
1717:
1712:
1708:
1681:
1678:
1675:
1671:
1661:during a time
1650:
1629:
1615:speed of light
1601:
1574:
1570:
1542:
1538:
1514:
1510:
1505:
1501:
1498:
1493:
1489:
1445:
1424:
1400:
1397:
1395:
1392:
1389:
1388:
1386:
1373:
1369:
1364:
1358:
1354:
1350:
1347:
1344:
1341:
1338:
1334:
1330:
1327:
1322:
1318:
1307:
1299:
1298:
1296:
1281:
1275:
1271:
1267:
1264:
1261:
1258:
1255:
1251:
1247:
1241:
1237:
1231:
1226:
1222:
1211:
1203:
1202:
1190:
1187:
1183:
1180:
1167:
1154:
1150:
1146:
1141:
1133:
1129:
1122:
1118:
1114:
1110:
1105:
1101:
1093:
1089:
1083:
1079:
1075:
1070:
1066:
1059:
1054:
1049:
1045:
1042:
1029:
1019:
1018:
1015:
1002:
997:
993:
987:
982:
978:
967:
959:
958:
955:
942:
938:
934:
931:
926:
922:
911:
903:
902:
899:
886:
881:
877:
871:
866:
862:
851:
843:
842:
839:
826:
822:
818:
815:
810:
806:
795:
787:
786:
784:
769:
766:
763:
758:
755:
752:
745:
738:
734:
728:
724:
718:
711:
707:
701:
697:
684:
682:Doppler effect
676:
675:
672:
669:
647:
643:
639:
636:
630:
626:
623:
620:
600:
580:
560:
557:
554:
550:
546:
543:
540:
528:
525:
490:Doppler effect
479:Doppler effect
445:
442:
441:
439:
438:
431:
424:
416:
413:
412:
411:
410:
397:
380:
379:
375:
374:
369:
364:
359:
354:
349:
344:
339:
334:
329:
324:
319:
314:
309:
304:
299:
294:
288:
287:
281:
280:
279:
276:
275:
272:
271:
266:
261:
255:
251:
250:
247:
241:
240:
239:
238:
235:
234:
231:
230:
225:
220:
214:
206:
205:
204:
201:
200:
197:
196:
191:
186:
181:
175:
167:
166:
165:
162:
161:
158:
157:
152:
147:
142:
137:
132:
127:
122:
117:
112:
106:
100:
99:
98:
95:
94:
91:
90:
85:
80:
78:Speed of light
75:
70:
64:
58:
57:
56:
53:
52:
51:
50:
45:
40:
32:
31:
23:
22:
9:
6:
4:
3:
2:
8369:
8358:
8355:
8353:
8350:
8349:
8347:
8337:
8332:
8327:
8325:
8320:
8315:
8313:
8308:
8303:
8301:
8296:
8291:
8289:
8279:
8278:
8275:
8264:
8254:
8248:
8247:
8243:
8241:
8238:
8236:
8233:
8231:
8228:
8226:
8223:
8221:
8218:
8216:
8213:
8211:
8208:
8206:
8203:
8201:
8198:
8196:
8193:
8191:
8188:
8186:
8183:
8181:
8178:
8176:
8173:
8171:
8168:
8166:
8163:
8161:
8158:
8156:
8155:Chandrasekhar
8153:
8151:
8148:
8146:
8143:
8141:
8138:
8136:
8133:
8131:
8128:
8126:
8123:
8121:
8118:
8116:
8115:Schwarzschild
8113:
8111:
8108:
8106:
8103:
8101:
8098:
8096:
8093:
8092:
8090:
8086:
8076:
8072:
8071:
8068:
8065:
8063:
8060:
8058:
8054:
8053:
8050:
8047:
8045:
8042:
8040:
8037:
8035:
8032:
8029:
8025:
8021:
8020:
8017:
8014:
8011:
8008:
8006:
8002:
8001:Schwarzschild
7998:
7997:
7994:
7991:
7989:
7986:
7984:
7981:
7979:
7976:
7974:
7971:
7968:
7964:
7960:
7959:
7957:
7955:
7951:
7945:
7942:
7940:
