Is General Relativity compatible with relative speeds bigger than $c$ between two inertial frames?

Question

My question is motivated by a remark done by Tegmark in his book "Our Mathematical Universe". He says that GRT does not prove that relative speeds between material points are always smaller than c. It would prove it only for material points traveling through the same point. He needs it for the consistence of a supposition that in the frame-work of the expanding universe, galaxies from opposite sides can achieve relative speeds bigger than c one relatively to the other. So a first question is if GRT is really compatible with distant masses in relative movement with relative speed bigger than c.

My interest is not centered on cosmology, but in the logic study of formal theories. So I change a little bit the question, and I ask (as second question) if GRT is compatible with relative speeds bigger than c for pairs of inertial frames, which are necessarily distant. This question is important in the following context: if it is true, then GRT is not a conservative extension of SRT (in the sense that not all theorems proven by SRT are also true in GRT). [This would be because SRT sharply prohibits such a situation.] Examples of theories which do not exactly include other theories are well known. For example SRT is not a conservative extension of Galilean Relativity, despite the fact that both of them contain the principle of relativity.

However both questions are important and a positive answer to any of them can be interpreted as GRT not being conservative extension (from the logical point of view) of SRT.

Answer 1

First, a remark just in case. Relative velocity can mean two different things. If you mean the relative velocity of two objects as seen by a third observer, then this can be larger than $c$. Picture yourself standing still and two photons going past you in opposite directions: their relative velocity according to you is $2c$. This is allowed by relativity, because we're going to use the other definition, which is the velocity of one object as seen from another. Special Relativity says that this is never higher than $c$.

So what happens in GR? Well, the problem is that in general you cannot define a relative velocity unless the two objects are at the same spacetime point (or close enough). You can use coordinate velocity $d\mathbf{x}/dt$, but this can perfectly well be higher than $c$. This is what happens with the expansion of the universe; I'll get to that in a second.

This doesn't mean that things can go faster than light, because as we know coordinate-dependent quantities in GR don't have much meaning; you can deform your coordinates to make $d\mathbf{x}/dt$ have any value you want. What we should be looking at is the four momentum of the object, which is always timelike (for massive particles) or null (for light). But this is only relevant for an observer whose worldline crosses the object's.

So in general you can't even define a relative velocity for distant objects, because in GR there are no global inertial frames. So why do people say that galaxies move away faster than $c$? Because in cosmology there is a particularly useful coordinate system, called co-moving coordinates, which makes a definite separation between space and time, and assigns spatial coordinates by asking that galaxies remain at fixed coordinates. In this frame it is possible to define relative velocities between far away galaxies, and they can be higher than $c$, but like I said before this is no problem because this is simply an artifact of the coordinates. Locally, nothing moves faster than light, ever.

You also ask what happens in GR between two inertial frames. I'm not sure what you mean by this; if you have a global inertial frame you're back in SR, so there's not much to say there.

Answer 2

My understanding is that the expansion of space in the cosmological model is not limited by the velocity c. This is because expansion of space is a different phenomenon than acceleration of a massive object.

This is not my field so I will quote

The expansion of the universe causes distant galaxies to recede from us faster than the speed of light, if proper distance and cosmological time are used to calculate the speeds of these galaxies. However, in general relativity, velocity is a local notion, so velocity calculated using comoving coordinates does not have any simple relation to velocity calculated locally. (See comoving distance for a discussion of different notions of 'velocity' in cosmology.) Rules that apply to relative velocities in special relativity, such as the rule that relative velocities cannot increase past the speed of light, do not apply to relative velocities in comoving coordinates, which are often described in terms of the "expansion of space" between galaxies.

It is not only in cosmology where apparent velocity can be larger than the speed of light without violating the speed of light bound ;). See this discussion.

Answer 3

You can have objects moving faster than $c$ in general relativity. This will be the case for point-like tachyons (tachyon fields are more complicated), or, I think, phantom fields (fields with a negative kinetic energy term). Also any spacelike curve in general will, if the object doesn't have to be real.

What you cannot have is an object moving faster than $c$ in one coordinate system, and slower than $c$ in another. The speed of an object (with respect to whether or not it goes faster than $c$) is dictated by the sign of its momentum squared, which is invariant under coordinate transformation.