# Time dilation

Time dilation

In the theory of relativity, time dilation is an observed difference of elapsed time between two events as measured by observers either moving relative to each other or differently situated from gravitational masses. An accurate clock at rest with respect to one observer may be measured to tick at a different rate when compared to a second observer's own equally accurate clocks. This effect arises not from technical aspects of the clocks nor from the fact that signals need time to propagate, but from the nature of space-time itself.

## Overview

Time dilation can arise from:

1. the relative velocity of motion between two observers, or
2. the difference in their distance from a gravitational mass.

### Relative velocity time dilation

When two observers are in relative uniform motion and uninfluenced by any gravitational mass, the point of view of each will be that the other's (moving) clock is ticking at a slower rate than the local clock. The faster the relative velocity, the greater the magnitude of time dilation. This case is sometimes called special relativistic time dilation. It is often interpreted as time "slowing down" for the other (moving) clock. But that is only true from the physical point of view of the local observer, and of others at relative rest (i.e. in the local observer's frame of reference). The point of view of the other observer will be that again the local clock (this time the other clock) is correct and it is the distant moving one that is slow. From a local perspective, time registered by clocks that are at rest with respect to the local frame of reference (and far from any gravitational mass) always appears to pass at the same rate.[1]

### Gravitational time dilation

There is another case of time dilation, where both observers are differently situated in their distance from a significant gravitational mass, such as (for terrestrial observers) the Earth or the Sun. One may suppose for simplicity that the observers are at relative rest (which is not the case of two observers both rotating with the Earth—an extra factor described below). In the simplified case, the general theory of relativity describes how, for both observers, the clock that is closer to the gravitational mass, i.e. deeper in its "gravity well", appears to go slower than the clock that is more distant from the mass (or higher in altitude away from the center of the gravitational mass). That does not mean that the two observers fully agree: each still makes the local clock to be correct; the observer more distant from the mass (higher in altitude) measures the other clock (closer to the mass, lower in altitude) to be slower than the local correct rate, and the observer situated closer to the mass (lower in altitude) measures the other clock (farther from the mass, higher in altitude) to be faster than the local correct rate. They agree at least that the clock nearer the mass is slower in rate and on the ratio of the difference.

### Time dilation: special vs. general theories of relativity

In Albert Einstein's theories of relativity, time dilation in these two circumstances can be summarized:

Special and general relativistic effects can combine, for example in some time-scale applications mentioned below.

In special relativity, the time dilation effect is reciprocal: as observed from the point of view of either of two clocks which are in motion with respect to each other, it will be the other clock that is time dilated. (This presumes that the relative motion of both parties is uniform; that is, they do not accelerate with respect to one another during the course of the observations.)

In contrast, gravitational time dilation (as treated in general relativity) is not reciprocal: an observer at the top of a tower will observe that clocks at ground level tick slower, and observers on the ground will agree about the direction and the ratio of the difference. There is not full agreement, as all the observers make their own local clocks out to be correct, but the direction and ratio of gravitational time dilation is agreed by all observers, independent of their altitude.

## Simple inference of time dilation due to relative velocity

Observer at rest sees time 2L/c.
Observer moving parallel relative to setup, sees longer path, time > 2L/c, same speed c.

Time dilation can be inferred from the observed fact of the constancy of the speed of light in all reference frames.[2][3][4][5]

This constancy of the speed of light means, counter to intuition, that speeds of material objects and light are not additive. It is not possible to make the speed of light appear faster by approaching at speed towards the material source that is emitting light. It is not possible to make the speed of light appear slower by receding from the source at speed. From one point of view, it is the implications of this unexpected constancy that take away from constancies expected elsewhere.

Consider a simple clock consisting of two mirrors A and B, between which a light pulse is bouncing. The separation of the mirrors is L and the clock ticks once each time it hits a given mirror.

