Time dilation in popular culture

Time dilation

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Time dilation explains why two working clocks will report different times after different accelerations. For example, at the ISS time goes slower, lagging approximately 0.01 seconds behind for every 12 earth months passed. For GPS satellites to work, they must adjust for similar bending of spacetime to coordinate properly with systems on Earth.

Time dilation is a difference in the elapsed time measured by two clocks, either due to them having a velocity relative to each other, or by there being a gravitational potential difference between their locations. After compensating for varying signal delays due to the changing distance between an observer and a moving clock (i.e. Doppler effect), the observer will measure the moving clock as ticking slower than a clock that is at rest in the observer's own reference frame. A clock that is close to a massive body (and which therefore is at lower gravitational potential) will record less elapsed time than a clock situated further from the said massive body (and which is at a higher gravitational potential).

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. Time dilation has also been the subject of science fiction works.

History

Time dilation by the Lorentz factor was predicted by several authors at the turn of the 20th century. Joseph Larmor (1897), at least for electrons orbiting a nucleus, wrote "... individual electrons describe corresponding parts of their orbits in times shorter for the [rest] system in the ratio: ${\displaystyle \scriptstyle {\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}$".^[4] Emil Cohn (1904) specifically related this formula to the rate of clocks. 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. Subsequently, Hermann Minkowski (1907) introduced the concept of proper time which further clarified the meaning of time dilation.

Velocity time dilation

From the local frame of reference of the blue clock, the red clock, being in motion, is perceived as ticking slower (exaggerated).

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

Theoretically, time dilation would make it possible for passengers in a fast-moving vehicle to advance further into the future in a short period of their own time. For sufficiently high speeds, the effect is 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.

With current technology severely limiting the velocity of space travel, however, 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 those on Earth. The cosmonauts Sergei Krikalev and Sergei Avdeyev both experienced time dilation of about 20 milliseconds compared to time that passed on Earth.

Simple inference of velocity 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.

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.

Reciprocity

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.

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.

Doppler effect

Main article: Ives–Stilwell experiment

• 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:

 

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

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

• Hasselkamp, Mondry, and Scharmann (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). 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.

Moving particles

• 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.

• 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.

Proper time and Minkowski diagram

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. 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.

Derivation and formulation

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. 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.

Hyperbolic motion

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:

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:

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:

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

Clock hypothesis

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.

Gravitational time dilation

Time passes more quickly further from a center of gravity, as is witnessed with massive objects (like the Earth)

Gravitational time dilation is experienced by an observer that, at a certain altitude within a gravitational potential well, finds that their local clocks measure less elapsed time than identical clocks situated at higher altitude (and which are therefore at higher gravitational potential).

Gravitational time dilation is at play e.g. for ISS astronauts. While the astronauts' relative velocity slows down their time, the reduced gravitational influence at their location speeds it up, although at a lesser degree. Also, a climber's time is theoretically passing slightly faster at the top of a mountain compared to people at sea level. It has also been calculated that due to time dilation, the core of the Earth is 2.5 years younger than the crust. "A clock used to time a full rotation of the earth will measure the day to be approximately an extra 10 ns/day longer for every km of altitude above the reference geoid." Travel to regions of space where extreme gravitational time dilation is taking place, such as near a black hole, could yield time-shifting results analogous to those of near-lightspeed space travel.

Contrarily to velocity time dilation, in which both observers measure the other as aging slower (a reciprocal effect), gravitational time dilation is not reciprocal. This means that with gravitational time dilation both observers agree that the clock nearer the center of the gravitational field is slower in rate, and they agree on the ratio of the difference.

Experimental testing

• 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.
• In 2010 gravitational time dilation was measured at the earth's surface with a height difference of only one meter, using optical atomic clocks.

Combined effect of velocity and gravitational time dilation

Daily time dilation (gain or loss if negative) in microseconds as a function of (circular) orbit radius r = rs/re, where rs is satellite orbit radius and re is the equatorial Earth radius, calculated using the Schwarzschild metric. At r ≈ 1.497 there is no time dilation. Here the effects of motion and reduced gravity cancel. ISS astronauts fly below, whereas GPS and geostationary satellites fly above.

