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Thought experiment in special relativity

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, 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 of looking at it is to realize the travelling twin is undergoing acceleration, which makes them 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. There is still debate as to the resolution of the twin paradox. 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 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 synchronize 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.

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):

Einstein: 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. Resnick: 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 (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
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.

Consider a space ship traveling 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.

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 aging of the travelers during their trip will be reduced by the factor , the reciprocal of the Lorentz factor (time dilation). In this case and the travelers will have aged only when they return.

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 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.

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.

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 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, (6.4 years in the example above).

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,

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.

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. 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 where Φ is the difference in gravitational potential. In this case, 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.