In the frame where the clock is at rest (diagram at right), the light pulse traces out a path of length 2L and the period of the clock is 2L divided by the speed of light:

$\Delta t = \frac{2 L}{c}.$

From the frame of reference of a moving observer traveling at the speed v (diagram at lower right), the light pulse traces out a longer, angled path. The second postulate of special relativity states that the speed of light is constant in all frames, which implies a lengthening of the period of this clock from the moving observer's perspective. That is to say, in a frame moving relative to the clock, the clock appears to be running more slowly. Straightforward application of the Pythagorean theorem leads to the well-known prediction of special relativity:

The total time for the light pulse to trace its path is given by

$\Delta t' = \frac{2 D}{c}.$

The length of the half path can be calculated as a function of known quantities as

$D = \sqrt{\left (\frac{1}{2}v \Delta t'\right )^2+L^2}.$

Substituting D from this equation into the previous and solving for Δt' gives:

$\Delta t' = \frac{2L/c}{\sqrt{1-v^2/c^2}}$

and thus, with the definition of Δt:

$\Delta t' = \frac{\Delta t}{\sqrt{1-v^2/c^2}}$

which expresses the fact that for the moving observer the period of the clock is longer than in the frame of the clock itself.

## Time dilation due to relative velocity symmetric between observers

Common sense would dictate that if time passage has slowed for a moving object, the moving object would observe the external world to be correspondingly "sped up". Counterintuitively, special relativity predicts the opposite.

A similar oddity occurs in everyday life. If Sam sees Abigail at a distance she appears small to him and at the same time Sam appears small to Abigail. Being very familiar with the effects of perspective, we see no mystery or a hint of a paradox in this situation.[6]

One is accustomed to the notion of relativity with respect to distance: the distance from Los Angeles to New York is by convention the same as the distance from New York to Los Angeles. On the other hand, when speeds are considered, one thinks of an object as "actually" moving, overlooking that its motion is always relative to something else — to the stars, the ground or to oneself. If one object is moving with respect to another, the latter is moving with respect to the former and with equal relative speed.

In the special theory of relativity, a moving clock is found to be ticking slowly with respect to the observer's clock. If Sam and Abigail are on different trains in near-lightspeed relative motion, Sam measures (by all methods of measurement) clocks on Abigail's train to be running slowly and similarly, Abigail measures clocks on Sam's train to be running slowly.

Note that in all such attempts to establish "synchronization" within the reference system, the question of whether something happening at one location is in fact happening simultaneously with something happening elsewhere, is of key importance. Calculations are ultimately based on determining which events are simultaneous. Furthermore, establishing simultaneity of events separated in space necessarily requires transmission of information between locations, which by itself is an indication that the speed of light will enter the determination of simultaneity.

It is a natural and legitimate question to ask how, in detail, special relativity can be self-consistent if clock A is time-dilated with respect to clock B and clock B is also time-dilated with respect to clock A. It is by challenging the assumptions built into the common notion of simultaneity that logical consistency can be restored. Simultaneity is a relationship between an observer in a particular frame of reference and a set of events. By analogy, left and right are accepted to vary with the position of the observer, because they apply to a relationship. In a similar vein, Plato explained that up and down describe a relationship to the earth and one would not fall off at the antipodes.

Within the framework of the theory and its terminology there is a relativity of simultaneity that affects how the specified events are aligned with respect to each other by observers in relative motion. Because the pairs of putatively simultaneous moments are identified differently by different observers (as illustrated in the twin paradox article), each can treat the other clock as being the slow one without relativity being self-contradictory. This can be explained in many ways, some of which follow.

### Temporal coordinate systems and clock synchronization

In Relativity, temporal coordinate systems are set up using a procedure for synchronizing clocks, discussed by Poincaré (1900) in relation to Lorentz's local time (see relativity of simultaneity). It is now usually called the Einstein synchronization procedure, since it appeared in his 1905 paper.