7937:
7935:
7932:
7931:
7929:
7923:
7917:
7914:
7911:
7907:
7903:
7900:
7898:
7897:Shapiro delay
7895:
7893:
7890:
7887:
7883:
7879:
7876:
7873:
7869:
7866:
7865:
7862:
7859:
7856:
7853:
7851:
7848:
7846:
7843:
7841:
7840:collaboration
7837:
7833:
7830:
7828:
7824:
7821:
7820:
7817:
7814:
7812:
7809:
7807:
7806:Event horizon
7804:
7802:
7799:
7798:
7796:
7792:
7786:
7783:
7781:
7778:
7776:
7773:
7771:
7768:
7766:
7763:
7761:
7758:
7756:
7753:
7751:
7750:ADM formalism
7748:
7747:
7745:
7741:
7735:
7732:
7730:
7727:
7725:
7722:
7720:
7717:
7715:
7712:
7711:
7709:
7703:
7697:
7694:
7692:
7689:
7688:
7686:
7682:
7679:
7677:
7671:
7661:
7658:
7656:
7655:Biquaternions
7653:
7651:
7648:
7646:
7643:
7641:
7638:
7637:
7635:
7633:
7629:
7623:
7620:
7618:
7615:
7613:
7610:
7608:
7605:
7603:
7600:
7598:
7595:
7593:
7590:
7588:
7585:
7583:
7582:Time dilation
7580:
7579:
7577:
7573:
7567:
7564:
7563:
7561:
7557:
7551:
7548:
7546:
7543:
7541:
7538:
7536:
7535:Proper length
7533:
7531:
7528:
7526:
7523:
7521:
7518:
7516:
7513:
7511:
7508:
7507:
7505:
7499:
7493:
7490:
7488:
7485:
7482:
7479:
7477:
7473:
7470:
7469:
7467:
7463:
7460:
7458:
7452:
7448:
7441:
7436:
7434:
7429:
7427:
7422:
7421:
7418:
7402:
7398:
7393:
7390:
7387:
7386:
7376:
7372:
7368:
7364:
7360:
7356:
7351:
7346:
7342:
7338:
7333:
7332:
7311:
7307:
7300:
7292:
7286:
7282:
7277:
7276:
7267:
7251:
7247:
7240:
7225:
7221:
7217:
7213:
7209:
7205:
7198:
7191:
7176:on 7 May 2021
7175:
7171:
7167:
7160:
7145:on 2008-08-03
7141:
7137:
7133:
7132:The Physicist
7126:
7119:
7113:
7108:
7102:
7098:
7097:
7089:
7081:
7079:9780750627689
7075:
7071:
7067:
7063:
7057:
7041:
7034:
7032:
7030:
7028:
7011:
7010:
7002:
6994:
6990:
6986:
6982:
6978:
6974:
6967:
6965:
6963:
6954:
6948:
6940:
6934:
6930:
6923:
6907:
6900:
6898:
6896:
6884:
6880:
6874:
6870:
6866:
6859:
6852:
6850:
6848:
6846:
6844:
6842:
6833:
6829:
6825:
6823:9780201021165
6819:
6815:
6811:
6807:
6803:
6799:
6795:
6789:
6787:
6778:
6774:
6770:
6763:
6761:
6745:
6741:
6737:
6733:
6729:
6722:
6716:
6711:
6705:
6701:
6700:
6692:
6688:
6673:
6669:
6665:
6661:
6657:
6653:
6649:
6645:
6638:
6630:
6626:
6622:
6618:
6614:
6610:
6606:
6602:
6595:
6588:
6580:
6576:
6572:
6568:
6564:
6560:
6553:
6545:
6541:
6537:
6533:
6529:
6525:
6518:
6510:
6506:
6502:
6498:
6494:
6490:
6486:
6482:
6475:
6467:
6463:
6459:
6455:
6451:
6447:
6440:
6432:
6428:
6424:
6420:
6416:
6412:
6405:
6397:
6393:
6389:
6385:
6381:
6377:
6370:
6362:
6358:
6353:
6348:
6343:
6338:
6334:
6330:
6326:
6322:
6318:
6311:
6303:
6299:
6294:
6289:
6285:
6281:
6278:(4881): 679.
