Twin paradox

thumb|upright=1.15|During the [[ISS year-long mission, astronaut Scott Kelly (right) aged about less than his twin brother Mark (left), on Earth, due to relativistic effects.]]

In physics, the twin paradox is a thought experiment in special relativity involving twins, one of whom takes a space voyage at relativistic speeds 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, as a consequence of 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 to understand the paradox is to realise the travelling twin is undergoing acceleration, thus becoming a non-inertial observer. In both views there is no symmetry between the spacetime paths of the twins. Therefore, the twin paradox is not actually 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 travelling 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 ageing difference. Explanations put forth by Albert Einstein and Max Born invoked gravitational time dilation to explain the ageing as a direct effect of acceleration. However, it has been proven that neither general relativity, nor even acceleration, are necessary to explain the effect, as the effect still applies if two astronauts pass each other at the turnaround point and synchronise their clocks at that point. The situation at the turnaround point can be thought of as where a pair of observers, one travelling away from the starting point and another travelling toward it, pass by each other, and where the clock reading of the first observer is transferred to that of 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 for two stationary and synchronous clocks that are placed at points A and B, if the clock at A is moved along the line AB and stops at B, the clock that moved from A would lag behind the clock at B. He stated that this result would also apply if the path from A to B was polygonal or circular. 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):

In 1911, Paul Langevin derived differential aging in round-trip experiments using portions of matter such as radium, showing by integration of the proper time of a co-moving accelerated observer, that the time experienced by that observer is shorter than for an observer in uniform motion, thus radium in motion is decaying at a slower rate. He went on to give a "striking example" by describing the story of a traveller making a trip at a Lorentz factor of (99.995% the speed of light). The traveller remains in a projectile for one year of their time, and then reverses direction. Upon return, the traveller will find that, having aged two years, 200 years have passed on Earth. During the trip, both the traveller 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 ageing rates. The asymmetry that occurred because only the traveller 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". Langevin's acceleration argument was subsequently also used by others such as Arnold Sommerfeld (1913), Einstein (1918) They argued that, from the time differential illustrated by the story of the twins, no self-contradiction could be constructed. In other words, none of them saw the story of the twins as constituting a challenge to the self-consistency of relativistic physics. Max von Laue (1911) was apparently the first to use the expression "paradox" in relation to this experiment by writing:

Laue (1911, 1913) went on to elaborate on Langevin's explanation:

Using Hermann Minkowski's spacetime formalism, he demonstrated that the world lines of the inertially moving bodies maximise the proper time elapsed between two events, which was also pointed out in the textbooks of Hermann Weyl (1918) Even before Laue, Emil Wiechert (1911) pointed out that there can be a time difference at reunion even when both clocks experience the same velocity changes or accelerations, and in 1921 Wiechert removed any acceleration by introducing the "relay" or "three-brother" approach, in which the travelling twin transfers their clock reading to a third one, travelling in the opposite direction.

In addition to Langevin's Doppler analysis, several methods have been invented to specifically show in which way the symmetry of time dilation is broken during turnaround from the perspective of the traveler: Hendrik Antoon Lorentz (1913) described that traveler's perspective in terms of two-way signals (radar) using three periods: In the first and the last, the stay-at-home clock moves at constant velocity and is therefore time dilated, yet in the middle period it apparently ticks faster which overcompensates the other periods and explains, why the traveler's clock is retarded at reunion with respect to the stay-at-home clock even from the traveler's own perspective. Alternatively, Hans Thirring (1921) used the Lorentz transformation to show that relativity of simultaneity leads to desynchronization of all the traveler's clocks during turnaround; this leads to an apparent forward jump of the stay-at-home clock which overcompensates its time dilation during constant velocity motion, all of which was visualized by Thirring using Minkowski diagrams. The same result was already before obtained by Einstein (1918), who used the equivalence principle to show that a homogeneous gravitational field appears in the rest frame of the traveler at turnaround, which is accompanied by gravitational time dilation; since the "free-falling" stay-at-home clock is at a location of higher potential during turnaround, it advances at a faster rate which overcompensates its retardation during constant velocity motion. Einstein's explanation was adopted in 1921 in the textbooks of Wolfgang Pauli and Max Born.

Eventually, Hermann Weyl (1918) replaced the clocks and travelers explicitly by "twins", writing:

Specific example

Consider a space ship travelling from Earth to the nearest star system: a distance years away, at a speed (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.

