Theoretical question here on relativity and black hole collisions. Suppose a ver

Question

Theoretical question here on relativity and black hole collisions.

Suppose a very small black hole (A) is at rest and an observer is within its event horizon.
Now consider that a much larger black hole (B) is approaching singularity A at a highly relativistic speed (hence very highly relativistic momentum), and the two singularities collide head-on (extreme precision).

Now, by Conservation of Relativistic Momentum, the two singularities (whether they fuse together or not) should travel in the direction of initial motion of the larger black hole (B) because it had much more momentum.

Q: Is it possible for the observer, initially suspended within event horizon of Black hole (A), to escape both black holes because the collision would cause the 2 singularities to recede away from her at a very high relativistic speed?

That is to say, would this bizarre black hole collision allow an observer to escape a black hole and violate the concept of the event horizon? If so, since not even light can escape a black hole from within the horizon, would this imply that the observer is receding away from the two singularities at a relative speed greater than c? What sort of physical implications would this have, what would happen to the local spacetime? What strange events might occur? If this event is not possible, please explain explicitly why not. If the observer did not escape, what sort of strange things might she see when being dragged by the receding 2 singularities?

Explanation / Answer

During a time T the crests ahead of the source move a distance cT,and the source moves a shorter distance uT in the same direction.The distance ? between successive crests--that is,the wavelength -- is thus ? = (c - u) * T,as measured in observer's frame.The frequency that he measures is (c/?).Therefore
f = (c/(c - u) * T) ----------(1) The time To is measured in the rest frame of the source,so it is a proper time.The relation between To and T is T = (To/(1 - u^2/c^2)^1/2) = (cTo/(c^2 - u^2)^1/2) or,since To = (1/fo), (1/T) = ((c^2 - u^2)^1/2/cTo) = ((c^2 - u^2)^1/2/c) * fo Remember,1/T is not equal to f.We must substitute this expression for 1/T into equation (1) to find f: f = (c/c - u) * ((c^2 - u^2)^1/2/c) * fo Using c^2 - u^2 = (c - u) * (c + u) gives f = (c + u/c - u)^1/2 * fo (Doppler effect,eletromagnetic waves) ----------(2) When (u/c) is much smaller than 1,the fractional shift (?f/f) is also small and is approximately equal to (u/c): (?f/f) = (u/c) When the source moves away from the observer,we change the sign of u in equation (2) to get f = (c - u/c + u)^1/2 * fo

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