Accelerating frames and black holes

[Iamu]

I have a question that's been bothering me for a couple of days, and none of my professors have been able to answer it.

Two observers, Alice and Bob, are initially at rest relative to each other. Bob begins a constant acceleration. I want to be specific; let's say his acceleration is g and the energy and momentum required for his acceleration is picked up along the way without appreciably changing his rest mass. So he's not already carrying a large amount of fuel (let's say he has a fusion device which he fuels by scooping ambient hydrogen as he travels, and then he ejects spent fuel). Alice sees him gaining kinetic energy. As his energy appears to increase, so does his gravitational field. Eventually, his energy becomes so great that his Schwarzschild radius is bigger then his vehicle and he appears to become a black hole.

As Bob accelerates, he sees the equivalent of a uniform gravitational field pulling the opposite direction, in which Alice is falling and gaining kinetic energy. I don't know if the concept of potential energy applies here, but let's say he stops accelerating after a long enough time and the apparent uniform gravitational field vanishes, but Alice already appears to have enough kinetic energy to fit inside her Schwarzschild radius.

I know little about GR, and I'm hoping someone could stop me here and point out a problem with my reasoning. It's actually not the black holes that bother me, but the fact this extreme of a boost appears irreversible. It seems like if the two were to approach each other's speeds again, both of them would see the other one becoming a smaller black hole.

Can someone please explain?

 

[JesseM]

Schwarzschild radius assumes an object at rest in Schwarzschild coordinates, for a moving object this would no longer tell you when an object becomes a black hole (which should be a coordinate-independent event). See this answer from the physics FAQ or the discussion on this thread (pervect's post #9 and Chris Hillman's post #10 are especially useful)

 

[bapowell]

Right. Bob would be surprised at the prospect of himself becoming a black hole because according to him, he's undergone no increase in mass. Of course, the seeming paradox arises with Alice, who sees Bob's mass increase to the point that would warrant his collapse into a black hole . But as JesseM says, the Schwarzschild solution is for a static body at rest -- to find out what kind of gravitational field Bob generates one would need to solve Einstein's Eqs for a massive body with momentum. I haven't done it, but presumably you won't get a black hole ;)

 

[Iamu]

That makes sense. I forgot about the off-diagonal terms in the stress-energy tensor. Thanks for the help.