 

Daily time dilation over circular orbit height split into its components

High-accuracy timekeeping, 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 ${\displaystyle dt_{\text{E}}}$ is given by:

${\displaystyle dt_{\text{E}}^{2}=\left(1-{\frac {2GM_{\text{i}}}{r_{\text{i}}c^{2}}}\right)dt_{\text{c}}^{2}-\left(1-{\frac {2GM_{\text{i}}}{r_{\text{i}}c^{2}}}\right)^{-1}{\frac {dx^{2}+dy^{2}+dz^{2}}{c^{2}}}\,}$

where:

${\displaystyle dt_{\text{E}}}$ is a small increment of proper time ${\displaystyle t_{\text{E}}}$ (an interval that could be recorded on an atomic clock),

${\displaystyle dt_{\text{c}}}$ is a small increment in the coordinate ${\displaystyle t_{\text{c}}}$ (coordinate time),

${\displaystyle dx,dy,dz}$ are small increments in the three coordinates ${\displaystyle x,y,z}$ of the clock's position,

${\displaystyle {\frac {-GM_{i}}{r_{i}}}}$ represents the sum of the Newtonian gravitational potentials due to the masses in the neighborhood, based on their distances ${\displaystyle r_{i}}$ from the clock. This sum includes any tidal potentials.

The coordinate velocity of the clock is given by:

${\displaystyle v^{2}={\frac {dx^{2}+dy^{2}+dz^{2}}{dt_{\text{c}}^{2}}}\,}$

The coordinate time ${\displaystyle t_{c}}$ is the time that would be read on a hypothetical "coordinate clock" situated infinitely far from all gravitational masses (${\displaystyle U=0}$), and stationary in the system of coordinates (${\displaystyle v=0}$). 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:

${\displaystyle {\frac {dt_{\text{E}}}{dt_{\text{c}}}}={\sqrt {1+{\frac {2U}{c^{2}}}-{\frac {v^{2}}{c^{2}}}+\left({\frac {c^{2}}{2U}}+1\right)^{-1}{\frac {{v_{\shortparallel }}^{2}}{c^{2}}}}}={\sqrt {1-\left(\beta ^{2}+\beta _{e}^{2}+{\frac {\beta _{\shortparallel }^{2}\beta _{e}^{2}}{1-\beta _{e}^{2}}}\right)}}\,}$

where:

${\displaystyle v_{\shortparallel }}$ is the radial velocity,

${\displaystyle v_{e}={\sqrt {\frac {2GM_{i}}{r_{i}}}}}$ is the escape speed,

${\displaystyle \beta =v/c}$, ${\displaystyle \beta _{e}=v_{e}/c}$ and ${\displaystyle \beta _{\shortparallel }=v_{\shortparallel }/c}$ are velocities as a percentage of speed of light c,

${\displaystyle U={\frac {-GM_{i}}{r_{i}}}}$ is the Newtonian potential; hence ${\displaystyle -U}$ equals half the square of the escape speed.

The above equation is exact under the assumptions of the Schwarzschild solution. It reduces to velocity time dilation equation in the presence of motion and absence of gravity, i.e. ${\displaystyle \beta _{e}=0}$. It reduces to gravitational time dilation equation in the absence of motion and presence of gravity, i.e. ${\displaystyle \beta =0=\beta _{\shortparallel }}$.

Experimental testing

• 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). In 2005, the National Physical Laboratory in the United Kingdom reported their limited replication of this experiment. 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.

Time dilation in popular culture

Velocity and gravitational time dilation has been the subject of science fiction works, as it technically provides the means for forward time travel. Interstellar and Tau Zero are examples.

