Article 17242 of sci.physics:
From: red...@convex.csd.uwm.edu (ian redmount)
Newsgroups: sci.physics
Subject: Re: A question about Black Holes
Summary: Mary, Mary, quite contrary, how does your Black Hole grow?
Keywords: Black Holes
Date: 22 Oct 91 00:24:42 GMT
Organization: University of Wisconsin - Milwaukee
Lines: 60

In article <1991Oct21.210018.18947@galois.mit.edu>
        jb...@nevanlinna.mit.edu (John C. Baez) writes:
> [discussion of previous post, concerning FAQ about matter falling into
> black holes, omitted for brevity.]
>I think the poster got *that* point; as far as I could tell, he's asking,
>"If it takes forever for us to *see* the fellow fall past the event horizon,
>and only after he's in does the black hole appear to grow (i.e., the event
>horizon grows in radius), then why doesn't it take forever for the black hole
>to grow." 
>
>That is an interesting question, anyway: drop a relatively puny mass into a
>black hole.  As a function of time (for us observers far away), describe the
>black hole's event horizon.  Presumably as t -> infinity it approaches
>a sphere of slightly larger radius.   Anyone know the manner in which it gets
>there?

Yes, someone knows.  The actual manner of the horizon's growth is
rather intriguing, and illuminating.  Let's start with a simple example:
Consider a Schwarzschild black hole of mass M, into which falls a thin
spherical shell of dust (concentric with the hole) of mass dM.  The spacetime
of the region between the hole and the infalling shell is the Schwarzschild
geometry with mass M; that outside the shell is Schwarzschild spacetime
with mass M+dM.  When the shell has fallen to a radius of 2G(M+dM)/c^2,
which incidentally takes a finite amount of time as measured by observers
stationary near the hole, it is engulfed by the horizon of the hole
WHICH HAS GROWN OUT TO MEET IT!  Thereafter the horizon remains at its
new radius.

That is, the hole's event horizon grows before and up to the infall of the
shell.   Counter-intuitive though it seems, the horizon evolves
``telelogically,'' in anticipation of future events rather that in
response to past ones.  This is because the event horizon is a global,
not a local, geometric feature.  It is defined in terms of the behavior
of signals in the asymptotic future:  It is the boundary between the regions
from which signals can and cannot escape to infinity.  But escape depends
on events to the future of a signal's emission, not its past, hence the
``future-anticipating'' behavior of the horizon. 

The evolution thus described is in terms of a time coordinate which is
regular on the horizon.  A description in terms of distant-observer time
is hampered by the same t->infinity coordinate singularity which pops up
in every discussion of infall into black holes, which prompted the original
posting in this thread, and which has been discussed here on numerous prior
occasions.  Basically, distant observers outside the shell, in Schwarzschild
spacetime of mass M+dM, see the shell contract and fade away (fast) as it
is engulfed by the horizon.

If we just drop a small mass into a hole, instead of a spherical shell,
things get even more interesting.  As the mass approaches (again, working
in a sensible, i.e, regular time coordinate) the horizon grows out to
meet it, developing a tidal bulge in the process.  Once the mass is
engulfed, the bulge falls back, the horizon oscillates or ``ripples,''
and gravitational radiation is emitted as the hole relaxes to a spherical
shape again.  Part of the mass-energy of the dropped body goes into making
the hole grow, part into radiation, and part into center-of-mass motion.

For further detail see, e.g., ``Black Holes:  The Membrane Paradigm,''
Thorne, Price, and Macdonald, eds. (Yale, 1986), and references therein.

Ian H. Redmount