An observer with a clock sends a light signal out at time t1 according to his clock. At a distant event, that light signal is reflected back, and arrives back at the observer at time t2 according to his clock. Since the light travels the same path at the same rate going both out and back for the observer in this scenario, the coordinate time of the event of the light signal being reflected for the observer tE is tE = (t1 + t2) / 2. In this way, a single observer's clock can be used to define temporal coordinates which are good anywhere in the universe.

Symmetric time dilation occurs with respect to temporal coordinate systems set up in this manner. It is an effect where another clock is being viewed as running slowly by an observer. Observers do not consider their own clock time to be time-dilated, but may find that it is observed to be time-dilated in another coordinate system.

## Overview of formulae

### Time dilation due to relative velocity

Lorentz factor as a function of speed (in natural units where c=1). Notice that for small speeds (less than 0.1), γ is approximately 1

The formula for determining time dilation in special relativity is:

$\Delta t' = \gamma \, \Delta t = \frac{\Delta t}{\sqrt{1-v^2/c^2}} \,$

where Δt is the time interval between two co-local events (i.e. happening at the same place) for an observer in some inertial frame (e.g. ticks on his clock) – this is known as the proper time, Δt ' is the time interval between those same events, as measured by another observer, inertially moving with velocity v with respect to the former observer, v is the relative velocity between the observer and the moving clock, c is the speed of light, and

$\gamma = \frac{1}{\sqrt{1-v^2/c^2}} \,$

is the Lorentz factor. Thus the duration of the clock cycle of a moving clock is found to be increased: it is measured to be "running slow". The range of such variances in ordinary life, where vc, even considering space travel, are not great enough to produce easily detectable time dilation effects and such vanishingly small effects can be safely ignored. It is only when an object approaches speeds on the order of 30,000 km/s (1/10 the speed of light) that time dilation becomes important.

Time dilation by the Lorentz factor was predicted by Joseph Larmor (1897), at least for electrons orbiting a nucleus. Thus "... individual electrons describe corresponding parts of their orbits in times shorter for the [rest] system in the ratio :$\scriptstyle \sqrt{1 - v^2/c^2}$" (Larmor 1897). Time dilation of magnitude corresponding to this (Lorentz) factor has been experimentally confirmed, as described below.

### Time dilation due to gravitation and motion together

High accuracy time keeping, low earth orbit satellite tracking, and pulsar timing are applications that require the consideration of the combined effects of mass and motion in producing time dilation. Practical examples include the International Atomic Time standard and its relationship with the Barycentric Coordinate Time standard used for interplanetary objects.

Relativistic time dilation effects for the solar system and the Earth can be modeled very precisely by the Schwarzschild solution to the Einstein field equations. In the Schwarzschild metric, the interval dtE is given by:[7][8]

$dt_E^2 = \left( 1-\frac{2GM_i}{r_i c^2} \right) dt_c^2 - \frac{dx^2+dy^2+dz^2}{c^2}, \,$

(1)

where:

dtE is a small increment of proper time tE (an interval that could be recorded on an atomic clock);
dtc is a small increment in the coordinate tc (coordinate time);
dx, dy and dz are small increments in the three coordinates x, y, z of the clock's position; and
GMi/ri represents the sum of the Newtonian gravitational potentials due to the masses in the neighborhood, based on their distances ri from the clock. This sum GMi/ri includes any tidal potentials, and is represented as U (using the positive astronomical sign convention for gravitational potentials).

The coordinate velocity of the clock is

$v^2 = \frac{dx^2+dy^2+dz^2}{dt_c^2} , \,$

(2)

The coordinate time tc is the time that would be read on a hypothetical "coordinate clock" situated infinitely far from all gravitational masses (U=0), and stationary in the system of coordinates (v=0). The exact relation between the rate of proper time and the rate of coordinate time for a clock with a radial component of velocity is:

$\frac{dt_E}{dt_c} = \sqrt{ 1 - \frac{2U}{c^2} - \frac{v^2}{c^2} - \left( \frac{c^2}{2U} - 1 \right)^{-1} \frac{{v_\shortparallel}^2}{c^2} } \,$

(3)

where:

$v_{\shortparallel}$ is the radial velocity, and
U = GMi/ri is the Newtonian potential, equivalent to half of the escape velocity squared.