Interstellar is a film example. One key plot point in this movie is gravitational time dilation. The film depicts a planet close to a rotating black hole, on the surface of which one hour is equivalent to seven years on Earth, which is not realistic, although it is possible in theory. The rendering of the black hole itself was considered to be very realistic; its simulation was the result of a year of work by company Double Negative. Physicist Kip Thorne collaborated in making the film and explained its scientific concepts in the book The Science of Interstellar.

There are more film examples, such as Passengers, Planet of the Apes, and Time Trap. Episode "A Matter of Time" of Stargate SG-1 is one sample from television series. Episode "The Return Part 1" of Stargate Atlantis is another sample from television series. Here a spaceship traveling at a speed very close to that of light, suffers from time passing very slow against the time passing for the rest of the universe.

Tau Zero is a book example. At one point, the novel discusses velocity time dilation in terms of the tau factor, explaining that tau approximates zero when a spaceship moves at near speed of light; hence the title Tau Zero. Such high speed is attained by using a Bussard ramjet. The time dilation in this story is so extreme that the ship travels through galaxies in a blink and eventually outlives the universe. Scientific works related to time dilation have included this book in their references.

There are more cases in literature, such as Mr Tompkins in Wonderland, Rocannon's World, and The Forever War. The story of X-O Manowar is one sample from comics.

 

Twin paradox (partial)

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"Clock problem" redirects here. For mathematical problems involving the positions of the hands on a clock face, see Clock angle problem.

In physics, the twin paradox is a thought experiment in special relativity involving identical twins, one of whom makes a journey into space in a high-speed rocket and returns home to find that the twin who remained on Earth has aged more. This result appears puzzling because each twin sees the other twin as moving, and so, according to an incorrect and naive application of time dilation and the principle of relativity, each should paradoxically find the other to have aged less. However, this scenario can be resolved within the standard framework of special relativity: the travelling twin's trajectory involves two different inertial frames, one for the outbound journey and one for the inbound journey. Another way of looking at it is by realising that the travelling twin is undergoing acceleration, which makes him a non-inertial observer. In both views there is no symmetry between the spacetime paths of the twins. Therefore, the twin paradox is not a paradox in the sense of a logical contradiction.

Starting with Paul Langevin in 1911, there have been various explanations of this paradox. These explanations "can be grouped into those that focus on the effect of different standards of simultaneity in different frames, and those that designate the acceleration [experienced by the travelling twin] as the main reason". Max von Laue argued in 1913 that since the traveling twin must be in two separate inertial frames, one on the way out and another on the way back, this frame switch is the reason for the aging difference. Explanations put forth by Albert Einstein and Max Born invoked gravitational time dilation to explain the aging as a direct effect of acceleration.  However, it has been proven that general relativity, or even acceleration, are not necessary to explain the effect, as the effect still applies to a theoretical observer that can invert the direction of motion instantly, maintaining constant speed all through the two phases of the trip. Such observer can be thought of as a pair of observers, one travelling away from the starting point and another travelling toward it, passing by each other where the turnaround point would be. At this moment, the clock reading in the first observer is transferred to the second one, both maintaining constant speed, with both trip times being added at the end of their journey.

History

In his famous paper on special relativity in 1905, Albert Einstein deduced that when two clocks were brought together and synchronized, and then one was moved away and brought back, the clock which had undergone the traveling would be found to be lagging behind the clock which had stayed put. Einstein considered this to be a natural consequence of special relativity, not a paradox as some suggested, and in 1911, he restated and elaborated on this result as follows (with physicist Robert Resnick's comments following Einstein's):

If we placed a living organism in a box ... one could arrange that the organism, after any arbitrary lengthy flight, could be returned to its original spot in a scarcely altered condition, while corresponding organisms which had remained in their original positions had already long since given way to new generations. For the moving organism, the lengthy time of the journey was a mere instant, provided the motion took place with approximately the speed of light. If the stationary organism is a man and the traveling one is his twin, then the traveler returns home to find his twin brother much aged compared to himself. The paradox centers on the contention that, in relativity, either twin could regard the other as the traveler, in which case each should find the other younger—a logical contradiction. This contention assumes that the twins' situations are symmetrical and interchangeable, an assumption that is not correct. Furthermore, the accessible experiments have been done and support Einstein's prediction.

