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December 6

Geodesics for Massive and Massless Particles

In general relativity, do massive and massless particles follow the same geodesic? Why or why not? Malypaet (talk) 23:19, 6 December 2024 (UTC)

According to the Einstein field equations, the worldline traced by a particle not subject to external, non-gravitational forces is a geodesic. Each particle follows its own worldline. Two particles that share their worldline are at all times at the same location and so have identical velocities.  --Lambiam 08:46, 7 December 2024 (UTC)

A massless particle must follow a null geodesic and massive particle must follow a time-like geodesic (in my limited understanding). catslash (talk) 22:20, 7 December 2024 (UTC)

So a massive particle with a velocity infinitely close to that of a photon (under the influence of a massive object) will have a geodesic infinitely close to that of the photon, right? Or is there another explanation and which one? Malypaet (talk) 22:11, 9 December 2024 (UTC)

I believe that is correct (perhaps there is an expert to hand who could confirm this?). catslash (talk) 23:42, 9 December 2024 (UTC)

In some frame of reference, the massive particle is at rest and so its spacetime interval along its geodesic is as spacelike time-like as can be (and thereby as non-null-like as can be for a non-tachyonic particle). So it depends on the point of view of the observer. Simplifying the case to special relativity and considering a particle traveling with speed $v$ in the x-direction, the spacetime interval $\mathrm{\Delta }s$ between two events separated by a time $\mathrm{\Delta }t$ is given by:

$(\mathrm{\Delta }s{)}^2=(\mathrm{\Delta }ct{)}^2-(\mathrm{\Delta }x{)}^2=(\mathrm{\Delta }ct{)}^2-(\mathrm{\Delta }vt{)}^2=(c^2-v^2)(\mathrm{\Delta }t{)}^2.$

In frames of reference in which $v$ approaches $c,$ the interval can become arbitrarily small, making it experimentally indistinguishable from that of a massless particle.  --Lambiam 07:40, 12 December 2024 (UTC)

@User:Lambian, could you re-read the spacetime interval section? I reckon that if there exists a frame of reference in which an interval is purely a time difference, then it is time-like, and if there exists a frame of reference in which the interval is purely a difference in location, then it is space-like. catslash (talk) 10:14, 12 December 2024 (UTC)

Yes, I used the wrong term, now corrected.  --Lambiam 07:30, 13 December 2024 (UTC)