Twin Paradox, GR

Twin Paradox, GR

Twin Paradox, GR

The Twin Paradox of Einstein is an interesting thought experiment involving two twins (who are nearly the same age of course), one of whom sets out on a journey into space and back. In the theory of relativity, the twin who left experiences a slowing down of time and will actually be much younger than the twin that stayed behind. The reason that this is considered a paradox is that the theory of relativity seems to imply that either one can be considered at rest, with the other one moving. In order to resolve the paradox, it is crucial that we pay strict attention to the reference frame in which we are calculating.

For special relativity, which deals with inertial reference frames, there are three equally valid inertial rest frames within the twin paradox. The three inertial frames are 1) at-home twin 2) the going-away twin and 3) the coming-back twin. It doesn't make any difference that the last two are physically the same twin--they still define different inertial frames. The result of applying special relativity to any of the inertial frames is that all observers must see and feel the same sensations, no matter which inertial frame is used as the reference frame. An example using twins Ann and Bob illustrates this point. Ann stays at home and Bob rockets away at 3/5 light speed. Time dilation is 80%. Bob lets 4 years pass. Bob returns at 3/5 light speed, again taking 4 years. Ann thinks 10 years have passed, and Ann and Bob agree that Bob is two years younger. The example shows that Ann's and Bob's measurements remain the same, no matter which of the inertial rest frames are used to perform the calculations.

What if we make the step to general relativity, and insist on analyzing the situation as if Bob did not move at all? Clearly, this is not an inertial rest frame. Bob will have to experience tremendous acceleration, while not moving. How is this possible?

The following analysis is greatly oversimplified, as it relies upon approximations, but the computations work out exactly in a fashion similar to more complicated analysis.

Bob will perceive that Ann speeds away at 3/5c. After 4 years of waiting, Bob must experience a force related to acceleration, without moving. In the "meantime," Ann will have experienced only 80% as much time, or 3.2 years. On the "return" trip, she again experiences 3.2 years while Bob ages 4 years. But in the original example, Bob aged 8 years while Ann aged 10--what happened to the extra 3.6 years that seem to be missing from Ann's total? As relativity claims, Ann will experience those years, but we must discover when, and why.

The resolution comes from the general relativity principle that objects within a gravity well experience time dilation, relative to other observers. The effect is equal to gLT/c², where g is the acceleration due to gravity, L is the distance between the observers, and c is the speed of light, and T is the length of time that the gravitational effects are experienced. Bob doesn't sit in a gravity field attributable to matter, but for the length of time T, he feels an acceleration of (6/5)c/T. Since he considers himself at rest, he must attribute the acceleration to a gravity field. Ann will start to move back towards Bob, and he will attribute this to Ann's freefall within the pervasive gravity field. Ann will not feel acceleration, as she is in free fall, consistent with her experiences in any of the other frames. The distance between Ann and Bob is approximately 3 light-years. Thus the difference in time perception between them is

gLT/c² = [(6/5)c/T][3 c years]T/c² = 3.6 years.
Dynamic calculations, which take account of the varying distance between Bob and Ann, would produce the same value, but would be much more complicated. Interestingly, the value 3.6 years does not depend upon the time T, as both factors cancel. If Bob were to accelerate very quickly, the time dilation would be more extreme, but it would not last as long. We see that there is no internal contradiction within general relativity, but we do have to be careful to model the physical effects that each twin experiences.

Special relativity claims that all inertial frames are equivalent. General relativity claims that all frames are equivalent. In other words, the laws of physics produce the same results no matter which coordinate system is used. No contradictions to these claims have been found. And we see that the twin paradox is not really a paradox--Bob does actually age less, no matter which frame is used. In order to resolve the paradox where we assume Bob is completely at rest and doesn't move during the whole trip, we must apply the principles of general relativity. But we can resolve it.

Special thanks to the sci.physics.relativity newsgroup fellows, especially for their twin paradox FAQ and GR twin paradox FAQ, which were both written by Michael Weiss.