In 1911, Paul Langevin gave a "striking example" by describing the story of a traveler making a trip at a Lorentz factor of γ = 100 (99.995% the speed of light). The traveler remains in a projectile for one year of his time, and then reverses direction. Upon return, the traveler will find that he has aged two years, while 200 years have passed on Earth. During the trip, both the traveler and Earth keep sending signals to each other at a constant rate, which places Langevin's story among the Doppler shift versions of the twin paradox. The relativistic effects upon the signal rates are used to account for the different aging rates. The asymmetry that occurred because only the traveler underwent acceleration is used to explain why there is any difference at all, because "any change of velocity, or any acceleration has an absolute meaning".

Max von Laue (1911, 1913) elaborated on Langevin's explanation. Using Hermann Minkowski's spacetime formalism, Laue went on to demonstrate that the world lines of the inertially moving bodies maximize the proper time elapsed between two events. He also wrote that the asymmetric aging is completely accounted for by the fact that the astronaut twin travels in two separate frames , while the Earth twin remains in one frame, and the time of acceleration can be made arbitrarily small compared with the time of inertial motion.  Eventually, Lord Halsbury and others removed any acceleration by introducing the "three-brother" approach. The traveling twin transfers his clock reading to a third one, traveling in the opposite direction. Another way of avoiding acceleration effects is the use of the relativistic Doppler effect (see What it looks like: the relativistic Doppler shift below).

Neither Einstein nor Langevin considered such results to be problematic: Einstein only called it "peculiar" while Langevin presented it as a consequence of absolute acceleration. Both men argued that, from the time differential illustrated by the story of the twins, no self-contradiction could be constructed. In other words, neither Einstein nor Langevin saw the story of the twins as constituting a challenge to the self-consistency of relativistic physics.

Specific example

Consider a space ship traveling from Earth to the nearest star system: a distance d = 4 light years away, at a speed v = 0.8c (i.e., 80% of the speed of light).

To make the numbers easy, the ship is assumed to attain full speed in a negligible time upon departure (even though it would actually take about 9 months accelerating at 1 g to get up to speed). Similarly, at the end of the outgoing trip, the change in direction needed to start the return trip is assumed to occur in a negligible time. This can also be modelled by assuming that the ship is already in motion at the beginning of the experiment and that the return event is modelled by a Dirac delta distribution acceleration.

The parties will observe the situation as follows:

Earth perspective

The Earth-based mission control reasons about the journey this way: the round trip will take t = 2d/v = 10 years in Earth time (i.e. everybody on Earth will be 10 years older when the ship returns). The amount of time as measured on the ship's clocks and the aging of the travelers during their trip will be reduced by the factor ${\displaystyle \varepsilon =\scriptstyle {\sqrt {1-v^{2}/c^{2}}}}$, the reciprocal of the Lorentz factor (time dilation). In this case ε = 0.6 and the travelers will have aged only 0.6 × 10 = 6 years when they return.

Travellers' perspective

The ship's crew members also calculate the particulars of their trip from their perspective. They know that the distant star system and the Earth are moving relative to the ship at speed v during the trip. In their rest frame the distance between the Earth and the star system is εd = 0.6 × 4 = 2.4 light years (length contraction), for both the outward and return journeys. Each half of the journey takes εd / v = 2.4 / 0.8 = 3 years, and the round trip takes twice as long (6 years). Their calculations show that they will arrive home having aged 6 years. The travelers' final calculation about their aging is in complete agreement with the calculations of those on Earth, though they experience the trip quite differently from those who stay at home.

Conclusion

Readings on Earth's and spaceship's clocks

Event	Earth (years)	Spaceship (years)
Departure	0	0
End of outgoing trip = Beginning of ingoing trip	5	3
Arrival	10	6

No matter what method they use to predict the clock readings, everybody will agree about them. If twins are born on the day the ship leaves, and one goes on the journey while the other stays on Earth, they will meet again when the traveler is 6 years old and the stay-at-home twin is 10 years old.

