Bondi K-Calculus

The twin paradox is related to one of the first questions people ask when learning special relativity, and one that few teachers know how to answer:

Suppose there are two twins working in the space program. Twin A works in mission control and stays on a space station. Twin B is an astronaut and flies a spaceship at relativistic speeds. Then relativity predicts that when the two reunite, twin B will be younger than twin A.

But wait! Twin B says that this is not what happened at all. Twin B claims that the spaceship remained stationary, and the spacestation flew in the other direction at relativistic speeds. Thus Twin B is certain that when the twins meet, Twin A will be younger. Who is right?

Clearly one of the twins is wrong, but the 'traditional' derivation of special relativity does not clearly show what is wrong.

Suppose that when B leaves, A and B reset their clocks to time 0. Then B travels for a time T on his own clock, and at point e in the diagram, suddenly reverses and travels back at the same speed.

Then using the equations derived in lesson one, the time when B reverses is seen to be (k+1/k)T/2 on A's clock. Since B travels at the same speed in both directions, it takes another period of time T for B to return. Hence when A and B reunite, the time on B's clock is 2T. But Similarly, it takes (k+1/k)T/2 for B to return as measured by A. Thus the total time A has experienced is (k+1/k)T.

THEOREM- For any positive value of k not equal to 1, (k+1/k) > 2.
PROOF- (If you are unfamiliar with calculus, accept this as true).
The derivative of (k+1/k) is 1 - 1/k². Thus the extremum is at k = 1, which corresponds to (k+1/k) = 2. Choosing any other value of k indicates that k=1 is a minimum. Thus k + 1/k > 2 for all k not equal to 1.

And so regardless of the speed of B, A has aged more than B.

From the diagram, it is clear where Twin B made the mistake. Although B and A cannot determine which one is moving, only B will experience the acceleration at point e. It is this acceleration which breaks the symmetry, and thus solves the twin paradox.