Interval 29 Sep 2008

Peter Fisher

Nov. 27, 2008

8.033

In lecture, we saw the twin paradox is not really a paradox at all, it arises because traveling twin changes reference frames at the turn around point (at t=T/2) by the Earth bound clock). Still, it seems odd that the traveling twin is younger when they meet up again at time t=T.

The reason for the age difference comes from the way we measure "distance" in spacetime. Recall, the invariant interval between two events

remains the same in all reference frames. If we make the two events infinitesimally close, we can write .

Suppose we are at rest in a frame S and there is an object accelerating with respect to our rest frame. Over any time interval dt, there is a frame S’ in which the object is at rest. Since the object accelerates, it will not longer be at rest in S’ in the next instant in time, but for the instant dt, S and S’ are relatively inertial and S’ is the rest frame of the object. In S’, the interval interval is invariant, so

and the clock time to go from A to B is

. The

and

Remember we are working out the clock time for a clock in the rest frame of the moving object while we are in our rest frame S. Let’s work out the clock time for the twins. The Earth bound twin is at rest, so dx/dt=0 and

. For the traveling twin, the

outbound part is equal to the inbound part and

From the figure to the left, you can see the traveling clock always reads a shorter time than a stationary

clock: if then . The minus sign1 in the interval means a moving object always takes a longer path through space-time than a stationary object. This is just a result of the way we measure “distance” (interval) in space-time.

Recall the minus sign comes from the Lorentz transformations and they come from the need to keep c constant, which comes from Einstein’s second postulate which comes from the fact there is no ether as Michelson and Morley showed.

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