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Black holes/2 body problem



Chris Jones wrote:
Thanks for all the replies. However there are a couple of points I'm still
unsure on.

1. If an observer fell into a black hole, would it be possible for him to
miss the singularity and pass out the other side? Or perhaps go into orbit
around it, within the event horizon? Ignoring differential gravitational
forces of course.....
What do you mean by the other side? If you mean the part of the event horizon
on the side of the BH around behind the singularity, the answer is no the
observer will not pass out on the other side. If you mean will the observer
miss the singularity and pass out of this spacetime (region) and into another
one connected to it via a sort of "wormhole" the answer is that it may depend
on the type of BH you are discussing. If we restrict ourselves to nonrotating
and uncharged black holes the answer is no. The observer must encounter the
origin and be destroyed at the singularity. No orbiting of the singularity is
possible inside the event horizon. If a more complicated type of BH is being
considered (i.e. one with sufficient angular momentum and/or charge) then
there are analytic extensions to the home spacetime (region) of the observer
which can be continued on through to a different asymptotic spacetime (region)
and in such a BH an observer could, in principle, pass through such a
wormhole. These kinds of black holes are very mathematically difficult and
complicated to describe, however, and I would very soon be in over my head if
I attempted to do so. Perhaps someone else on the list whose actual area of
expertise is in general relativity (P. Camp perhaps?) could explain all the
possibilities better than I could having only done recreational reading in GR.

2. A nonrotating uncharged BH is a pretty idealised object. What are the
effects of angular momentum and charge on its characteristics? Can anyone
recommend a good reference for this, the books and articles I've looked at
seem to dismiss it as a problem seemingly too difficult to compute to bother
with!
I understand why they dismiss the problem. See my answer above. What little
I know about the various regions describing rotating and charged black holes
comes from Hawking & Ellis' book _The_Large_Scale_Structure_of_Space-time_
published by Cambridge University Press (1973). Also Misner, Wheeler, &
Thorne's opus _Gravitation_, Freeman (1973) also has extended discussions of
these things. I'm sure you could probably get an as detailed discussion as
you want from any more modern book on general relativity than these above if
its index contains extended references to the Kerr metric (rotating BH), the
Reissner-Nordstrom metric (charged unrotating BH), and the Kerr-Newman metric
(BH which is both charged and rotating). Again maybe someone whose research
field is general relativity could suggest better references--maybe even
popular references for the layman?

3. Dave Bowman's reply is very in-depth (thanks!). But how does it affect
the idea that if 2 BHs were to merge, the event horizon afterwards would
have an area equal to the sum of the two previous event horizons' areas?
Actually, as I understand it, if two BHs were to merge the new event horizon
area would be greater than the sum of the areas of the two originally separate
BHs. In general this is an artifact of the generalized 2nd law of "Black Hole
Thermodynamics" which relates the entropy of a BH to the proper area if its
event horizon. In the Schwarzschild (nonrotating uncharged) case we can
recognize that for such a BH the horizon area A = 16_pi_((GM)^2)/c^4. Since
for Holes 1 & 2 we see that A_1 = kM_1^2 and A_2 = kM_2^2 where k is a simple
constant. Thus A_1 + A_2 = k(M_1^2 + M_2^2). After they merge the combined
mass is a little less than M_1 + M_2 due to the inevitable gravitational
radiation that must have been emitted during the collision. Thus the area
of the event horizon of the combined BH must be a little less than
k(M_1 + M_2)^2 = k(M_1^2 + M_2^2 + 2M_2M_2) > k(M_1^2 + M2_^2) = A_1 + A_2.

Al Clark wrote:
All the recent discussion of GR has generated a question in my mind. Can
anyone direct me references regarding the 2-body problem in general
relativity, ie. the evolution of the metric and positions of 2 "small",
isolated masses?
I don't understand what you mean here. If one is to describe the "evolution
of the metric" for a 2-body problem in GR such as the BH collision above or
the classic case of a close-orbiting binary pulsar system, then the isolated
masses are not "small" as they BOTH cause sufficient distortion of spacetime.
If only one of the masses is "small" we have the case of a test particle
(observer maybe) following a geodesic on the Schwarzschild metric of the
massive particle. This metric does not "evolve"; it is static. If both
isolated masses are "small" then we have the Newtonian 2-body problem
(like the one A. Marlow has J. Rauber solving in a rotating reference frame).
Unfortunately, I don't have handy with me a good reference to the GR 2-body
problem as used in the gravitationally radiating binary pulsar problem. Again
maybe a list-resident general relativist could supply such a reference.

David Bowman
Georgetown College
dbowman@gtc.georgetown.ky.us