# Time dilation

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Time dilation is the phenomenon where the observed time rate of an observer's reference frame is different from that of a different reference frame. In Albert Einstein's theories of relativity the effect is manifested in two ways:

The formula for determining time dilation in special relativity is:

$\Delta T = \gamma \Delta T_0 \!$

where

ΔT is the time period as measured by a stationary observer,
ΔT0 is the time period of the moving object as seen by the stationary observer,
$\gamma \equiv \frac{1}{\sqrt{1 - \frac{u}{c}}}$ is the Lorentz factor,
u is the relative velocity between the observer and the object, and
c is the speed of light.

This effect is extremely small at ordinary speeds, and can then be safely ignored. It is only when an object approaches speeds on the order of 30,000 km/s (1/10 of the speed of light) that it 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 :$\sqrt{1 - v^2/c^2}$" (Larmor 1897). Time dilation of magnitude corresponding to this (Lorentz) factor has been experimentally confirmed.

## Experimental confirmations Edit

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 testsEdit

• Ives and Stilwell (1938, 1941), “An experimental study of the rate of a moving clock”, in two parts. 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 (see Doppler effect)
$f_\mathrm{detected} = \frac{f_\mathrm{moving}}{1 - v/c}$ 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
$f_\mathrm{detected} = f_\mathrm{rest}\sqrt{\left(1 + \frac{v}{c}\right)/\left(1 - \frac{v}{c}\right) }$ and $f_\mathrm{rest}\sqrt{\left(1 - \frac{v}{c}\right)/\left(1 + \frac{v}{c}\right)}$
as deduced by Einstein (1905) from the Lorentz transformation, when the source is running slow by the Lorentz factor. Thus, the experiment shows that the frequency of the moving sources is reduced and given by
$f_\mathrm{moving} = f_\mathrm{rest}\sqrt{1 - v^2/c^2}.$
• 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 due to the time-dilation because of their high-speed relative to the experimenters. That is to say, the muons are decaying about 10 times slower than they would in a rest frame.
• Hasselkamp, Mondry, and Scharmann (1979) measured the Doppler shift from a source moving at right angles to the line of sight (the transverse Doppler shift). For an invariant source frequency, there is no classical transverse Doppler shift, so, unlike the Ives-Stillwell experiment, the lower frequency of the moving source can be attributed to the time dilation effect alone.

### Gravitational red shift testsEdit

• Pound, Rebka in 1959 measured the gravitational red shift of photons falling in the Earth's gravitational field. The results were within 10% of general relativity. Later Pound and Snider (in 1964) improved this to 1%.

### Velocity and gravitational time dilation combined testsEdit

• Hefele and Keating, in 1971, flew caesium clocks east and west around the Earth in commercial airliners, to compare the elapsed time to that for a clock that remained at the US Naval Observatory. Two opposite effects came in to play. The clocks were expected to age quicker (show a larger elapsed time) than the reference clock, since they were in a higher gravitational potential for most of the trip (c.f. Pound, Rebka). The clocks were expected to age slower because of the speed of the travel. The gravitational effect was the larger, and the clocks suffered a net gain in elapsed time. To within experimental error, the net gain was consistent with the difference between the predicted gravitational gain and the predicted velocity time loss. In 2005, the National Physical Laboratory in the United Kingdom, report their limited replication of this experiment. The NPL experiment differed from the original in that the ceasium clocks were sent on a shorter trip (London-Washingon 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 on-orbit clocks are corrected for both special and general relativistic time-dilation effects so they appear to run at the same (average) rate as clocks at the surface of the Earth. In addition, but not directly time-dilation related, general relativistic correction terms are built into the model of motion that the satellites broadcast to receivers -- uncorrected, these eccentric terms would amount to a 12-hour, approximately 7 metre, oscillation in the pseudo-ranges measured by a receiver.

## Time dilation and space flight Edit

Time dilation would make it possible for a fast moving clock to travel into the future, while aging very little. That is, the clock (and according to relativity, any human travelling with it) shows less elapsed time than stationary clocks. For sufficiently high speed the effect could be dramatic. For example one year of travel might correspond to ten years at home. Indeed, a constant 1g acceleration would permit humans to circumnavigate the known universe (with a radius of some 15 billion light years) in one human lifetime. The space-travellers could return to earth billions of years in the future (provided the Universe hadn't collapsed and our solar system was still around, of course). 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 use of this effect would require an entirely new method of propulsion. A further problem with relativistic travel is that the interstellar medium would turn into a stream of cosmic rays that would destroy the ship unless stark radiation protection measures were taken.

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. As a result, time dilation is not currently a major factor in space travel.

## Simple inference of time dilation Edit

Time dilation can be inferred from the constancy of the speed of light in all reference frames as follows:

Consider a simple clock consisting of two mirrors A and B, between which a photon 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 photon traces out a path of length 2L and the period of the clock is 2L divided by the speed of light.

From the frame of reference of a moving observer (diagram at lower right), the photon traces out a longer, triangular, 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 slower. Straightforward application of the Pythagorean theorem leads to the well-known prediction of special relativity.

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

$\Delta = \sqrt{\left (\frac{1}{2}vt\right )^2+L^2}$

$ct = 2\sqrt{\left (\frac{1}{2}vt\right )^2+L^2}$

$c^2t^2 = v^2t^2+4L^2$

$t^2 = \frac{4L^2}{c^2-v^2}$

$t = \frac{2L/c}{\sqrt{1-(v/c)^2}}$

It is interesting to note that in Voigt's transformation, the distance between the mirrors (length in the transverse direction) is not independent of speed, but given by $L/\sqrt{1-v^2/c^2}$, and the corresponding time dilation is even greater. The experimental evidence cited above agrees with the Lorentz transformation.

## Time dilation is symmetric between two inertial observers Edit

A measurement of relative time must regard one clock as being "stationary" in spacetime, and that clock is the basis of a temporal coordinate system where time is represented as synchronized with the stationary clock. The traveler's "moving" clock is in motion with respect to this stationary clock. In the special theory of relativity, the moving clock is found to be ticking slow with respect to the temporal coordinate system of the stationary clock. This effect is symmetrical: In a coordinate system synchronized with the "moving" clock, it is the "stationary" clocks that seem to be running slow. (A misunderstanding about this symmetry leads to the so-called twin paradox.)

A legitimate question is how 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. The short answer is that the relativity of simultaneity affects how the moments of simultaneous time are placed with respect to each other by observers who are in motion with respect to each other. Because the simultaneous moments are different for the different observers (as illustrated in the twin paradox article), each can treat the other clock as being slow without relativity being self-contradictory. This can be explained at greater length in many ways, some of which follow.

### Temporal coordinate systems and clock synchronization Edit

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 to, and arrives back to 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 photon 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 slow by an observer. Observers in rest in their coordinate system do not consider their own clock time to be time dilated, but may find that it is seen as being time dilated in another coordinate system.

### The Space-time geometry of velocity time dilation Edit

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 lightpulses 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 lightpulses 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 as "non-moving" in their own frame of reference.

## Time dilation in popular cultureEdit

Time dilation caused by long distance space flight is central to the novel The Forever War.

## ReferencesEdit

• Template:Cite book
• Einstein, A. (1905) "Zur Elektrodynamik bewegter Körper", Annalen der Physik, 17, 891. English translation:
• 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
• 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).
• Poincare, H. (1900) "La theorie de Lorentz et la Principe de Reaction", Archives Neerlandaies, V, 253-78.
• 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 den Wissenschaft zu Göttingen, 2, 41-51.

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