Time dilation

Left: Observer at rest measures time 2L/c between co-local events of light signal generation at A and arrival at A.
Right: Events according to an observer moving to the left of the setup: bottom mirror A when signal is generated at time t'=0, top mirror B when signal gets reflected at time t'=D/c, bottom mirror A when signal returns at time t'=2D/c

Time dilation can be inferred from the observed constancy of the speed of light in all reference frames dictated by the second postulate of special relativity.[13][14][15][16]

This constancy of the speed of light means that, counter to intuition, speeds of material objects and light are not additive. It is not possible to make the speed of light appear greater by moving towards or away from the light source.

Consider then, a simple vertical 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 the light pulse hits either of the mirrors.

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

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

From the frame of reference of a moving observer traveling at the speed v relative to the resting frame of the clock (diagram at right), the light pulse is seen as tracing out a longer, angled path. Keeping the speed of light constant for all inertial observers, requires 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 local clock, this clock will appear 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:

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

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

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

Elimination of the variables D and L from these three equations results in:

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

which expresses the fact that the moving observer's period of the clock ${\displaystyle \Delta t'}$ is longer than the period ${\displaystyle \Delta t}$ in the frame of the clock itself.

Because all clocks that have a common period in the resting frame should have a common period when observed from the moving frame, all other clocks—mechanical, electronic, optical (such as an identical horizontal version of the clock in the example)—should exhibit the same velocity-dependent time dilation.[17]

Time UV of a clock in S is shorter compared to Ux′ in S′, and time UW of a clock in S′ is shorter compared to Ux in S

Transversal time dilation. The blue dots represent a pulse of light. Each pair of dots with light "bouncing" between them is a clock. For each group of clocks, the other group appears to be ticking more slowly, because the moving clock's light pulse has to travel a larger distance than the stationary clock's light pulse. That is so, even though the clocks are identical and their relative motion is perfectly reciprocal.

Given a certain frame of reference, and the "stationary" observer described earlier, if a second observer accompanied the "moving" clock, each of the observers would perceive the other's clock as ticking at a slower rate than their own local clock, due to them both perceiving the other to be the one that is in motion relative to their own stationary frame of reference.

Common sense would dictate that, if the passage of time has slowed for a moving object, said object would observe the external world's time to be correspondingly sped up. Counterintuitively, special relativity predicts the opposite. When two observers are in motion relative to each other, each will measure the other's clock slowing down, in concordance with them being in motion relative to the observer's frame of reference.

While this seems self-contradictory, a similar oddity occurs in everyday life. If two persons A and B observe each other from a distance, B will appear small to A, but at the same time A will appear small to B. Being familiar with the effects of perspective, there is no contradiction or paradox in this situation.[18]

The reciprocity of the phenomenon also leads to the so-called twin paradox where the aging of twins, one staying on Earth and the other embarking on a space travel, is compared, and where the reciprocity suggests that both persons should have the same age when they reunite. On the contrary, at the end of the round-trip, the traveling twin will be younger than their sibling on Earth. The dilemma posed by the paradox, however, can be explained by the fact that the traveling twin must markedly accelerate in at least three phases of the trip (beginning, direction change, and end), while the other will only experience negligible acceleration, due to rotation and revolution of Earth. During the acceleration phases of the space travel, time dilation is not symmetric.

• The stated purpose by Ives and Stilwell (1938, 1941) of these experiments was to verify the time dilation effect, predicted by Larmor–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 classically predicted values:

${\displaystyle {\frac {f_{0}}{1-v/c}}\qquad {\text{and}}\qquad {\frac {f_{0}}{1+v/c}}\,}$

The high and low frequencies of the radiation from the moving sources were measured as:[19]

${\displaystyle {\sqrt {\frac {1+v/c}{1-v/c}}}f_{0}=\gamma \left(1+v/c\right)f_{0}\qquad {\text{and}}\qquad {\sqrt {\frac {1-v/c}{1+v/c}}}f_{0}=\gamma \left(1-v/c\right)f_{0}\,}$

as deduced by Einstein (1905) from the Lorentz transformation, when the source is running slow by the Lorentz factor.

• Hasselkamp, Mondry, and Scharmann[20] (1979) measured the Doppler shift from a source moving at right angles to the line of sight. The most general relationship between frequencies of the radiation from the moving sources is given by:

${\displaystyle f_{\mathrm {detected} }=f_{\mathrm {rest} }{\left(1-{\frac {v}{c}}\cos \phi \right)/{\sqrt {1-{v^{2}}/{c^{2}}}}}}$

as deduced by Einstein (1905).[21] For ϕ = 90° (cos ϕ = 0) this reduces to f_detected = f_restγ. This lower frequency from the moving source can be attributed to the time dilation effect and is often called the transverse Doppler effect and was predicted by relativity.

• 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.[22]

• 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 2.197 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. An example is Rossi and Hall (1941), who compared the population of cosmic-ray-produced muons at the top of a mountain to that observed at sea level.[23]

• The lifetime of particles produced in particle accelerators appears longer due to time dilation. In such experiments 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. This is routinely taken into account in particle physics, and many dedicated measurements have been performed. For instance, in the muon storage ring at CERN the lifetime of muons circulating with γ = 29.327 was found to be dilated to 64.378 μs, confirming time dilation to an accuracy of 0.9 ± 0.4 parts per thousand.[24]

Minkowski diagram and twin paradox

Clock C in relative motion between two synchronized clocks A and B. C meets A at d, and B at f.

