Einstein’s twin paradox is more familiar to us than we think. Then why does it feel so strange? I promised my students some extra guidance, so I wrote this post.

[For a course I am currently teaching at a university in Amsterdam]

## Einstein’s light clock

Einstein’s thought experiment with a light clock (see picture) shows that a moving clock must slow down if the speed of light is always the same. But that does not only go for clocks, *any *physical process in a moving reference frame slows down with respect to an observer who is at rest. This is a real effect: ageing – being a physical process – also slows down.

From this follows the famous twin paradox: if there are two people of the same age on Earth, and one of them makes a trip in a rocket at nearly the speed of light, then the person who has stayed on Earth will have aged more. This is a paradox because Einstein’s thought experiment is *symmetrical*. This sounds abstract, but the idea is pretty simple: from the perspective of the moving clock, it is our clock that seems to be moving, and so our clock slows down from its perspective. If there is this symmetry, how come there is an asymmetry in the ageing of the twins? If both of the twins see the other’s clock slow down, then why does only one of the twins actually age less?

The point of the paradox is just that: there is a difference in the time elapsed, even though there is symmetry in terms of perspective. Have you ever encountered something as mind-boggling as this? Actually, you have. Whenever you go to the post office.

## The postmen paradox

Suppose two postmen walk from point A to point B (see picture). The first person walks in a straight line from A to B (the black line), while the second person begins at a slight angle and turns halfway so that he also arrives at point B (red line).

When the postmen start walking, both see the other moving away (in the vertical direction)- there is symmetry in terms of perspective. And yet one of the postmen walks a larger distance than the other – there is an asymmetry in path length. The reason why there is this difference is of course that the postmen have different velocities (the reason for the difference is *not *the turn made by one of the postmen. Just think about it: even if the red line were straight, the postman walking it would still be going faster than the other).

The solution to the twin paradox in relativity is that the twins, too, have different velocities. But these velocities are velocities through spacetime and not just through space, as we are used to. As a result, there is a difference in the elapsed time.

note:

It is often stated that the twins’ difference in ageing is due to either acceleration or the presence of a gravitational field. But the twin paradox is purely about

specialrelativity – the thought experiment about the light clock shows that acceleration and gravity are not needed to explain what is going on.Special relativity tells us that the speed at which we are moving through space is intimately connected with the speed with which we are moving through time: the faster we move through space, the slower we move through time.

EDIT (Nov 17, 2023):

The point of the paradox is a little more subtle than stated above: the reason why there is a difference in path length is *not* that the postmen have different velocities, as those would be velocities with respect to absolute space (their *relative* velocities are the same). Instead, we follow Leibniz in rejecting absolute space and time, and only talk about the path itself – there is a difference in both length, and that’s it.