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While using Minkowski diagrams for studying spacetime and time dilation, I came across a problem that seems like a paradox to me, but may actually just be from a lvl of understanding.

Imagining two objects, A and B, where B travels at a speed relative to A such that, after synchronizing clocks from a simultaneous starting position, B’s clock appears to run twice as fast as A’s from A’s reference frame.

Now, make a Minkowski diagram for the reference frame of A. Imagine B travels away from A and then back towards A. At 0t, A and B are at the same point. At 2t, B’s clock appears to A to be at 1t. At 4t, B’s clock appears to be at 2t and B turns around. At 8t, A and B reconvene and, on this Minkowski diagram, A’s clock reads 8t and B’s reads 4t.

However, when making the Minkowski diagram for B, since A seems to move away from and then toward B at the same rate B appears to move away from and toward A in the first diagram, the diagrams would be identical, but B would change course at 2t and reconvene at 4t. So that means in the second diagram B’s clock would read 4t and A’s would read 2t.

So in the first diagram when B returns when A’s clock reads 8t but in the second B returns when A’s clock reads 2t. How is this possible? When B turns around, does the time relationship with A change even though its speed is still constant relative to A, just in a different direction? Is there a property of Minkowski diagrams that I am missing?

Qmechanic
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ikrtheblogger
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1 Answers1

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Showing us the diagrams would help us understand what you have done wrong, but it seems to me that you are describing two unequal scenarios, one in which B turns round and one in which A does.

Remember that acceleration is absolute, not relative.

Take the scenario in which B accelerates away, coasts for a time, then comes back to rejoin A. In the spacetime diagram for that scenario from A's perspective, A's world line is straight and vertical, while B's goes off at an angle, then tilts back towards A at another angle, so that B's line has a kink in it at the turnaround point.

If you draw that exact scenario from B's perspective, B's world line starts vertical and A goes off at an angle. However, at the turnaround point it is B's world line that tips over at an angle, while A's carries on unkinked at its original angle. That is because it is B who has accelerated, not A.

If you draw a diagram in which B's world line is straight throughout while A's is kinked at the turnaround point, you are representing a quite different scenario in which A accelerates not B.

As John Rennie suggests in his comment, what you are effectively doing is depicting the 'twin paradox' in two separate ways, in each of which the roles of twins are reversed.

Marco Ocram
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