From: ramsay@unixg.ubc.ca (Keith Ramsay)
Subject: Re: An easy problem in general relativity
Organization: University of British Columbia, Vancouver, B.C., Canada
Date: Wed, 12 Aug 1992 23:21:30 GMT

In article <1992Aug10.060505.18204@nuscc.nus.sg> 
matmcinn@nuscc.nus.sg (Mcinnes B T (Dr)) writes:

>I f you want to find out how much matter [you know, matter: this stuff
>here, not that interesting but irrelevant [here] parameter with which
>Schoen and Yau concerned themselves] there is in a region, you just
>integrate the appropriate component of the stress-energy tensor over the
>appropriate spacelike hypersurface. In the Schwarzschild case, the
>answer is zero.

The "appropriate spacelike hypersurface" which I suspect you are
talking about is not closed, however, but punctured. Hence I think
one should consider well what kind of integration it is we are doing.

I sense a certain analogy to Stokes' theorem in differential geometry.
It is possible to have a vector field which has zero curl, and yet
which has a non-zero circulation around a closed loop. The domain on
which the field is defined can't include a (closed) surface which has
the loop as its boundary: a "singularity" passes through the middle of
the loop. Otherwise, we could apply Stokes' theorem to that surface
and show the circulation is zero. In Stokes' theorem, we are not
allowed to integrate across a singularity in such a way. If it is
possible to stretch the surface so as to go past the end of the
singularity, then somewhere along the way the flux of the curl through
it has to be non-zero. Even if it is not, one can still define the
"curl" of the vector field "in the singularity" as a distribution (as
you probably know).

I think there is a certain poetic justice, then, which I expect can be
rigorously founded, to assigning a mass to a singularity, even without
there being matter there (or being a "there" there even!). Mass should
be a property of regions, not of places, and if our notion of "region"
allows a "region" to include singularities, then its mass may not be
equal to the limit of the mass of singularity-free portions of the
region, as we allow the bounds to approach the singularity.

If you want to avoid poetry of this kind, then for consistency, I
think you probably also want to avoid integrating over surfaces
passing through singularities. Let your "regions" be well-defined
closed space-like surfaces in space-time, then.

If you extend the hypersurface in such a way as to close it, you have
to extend it "back" before the point at which the singularity formed,
in which case the integral will include the mass which formed or fell
into the singularity along the way.

Yes, I am pretty sure that for the Schwarzschild black-hole (including
its process of formation) one can find such a space-like surface. This
extension is only "back in time" in the sense that our usual
time-coordinate places the collapsing "before" the present day.
General relativity has no respect for our sense that the event horizon
formed "long ago". As far as GTR is concerned, a space-like surface is
a space-like surface.

So in some sense we are still not really "after" the formation of the
event horizon: the mass which fell in, though beyond rescue, is not
yet wholly a thing of the past! This is almost by definition of "black
hole"; future-directed world-lines from the event horizon only go into
the hole, not out to us. This may also serve to relieve those who
worry about the source of the gravitational field (let along
"gravitons") which remain "after" the mass is "gone".

                       "...who needs the goddamn body? It's a pain in
                       the ass. If only I could get my mind clear again,
Keith Ramsay           I'd be able to get down and do some work."
ramsay@unixg.ubc.ca         _The_Thread_, Phillip Davis