The above equation is exact under the assumptions of the Schwarzschild solution.

## Experimental confirmation

Time dilation has been tested a number of times. The routine work carried on in particle accelerators since the 1950s, such as those at CERN, is a continuously running test of the time dilation of special relativity. The specific experiments include:

### Velocity time dilation tests

• Ives and Stilwell (1938, 1941). The stated purpose of these experiments was to verify the time dilation effect, predicted by Lamor-Lorentz ether theory, due to motion through the ether using Einstein's suggestion that Doppler effect in canal rays would provide a suitable experiment. These experiments measured the Doppler shift of the radiation emitted from cathode rays, when viewed from directly in front and from directly behind. The high and low frequencies detected were not the classical values predicted.
$f_\mathrm{detected} = \frac{f_\mathrm{moving}}{1 - v/c}$ and $\frac{f_\mathrm{moving}}{1+v/c}\,=\, \frac{f_\mathrm{rest}}{1 - v/c}$ and $\frac{f_\mathrm{rest}}{1+v/c}$
i.e. for sources with invariant frequencies $f_\mathrm{moving}\, = f_\mathrm{rest}$ The high and low frequencies of the radiation from the moving sources were measured as
$f_\mathrm{detected} = f_\mathrm{rest}\sqrt{\left(1 + v/c\right)/\left(1 - v/c\right) }$ and $f_\mathrm{rest}\sqrt{\left(1 - v/c\right)/\left(1 + v/c\right)}$
as deduced by Einstein (1905) from the Lorentz transformation, when the source is running slow by the Lorentz factor.
• Rossi and Hall (1941) compared the population of cosmic-ray-produced muons at the top of a mountain to that observed at sea level. Although the travel time for the muons from the top of the mountain to the base is several muon half-lives, the muon sample at the base was only moderately reduced. This is explained by the time dilation attributed to their high speed relative to the experimenters. That is to say, the muons were decaying about 10 times slower than if they were at rest with respect to the experimenters.
• Hasselkamp, Mondry, and Scharmann[9] (1979) measured the Doppler shift from a source moving at right angles to the line of sight (the transverse Doppler shift). The most general relationship between frequencies of the radiation from the moving sources is given by:
$f_\mathrm{detected} = f_\mathrm{rest}{\left(1 - \frac{v}{c} \cos\phi\right)/\sqrt{1 - {v^2}/{c^2}} }$
as deduced by Einstein (1905)[1]. For $\phi = 90^\circ$ ($\cos\phi = 0\,$) this reduces to fdetected = frestγ. Thus there is no transverse Doppler shift, and the lower frequency of the moving source can be attributed to the time dilation effect alone.
• In 2010 time dilation was observed at speeds of less than 10 meters per second using optical atomic clocks connected by 75 meters of optical fiber.[10]

### Gravitational time dilation tests

• In 1959 Robert Pound and Glen A. Rebka measured the very slight gravitational red shift in the frequency of light emitted at a lower height, where Earth's gravitational field is relatively more intense. The results were within 10% of the predictions of general relativity. Later Pound and Snider (in 1964) derived an even closer result of 1%. This effect is as predicted by gravitational time dilation. (See Pound–Rebka experiment)
• In 2010 gravitational time dilation was measured at the Earth's surface with a height difference of only one meter, using optical atomic clocks.[10]