Resolution of the paradox in special relativity

The paradoxical aspect of the twins' situation arises from the fact that at any given moment the travelling twin's clock is running slow in the earthbound twin's inertial frame, but based on the relativity principle one could equally argue that the earthbound twin's clock is running slow in the travelling twin's inertial frame. One proposed resolution is based on the fact that the earthbound twin is at rest in the same inertial frame throughout the journey, while the travelling twin is not: in the simplest version of the thought-experiment, the travelling twin switches at the midpoint of the trip from being at rest in an inertial frame which moves in one direction (away from the Earth) to being at rest in an inertial frame which moves in the opposite direction (towards the Earth). In this approach, determining which observer switches frames and which does not is crucial. Although both twins can legitimately claim that they are at rest in their own frame, only the traveling twin experiences acceleration when the spaceship engines are turned on. This acceleration, measurable with an accelerometer, makes his rest frame temporarily non-inertial. This reveals a crucial asymmetry between the twins' perspectives: although we can predict the aging difference from both perspectives, we need to use different methods to obtain correct results.

Role of acceleration

Although some solutions attribute a crucial role to the acceleration of the travelling twin at the time of the turnaround, others note that the effect also arises if one imagines two separate travellers, one outward-going and one inward-coming, who pass each other and synchronize their clocks at the point corresponding to "turnaround" of a single traveller. In this version, physical acceleration of the travelling clock plays no direct role; "the issue is how long the world-lines are, not how bent". The length referred to here is the Lorentz-invariant length or "proper time interval" of a trajectory which corresponds to the elapsed time measured by a clock following that trajectory (see Section Difference in elapsed time as a result of differences in twins' spacetime paths below). In Minkowski spacetime, the travelling twin must feel a different history of accelerations from the earthbound twin, even if this just means accelerations of the same size separated by different amounts of time, however "even this role for acceleration can be eliminated in formulations of the twin paradox in curved spacetime, where the twins can fall freely along space-time geodesics between meetings".

Relativity of simultaneity

Minkowski diagram of the twin paradox. There is a difference between the trajectories of the twins: the trajectory of the ship is equally divided between two different inertial frames, while the Earth-based twin stays in the same inertial frame.

For a moment-by-moment understanding of how the time difference between the twins unfolds, one must understand that in special relativity there is no concept of absolute present. For different inertial frames there are different sets of events that are simultaneous in that frame. This relativity of simultaneity means that switching from one inertial frame to another requires an adjustment in what slice through spacetime counts as the "present". In the spacetime diagram on the right, drawn for the reference frame of the Earth-based twin, that twin's world line coincides with the vertical axis (his position is constant in space, moving only in time). On the first leg of the trip, the second twin moves to the right (black sloped line); and on the second leg, back to the left. Blue lines show the planes of simultaneity for the traveling twin during the first leg of the journey; red lines, during the second leg. Just before turnaround, the traveling twin calculates the age of the Earth-based twin by measuring the interval along the vertical axis from the origin to the upper blue line. Just after turnaround, if he recalculates, he will measure the interval from the origin to the lower red line. In a sense, during the U-turn the plane of simultaneity jumps from blue to red and very quickly sweeps over a large segment of the world line of the Earth-based twin. When one transfers from the outgoing inertial frame to the incoming inertial frame there is a jump discontinuity in the age of the Earth-based twin (6.4 years in the example above).

A non space-time approach

As mentioned above, an "out and back" twin paradox adventure may incorporate the transfer of clock reading from an "outgoing" astronaut to an "incoming" astronaut, thus entirely eliminating the effect of acceleration. Also, according to the so-called "clock postulate", physical acceleration of clocks doesn't contribute to the kinematical effects of special relativity. Rather, the time differential between two reunited clocks is produced purely by uniform inertial motion, as discussed in Einstein's original 1905 relativity paper, as well as in all subsequent kinematical derivations of the Lorentz transformations.

