Time dilation

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Time dilation is the difference in elapsed time as measured by two clocks, either because of a relative velocity between them (special relativity), or a difference in gravitational potential between their locations (general relativity). When unspecified, "time dilation" usually refers to the effect due to velocity. The dilation compares "wristwatch" clock readings between events measured in different inertial frames and is not observed by visual comparison of clocks across moving frames.

These predictions of the theory of relativity have been repeatedly confirmed by experiment, and they are of practical concern, for instance in the operation of satellite navigation systems such as GPS and Galileo.^[1]

Time dilation is a relationship between clock readings. Visually observed clock readings involve delays due to the propagation speed of light from the clock to the observer. Thus there is no direct way to observe time dilation. As an example of time dilation, two experimenters measuring a passing train traveling at .86 light speed may see a 2 second difference on their clocks while on the train the engineer reports only one second elapsed when the experimenters went by. Observations of a clock on the front of the train would give completely different results: the light from the train would not reach the second experimenter only 0.27s before the train passed. This effect of moving objects on observations is associated with the Doppler effect.^[2]

Time dilation by the Lorentz factor was predicted by several authors at the turn of the 20th century.^[3][4] Joseph Larmor (1897) wrote that, at least for those orbiting a nucleus, individual electrons describe corresponding parts of their orbits in times shorter for the [rest] system in the ratio: ${\textstyle {\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}$.^[5] Emil Cohn (1904) specifically related this formula to the rate of clocks.^[6] In the context of special relativity it was shown by Albert Einstein (1905) that this effect concerns the nature of time itself, and he was also the first to point out its reciprocity or symmetry.^[7] Subsequently, Hermann Minkowski (1907) introduced the concept of proper time which further clarified the meaning of time dilation.^[8]

Special relativity indicates that, for an observer in an inertial frame of reference, a clock that is moving relative to the observer will be measured to tick more slowly than a clock at rest in the observer's frame of reference. This is sometimes called special relativistic time dilation. The faster the relative velocity, the greater the time dilation between them, with time slowing to a stop as one clock approaches the speed of light (299,792,458 m/s).

In theory, time dilation would make it possible for passengers in a fast-moving vehicle to advance into the future in a short period of their own time. With sufficiently high speeds, the effect would be dramatic. For example, one year of travel might correspond to ten years on Earth. Indeed, a constant 1 g acceleration would permit humans to travel through the entire known Universe in one human lifetime.^[10]

With current technology severely limiting the velocity of space travel, the differences experienced in practice are minuscule. After 6 months on the International Space Station (ISS), orbiting Earth at a speed of about 7,700 m/s, an astronaut would have aged about 0.005 seconds less than he would have on Earth.^[11] The cosmonauts Sergei Krikalev and Sergey Avdeev both experienced time dilation of about 20 milliseconds compared to time that passed on Earth.^[12][13]

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. This constancy of the speed of light means that, counter to intuition, the 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.^[15][16]

The relativity of time can be illustrated with a thought experiment based on an abstract vertical clock consisting of two mirrors A and B, between which a light pulse is bouncing.^[17] The separation of the mirrors is L and the clock ticks once each time the light pulse hits mirror A. In the frame in which the clock is at rest (see left part of the diagram), the light pulse traces out a path of length 2L and the time period between the ticks of the clock ${\displaystyle \Delta t}$ is equal to 2L divided by the speed of light c:

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

From the frame of reference of a moving observer traveling at the speed v relative to the resting frame of the clock (right part of diagram), the light pulse is seen as tracing out a longer, angled path 2D. Keeping the speed of light constant for all inertial observers requires a lengthening (that is dilation) of the time period between the ticks of this clock ${\displaystyle \Delta t'}$ from the moving observer's perspective. That is to say, as measured in a frame moving relative to the local clock, this clock will be running (that is ticking) more slowly, since tick rate equals one over the time period between ticks 1/${\displaystyle \Delta t'}$.

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:

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. The Lorentz factor gamma (γ) is defined as^[18]

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

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.^[19]

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 measure the other's clock as ticking at a slower rate than their own local clock, due to them both measuring 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.^[20]

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 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 the sibling on Earth. The dilemma posed by the paradox can be explained by the fact that situation is not symmetric. The twin staying on Earth is in a single inertial frame, and the traveling twin is in two different inertial frames: one on the way out and another on the way back. See also Twin paradox § Role of acceleration.

• 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.^[21]
• The lifetime of particles produced in particle accelerators are 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.^[22]

• 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:^[23]$${\displaystyle {\sqrt {\frac {1+v/c}{1-v/c}}}f_{0}=\gamma \left(1+v/c\right)f_{0}\qquad {\text{and}}\qquad {\sqrt {\frac {1-v/c}{1+v/c}}}f_{0}=\gamma \left(1-v/c\right)f_{0}\,}$$as deduced by Einstein (1905) from the Lorentz transformation, when the source is running slow by the Lorentz factor.
• Hasselkamp, Mondry, and Scharmann^[24] (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).^[25] 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 metres per second using optical atomic clocks connected by 75 metres of optical fiber.^[26]

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.^[27] 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.^[27]

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.^[28] 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. As an approximate threshold, time dilation of 0.5% may become important when an object approaches speeds on the order of 30,000 km/s (1/10 the speed of light).^[29]

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:^[30]

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:^[31]

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:

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

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 light of experimental verification up to very high accelerations in particle accelerators.^[32][33]