6277:
6273:
6269:
6262:
6254:
6250:
6246:
6242:
6238:
6234:
6227:
6219:
6215:
6211:
6207:
6203:
6199:
6192:
6190:
6183:
6177:
6172:
6168:
6164:
6160:
6157:(in German).
6156:
6155:
6150:
6143:
6141:
6132:
6128:
6124:
6120:
6116:
6112:
6111:
6106:
6099:
6097:
6092:
6078:
6074:
6070:
6064:
6054:
6050:
6040:
6037:
6035:
6032:
6030:
6029:Time dilation
6027:
6025:
6022:
6020:
6017:
6015:
6012:
6010:
6007:
6005:
6002:
6001:
5995:
5993:
5992:
5975:
5972:
5967:
5963:
5954:
5953:
5936:
5933:
5928:
5924:
5894:
5886:
5882:
5876:
5872:
5865:
5862:
5855:
5847:
5843:
5837:
5833:
5826:
5823:
5789:
5782:
5779:
5733:
5730:
5701:
5694:
5691:
5645:
5642:
5636:
5629:
5625:
5619:
5615:
5605:
5601:
5600:
5599:
5596:
5477:
5473:
5470:
5468:
5467:
5448:
5444:
5421:
5417:
5396:
5374:
5370:
5360:
5346:
5324:
5320:
5297:
5293:
5263:
5255:
5251:
5245:
5241:
5234:
5231:
5224:
5216:
5212:
5206:
5202:
5195:
5192:
5180:
5176:
5171:
5165:
5161:
5157:
5154:
5147:
5143:
5138:
5132:
5128:
5124:
5121:
5115:
5088:
5085:
5070:
5067:
5056:
5049:
5045:
5039:
5035:
5025:
5021:
5020:
5019:
5005:
5000:
4996:
4973:
4969:
4946:
4942:
4915:
4907:
4903:
4897:
4893:
4886:
4881:
4876:
4872:
4866:
4858:
4855:
4843:
4825:
4817:
4813:
4807:
4803:
4796:
4791:
4786:
4782:
4776:
4768:
4765:
4753:
4734:
4730:
4724:
4720:
4714:
4709:
4705:
4680:
4676:
4670:
4666:
4660:
4657:
4652:
4648:
4640:
4620:
4616:
4612:
4607:
4603:
4595:
4591:
4587:
4582:
4578:
4571:
4566:
4562:
4554:
4553:
4552:
4550:
4546:
4542:
4519:
4516:
4507:
4502:
4497:
4493:
4464:
4461:
4452:
4447:
4442:
4438:
4417:
4407:
4403:
4395:
4386:
4377:
4375:
4371:
4367:
4362:
4360:
4355:
4345:
4341:
4338:
4333:
4329:
4327:
4318:
4309:
4304:
4294:
4292:
4288:
4283:
4281:
4265:
4245:
4222:
4219:
4212:
4209:
4204:
4201:
4195:
4188:
4184:
4173:
4169:
4164:
4159:
4149:
4146:
4137:
4133:
4129:
4128:
4127:
4125:
4121:
4117:
4113:
4104:
4098:Visualization
4081:
4076:
4072:
4067:
4061:
4057:
4053:
4050:
4047:
4044:
4041:
4037:
4033:
4030:
4025:
4021:
4013:
3997:
3993:
3988:
3982:
3978:
3974:
3971:
3968:
3965:
3962:
3958:
3954:
3951:
3946:
3942:
3934:
3933:
3932:
3930:
3926:
3921:
3918:
3916:
3915:
3910:
3909:
3892:
3889:
3884:
3880:
3871:
3870:
3853:
3850:
3845:
3841:
3831:
3829:
3828:
3823:
3822:
3817:
3816:
3790:
3786:
3782:
3779:
3776:
3773:
3770:
3765:
3762:
3757:
3753:
3749:
3746:
3740:
3735:
3731:
3727:
3724:
3717:
3713:
3712:
3711:
3709:
3691:
3687:
3664:
3660:
3637:
3633:
3609:
3604:
3600:
3595:
3589:
3585:
3581:
3578:
3575:
3572:
3569:
3565:
3561:
3558:
3553:
3549:
3541:
3537:
3536:
3535:
3516:
3511:
3506:
3502:
3496:
3491:
3487:
3479:
3478:
3477:
3475:
3457:
3453:
3449:
3444:
3440:
3416:
3409:
3403:
3399:
3395:
3392:
3389:
3386:
3383:
3379:
3375:
3369:
3365:
3359:
3354:
3350:
3342:
3338:
3337:
3336:
3323:
3320:
3311:
3298:
3292:
3288:
3283:
3280:
3277:
3255:
3251:
3230:
3221:
3212:
3203:
3189:
3164:
3160:
3156:
3151:
3143:
3139:
3132:
3128:
3124:
3120:
3115:
3111:
3103:
3099:
3093:
3089:
3085:
3080:
3076:
3069:
3064:
3059:
