Re: Does a black hole actually change the geometry of surrounding space?

Very interesting question! Clearly you've been pondering relativity.

First off, I'd like to get a bit picky and point out that mass does not actually change with velocity. Energy changes, and, indeed, heads off to infinity as you approach the speed of light. Physicists sometimes think of mass as increasing, but this is really rather dangerous, since it doesn't (it looks like it does in some ways because of how the momentum looks at high velocities, but that's just our brains trying to use Newton's mechanics somewhere were they break down).

I'd also like to try to emphasize the line between proving something based on a theory (which is what I imagine you meant) and actually doing an experiment to "prove" something. The former is only as good at the theory behind it, which could well be flawed (so many theories have been). If you could do an experiment, however, you'd have something much stronger.

Now that I've been really picky, I'll try to address your question. Could, say a black hole, warp spacetime so that faster than light travel is possible? Not if relativity is correct. Space-time can bend all kinds of weird ways, but light speed is the still the speed limit for us massive objects. In fact, this is why black holes are black in the first place: at some point, the escape velocity becomes equal or greater than the speed of light, so nothing, not even light, can escape it.

It's rather unfortunate the Gene Rodenberry decided that the Enterprise was to fly around at "warp" speed, because the term has basically sunk in to our collective consciouses, but warping space-time doesn't let you sneak around the light-speed limit, at least not the way that ships on Star Trek do.

Here is a way to circumvent the speed limit, however. It's purely theoretical. It may well be impossible to do. But if you imagine space-time is a great rubber sheet (except 4-dimensional), you can imagine pushing two points that are in different areas until they meet up. Then you could step from one are to another without ever going faster than light at any point on your trip. However, if the points are quite far apart, you could get there before a photon which took the normal, longer route arrives at your destination.

If you want to read a really good book on this sort of thing, try Kip Thorne's Black Holes and Time Warps, Einstein's Outrageous Legacy (Kip one of the leaders in the field of relativity, and the book is both fascinating and quite well-written).