Q & A: twin paradox

Something seems wrong with the Twins Paradox of Special Relativity. Consider the standard version of this thought experiment which lasts 20 years. But let’s assume the space traveling twin accelerates at a constant 1 g for the entire trip. This can be done as follows: his rocket ship has a main burner that accelerates him at a constant 1 g. He travels for 5 years with his main burner facing Earth; then he travels the next 10 years with his main burner facing away from Earth; and finally he travels the last 5 years with his main burner facing Earth. This procedure bring him to rest on Earth after being accelerated at 1 g for all 20 years of the trip. For 15 of the 20 years his velocity will be well above 50% the speed of light, so there should be plenty of time dilation between himself and his Earthbound twin. But how can this be? Both twins have the same relative velocity with repect to one another at all times during the trip. And both have the same acceleration with respect to each other at all times. Their two frames of reference a exactly equivalent. Consequently there should be no difference between the age of the twins when they meet again 20 years later.
- Pat Dolan (age 43)
Seattle, WA

Pat- That's a nice question. As you say, the relative velocities and accelerations of the two objects seem at first like they should just be exact opposites of each other. Special relativity describes how space and time are related for observers who do not accelerate and are not in the presence of gravitational fields.

It is most convenient to analyze situations like these in just one nonaccelerating frame of reference. Only one twin, the stay-at-home one, is stationary in a single frame of reference. In this frame, one can add up all the time on his watch, and compare it with the time that accumulates on the traveling twin's watch, and compare it when they meet up. You can also do the calculation in any other nonaccelerating frame of reference, and you will always get the same answer, that the twin that accelerates will have less time on his watch when he meets his brother.

Special relativity also gives you the tools to analyze the situation from the traveling twin's perspective.
The traveling twin does not stay put in a single nonaccelerating frame of reference during his journey, however, so life is more complicated for him. Instead, at each moment of his journey, we can define a frame of reference that is moving along with the traveling twin. In this frame of reference, we can ask the question what time is read on his brother's watch. Then a moment later, we have to pick another frame of reference moving along with the traveling twin's new speed, and ask what time is read on his brother's watch. If we make a graph of the time read on the stay-at-home twin's watch in all of these different frames of reference as a function of the traveling twin's watch's reading, we will find it's not a straight line. On out outgoing part, the traveling twin thinks his stay-at-home twin's watch is running slow, just like his stay-at-home twin thinks the traveling twin's watch is runnning slow. On the turnaround part, the traveling twin thinks his stay-at-home brother's watch speeds up, and then on the return trip, he thinks the stay-at-home twin's watch runs slow again. It's a mess, but it gives the same answer as doing everything in just one frame.

In General Relativity. a broader class of reference frames is considered. These accelerating frames, like that of the traveler, have additional peculiar effects on time, but must agree with the Special Relativity answers when they are applicable (for example, in cases with no gravitational fields).

Your argument about the symmetry between the two twins assumes that only their relative motion, including acceleration, matters. It turns out that acceleration is not only a relative quantity -- you can devise experiments which give you different outcomes depending on the acceleration of the apparatus, without regard to the acceleration of anything else.

The idea that only relative motion can matter is known as Mach's principle. Einstein tried to follow it for a while, but ultimately dropped it in General Relativity.

Mike W. and Tom J.

(published on 10/22/2007)