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> When we say in general relativity that particles can't travel faster than light, what is that speed measured relative to?

Relative to a local inertial frame, i.e., relative to an inertial observer (i.e., an observer in free fall, feeling no force) who is at the same spatial location as the particle whose speed is being measured. That is the only context in which the concept of "relative velocity", as it appears in the "can't travel faster than light" condition, has any physical meaning.

All of your suggested examples attempt to compare "relative velocity" between objects that are not at the same spatial location. That has no physical meaning in GR.

Think of it this way: c isn't a speed limit, it's the only speed that anything can travel at. Specifically, time is only one component of four-dimensional spacetime. If you move in x, y, and/or z, the vector length of that spatial movement has to come out of your velocity in t.

So it's not that you can't travel at c meters per second; it's that if you do, you'll zero out your velocity in the "seconds" component.

To cite an example from one of Neal DeGrasse Tyson's lectures, that's one of the funky things about photons from the most distant stars: they don't experience the passage of time at all from their own point of view. They arrive at our eyes/telescopes/radios as soon as they're emitted. Meanwhile, their velocity is limited to c from our point of view. The same thing would presumably happen to you if you could travel at c... but there would be other inconvenient effects in that case, such as an infinite increase in mass.

It's perhaps not altogether useful to regard c as a speed. Or rather you should redefine your idea of speed. It's a universal constant relating matter to spacetime. Time is a length measurement, and you're already moving at c in that dimension. If you start accelerating then you are trading time velocity for spacial velocity, so as you approach c your movement through time approaches zero.

> does that really mean that space is really so constantly twisting and turning that SOL can't be exceeded?

The speed of light stays the same and both time and space twist and turn around it. It's weird, but having a preferred reference frame would be just as weird. c is a dimensionless constant, and light happens to travel at that speed because it has no rest mass.

"When we say in general relativity that particles can't travel faster than light, what is that speed measured relative to?"

It's a theory postulating that a particle can't travel faster than the speed of light in relationship to a particle that isn't moving. In practice all particles are moving in relation to some other particle thus it's not something you can visibly measure or see. At least that is the way I always viewed it.

does that really mean that space is really so constantly twisting and turning that SOL can't be exceeded?

IANAP, but I've never read that space twists to prevent the SOL from being exceeded. I think you're thinking of time and mass. It would require an infinite amount of time or infinite acceleration to for something to accelerate to the speed of light relative to something else.

Space is twisted by the presence of matter, and we call that gravity.

If it's photons emitting from flashlights pointed in opposite directions relative to each other, then are they compressing space as they travel so as not to exceed the speed of light

No. My understanding is that from the point of view of a photon time is completely stopped. Nothing moves relative to it because of this. Since nothing moves relative to it, nothing is exceeding the speed of light relative to it.

> The underlying question is: When we say in general relativity that particles can't travel faster than light, what is that speed measured relative to?

It's based on the observation of two points of references moving a different speeds shining light and the light from each traveling at the same speed. So the speed of light is measured relative to other light. The speed of light is constant, and it is so regardless of point of reference. Flashlights pointed in opposite directions has nothing to do with the speed of light. The light is traveling at the same speed in opposite directions.

I found this [1] PBS video useful.

[1] https://www.youtube.com/watch?v=msVuCEs8Ydo