### Velocity and gravitational time dilation combined-effect tests

• Hafele and Keating, in 1971, flew caesium atomic clocks east and west around the Earth in commercial airliners, to compare the elapsed time against that of a clock that remained at the US Naval Observatory. Two opposite effects came into play. The clocks were expected to age more quickly (show a larger elapsed time) than the reference clock, since they were in a higher (weaker) gravitational potential for most of the trip (c.f. Pound, Rebka). But also, contrastingly, the moving clocks were expected to age more slowly because of the speed of their travel. From the actual flight paths of each trip, the theory predicted that the flying clocks, compared with reference clocks at the U.S. Naval Observatory, should have lost 40+/-23 nanoseconds during the eastward trip and should have gained 275+/-21 nanoseconds during the westward trip. Relative to the atomic time scale of the U.S. Naval Observatory, the flying clocks lost 59+/-10 nanoseconds during the eastward trip and gained 273+/-7 nanoseconds during the westward trip (where the error bars represent standard deviation). [11] In 2005, the National Physical Laboratory in the United Kingdom reported their limited replication of this experiment.[12] The NPL experiment differed from the original in that the caesium clocks were sent on a shorter trip (London–Washington D.C. return), but the clocks were more accurate. The reported results are within 4% of the predictions of relativity.
• The Global Positioning System can be considered a continuously operating experiment in both special and general relativity. The in-orbit clocks are corrected for both special and general relativistic time dilation effects as described above, so that (as observed from the Earth's surface) they run at the same rate as clocks on the surface of the Earth.

A comparison of muon lifetimes at different speeds is possible. In the laboratory, slow muons are produced, and in the atmosphere very fast moving muons are introduced by cosmic rays. Taking the muon lifetime at rest as the laboratory value of 2.22 μs, the lifetime of a cosmic ray produced muon traveling at 98% of the speed of light is about five times longer, in agreement with observations.[13] In this experiment the "clock" is the time taken by processes leading to muon decay, and these processes take place in the moving muon at its own "clock rate", which is much slower than the laboratory clock.

## Time dilation and space flight

Time dilation would make it possible for passengers in a fast-moving vehicle to travel further into the future while aging very little, in that their great speed slows down the rate of passage of on-board time. That is, the ship's clock (and according to relativity, any human travelling with it) shows less elapsed time than the clocks of observers on Earth. For sufficiently high speeds the effect is dramatic. For example, one year of travel might correspond to ten years at home. Indeed, a constant 1 g acceleration would permit humans to travel through the entire known Universe in one human lifetime.[14] The space travellers could return to Earth billions of years in the future. A scenario based on this idea was presented in the novel Planet of the Apes by Pierre Boulle.

A more likely use of this effect would be to enable humans to travel to nearby stars without spending their entire lives aboard the ship. However, any such application of time dilation during interstellar travel would require the use of some new, advanced method of propulsion. The Orion Project has been the only major attempt toward this idea.

Current space flight technology has fundamental theoretical limits based on the practical problem that an increasing amount of energy is required for propulsion as a craft approaches the speed of light. The likelihood of collision with small space debris and other particulate material is another practical limitation. At the velocities presently attained, however, time dilation is not a factor in space travel. Travel to regions of space-time where gravitational time dilation is taking place, such as within the gravitational field of a black hole but outside the event horizon (perhaps on a hyperbolic trajectory exiting the field), could also yield results consistent with present theory.

### Time dilation at constant acceleration

In special relativity, time dilation is most simply described in circumstances where relative velocity is unchanging. Nevertheless, the Lorentz equations allow one to calculate proper time and movement in space for the simple case of a spaceship whose acceleration, relative to some referent object in uniform (i.e. constant velocity) motion, equals g throughout the period of measurement.