Earth perspective

The Earth-based mission control reasons about the journey this way: the round trip will take in Earth time (i.e. everybody who stays on Earth will be 10 years older when the ship returns). The amount of time as measured on the ship's clocks and the ageing of the travellers during their trip will be reduced by the factor <math>\alpha = \scriptstyle{\sqrt{1 - v^2/c^2</math>, the reciprocal of the Lorentz factor (time dilation). In this case and the travellers will have aged only 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 years (length contraction), for both the outward and return journeys. Each half of the journey takes , and the round trip takes twice as long (6 years). Their calculations show that they will arrive home having aged 6 years. The travellers' final calculation about their ageing 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

{| class="wikitable"

|+ 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 traveller 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 travelling twin experiences acceleration when the spaceship engines are turned on. This acceleration, measurable with an accelerometer, makes their rest frame temporarily non-inertial. This reveals a crucial asymmetry between the twins' perspectives: although we can predict the ageing difference from both perspectives, we need to use different methods to obtain correct results.

Role of acceleration

180px|left|thumb|"Relay" experiment without acceleration, in which the traveler is replaced by three non-accelerated B-clocks, whose combined time is retarded with respect to the time of the single A-clock ([[Emil Wiechert|Wiechert, 1921). 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 synchronise their clocks at the point corresponding to "turnaround" of a single traveller. In this "relay" or "three-brother" version (see figure on the left), physical acceleration of the travelling clock plays no direct role; 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 ' 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 (see figure on the right), (6.4 years in the example above).

A non-spacetime 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 eliminating the effect of acceleration. Also, the physical acceleration of clocks does not contribute to the kinematical effects of special relativity. Rather, in special relativity, the time differential between two reunited clocks is produced purely by uniform inertial motion, as discussed in Einstein's original 1905 relativity paper,

Equivalence of biological ageing 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 ageing, therefore, is in no way different from clock time-keeping. This means that biological ageing 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.

Other calculations have been done for the travelling 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 travelling twin measures the elapsed time on the stay-at-home clock. An outline of this second approach is given in a separate section below.

Difference in elapsed time as a result of differences in twins' spacetime paths

thumb|right|250px|Twin paradox employing a rocket following an acceleration profile in terms of coordinate time T and by setting c=1: Phase 1 (a=0.6, T=2); Phase 2 (a=0, T=2); Phase 3-4 (a=-0.6, 2T=4); Phase 5 (a=0, T=2); Phase 6 (a=0.6, T=2). The twins meet at T=12 and τ=9.33. The blue numbers indicate the coordinate time T in the inertial frame of the stay-at-home-twin, the red numbers the proper time τ of the rocket-twin, and "a" is the proper acceleration. The thin red lines represent lines of simultaneity in terms of the different momentary inertial frames of the rocket-twin. The points marked by blue numbers 2, 4, 8 and 10 indicate the times when the acceleration changes direction.

The following paragraph shows several things:

how to employ a precise mathematical approach in calculating the differences in the elapsed time
how to prove exactly the dependency of the elapsed time on the different paths taken through spacetime by the twins
how to quantify the differences in elapsed time
how to calculate proper time as a function (integral) of coordinate time

Let clock K be associated with the "stay at home twin".

Let clock <var>K'</var> be associated with the rocket that makes the trip.

At the departure event both clocks are set to 0.

: Phase 1: Rocket (with clock <var>K'</var>) embarks with constant proper acceleration a during a time Ta as measured by clock K until it reaches some velocity V.

: Phase 2: Rocket keeps coasting at velocity V during some time Tc according to clock K.

: Phase 3: Rocket fires its engines in the opposite direction of K during a time Ta according to clock K until it is at rest with respect to clock K. The constant proper acceleration has the value −a, in other words the rocket is decelerating.

: Phase 4: Rocket keeps firing its engines in the opposite direction of K, during the same time Ta according to clock K, until <var>K'</var> regains the same speed V with respect to K, but now towards K (with velocity −V).

: Phase 5: Rocket keeps coasting towards K at speed V during the same time Tc according to clock K.

: Phase 6: Rocket again fires its engines in the direction of K, so it decelerates with a constant proper acceleration a during a time Ta, still according to clock K, until both clocks reunite.