3055:
3052:
3042:
3038:
3037:
3036:
3022:
3019:
2999:
2995:
2992:
2983:
2968:
2947:
2944:
2934:
2932:
2927:
2925:
2921:
2915:
2913:
2909:
2904:
2896:
2887:
2885:
2880:
2878:
2876:
2869:
2855:
2841:
2832:
2828:
2825:
2815:
2806:
2804:
2783:
2778:
2774:
2768:
2763:
2759:
2751:
2747:
2746:
2745:
2738:
2733:
2726:
2704:
2700:
2696:
2693:
2688:
2684:
2676:
2672:
2671:
2666:
2662:
2661:
2660:
2652:
2643:
2641:
2636:
2634:
2628:
2626:
2622:
2618:
2615:predicted by
2614:
2610:
2600:
2598:
2594:
2590:
2571:
2564:
2561:
2558:
2553:
2550:
2547:
2540:
2533:
2529:
2523:
2519:
2513:
2506:
2502:
2496:
2492:
2482:
2478:
2477:
2476:
2474:
2469:
2467:
2463:
2459:
2436:
2433:
2430:
2425:
2422:
2419:
2412:
2405:
2401:
2395:
2391:
2381:
2380:
2379:
2362:
2357:
2353:
2344:
2341:
2338:
2333:
2330:
2327:
2320:
2299:
2295:
2286:
2282:
2278:
2275:
2270:
2267:
2264:
2258:
2239:
2234:
2231:
2228:
2224:
2220:
2215:
2211:
2203:
2199:
2198:
2197:
2181:
2178:
2175:
2171:
2162:
2142:
2138:
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2032:
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2020:
2012:
2008:
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1993:
1989:
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1980:
1977:
1972:
1969:
1966:
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1939:
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1464:
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1406:time dilation
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1367:
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1027:common center
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495:
494:time dilation
491:
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270:
267:
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248:
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242:
237:
236:
229:
228:Four-momentum
226:
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216:
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195:
192:
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187:
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143:
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131:
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126:
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118:
116:
113:
111:
110:Time dilation
108:
107:
97:
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86:
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39:
36:
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34:
33:
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24:
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8245:
7939:KaluzaâKlein
7691:Introduction
7617:Twin paradox
7601:
7405:. Retrieved
7400:
7340:
7336:
7314:. Retrieved
7309:
7299:
7274:
7266:
7254:. Retrieved
7249:
7239:
7227:. Retrieved
7210:(1): 56â60.
7207:
7203:
7190:
7178:. Retrieved
7174:the original
7169:
7159:
7147:. Retrieved
7140:the original
7135:
7131:
7118:
7095:
7088:
7069:
7062:Landau, L.D.