Twin paradox. One twin has to change frames, leading to different proper times in the twin's world lines.

In the Minkowski diagram from the first image on the right, clock C resting in inertial frame S′ meets clock A at d and clock B at f (both resting in S). All three clocks simultaneously start to tick in S. The worldline of A is the ct-axis, the worldline of B intersecting f is parallel to the ct-axis, and the worldline of C is the ct′-axis. All events simultaneous with d in S are on the x-axis, in S′ on the x′-axis.

The proper time between two events is indicated by a clock present at both events.[25] It is invariant, i.e., in all inertial frames it is agreed that this time is indicated by that clock. Interval df is therefore the proper time of clock C, and is shorter with respect to the coordinate times ef=dg of clocks B and A in S. Conversely, also proper time ef of B is shorter with respect to time if in S′, because event e was measured in S′ already at time i due to relativity of simultaneity, long before C started to tick.

From that it can be seen, that the proper time between two events indicated by an unaccelerated clock present at both events, compared with the synchronized coordinate time measured in all other inertial frames, is always the minimal time interval between those events. However, the interval between two events can also correspond to the proper time of accelerated clocks present at both events. Under all possible proper times between two events, the proper time of the unaccelerated clock is maximal, which is the solution to the twin paradox.[25]

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.

In addition to the light clock used above, the formula for time dilation can be more generally derived from the temporal part of the Lorentz transformation.[26] Let there be two events at which the moving clock indicates ${\displaystyle t_{a}}$ and ${\displaystyle t_{b}}$, thus:

${\displaystyle t_{a}^{\prime }={\frac {t_{a}-{\frac {vx_{a}}{c^{2}}}}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}},\ t_{b}^{\prime }={\frac {t_{b}-{\frac {vx_{b}}{c^{2}}}}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}}$

Since the clock remains at rest in its inertial frame, it follows ${\displaystyle x_{a}=x_{b}}$, thus the interval ${\displaystyle \Delta t^{\prime }=t_{b}^{\prime }-t_{a}^{\prime }}$ is given by:

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

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 their clock), 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 the Lorentz factor (conventionally denoted by the Greek letter gamma or γ) is:

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

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 v ≪ c, even considering space travel, are not great enough to produce easily detectable time dilation effects and such vanishingly small effects can be safely ignored for most purposes. 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.[27]

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 which is applied with a force per unit mass, relative to some reference object in uniform (i.e. constant velocity) motion, equal to 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:

${\displaystyle \gamma _{0}={\frac {1}{\sqrt {1-v_{0}^{2}/c^{2}}}}}$

the following formulas hold:[28]

Position:

${\displaystyle x(t)={\frac {c^{2}}{g}}\left({\sqrt {1+{\frac {\left(gt+v_{0}\gamma _{0}\right)^{2}}{c^{2}}}}}-\gamma _{0}\right)}$

Velocity:

${\displaystyle v(t)={\frac {gt+v_{0}\gamma _{0}}{\sqrt {1+{\frac {\left(gt+v_{0}\gamma _{0}\right)^{2}}{c^{2}}}}}}}$

Proper time as function of coordinate time:

${\displaystyle \tau (t)=\tau _{0}+\int _{0}^{t}{\sqrt {1-\left({\frac {v(t')}{c}}\right)^{2}}}dt'}$

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:

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

As functions of the proper time ${\displaystyle \tau }$ of the ship, the following formulae hold:[29]

Position:

${\displaystyle x(\tau )={\frac {c^{2}}{g}}\left(\cosh {\frac {g\tau }{c}}-1\right)}$

Velocity:

${\displaystyle v(\tau )=c\tanh {\frac {g\tau }{c}}}$

Coordinate time as function of proper time:

${\displaystyle t(\tau )={\frac {c}{g}}\sinh {\frac {g\tau }{c}}}$

The clock hypothesis is the assumption that the rate at which a clock is affected by time dilation does not depend on its acceleration but only on its instantaneous velocity. This is equivalent to stating that a clock moving along a path ${\displaystyle P}$ measures the proper time, defined by:

${\displaystyle d\tau =\int _{P}{\sqrt {dt^{2}-dx^{2}/c^{2}-dy^{2}/c^{2}-dz^{2}/c^{2}}}}$

The clock hypothesis was implicitly (but not explicitly) included in Einstein's original 1905 formulation of special relativity. Since then, it has become a standard assumption and is usually included in the axioms of special relativity, especially in the light of experimental verification up to very high accelerations in particle accelerators.[30][31]

• In 1959 Robert Pound and Glen A. Rebka measured the very slight gravitational redshift 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. In 1964, Pound and J. L. Snider measured a result within 1% of the value predicted by gravitational time dilation.[34] (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.[22]

• 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 U.S. 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 experiment). 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).[38] In 2005, the National Physical Laboratory in the United Kingdom reported their limited replication of this experiment.[39] 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, within the uncertainty of the measurements.
• 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.[40]