Re: On spherical gravitational collapse and the stability of classical black holes.

LEJ Brouwer says...

Now, if the shell starts collapsing inward, as long as the
collapse is spherically symmetrical, the exterior metric
doesn't change. The parameter M counts *all* the matter that
has fallen into the black hole.

I agree that the above metric remains unchanged outside of the
horizon, and that the mass is M, but I disagree that matter
subsequently crossing the horizon increases M

It certainly does. Instead of a single shell of matter, consider
two shells of matter, one of mass M and another of mass m. The
radius of the second shell is larger than the radius of the
first shell, and represents matter falling into the black
hole created by the first shell.

For someone outside *both* shells, there is no observable
difference between having two shells of mass M and m and
having one shell of mass m+M, which is the same as having
a single point-mass of mass m+M. So the mass parameter in
the Schwarzschild metric includes both shells even *before*
the second shell falls into the black hole. Even before the
*first* shell falls into the black hole. The subsequent
history of the two shells is irrelevant as far as external
observers are concerned.

The mass parameter in the exterior Schwarzschild metric
counts *all* the mass.

- it does fall into the
black hole and hit the singularity, but it does NOT enter the region
which contains the mass M which was responsible for the formation of
the horizon in the first place.

I don't see how that makes any sense. If you have two shells
that are both collapsing inward under gravity, then someone
with a rocket can easily jet back and forth between the two
shells. If they went different places, then there would be
some moment where jetting back and forth would be impossible.
Which moment is that, and what makes it impossible? The only
moment that is special, from the point of view of the infalling
observers, is the moment at which infalling matter hits the
central singularity.

Try to describe what you think is "magical" in observational
terms, *NOT* in terms of coordinates. What weird thing do you
think happens as the shell crosses the event horizon? If someone
is standing on the spherical shell as it collapses, assuming
that the tidal forces are not too great (that is, assuming that
M is large enough) he won't notice anything special as the
shell passes the event horizon. *Nothing* magical happens,
as far as he can see by looking around him or by performing
local measurements. So there is nothing magical to explain.

Yes, it is true, for that particular observer that nothing special
appears to happen as he crosses the event horizon. However, an
indication that something magical _has_ happened is that after a
specific amount of time, depending upon the mass M of the black hole
into which he has fallen, he will be crushed by a singularity (or fall
off the universe, or bounce off the singularity, or maybe something
else), and there is nothing in his power that he can do to avoid or
even delay this inevitability.

But that doesn't seem very weird to me. To escape requires
rockets that are capable of providing enough force to overcome
gravity. If the gravity is too strong, then no rocket is
powerful enough. Why is that strange?

Yes, I suppose I do think the timelike<->spacelike transformation is
magical.

Yes, but it isn't *physical*; it's not something observable.
The Schwarzschild coordinates t and r are not observable.
As I have pointed out before, a similar timelike <-> spacelike
transformation happens in noninertial coordinates in ordinary
Minkowsky space. If instead of inertial coordinates x and t,
you use coordinates X and T defined by

X = (x^2 - t^2)/8k
T = 4k arctanh(t/x)

then the Minkowsky metric looks like this:

ds^2 = X/2m dT^2 - 2m/X dX^2

Crossing from X > 0 to X < 0 causes T to switch from timelike
to spacelike, and causes X to switch from spacelike to timelike.
But that's only an artifact of the accelerated coordinate system.
In Minkowsky coordinates, this transition corresponds to the
transition from

x > ct
to
x < ct

Nothing physically meaningful is going on at the boundary.

I recall you mentioned that Steve Carlip introduced the example we are
currently discussing of the shell of matter/stars being boosted
inwards in a spherically symmetric matter so that they all reach their
Schwarzschild radius at the same time, as an illustration of spherical
collapse to form a black hole.

Let me adapt this scenario slightly to make it clearer why I think
something unusual is happening.

Let us agree that there are indeed boosters attached to the
constitutents of the spherical shell of matter which keep them at a
constant radius from the centre. And let us assume that these boosters
are oriented facing outwards so that they can control the speed at
which the shell falls inwards, and in particular so that the exhaust
emitted from the boosters is ejected towards the spatial origin.

Let us also add a number of (negligible mass) observers inside the
infalling spherical shell (where spacetime we agree is flat), which
remain relatively stationary. We are particularly interested in the
observations of the observers lying within the Schwarzschild radius of
the infalling matter.

