Re: System of black holes

From: Tom Roberts (tjroberts_at_lucent.com)
Date: 08/25/04 

Date: Wed, 25 Aug 2004 03:40:35 GMT

Satya Das wrote:
> If the two black holes are not making any kind of system analogous to
> a system of a pair of star then they will probably merge into one
> black hole. But what if they remain as two separate black holes?

I don't think it is possible for there to be two black holes in any
manifold forever. I think either they will eventually "merge", or the
manifold will not have an infinite timelike dimension.

        See my earlier post, in which I discuss the ambiguity of
        saying they "merge"....

> Will
> there be two point of singularity in space?

No black hole is a "point of singularity in space". For instance, the
Schwarzschild manifold has two spacelike singularities (i.e. the limit
surfaces surrounding each are spacelike; neither could possibly be
described as a "point in space").

        As I said before, just because the singularity is at r=0
        does NOT mean it is a "point in space". In fact, in the region
        r<2M, r is timelike, not spacelike....

> Will the singularities
> itself move in space time under gravity?

First you have to decide what you mean by "move" -- i.e. you must select
coordinates wrt which to define motion. As soon a you do that, however,
you end up with a coordinate-dependent quantity, which is not very
useful in a general and abstract discussion like this....

But in the usual coordinates which are inertial at spatial infinity, a
black hole acts like an ordinary mass far from its horizon; so the black
hole will orbit another mass (etc.). Note I discussed the black hole,
not the singularity (here there be dragons)....

        And using those "usual coordinates" implies weak gravitation
        (i.e. an approximation in which we don't look too close at any
        horizon)....

> What will be the metric in
> that case?

Complicated (:-)).

> I guess that the problem of black hole system can be solved if we find the
> metric of an accelerating BH.

As stated, this is easy: simply project the Schwarzschild metric onto a
set of accelerating coordinates. Bingo you have the "metric [components]
of an accelerating black hole".

At a deeper level, "acceleration" has the same problem as "moves" above
-- you need to define coordinates wrt which to measure acceleration, and
then you have a mere coordinate-dependent quantity, which is essentially
useless....

I think what you want is the metric for two orbiting black holes. That
is indeed an extremely difficult problem (I believe there is a prize
offered for a solution). There have been numerical solutions attempted,
but AFAIK none are fully believable; one not-so-small problem is simply
selecting coordinates (which becomes VERY tricky when they inspiral
quite close). But I am not up-to-date on this.

I hope I'm giving you a flavor of the subtleties of GR. Things are not
always as they seem.... I am not fully up-to-speed here....

Tom Roberts tjroberts@lucent.com

Relevant Pages

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