7056:
7044:. Retrieved
7014:. Retrieved
7008:
7001:
6976:
6972:
6928:
6922:
6910:. Retrieved
6883:the original
6864:
6809:
6768:
6747:. Retrieved
6735:
6731:
6721:
6698:
6691:
6647:
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6637:
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6558:
6552:
6527:
6523:
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6439:
6414:
6411:Phys. Rev. D
6410:
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6379:
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6320:
6310:
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6271:
6261:
6236:
6232:
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4369:
4363:
4358:
4351:
4342:
4334:
4330:
4323:
4306:
4284:
4279:
4168:Planck's law
4165:
4161:
4142:
4123:
4109:
3928:
3924:
3922:
3919:
3912:
3906:
3867:
3832:
3825:
3819:
3813:
3811:
3624:
3532:
3431:
3312:
3243:at an angle
3217:
3181:
3035:is given by
2981:
2935:
2930:
2928:
2916:
2910:conducted a
2905:
2901:
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2870:
2846:
2829:
2823:
2820:
2802:
2800:
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2736:
2724:
2657:
2637:
2629:
2624:
2608:
2606:
2596:
2588:
2586:
2470:
2462:astrophysics
2457:
2455:
2377:
2052:time dilated
2049:
2046:
1695:
1619:
1475:Suppose one
1474:
1467:
1462:
1458:
1412:
1410:
1402:
530:
506:
487:
462:
460:
454:
134:
102:Consequences
48:Formulations
8312:Mathematics
8028:KerrâNewman
7999:Spherical:
7868:Other tests
7811:Singularity
7743:Formulation
7705:Fundamental
7559:Formulation
7540:Proper time
7501:Fundamental
7204:Am. J. Phys
7170:Physics FAQ
7042:. Mathpages
6929:Gravitation
6908:. Mathpages
6738:: 105â111.
6327:(4): 1340.
5389:approaches
2473:wavelengths
223:Proper mass
218:Proper time
60:Foundations
8346:Categories
8180:Zel'dovich
8088:Scientists
8067:Alcubierre
7874:of Mercury
7872:precession
7801:Black hole
7684:Background
7676:relativity
7645:World line
7640:Light cone
7465:Background
7457:relativity
7447:Relativity
7407:17 October
7316:16 October
7256:17 October
7229:17 October
7180:17 October
7149:17 October
7016:13 October
6912:12 October
6832:2010938208
6749:11 October
6683:References
6382:(7): 215.
6077:wavelength
4326:canal rays
4145:black body
2633:canal rays
2378:The ratio
1558:wavelength
1394:Derivation
957:Blueshift
841:Blueshift
496:effect of
471:wavelength
312:FitzGerald
297:Sommerfeld
249:Precursors
194:Light cone
189:World line
8288:Astronomy
8150:Robertson
8135:Friedmann
8130:Eddington
8120:de Sitter
7954:Solutions
7832:detectors
7827:astronomy
7794:Phenomena
7729:Geodesics
7632:Spacetime
7575:Phenomena
6947:cite book
6509:120963034
6073:frequency
6024:Blueshift
5866:−
5827:−
5790:θ
5783:
5734:−
5702:θ
5695:
5646:−
5235:−
5196:−
5125:−
4887:−
4797:−
4661:−
4613:−
4588:−
4220:−
4058:θ
4054:
4048:β
4045:−
4034:γ
3979:θ
3975:
3969:β
3966:−
3955:γ
3881:θ
3842:θ
3787:θ
3783:
3777:β
3774:−
3766:β
3763:−
3754:θ
3750:
3732:θ
3728:
3688:θ
3661:θ
3634:θ
3586:θ
3582:
3576:β
3573:−
3562:γ
3512:γ
3458:∘
3441:θ
3400:θ
3396:
3390:β
3376:γ
3321:γ
3289:θ
3284:
3252:θ
3190:ω
3140:ω
3121:−
3100:ω
3086:−
2856:γ
2784:γ
2723:Receiver
2697:γ
2613:blueshift
2565:β
2562:−
2554:β
2503:λ
2493:λ
2437:β
2434:−
2426:β
2345:β
2334:β
2331:−
2283:β
2279:−
2271:β
2268:−
2240:γ
2139:β
2135:−
2116:γ
2096:γ
2027:β
2024:−
1894:β
1865:β
1862:−
1818:⟺
1801:−
1764:λ
1760:⟺
1707:λ
1590:frequency
1537:λ
1488:λ
1477:wavefront
1353:θ
1349:
1343:β
1340:−
1329:γ
1270:θ
1266:
1260:β
1246:γ
1128:ω
1109:−
1088:ω
1074:−
1017:Redshift
1001:γ
933:γ
901:Redshift
885:γ
817:γ
768:β
765:−
757:β
706:λ
696:λ
642:β
638:−
619:γ
539:β
513:blueshift
475:amplitude
467:frequency
362:Ehrenfest
332:Minkowski
302:Michelson
170:Spacetime
8263:Category
8140:LemaĂźtre
8105:Einstein
8095:Poincaré
8055:Others:
8039:TaubâNUT
8005:interior
7927:theories
7925:Advanced
7892:redshift
7707:concepts
7525:Rapidity
7503:concepts
7375:11347287
7068:(2005).