Because spacetime diagrams incorporate Einstein's clock synchronization (with its lattice of clocks methodology), there will be a requisite jump in the reading of the Earth clock time made by a "suddenly returning astronaut" who inherits a "new meaning of simultaneity" in keeping with a new clock synchronization dictated by the transfer to a different inertial frame, as explained in Spacetime Physics by John A. Wheeler.

If, instead of incorporating Einstein's clock synchronization (lattice of clocks), the astronaut (outgoing and incoming) and the Earth-based party regularly update each other on the status of their clocks by way of sending radio signals (which travel at light speed), then all parties will note an incremental buildup of asymmetry in time-keeping, beginning at the "turn around" point. Prior to the "turn around", each party regards the other party's clock to be recording time differently from his own, but the noted difference is symmetrical between the two parties. After the "turn around", the noted differences are not symmetrical, and the asymmetry grows incrementally until the two parties are reunited. Upon finally reuniting, this asymmetry can be seen in the actual difference showing on the two reunited clocks.

The equivalence of biological aging and clock time-keeping

All processes—chemical, biological, measuring apparatus functioning, human perception involving the eye and brain, the communication of force—are constrained by the speed of light. There is clock functioning at every level, dependent on light speed and the inherent delay at even the atomic level. Biological aging, therefore, is in no way different from clock time-keeping. This means that biological aging would be slowed in the same manner as a clock.

What it looks like: the relativistic Doppler shift

In view of the frame-dependence of simultaneity for events at different locations in space, some treatments prefer a more phenomenological approach, describing what the twins would observe if each sent out a series of regular radio pulses, equally spaced in time according to the emitter's clock. This is equivalent to asking, if each twin sent a video feed of themselves to each other, what do they see in their screens? Or, if each twin always carried a clock indicating his age, what time would each see in the image of their distant twin and his clock?

Shortly after departure, the traveling twin sees the stay-at-home twin with no time delay. At arrival, the image in the ship screen shows the staying twin as he was 1 year after launch, because radio emitted from Earth 1 year after launch gets to the other star 4 years afterwards and meets the ship there. During this leg of the trip, the traveling twin sees his own clock advance 3 years and the clock in the screen advance 1 year, so it seems to advance at ​¹⁄₃ the normal rate, just 20 image seconds per ship minute. This combines the effects of time dilation due to motion (by factor ε=0.6, five years on Earth are 3 years on ship) and the effect of increasing light-time-delay (which grows from 0 to 4 years).

Of course, the observed frequency of the transmission is also ​¹⁄₃ the frequency of the transmitter (a reduction in frequency; "red-shifted"). This is called the relativistic Doppler effect. The frequency of clock-ticks (or of wavefronts) which one sees from a source with rest frequency f_rest is

${\displaystyle f_{\mathrm {obs} }=f_{\mathrm {rest} }{\sqrt {\left({1-v/c}\right)/\left({1+v/c}\right)}}}$

when the source is moving directly away. This is f_obs = ​¹⁄₃f_rest for v/c = 0.8.

As for the stay-at-home twin, he gets a slowed signal from the ship for 9 years, at a frequency ​¹⁄₃ the transmitter frequency. During these 9 years, the clock of the traveling twin in the screen seems to advance 3 years, so both twins see the image of their sibling aging at a rate only ​¹⁄₃ their own rate. Expressed in other way, they would both see the other's clock run at ​¹⁄₃ their own clock speed. If they factor out of the calculation the fact that the light-time delay of the transmission is increasing at a rate of 0.8 seconds per second, both can work out that the other twin is aging slower, at 60% rate.

Then the ship turns back toward home. The clock of the staying twin shows "1 year after launch" in the screen of the ship, and during the 3 years of the trip back it increases up to "10 years after launch", so the clock in the screen seems to be advancing 3 times faster than usual.