Let t be the time in an inertial frame subsequently called the rest frame. Let x be a spatial coordinate, and let the direction of the constant acceleration as well as the spaceship's velocity (relative to the rest frame) be parallel to the x-axis. Assuming the spaceship's position at time t = 0 being x = 0 and the velocity being v0 and defining the following abbreviation

$\gamma_0 := \frac{1}{\sqrt{1-v_0^2/c^2}},$

the following formulas hold:[15]

Position:

$x(t) = \frac {c^2}{g} \left( \sqrt{1 + \frac{\left(gt + v_0\gamma_0\right)^2}{c^2}} -\gamma_0 \right).$

Velocity:

$v(t) =\frac{gt + v_0\gamma_0}{\sqrt{1 + \frac{ \left(gt + v_0\gamma_0\right)^2}{c^2}}}.$

Proper time:

$\tau(t) = \tau_0 + \int_0^t \sqrt{ 1 - \left( \frac{v(t')}{c} \right)^2 } dt'$

In the case where v(0) = v0 = 0 and τ(0) = τ0 = 0 the integral can be expressed as a logarithmic function or, equivalently, as an inverse hyperbolic function:

$\tau(t) = \frac{c}{g} \ln \left( \frac{gt}{c} + \sqrt{ 1 + \left( \frac{gt}{c} \right)^2 } \right) = \frac{c}{g} \operatorname {arsinh} \left( \frac{gt}{c} \right) .$

### Spacetime geometry of velocity time dilation

Time dilation in transverse motion

The green dots and red dots in the animation represent spaceships. The ships of the green fleet have no velocity relative to each other, so for the clocks onboard the individual ships the same amount of time elapses relative to each other, and they can set up a procedure to maintain a synchronized standard fleet time. The ships of the "red fleet" are moving with a velocity of 0.866 of the speed of light with respect to the green fleet.

The blue dots represent pulses of light. One cycle of light-pulses between two green ships takes two seconds of "green time", one second for each leg.

As seen from the perspective of the reds, the transit time of the light pulses they exchange among each other is one second of "red time" for each leg. As seen from the perspective of the greens, the red ships' cycle of exchanging light pulses travels a diagonal path that is two light-seconds long. (As seen from the green perspective the reds travel 1.73 ($\sqrt{3}$) light-seconds of distance for every two seconds of green time.)

One of the red ships emits a light pulse towards the greens every second of red time. These pulses are received by ships of the green fleet with two-second intervals as measured in green time. Not shown in the animation is that all aspects of physics are proportionally involved. The light pulses that are emitted by the reds at a particular frequency as measured in red time are received at a lower frequency as measured by the detectors of the green fleet that measure against green time, and vice versa.

The animation cycles between the green perspective and the red perspective, to emphasize the symmetry. As there is no such thing as absolute motion in relativity (as is also the case for Newtonian mechanics), both the green and the red fleet are entitled to consider themselves motionless in their own frame of reference.

Again, it is vital to understand that the results of these interactions and calculations reflect the real state of the ships as it emerges from their situation of relative motion. It is not a mere quirk of the method of measurement or communication.

