Greg Egan on Nostr: Along the world line of a massive particle, we can talk about the “proper time” ...

Along the world line of a massive particle, we can talk about the “proper time” that would be measured by a clock at rest with respect to that particle.

Along the world line of a photon, the spacetime interval between events is zero. This sometimes causes people who are aware of this from pop science coverage to jump to all kinds of weird conclusions, such as “everything happens all at once for a photon!”

Statements like this are wrong in a number of ways. Firstly, saying “for a photon” is a kind of category error. It is literally impossible to have a clock at rest with respect to a photon, so it is meaningless to talk about proper time here the way you can for a massive particle.

But also, not only are the events along a photon’s world line distinct, you can order them (A, B, C, ...) and you can even say “event C is twice as far along the world line from event A as event B”, notwithstanding zero spacetime intervals all the way.

In special relativity this is simple: just pick any reference frame in which the photon’s world line lies in the (x,t) plane, and use either the x or t coordinates of A, B, C. Different reference frames meeting this condition will assign different values to these x and t coordinates, but they will be connected by linear functions that preserve “C is twice as far from A as B is.”

In general relativity, there is a number called the “affine parameter” along every geodesic, even null geodesics, which can only be changed by a *linear* function if it is to satisfy the geodesic equation. So, like the x or t coordinate in SR, we can use this to order and compare different events on a photon’s world line.