When the source is moving towards the observer, the observed frequency is higher ("blue-shifted") and given by

${\displaystyle f_{\mathrm {obs} }=f_{\mathrm {rest} }{\sqrt {\left({1+v/c}\right)/\left({1-v/c}\right)}}}$

This is f_obs = 3f_rest for v/c = 0.8.

As for the screen on Earth, it shows that trip back beginning 9 years after launch, and the traveling clock in the screen shows that 3 years have passed on the ship. One year later, the ship is back home and the clock shows 6 years. So, during the trip back, both twins see their sibling's clock going 3 times faster than their own. Factoring out the fact that the light-time-delay is decreasing by 0.8 seconds every second, each twin calculates that the other twin is aging at 60% his own aging speed.

Light paths for images exchanged during trip
Left: Earth to ship. Right: Ship to Earth.
Red lines indicate low frequency images are received, blue lines indicate high frequency images are received

The x–t (space–time) diagrams at left show the paths of light signals traveling between Earth and ship (1st diagram) and between ship and Earth (2nd diagram). These signals carry the images of each twin and his age-clock to the other twin. The vertical black line is the Earth's path through spacetime and the other two sides of the triangle show the ship's path through spacetime (as in the Minkowski diagram above). As far as the sender is concerned, he transmits these at equal intervals (say, once an hour) according to his own clock; but according to the clock of the twin receiving these signals, they are not being received at equal intervals.

After the ship has reached its cruising speed of 0.8c, each twin would see 1 second pass in the received image of the other twin for every 3 seconds of his own time. That is, each would see the image of the other's clock going slow, not just slow by the ε factor 0.6, but even slower because light-time-delay is increasing 0.8 seconds per second. This is shown in the figures by red light paths. At some point, the images received by each twin change so that each would see 3 seconds pass in the image for every second of his own time. That is, the received signal has been increased in frequency by the Doppler shift. These high frequency images are shown in the figures by blue light paths.

The asymmetry in the Doppler shifted images

The asymmetry between the Earth and the space ship is manifested in this diagram by the fact that more blue-shifted (fast aging) images are received by the ship. Put another way, the space ship sees the image change from a red-shift (slower aging of the image) to a blue-shift (faster aging of the image) at the midpoint of its trip (at the turnaround, 5 years after departure); the Earth sees the image of the ship change from red-shift to blue shift after 9 years (almost at the end of the period that the ship is absent). In the next section, one will see another asymmetry in the images: the Earth twin sees the ship twin age by the same amount in the red and blue shifted images; the ship twin sees the Earth twin age by different amounts in the red and blue shifted images.

Calculation of elapsed time from the Doppler diagram

The twin on the ship sees low frequency (red) images for 3 years. During that time, he would see the Earth twin in the image grow older by 3/3 = 1 years. He then sees high frequency (blue) images during the back trip of 3 years. During that time, he would see the Earth twin in the image grow older by 3 × 3 = 9 years. When the journey is finished, the image of the Earth twin has aged by 1 + 9 = 10 years.

The Earth twin sees 9 years of slow (red) images of the ship twin, during which the ship twin ages (in the image) by 9/3 = 3 years. He then sees fast (blue) images for the remaining 1 year until the ship returns. In the fast images, the ship twin ages by 1 × 3 = 3 years. The total aging of the ship twin in the images received by Earth is 3 + 3 = 6 years, so the ship twin returns younger (6 years as opposed to 10 years on Earth).

The distinction between what they see and what they calculate

To avoid confusion, note the distinction between what each twin sees and what each would calculate. Each sees an image of his twin which he knows originated at a previous time and which he knows is Doppler shifted. He does not take the elapsed time in the image as the age of his twin now.

