The Twin Paradox

To fully understand the twin paradox let's first look at the Einstein's Theory of special relativity.  According to this theory, time isn't the same to all reference frames across the universe. From special relativity we obtain a general result which is: 

Δt0 is known as the proper time, Δt is the dilated time interval, u is the velocity of the reference frame and c is the velocity of light. 

    

    According to this equation, an observer O measures a longer time interval than O' measures. The observer O' is at rest relative to a device that produces a time interval Δt0. For this observer, the beginning and end of the time interval occur at the same location, and so the interval Δt0 is known as the proper time. An observer O, relative to whom O′ is in motion, measures a longer time interval Δt for the same device. The dilated time interval Δt is always longer than the proper time interval Δt0, no matter what the magnitude or direction of u.

    This is a real effect that applies not only to clocks based on light beams but also to time itself; all clocks run more slowly according to an observer in relative motion, biological clocks included. Even the growth, aging, and decay of living systems are slowed by the time dilation effect. However, note that under normal circumstances (u ≪ c), there is no measurable difference between Δt and Δt0, so we don’t notice the effect in our everyday activities. Time dilation has been verified experimentally with decaying elementary particles as well as with precise atomic clocks carried aboard aircraft.

    Given this let's imagine a pair of identical twins. What will happen if one stays on earth and the other goes on a trip throughout the cosmos? Let's call the first brother Ross and his twin sister Monica. Now let's analyse what will happen when Monica goes on a trip to the nearest star1 . Ross with his understanding of special relativity, knows that his sister´s clock will run slow relative to his, which means that by the time of her return, she may be younger than him, as our discussion of time  dilation would suggest. Now if he recalls that discussion, we know that for two observers in motion, each thinks the other's clocks is running slow. We could therefore study this problem from Monica's perspective. Given what as been told before, Monica will think that her brother's clock is running slow and will therefore expect her brother to be younger than she at the time of her return. 

Cartoon representation of the twin paradox. (image created by FICMYST on WordPress.com)

One can disagree with other over whose clocks is running slow relative to his or her, which is merely a problem of frames of reference, when Monica returns to Earth (or when Earth returns to Monica), all observers must agree as to which twin has aged less rapidly. This is the paradox- each twin expects the other to be younger. 

    In order to comprehend what happens in reality let's analyse this paradox more quantitative, with a numerical example. We assume that the acceleration and deceleration take negligible time intervals, so that all of Monica's aging is done during the coasting. For simplicity, we assume the distant planet is at rest relative to Earth2. In our problem, the planet is 8 light-years distant from Earth, supposing that Monica travels at a speed of 0.8c3. To Ross it takes 10 years (10 years x 0.8c = 8 light-years) for Monica to reach the planet and 10 years to return, and therefore she is gone for a total of 20 years. (Ross does not know that Monica arrived at the planet right away, we only know that after 18 years, it takes 8 years for light to make the journey. Two years later she returns to Earth). Let's analyse the frame of reference of Monica. From the special theory of relativity we know that the distance to the planet is contracted. The factor of contraction is √(1 − (0.8)2) = 0.6, and therefore 0.6*8 light-years = 4.8 light-years. At a speed of 0.8c, Monica will measure 6 years for the trip to the planet, for a total round trip time of 12 years. Thus Ross ages 20 years while Monica ages only 12 years and is indeed younger on her return. 

    

1- Alpha Centauri, 4.3 light-years away. 2- This does not change the problem, but it avoids the need to introduce yet another frame of reference. 3- 0.8c means 0.8 times the velocity of light, 0.8*c. 

References

Crowell, Benjamin (2000). The Modern Revolution in Physics (illustrated ed.). Light and Matter. p. 23. ISBN 978-0-9704670-6-5. Extract of page 23.

Einstein, A. (1961). Relativity: The Special and the General Theory. Princeton University Press and The Hebrew University of Jerusalem. 

John Simonetti. "Frequently Asked Questions About Special Relativity - The Twin Paradox". Virginia Tech Physics. Retrieved 25 May 2020.

Krane, K. S. (2020). Modern Physics (4th ed.). Hoboken, New Jersey : John Wiley & Sons, Inc.

Miller, Arthur I. (1981). Albert Einstein's special theory of relativity. Emergence (1905) and early interpretation (1905–1911). 

Twin paradox. (2021, August 30). In Wikipedia. https://en.wikipedia.org/wiki/Twin_paradox