# "The Twin Paradox" by Cho Yin Kong

It is an unnamed year, far in the future. Humans (or post-human creatures) have invented the technology to fly into space at a velocity comparable to the speed of light. There are two twins, with the sort of typical names found in every math word problem, Alice and Bob. Alice decides to fly far off into space with her new-age ship. At one point, she turns around to returns to Bob. According to Einstein’s special relativity equations, Bob sees that Alice has aged less than him. Nevertheless, according to Alice who feels as if she is standing still, Bob is the one who “flies” away from her, so she will see a Bob that has aged less than her. The paradox shows that as this cannot happen, and both Alice and Bob cannot see an older version of each other at the same time, relativity must be wrong. However, if we take into account how Alice would have to turn around her ship, we have to apply physical laws that reveals how Alice should be younger than Bob from both of their perspectives.

## The math of the paradox

On June 30th, 1905, Einstein formed his two postulates for his theory of special relativity: 1) laws of physics hold in all inertial frames; 2) in all inertial frames, the speed of light in a vacuum is always the finite, unchanging velocity of m/s, represented by c.[1] The significance of the first law suggests that there is no specific, preferred frame of reference; each inertial frame cannot be distinguished from each other. The second law implies that it was impossible for an observer to travel at a speed larger or equal to c.

One can see this though one of Einstein’s most famous thought experiments: An observer sits on a train at almost the speed of light, c, and holds a mirror in front of his face. In order for him to see his reflection, light would need to travel to the mirror, and reflect back to the observer's eyes at c, from the observer's point of view. This means that if you were standing on the still platform, the light from the mirror would be travelling at almost 2c. As light must travel at the same speed c everywhere, it must be the case that the time and distance components of speed are perceived differently by the observer in the train and the observer on the platform. Time and distance can be dilated depending on the frame of reference from which you observe the phenomena.

The observer doesn’t notice anything different about the light from normal. The person on the platform must see the light that hits the mirror travelling very slowly, and the light that is reflected back to the person as travelling very quickly. Intuitively, the light that hits the mirror must account for the difference between the actual velocity of the train, and the speed of light, and seems to travel more slowly to let the train catch up. Similarly, the light that is reflected from the mirror seems to travel a lot faster, because it’s happening opposite the train’s direction. From these principles, we can see the time dilation equation, which is given by,

where delta t' is the time duration in the frame moving away from the observer, and delta t is the time duration as seen from the still observer’s point of view.

In a case similar to Einstein’s thought experiment, if there are two observers, one of which is speeding away at a speed that is comparable to c, we could imagine that one inertial frame is speeding away at a certain constant velocity, similar to the train. If two events happen, the time interval measured by the observer at rest will be different to the observer on the frame that is speeding away. This difference in time intervals is the time dilation. There are two things we can notice: a) as v approaches c, the time dilation becomes more significant. When v is insignificant to c, the time dilation and difference is trivial; b) let

in the speeding frame, the measured time is longer than the resting frame, so time must pass slower than in the resting frame for it to match up. For any given delta t, Bob’s time waiting on earth, Bob believes that Alice will experience a time dilation of delta t’, so that she must have experienced a shorter time from her perspective.

Tying this back to the paradox, the first law implies that both observers in different frames can’t have different laws of physics, and time dilation can be seen from either frame’s point of view. If Alice is travelling at a constant velocity away from Bob, she sees the earth moving away, and her measured time is delta t, while Bob measures the slower delta t’. From Bob’s point of view, Alice is moving away while he is staying still, so Bob measures delta t while Alice is supposed to measure delta t’. Is the paradox right then? And Einstein wrong?

## The solutions to the paradox

The answer is has been an unequivocal “no” from the physics community, and there are several ways to show this. The two most prominent ways are through general relativity and acceleration. This paradox supposedly exists only in inertial frames, a reference frame where Newton’s first law holds. [2] This would be true if we removed the effects of earth’s gravitational field on Alice’s travelling, and suppose that Bob is a floating observer. In (x,y,z,t) coordinates, Bob is at (0,0,0,t). Bob still uses the same time dilation formula, where he measures the longer t, and Alice will be younger. However, at some point, has to turn around and return to Bob, which means she has to decelerate, stop, turn around, and accelerate back to Bob. Her relative coordinates change with time, and is at (f(t),g(t),h(t),t). Thus, she can’t be an observer in an inertial frame, and the equivalence principle can’t be applied.[3] Alice can’t use the same time dilation formula, and it makes sense that Alice returns younger than Bob.

The second way is explaining the paradox through general relativity. Two objects exert a gravitational force on each other, and this force is dependent on the mass and distance of both objects.

The key is that this force can act over any distance. Einstein created his general principles of relativity, and formed additional postulates with some consequences: a) Special relativity applies to all motion in any system that you define; b) space-time curvature (the linearity of space and time) can be affected by matter with mass, and this matter is also affected by space-time. Thus, the gravitational field causes a distortion or curvature in space-time, which means that any object with a mass causes this curvature, and that different gravitational fields can affect time, causing gravitational time dilation. The lower down the object in the potential well, the longer the time dilation.

The premises are the same as those from before; when Alice flies away from Bob, Alice sees that Bob’s time is slow. It is the same as if Bob is “flying” away from Alice; he also sees her time as slow. However, when she decelerates to start turning around, she is reducing her force. In order for her to feel as if she was still standing still, she imagines that there’s a gravitational field that acts against her, and that the universe is accelerating towards her. From her point of view, Bob is “higher” than her in the gravitational field, so his clock runs faster. As we can see from the equation, the further the distance, the higher Bob is. After Alice turns around, she will once again see Bob’s time as dilating. However, in the brief moment that Bob’s clock runs fast, Bob’s total time that passes by is longer than the total time that Alice sees that his clock runs slow. Thus, when Alice returns, she is younger than Bob.[4] We can also consider the acceleration that Alice goes through when she leaves from Earth, but as she is still very close to Earth, the effect is trivially small

We can see that this paradox came from assuming that Bob and Alice’s situation both were symmetrical. However, once the situation is examined carefully, it is clear that the deceleration and acceleration that Alice necessarily goes through means that the situations are asymmetrical. Once we treat each situation as it is, Alice will be younger than Bob from her point of view, and Bob will be older than Alice from his point of view; there is no paradox. In addition, there has been variations on the paradox that shows that Alice is older than Bob [5], depending on the properties of space-time that one chooses to apply to this situation. The moral of the story is not that Alice is younger than Bob, but that even in theoretical thought problems, we should account for all the situations that might happen realistically.

### References

[1] Nobel Media AB. “The Postulates of Special Relativity.” Accessed November 23, 2016. https://www.nobelprize.org/educational/physics/relativity/postulates-1.html.

[2] Pössel, Markus. 2010. “The Case of the Travelling Twins.” Einstein Online 04: 1007. Accessed November 23, 2016. http://www.einstein-online.info/spotlights/Twins.

[3] Weiss, Michael. “The Twin Paradox: The Spacetime Diagram Analysis.” Accessed November 23, 2016. http://math.ucr.edu/home/baez/physics/Relativity/SR/TwinParadox/twin_spacetime.html.

[4] Simonetti, John. “Special Relativity --- The Twin Paradox.” October 21, 1997. Accessed November 23, 2016. http://www.phys.vt.edu/~jhs/faq/twins.html.

[5] Abramowicz, M.A., and Bajtlik, S. 2009. “Adding to the paradox: the accelerated twin is older.” arXiv:0905.2428v1.