• If he wants to calculate when his twin was the age shown in the image (i.e. how old he himself was then), he has to determine how far away his twin was when the signal was emitted—in other words, he has to consider simultaneity for a distant event.
• If he wants to calculate how fast his twin was aging when the image was transmitted, he adjusts for the Doppler shift. For example, when he receives high frequency images (showing his twin aging rapidly) with frequency ${\displaystyle \scriptstyle {f_{\mathrm {rest} }{\sqrt {\left({1+v/c}\right)/\left({1-v/c}\right)}}}}$, he does not conclude that the twin was aging that rapidly when the image was generated, any more than he concludes that the siren of an ambulance is emitting the frequency he hears. He knows that the Doppler effect has increased the image frequency by the factor 1 / (1 − v/c). Therefore, he calculates that his twin was aging at the rate of

${\displaystyle f_{\mathrm {rest} }{\sqrt {\left({1+v/c}\right)/\left({1-v/c}\right)}}\times \left(1-v/c\right)=f_{\mathrm {rest} }{\sqrt {1-v^{2}/c^{2}}}\equiv \epsilon f_{\mathrm {rest} }}$

when the image was emitted. A similar calculation reveals that his twin was aging at the same reduced rate of εf_rest in all low frequency images.

Simultaneity in the Doppler shift calculation

It may be difficult to see where simultaneity came into the Doppler shift calculation, and indeed the calculation is often preferred because one does not have to worry about simultaneity. As seen above, the ship twin can convert his received Doppler-shifted rate to a slower rate of the clock of the distant clock for both red and blue images. If he ignores simultaneity, he might say his twin was aging at the reduced rate throughout the journey and therefore should be younger than he is. He is now back to square one, and has to take into account the change in his notion of simultaneity at the turnaround. The rate he can calculate for the image (corrected for Doppler effect) is the rate of the Earth twin's clock at the moment it was sent, not at the moment it was received. Since he receives an unequal number of red and blue shifted images, he should realize that the red and blue shifted emissions were not emitted over equal time periods for the Earth twin, and therefore he must account for simultaneity at a distance.

Viewpoint of the traveling twin

During the turnaround, the traveling twin is in an accelerated reference frame. According to the equivalence principle, the traveling twin may analyze the turnaround phase as if the stay-at-home twin were freely falling in a gravitational field and as if the traveling twin were stationary. A 1918 paper by Einstein presents a conceptual sketch of the idea. From the viewpoint of the traveler, a calculation for each separate leg, ignoring the turnaround, leads to a result in which the Earth clocks age less than the traveler. For example, if the Earth clocks age 1 day less on each leg, the amount that the Earth clocks will lag behind amounts to 2 days. The physical description of what happens at turnaround has to produce a contrary effect of double that amount: 4 days' advancing of the Earth clocks. Then the traveler's clock will end up with a net 2-day delay on the Earth clocks, in agreement with calculations done in the frame of the stay-at-home twin.

The mechanism for the advancing of the stay-at-home twin's clock is gravitational time dilation. When an observer finds that inertially moving objects are being accelerated with respect to themselves, those objects are in a gravitational field insofar as relativity is concerned. For the traveling twin at turnaround, this gravitational field fills the universe. In a weak field approximation, clocks tick at a rate of t' = t (1 + Φ / c²) where Φ is the difference in gravitational potential. In this case, Φ = gh where g is the acceleration of the traveling observer during turnaround and h is the distance to the stay-at-home twin. The rocket is firing towards the stay-at-home twin, thereby placing that twin at a higher gravitational potential. Due to the large distance between the twins, the stay-at-home twin's clocks will appear to be sped up enough to account for the difference in proper times experienced by the twins. It is no accident that this speed-up is enough to account for the simultaneity shift described above. The general relativity solution for a static homogeneous gravitational field and the special relativity solution for finite acceleration produce identical results.

Other calculations have been done for the traveling twin (or for any observer who sometimes accelerates), which do not involve the equivalence principle, and which do not involve any gravitational fields. Such calculations are based only on the special theory, not the general theory, of relativity. One approach calculates surfaces of simultaneity by considering light pulses, in accordance with Hermann Bondi's idea of the k-calculus. A second approach calculates a straightforward but technically complicated integral to determine how the traveling twin measures the elapsed time on the stay-at-home clock. An outline of this second approach is given in a separate section below.