Question

Explain the Twin’s Paradox and its resolution. Be sure to explain why the paradox is a paradox, that is: What is it about the Special Theory of Relativity that appears to make it fatally contradict itself and what the resolution is in terms of what the space faring twin experiences that the Earthbound twin does not.

Answer 1

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, and so 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.

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 percent 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 close to a year 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 negligible time.

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.

Travelers' perspective

he 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.

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.

The paradoxical aspect of the twins' situation arises from the fact that at any given moment the traveling 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 traveling twin's inertial frame.^[16][17][18] 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 traveling twin is not: in the simplest version of the thought-experiment, the traveling 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 be 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.