Now, let us assume that the power of the boosters can be adjusted so
that infalling shell of matter falls in very slowly. Obviously the
power will need to be increased as the Schwarzschild radius is
approached. The observers within the Schwarzschild radius will be able
to observe the exhaust from the boosters being ejected towards the
origin, and will no doubt be anxious about their fate as they watch
the shell of matter slowly converge radially upon them.

Once the shell of matter reaches its Schwarzschild radius (very slowly
remember), the boosters would need to provide an infinite boost to
prevent them falling through the event horizon (this is one of the
indications that the horizon is a strange place - objects held
stationary there appear to have infinite weight). Clearly they cannot,
so the shell of matter will proceed to cross the horizon despite the
boosters' efforts to prevent this from happening.

Now, the big question I put to you is this:- what exactly happens
next?

According to your account, locally, the infalling matter will not
notice anything special, and will continue to be boosted by the
boosters - but in what direction are they now being boosted?

Someone standing on the surface of the infalling shell will
use a local coordinate system with coordinates R and T that
will mix the Schwarzschild coordinates r and t in such a way
that R is spacelike and T is timelike. When the shell is
far outside its Schwarzschild radius, R will be approximately
r and T will be approximately t, but as he falls, the relationship
between (R,T) and (r,t) will change. The boosters will be pointing
in the direction of +R, whatever that is in terms of r and t.

They are not being boosted 'radially outwards' any more, as there
is no such concept in the Schwarzschild interior, and it certainly
doesn't make sense for them to be boosted backwards in time.
Moreover, the boosters can do nothing to prevent the (now temporally)
infalling matter from hitting the singularity which lies at a
fixed _time_ in the future.
(Note that the singularity corresponds to a _time_, not a position).

More importantly, what do the observers who were waiting patiently
inside the Schwarzschild radius see happen? Surely they will continue
to see the boosters firing away (presumably now unable to prevent the
matter shell from falling inwards), with the exhaust still being
ejected towards the origin? But how can this be consistent with the
shell of matter being in the Schwarzschild interior, which has no
spatial origin and which therefore renders meaningless/impossible that
the boosters can have a radial orientation?

The region *inside* the shell has a meaningful notion of radius.
They don't see anything special happen as the shell crosses the
event horizon. The shell just gets closer and closer.

Do you claim that the interior observers (i.e. those which are at
radius less than the Schwarzschild radius) will see the infalling
shell continue to converge upon them and then 'become' a singularity
at the spatial origin?

Yes.

But how is this compatible with the fact that
there is no concept of a spatial centre of symmetry in the
Schwarzschild interior and that the singularity is a moment in time
(relative to crossing the event horizon), and not a point in space?

Inside the shell, there continues to be a meaningful
notion of radius. The region for which there is *not*
a meaningful (global) notion of radius is the region
between the shell and the event horizon.

Can the effect of the boosters delay the inevitable for the observers
inside the shell of matter, while being powerless to delay the
inevitable for the shell of matter itself?

I'm not sure whether the boosters can change the time till
the big crunch, or not. I'll have to think about it.

Presumably you will also claim that the observers who were within the
Schwarzschild radius somehow end up hitting the singularity - could
you describe what happens to an observer initially at radius r <
R_Schwarzschild - what is his motion, where is the singularity that he
falls into, and how long after the formation of the event horizon does
he fall into it?

Let's take the special case of someone who is at r=0 long before
the shell crosses its event horizon. In that case, he doesn't
"fall" anywhere, he just sees a shell of matter come crashing
down on him.

I cannot see how the metric inside the Schwarzschild radius, which was
originally flat and has a well-defined (spatial) radial direction, can
possibly be transformed into something compatible with the interior
metric of a Schwarzschild black hole which does not have a radial
direction. The symmetries of the two regions are a complete mismatch
and it is nonsensical to try to convert one into other, or identify
them in some way.

To see whether two different patches of spacetime are a "mismatch",
you have to see if it is possible to define a patch that overlaps
the two regions such that the patch simultaneously agrees with both.

The perspective of the infalling matter and the perspective of the
observers within the horizon are completely incompatible, and the
scenario you describe seems inconsistent and wrong.

It's only inconsistent if there are two different ways to compute
the same *observable* result such that you get observably different
answers. It's not inconsistent in that sense.

--
Daryl McCullough
Ithaca, NY

.

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