7046:6 August
6777:5947329M
6672:10054065
6629:10031461
6361:16593162
6302:42033531
6019:Redshift
5998:See also
4485:, while
3833:Setting
3824:yields
3129:′
3056:′
2996:′
2982:anywhere
2948:′
2621:redshift
2589:receiver
2108:, where
1459:negative
1189:′
1117:′
1044:′
668:Scenario
509:redshift
407:Category
357:von Laue
327:Poincaré
317:Herglotz
292:Einstein
209:Dynamics
8336:Science
8324:Physics
8274:Portals
8205:Hawking
8200:Penrose
8185:Novikov
8165:Wheeler
8110:Hilbert
8100:Lorentz
8057:pp-wave
7878:lensing
7674:General
7455:Special
7355:Bibcode
7212:Bibcode
6981:Bibcode
6740:Bibcode
6652:Bibcode
6609:Bibcode
6567:Bibcode
6532:Bibcode
6489:Bibcode
6454:Bibcode
6419:Bibcode
6384:Bibcode
6329:Bibcode
6280:Bibcode
6241:Bibcode
6206:Bibcode
6163:Bibcode
6119:Bibcode
5603:Eq. 10:
4238:(where
3911:yields
2159:is the
1613:is the
1588:is the
1556:is the
1463:towards
671:Formula
342:Abraham
322:Lorentz
245:History
8246:others
8235:Thorne
8225:Misner
8210:Taylor
8195:Geroch
8190:Ehlers
8160:Zwicky
7978:Kasner
7373:
7287:
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6272:Nature
5023:Eq. 9:
4374:et al.
4370:et al.
3818:into
3715:Eq. 8:
3539:Eq. 7:
3340:Eq. 6:
3182:where
3040:Eq. 5:
2877:(MCRF)
2803:seeing
2749:Eq. 4:
2674:Eq. 3:
2480:Eq. 2:
2464:: see
2201:Eq. 1:
1886:where
674:Notes
405:
367:Tolman
352:Planck
337:Fizeau
307:Morley
283:People
8300:Stars
8240:Weiss
8220:Bondi
8215:Hulse
8145:Milne
8049:discs
7993:Milne
7988:Gödel
7845:Virgo
7371:S2CID
7345:arXiv
7283:â22.
7200:(PDF)
7143:(PDF)
7128:(PDF)
6886:(PDF)
6861:(PDF)
6597:(PDF)
6505:S2CID
6298:S2CID
6045:Notes
1171:when
372:Dirac
8175:Kerr
8125:Weyl
8024:Kerr
7884:and
7838:and
7836:LIGO
7409:2018
7318:2018
7285:ISBN
7258:2018
7231:2018
7182:2018
7151:2018
7101:ISBN
7074:ISBN
7048:2015
7018:2018
6953:link
6933:ISBN
6914:2018
6873:ISBN
6828:LCCN
6818:ISBN
6751:2018
6704:ISBN
6668:PMID
6625:PMID
6357:PMID
5436:and
5312:and
4961:and
3927:and
2908:Moon
2884:both
2824:sees
2737:sees
2725:sees
2625:sees
1698:So,
1413:away
556:>
473:and
461:The
347:Born
8230:Yau
7855:GEO
7363:doi
7220:doi
6989:doi
6660:doi
6617:doi
6575:doi
6540:doi
6497:doi
6485:289
6462:doi
6427:doi
6392:doi
6347:PMC
6337:doi
6288:doi
6276:198
6249:doi
6214:doi
6202:129
6171:doi
6159:322
6127:doi
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6071:in
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5692:cos
5285:If
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4608:B
4604:t
4596:A
4592:x
4583:B
4579:x
4572:=
4567:s
4563:c
4549:B
4541:;
4524:|
4520:B
4517:O
4513:|
4508:n
4503:=
4498:r
4494:f
4469:|
4465:A
4462:O
4458:|
4453:n
4448:=
4443:s
4439:f
4418:n
4359:H
4266:f
4246:T
4223:1
4213:T
4210:k
4205:f
4202:h
4196:e
4189:3
4185:f
4124:c
4082:.