Knowing that the clock K remains inertial (stationary), the total accumulated proper time Δτ of clock <var>K'</var> will be given by the integral function of coordinate time Δt

: <math>\Delta \tau = \int \sqrt{ 1 - (v(t)/c)^2 } \ dt \ </math>

where v(t) is the coordinate velocity of clock <var>K'</var> as a function of t according to clock K, and, e.g. during phase 1, given by

: <math>v(t) = \frac{a t}{ \sqrt{1+ \left( \frac{a t}{c} \right)^2.</math>

This integral can be calculated for the 6 phases:

: Phase 1 <math>:\quad c / a \ \text{arsinh}( a \ T_a/c )\,</math>

: Phase 2 <math>:\quad T_c \ \sqrt{ 1 - V^2/c^2 }</math>

: Phase 3 <math>:\quad c / a \ \text{arsinh}( a \ T_a/c )\,</math>

: Phase 4 <math>:\quad c / a \ \text{arsinh}( a \ T_a/c )\,</math>

: Phase 5 <math>:\quad T_c \ \sqrt{ 1 - V^2/c^2 }</math>

: Phase 6 <math>:\quad c / a \ \text{arsinh}( a \ T_a/c )\,</math>

where a is the proper acceleration, felt by clock <var>K'</var> during the acceleration phase(s) and where the following relations hold between V, a and Ta:

: <math>V = a \ T_a / \sqrt{ 1 + (a \ T_a/c)^2 }</math>

: <math>a \ T_a = V / \sqrt{ 1 - V^2/c^2 }</math>

So the travelling clock <var>K'</var> will show an elapsed time of

: <math>\Delta \tau = 2 T_c \sqrt{ 1 - V^2/c^2 } + 4 c / a \ \text{arsinh}( a \ T_a/c )</math>

which can be expressed as

: <math>\Delta \tau = 2 T_c / \sqrt{ 1 + (a \ T_a/c)^2 } + 4 c / a \ \text{arsinh}( a \ T_a/c )</math>

whereas the stationary clock K shows an elapsed time of

: <math>\Delta t = 2 T_c + 4 T_a\,</math>

which is, for every possible value of a, Ta, Tc and V, larger than the reading of clock <var>K'</var>:

: <math>\Delta t > \Delta \tau\,</math>

Difference in elapsed times: how to calculate it from the ship

thumb|right|250px|Twin paradox employing a rocket following an acceleration profile in terms of proper time τ and by setting c = 1: Phase 1 (a = 0.6, τ = 2); Phase 2 (a = 0, τ = 2); Phase 3-4 (a = -0.6, 2τ = 4); Phase 5 (a = 0, τ = 2); Phase 6 (a = 0.6, τ = 2). The twins meet at T = 17.3 and τ = 12. This is a different voyage than the one shown above, as both schemes take the same assumed total point-of-view time: T=12 (stay-at-home), resp τ=12 (ship), so the results of the calculated other-one's times must be different: τ = 9.33 (ship), resp T = 17.3 (stay at home).

In the standard proper time formula

: <math>\Delta \tau = \int_0^{\Delta t} \sqrt{ 1 - \left(\frac{v(t)}{c}\right)^2 } \ dt, \ </math>

Δτ represents the time of the non-inertial (travelling) observer <var>K'</var> as a function of the elapsed time Δt of the inertial (stay-at-home) observer K for whom observer <var>K'</var> has velocity v(t) at time t.

To calculate the elapsed time Δt of the inertial observer K as a function of the elapsed time Δτ of the non-inertial observer <var>K'</var>, where only quantities measured by <var>K'</var> are accessible, the following formula can be used:

: <math>\Delta t^2 = \left[ \int^{\Delta\tau}_0 e^{\int^{\bar{\tau_0 a(\tau')d \tau'} \, d \bar\tau\right] \,\left[\int^{\Delta \tau}_0 e^{-\int^{\bar\tau}_0 a(\tau')d \tau'} \, d \bar\tau \right], \ </math>

where a(τ) is the proper acceleration of the non-inertial observer <var>K'</var> as measured by themself (for instance with an accelerometer) during the whole round-trip. The Cauchy–Schwarz inequality can be used to show that the inequality follows from the previous expression:

: <math>\begin{align}

\Delta t^2 & = \left[ \int^{\Delta\tau}_0 e^{\int^{\bar{\tau_0 a(\tau')d\tau'} \, d \bar\tau\right] \,\left[\int^{\Delta \tau}_0 e^{-\int^{\bar\tau}_0 a(\tau')d \tau'} \, d \bar\tau \right] \\

& > \left[ \int^{\Delta\tau}_0 e^{\int^{\bar{\tau_0 a(\tau')d\tau'} \, e^{-\int^{\bar\tau}_0 a(\tau') \, d \tau'} \, d \bar\tau \right]^2 = \left[ \int^{\Delta\tau}_0 d \bar\tau \right]^2 = \Delta \tau^2.

