# Is it possible to have faster-than-light movement in General Relativity?

The speed of light as the maximal possible speed is build into Special Relativity as a premise of the theory. However I know of no such premise in General Relativity. When looking at two stars laying in opposite directions from earth, each moving away from us at the speed of light, their relative velocity will be twice the speed of light. However, here them "moving away" from each other is not really true, since they are each at rest, only the space between them expands. My question is: Is there a situation in General Relativity where particle and/or energy can actually move faster than light by its own propulsion? Or can objects only travel faster than c when aided by space expansion?

(I'm not looking for situations involving worm holes or black holes.)

The premise in SR isn't "there is a maximum speed, let's call it $c$." Rather, it's more like "the invariant interval between events is $\Delta s^2 = -c^2 \Delta t^2 + (\Delta x)^2$, where $c$ is the speed of light in vacuum." Once you crank through the math with all those $\gamma$'s, you see that nothing can move faster than $c$. So this speed limit is a conclusion, not an explicit assumption.

In GR, things are, needless to say, far more complicated. But one mathematical fact about GR to keep in mind is that it reduces to SR when considering a small enough region. That is, we might have a much more complicated expression for $\Delta s^2$ (or rather $\mathrm{d}s^2$, if we consider infinitesimal intervals), but, given a point, we can choose a small enough region around the point and make an appropriate change of coordinates to make $\mathrm{d}s^2$ arbitrarily close to the SR version in that region.

When we talk about physics excluding faster-than-light travel, we mean in this local sense, relative to objects so close we can disregard the curvature of spacetime. Indeed, standard GR, just like SR, does not allow anything to go faster than $c$ in this sense.

You are perfectly correct in noting that the separation between objects can grow faster than $c$ due to expansion of space.

• Things in GR are more complicated, but the premise is still $ds^2 = g_{\alpha\beta}dx^\alpha dx^\beta$ is still invariant. – kηives Sep 15 '12 at 17:11
• @kηives: this is equivalent to saying that the objects are so close we can disregard the curvature of spacetime. That's why what you wrote is a differential equation rather than a the algebraic one that the special relativistic version is. – Jerry Schirmer Sep 16 '12 at 0:02
• @JerrySchirmer What I wrote is entirely general. It's a line element for an arbitrary einstein metric. – kηives Sep 16 '12 at 5:01
• @kηives: yes, of course. Where do I say otherwise. That line element is arbitrarily close to the minkowski one if you choose a sufficently small neighborhood around a point. – Jerry Schirmer Sep 16 '12 at 5:32

Speed of light is also encoded in the general relativity, in the same way as in it done in special relativity (Minkowski space is essential) , anyway yes it allowable for space to be able expand in a speed higher than speed of light because that doesn't contradict special relativity, and this because (roughly speaking) SR says that no matter or information can be transmitted in speed bigger than speed of light relative to "space" it self (you may want to read about comoving frames of reference in GR).

So some very far galaxies, and even some cosmological models, uses this and redshift shows that they are actually running "away" by speed bigger than speed of light, so the answer on your question is no.