4077:s
4073:B
4068:)
4062:s
4042:1
4038:(
4031:=
4026:r
4022:B
3998:s
3994:E
3989:)
3983:s
3963:1
3959:(
3952:=
3947:r
3943:E
3929:B
3925:E
3893:0
3890:=
3885:s
3854:0
3851:=
3846:r
3791:s
3771:1
3758:s
3741:=
3736:r
3692:r
3665:s
3638:s
3610:.
3605:s
3601:f
3596:)
3590:s
3570:1
3566:(
3559:=
3554:r
3550:f
3517:.
3507:s
3503:f
3497:=
3492:r
3488:f
3450:=
3445:r
3417:.
3410:)
3404:r
3387:+
3384:1
3380:(
3370:s
3366:f
3360:=
3355:r
3351:f
3324::
3299:.
3293:r
3278:v
3256:r
3231:v
3165:2
3161:/
3157:1
3152:)
3144:2
3133:2
3125:R
3116:2
3112:c
3104:2
3094:2
3090:R
3081:2
3077:c
3070:(
3065:=
3060:f
3053:f
3023:,
3020:f
3000:,
2993:f
2969:R
2945:R
2779:s
2775:f
2769:=
2764:r
2760:f
2705:s
2701:f
2694:=
2689:r
2685:f
2631:"
2572:,
2559:1
2551:+
2548:1
2541:=
2534:r
2530:f
2524:s
2520:f
2514:=
2507:s
2497:r
2431:1
2423:+
2420:1
2413:=
2406:r
2402:f
2396:s
2392:f
2363:.
2358:s
2354:f
2342:+
2339:1
2328:1
2321:=
2300:s
2296:f
2287:2
2276:1
2265:1
2259:=
2235:s
2232:,
2229:r
2225:f
2221:=
2216:r
2212:f
2182:s
2179:,
2176:r
2172:t
2143:2
2132:1
2126:/
2122:1
2119:=
2092:/
2086:s
2083:,
2080:r
2076:t
2072:=
2067:r
2063:t
2033:.
2030:)
2021:1
2018:(
2013:s
2009:f
2005:=
2000:s
1997:,
1994:r
1990:t
1985:/
1981:1
1978:=
1973:s
1970:,
1967:r
1963:f
1940:s
1937:,
1934:r
1930:f
1908:c
1904:/
1900:v
1897:=
1874:,
1868:)
1859:1
1856:(
1851:s
1847:f
1842:1
1837:=
1832:s
1829:,
1826:r
1822:t
1815:)
1812:c
1808:/
1804:v
1798:1
1795:(
1790:s
1787:,
1784:r
1780:t
1776:c
1773:=
1768:s
1755:s
1752:,
1749:r
1745:t
1741:c
1738:=
1733:s
1730:,
1727:r
1723:t
1719:v
1716:+
1711:s
1680:s
1677:,
1674:r
1670:t
1649:v
1628:c
1600:c
1573:s
1569:f
1541:s
1513:s
1509:f
1504:/
1500:c
1497:=
1492:s
1444:v
1423:v
1372:s
1368:f
1363:)
1357:s
1337:1
1333:(
1326:=
1321:r
1317:f
1280:)
1274:r
1257:+
1254:1
1250:(
1240:s
1236:f
1230:=
1225:r
1221:f
1186:R
1182:=
1179:R
1153:2
1149:/
1145:1
1140:)
1132:2
1121:2
1113:R
1104:2
1100:c
1092:2
1082:2
1078:R
1069:2
1065:c
1058:(
1053:=
1048:f
1041:f
996:s
992:f
986:=
981:r
977:f
941:s
937:f
930:=
925:r
921:f
880:s
876:f
870:=
865:r
861:f
825:s
821:f
814:=
809:r
805:f
762:1
754:+
751:1
744:=
737:r
733:f
727:s
723:f
717:=
710:s
700:r
646:2
635:1
629:/
625:1
622:=
599:c
579:v
559:0
553:c
549:/
545:v
542:=
511:/
455:c
435:e
428:t
421:v
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