## References

1. ^ For sources on special relativistic time dilation, see Albert Einstein's own popular exposition, published in English translation (1920) as "Relativity: The Special and General Theory", especially at "8: On the Idea of Time in Physics", and in following sections 9–12. See also the articles Special relativity, Lorentz transformation and Relativity of simultaneity.
2. ^ Cassidy, David C.; Holton, Gerald James; Rutherford, Floyd James (2002), Understanding Physics, Springer-Verlag New York, Inc, ISBN 0-387-98756-8 , Chapter 9 §9.6, p. 422
3. ^ Cutner, Mark Leslie (2003), Astronomy, A Physical Perspective, Cambridge University Press, ISBN 0-521-82196-7 , Chapter 7 §7.2, p. 128
4. ^ Lerner, Lawrence S. (1996), Physics for Scientists and Engineers, Volume 2, Jones and Bertlett Publishers, Inc, ISBN 0-7637-0460-1 , Chapter 38 §38.4, p. 1051,1052
5. ^ Ellis, George F. R.; Williams, Ruth M. (2000), Flat and Curved Space-times, Second Edition, Oxford University Press Inc, New York, ISBN 0-19-850657-0 , Chapter 3 §1.3, p. 28-29
6. ^ Adams, Steve (1997), Relativity: an introduction to space-time physics, CRC Press, p. 54, ISBN 0-748-40621-2 , Section 2.5, page 54
7. ^ See T D Moyer (1981a), "Transformation from proper time on Earth to coordinate time in solar system barycentric space-time frame of reference", Celestial Mechanics 23 (1981) pages 33-56, equations 2 & 3 at pages 35-6 combined here and divided throughout by c2.
8. ^ A version of the same relationship can also be seen in Neil Ashby (2002), "Relativity and the Global Positioning System", Physics Today (May 2002), at equation (2).
9. ^ Hasselkamp, D.; Mondry, E.; Scharmann, A. (1979). "Direct observation of the transversal Doppler-shift". Zeitschrift fur Physik a Atoms and Nuclei 289 (2): 151–155. Bibcode 1979ZPhyA.289..151H. doi:10.1007/BF01435932.
10. ^ a b Chou, C. W.; Hume, D. B.; Rosenband, T.; Wineland, D. J. (2010). "Optical Clocks and Relativity". Science 329 (5999): 1630–1633. Bibcode 2010Sci...329.1630C. doi:10.1126/science.1192720. PMID 20929843.
11. ^ http://hyperphysics.phy-astr.gsu.edu/hbase/relativ/airtim.html
13. ^ JV Stewart (2001), Intermediate electromagnetic theory, Singapore: World Scientific, p. 705, ISBN 9810244703
14. ^ Calder, Nigel (2006). Magic universe: a grand tour of modern science. Oxford University Press. p. 378. ISBN 0-192-80669-6. , Extract of page 378
15. ^ See equations (3), (4), (6), (9) of Iorio, Lorenzo (27-Jun-2004). "An analytical treatment of the Clock Paradox in the framework of the Special and General Theories of Relativity". arXiv:physics/0405038.
• Callender, Craig & Edney, Ralph (2001), Introducing Time, Icon, ISBN 1-84046-592-1
• Einstein, A. (1905) "Zur Elektrodynamik bewegter Körper", Annalen der Physik, 17, 891. English translation: On the electrodynamics of moving bodies
• Einstein, A. (1907) "Über eine Möglichkeit einer Prüfung des Relativitätsprinzips", Annalen der Physik.
• Hasselkamp, D., Mondry, E. and Scharmann, A. (1979) "Direct Observation of the Transversal Doppler-Shift", Z. Physik A 289, 151–155
• Ives, H. E. and Stilwell, G. R. (1938), "An experimental study of the rate of a moving clock", J. Opt. Soc. Am, 28, 215–226
• Ives, H. E. and Stilwell, G. R. (1941), "An experimental study of the rate of a moving clock. II", J. Opt. Soc. Am, 31, 369–374
• Joos, G. (1959) Lehrbuch der Theoretischen Physik, 11. Auflage, Leipzig; Zweites Buch, Sechstes Kapitel, § 4: Bewegte Bezugssysteme in der Akustik. Der Doppler-Effekt.
• Larmor, J. (1897) "On a dynamical theory of the electric and luminiferous medium", Phil. Trans. Roy. Soc. 190, 205–300 (third and last in a series of papers with the same name).
• Poincaré, H. (1900) "La theorie de Lorentz et la Principe de Reaction", Archives Neerlandaies, V, 253–78.
• Reinhardt et al. Test of relativistic time dilation with fast optical atomic clocks at different velocities (Nature 2007)
• Rossi, B and Hall, D. B. Phys. Rev., 59, 223 (1941).
• NIST Two way time transfer for satellites
• Voigt, W. "Ueber das Doppler'sche princip" Nachrichten von der Königlicher Gesellschaft der Wissenschaften zu Göttingen, 2, 41–51.

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