\end{align}</math>

Using the Dirac delta function to model the infinite acceleration phase in the standard case of the traveller having constant speed v during the outbound and the inbound trip, the formula produces the known result:

: <math>\Delta t = \frac{1}{\sqrt{1-\tfrac{v^2}{c^2} \Delta\tau .\ </math>

In the case where the accelerated observer <var>K'</var> departs from K with zero initial velocity, the general equation reduces to the simpler form:

:<math>\Delta t = \int^{\Delta\tau}_0 e^{\pm\int^{\bar{\tau_0 a(\tau')d \tau'} \, d \bar\tau , \ </math>

which, in the smooth version of the twin paradox where the traveller has constant proper acceleration phases, successively given by a, −a, −a, a, results in In addition to rotational acceleration, Bob must decelerate to become stationary and then accelerate again to match the orbital speed of the space station.

No twin paradox in an absolute frame of reference

Einstein's conclusion of an actual difference in registered clock times (or ageing) between reunited parties caused Paul Langevin to posit an actual, albeit experimentally indiscernible, absolute frame of reference:

In 1911, Langevin wrote: "A uniform translation in the aether has no experimental sense. But because of this it should not be concluded, as has sometimes happened prematurely, that the concept of aether must be abandoned, that the aether is non-existent and inaccessible to experiment. Only a uniform velocity relative to it cannot be detected, but any change of velocity ... has an absolute sense."

In the relativity of Poincaré and Hendrik Lorentz, which assumes an absolute (though experimentally indiscernible) frame of reference, no paradox arises due to the fact that clock slowing (along with length contraction and velocity) is regarded as an actuality, hence the actual time differential between the reunited clocks.

In that interpretation, a party at rest with the totality of the cosmos (at rest with the barycenter of the universe, or at rest with a possible ether) would have the maximum rate of time-keeping and have non-contracted length. All the effects of Einstein's special relativity (consistent light-speed measure, as well as symmetrically measured clock-slowing and length-contraction across inertial frames) fall into place.

That interpretation of relativity, which John A. Wheeler calls "ether theory B (length contraction plus time contraction)", did not gain as much traction as Einstein's, which simply disregarded any deeper reality behind the symmetrical measurements across inertial frames. There is no physical test which distinguishes one interpretation from the other.

In 2005, Robert B. Laughlin (Physics Nobel Laureate, Stanford University), wrote about the nature of space: "It is ironic that Einstein's most creative work, the general theory of relativity, should boil down to conceptualizing space as a medium when his original premise [in special relativity] was that no such medium existed ... The word 'ether' has extremely negative connotations in theoretical physics because of its past association with opposition to relativity. This is unfortunate because, stripped of these connotations, it rather nicely captures the way most physicists actually think about the vacuum. ... Relativity actually says nothing about the existence or nonexistence of matter pervading the universe, only that any such matter must have relativistic symmetry (i.e., as measured)."

In Special Relativity (1968), A. P. French wrote: "Note, though, that we are appealing to the reality of A's acceleration, and to the observability of the inertial forces associated with it. Would such effects as the twin paradox (specifically – the time keeping differential between reunited clocks) exist if the framework of fixed stars and distant galaxies were not there? Most physicists would say no. Our ultimate definition of an inertial frame may indeed be that it is a frame having zero acceleration with respect to the matter of the universe at large."

Historical sources

Secondary sources

External links

Twin Paradox overview in the Usenet Physics FAQ
The twin paradox: Is the symmetry of time dilation paradoxical? From Einsteinlight: Relativity in animations and film clips.
FLASH Animations: from John de Pillis. (Scene 1): "View" from the Earth twin's point of view. (Scene 2): "View" from the travelling twin's point of view.
Relativity Science Calculator - Twin Clock Paradox