Re: cosmology and quantum gravity

[David Bowman <David_Bowman@GEORGETOWNCOLLEGE.EDU>]

I'm not sure it would be wise for me to butt into this dispute between
Joel and Jack, but I'm not very wise.

Regarding where Joel wrote:
> ...
> However, I must reiterate, there are no physical effects that blow up at the
> event horizon of a Schwarzchild black hole, at least in the usual physics
> sense of the words "blow up", which I interpret as loose language for
> diverging to infinity.  That does not mean interesting things don't happen
> at the event horizon, as several well known interesting things do occur at
> the event horizon.

I think I would have to side with Jack on this one.  Although it is
certainly true that all the components of the various curvature tensors
remain well behaved at the event horizon this does not necessarily mean
that there are no gravitational effects that "blow up" there, only that
those such effects that happen to transcend all coordinate systems remain
well-behaved.  There are some ideosyncratic gravitational effects that
*do* blow up at the event horizon in some coordinate systems.

For simplicity of illustration let's consider a nice spherical non-
rotating Schwarzschild black hole.  There is an important class of
coordinate systems for this system that have the property that the metric
is *static* in the region external to the event horizon.  These
coordinate systems are most useful in describing the properties of the
hole as observed by external observers that remain at rest relative to
the hole, because for such observers their spatial location remains fixed
independent of time all along their world lines.  For such observers
there *are* gravitational effects that *do* 'blow up' at the event
horizon.  Because these observers remain at rest in frames with a static
metric these observers are *not* freely falling and their world lines are
not geodesics.  These observers are station keeping w.r.t. the hole and
they do so via an external nongravitational thrust or force that opposes
the gravitational force that they feel that would otherwise tend to pull
them into the hole had their rocket motor of jet pack thruster been
turned off.  The amount of such station-keeping thrust or supporting
force needed by these observers to keep from falling into the hole
depends on their distance from the event horizon, and it represents the
'weight' of the observers at their own particular location in the gravity
field of the hole.  For any such stationary observer of any small finite
mass this 'weight'/thrust force tends to *diverge*, i.e. 'blow up' as the
observer is gradually lowered into position ever closer to the event
horizon.  Any stationary observer would have an *infinite* weight at the
event horizon, and would weigh *very much* just a little above the
horizon.  The event horizon in this case is also a stationary limit
surface in that it would require an infinite amount of station-keeping/
supporting force to keep from falling through the horizon when trying to
hover *at* the horizon.  I *would* call this a gravitational effect that
'blows up' at the horizon.

There are other such effects that 'blow up' there as well.  For instance,
the gravitational red shift for light signals from from a stationary
object just outside the horizon as observed by another stationary
observer farther out from from the hole tend to diverge to infinity as
the inner observer approaches the horizon (very gingerly).  Also, the
gravitational blue shift in the light from the outer observer as observed
by the inner observer *also* diverges as the inner observer gingerly
sneaks up (ever so slowly) on the horizon from above. Also the time
experienced by the inner fixed observer becomes infinitely dilated
relative to another fixed outer observer as the inner observer sneaks up
on the horizon infinitely slowly.  I would call these effects as
'blowing up' as well.

In the region inside the black hole there are *no* stationary metrics and
any observer in there *must* move w.r.t the singularity which *is* in the
*future* of all such observers.  The singularity is *not* a point in
space, but it is a point, i.e. the endpoint, in *time* for all internal
observers (whose world lines end at the singularity).

It certainly is possible to write the metric for the black hole in such a
way that the coordinate singularity at the event horizon vanishs.  But
all such coordinte systems are not stationary w.r.t. the hole in the
external world.  Any observer with fixed spatial coordintes in such a
coordinate system is an observer that is falling through the hole into
the interior region.  Since freely falling observers are weightless such
observers do not have the 'weight problem' that the stationary ones do
*even* as the world lines of the falling observers cross the horizon.
For such falling observers there are no 'blowing up' gravitational
effects at the horizon.  It's just that their experience of the hole is
most benign seeing that the gravitational field for a freely falling
observer vanishs *even* at the horizon.  It's just that such observers
falling are not the only ones of interest.  Sometimes the experiences of
stationary observers are of interest.  In fact, even on Earth we tend to
weigh objects that are at rest relative to the Earth's surface when we
wish to find out how strong gravitational effects are here.  We don't
usually weigh objects in freely falling elevators when we want to find
out how strong the Earth's gravity is.

> ...
> The Ricci tensor, a measure of gravitational effects in a local region of
> space, does not diverge at the event horizon of a Schwarzchild black hole.

Actually, for a black hole the Ricci tensor doesn't measure *anything*
because a black hole is a vacuum solution of Einstein's equations and the
Ricci tensor for a vacuum vanishes identically. In a region that is not a
vacuum the Ricci tensor measures a local encoding of the stress-energy of
the matter and radiation present into the curvature of spacetime.  I
suspect that what Joel means is that the *Weyl* tensor for the black hole
measures gravitational effects since even in a vacuum the Weyl tensor
measures the tidal distortions of spacetime (which certainly *are*
present around the hole).

David Bowman
David_Bowman@georgetowncollege.edu