- 3,770

- 1,384

To properly understand how this is not forbidden by GR, I wanted to focus on exactly what it was about GR, what 'rule', that said an object can't travel superluminally, and then compare that to the distant galaxy scenario to see why that doesn't contradict the rule.

I am trying to think about things in a coordinate independent way and it seems that an object travelling superluminally, say at 1.1c relative to the CMBR frame, would have a spacelike worldline. If we assume the object's acceleration through and beyond c was in the past then the 'current' piece of the worldline is a spacelike geodesic.

I assume such a thing cannot happen, in which case presumably GR says that no particle can travel along a spacelike geodesic. But what aspect of GR is it that mandates this? Is it something to do with mass and energy and/or being unable to accelerate (say via electromagnetic forces such as in a particle accelerator) from a timelike velocity vector to a spacelike one? If so, which aspect of the equations is it that prevents this?

I then moved on to thinking about the recession speed of distant galaxies and realised that the 'relative velocity' of the two galaxies (ours and the distant one) doesn't appear to be a clearly defined term. The two four-velocities are vectors in distinct tangent spaces so we cannot subtract them to obtain a relative velocity, unless spacetime is flat (in which case we can identify all the tangent spaces with one another, which seems to be what SR implicitly does). If that's right then there doesn't even seem to be a precise way to express a prohibition on superluminal relative velocities. Is that right?