Re: More on the controversy about the Schwarzschild radius and black holes.

On May 5, 5:49 am, stevendaryl3...@xxxxxxxxx (Daryl McCullough) wrote:
LEJ Brouwer says...

What the mathematics has shown is that the interior solution does not
have spherically symmetry about a point, and it is quite unclear as to
what physical matter configuration could give rise to the properties
described by the interior solution - it is certainly not due to a
point mass. Could you explain what matter configuration the interior
solution describes?

This has been discussed by Steven Carlip. Here's an intuitive answer:
Imagine sitting on Earth, and surrounded by a spherical shell of
distant stars. Now, imagine attaching powerful rockets to each star
in the shell so that at the same moment (according to the coordinate
system in which the Earth is at rest), the stars are all sent hurtling
towards the earth. The rockets are timed so that all the stars will
arrive simultaneously.

Now, what you can do is add up the masses of all the stars and compute
the Schwarzschild radius for that mass. At some point, the stars will
pass that radius. After that, you and the Earth and all those stars
are inside a black hole.

There still isn't any point mass anywhere except in the *future*.
The predictions of General Relativity are that all the mass inside
the spherical shell will eventually be concentrated into a single
point. That's the singularity, and it exists in the *future*, not
in any particular spatial location.

From the point of view of someone outside the spherical shell of
stars, the geometry will approach that of the Schwarzschild exterior
solution.

Apologies for the long delay in response. I thank you for bringing
this scenario to my attention, as it raises some interesting questions
which I think could help to allay my earlier misgivings on the matter.

Imagine that I am initially sitting at the origin (i.e. the centre of
spherical symmetry) when the stars start accelerating radially towards
me. At the time that the stars converge at their Schwarzschild radius
to form the event horizon, then according to the form of the interior
solution, the stars don't appear to cross the event horizon and pile
in towards me because the interior of the event horizon in which I am
sitting has a spatial spherical symmetry which is not reflected in the
form of the interior Schwarzschild solution. Rather, the solution in
which I am sitting, which is interior to the event horizon, is just a
standard Schwarzschild exterior solution since it is a spherically
symmetric vacuum solution with insufficient mass to form a black hole.
However, this interior solution also has a boundary at the event
horizon. I am not sure what the boundary conditions are, though
logically it seems that nothing can enter and nothing can leave. I say
this, because once the stars 'cross' the event horizon, they do not
enter into my little sphere of tranquility, but rather head off
towards some future singularity in some region described by the
Schwarzschild interior, which happens to be at a specific point in
(star proper) time due to the artificial way in which the scenario has
been constructed wherein all stars reach/form the event horizon
simultaneously.

There therefore appear to be three separate regions in the scenario
above:

(i) The exterior of the horizon, which is just an exterior
Schwarzschild solution.
(ii) The spherical region in which I sit, which also has a
Schwarzschild exterior-like metric, but with a spherical bounding
surface which shields me from infalling stars, and also prevents
anything from leaving.
(iii) The interior Schwarzschild solution, complete with temporal
singularity, into which the stars disappear upon crossing the horizon.

This is a scenario which I am happy with, though I am guessing that
this is not the picture you are trying to describe.

In a more realistic collapse scenario where we have a continuous
spherically symmetric distribution of radially infalling matter, this
is what I think would happen:

As the matter collapses inwards, there will eventually be a high
enough mass density to form an event horizon. Any matter external to
the horizon will subsequently be 'diverted' towards the singularity
lying in its future worldline, but this interior region is not the
same as the region which contains the matter which was responsible for
the formation of the horizon. Rather, the matter which was interior to
the event horizon has a spherically symmetric distribution, and
becomes isolated from the exterior, and also from the Schwarzschild
interior containing the temporal singularity. Matter can neither enter
nor escape from this region, so it has unusual boundary conditions,
but it remains otherwise well-behaved and is without any kind of
singularity.

So this thought experiment leads me to conclude that the initial
spherically symmetric distribution undergoes a kind of phase
transition when the event horizon forms to become three separate
regions, each with spherical symmetry, and one of which contains a
future singularity (which is neither pointlike, nor a mass).

I therefore disagree with your claim that all the mass inside the
spherical shell will eventually be concentrated into a single point.
As I mentioned before, and you confirmed, the singularity is not a
'point'. The singularity exists in the (relative) *future*, not at any
particular spatial location or particular time. The mass inside the
spherical shell does not eventually become concentrated into a single
point - rather it lives to evolve happily in a little spherically
symmetric world of its own. The singularity lies in an independent,
and as I have stated before, a rather hard to physically interpret,
region and upon which all matter crossing from the exterior converges
upon.

Here is an (admittedly rather poor attempt at) representation of this
scenario:

C
x
^
|
|
A ----o<---- B

The radially infalling matter is in region B. The trapped region which
arises once the horizon (indicated by 'o') has formed is labelled A,
which contains the matter responsible for creating the horizon in the
first place. The matter crossing the horizon from B falls towards the
singularity at 'x' which lies in region C, corresponding to the
Schwarzschild interior solution. The singularity lies at a fixed time
relative to the time at which an infalling particle crosses the
horizon at 'o'. The region A becomes physically isolated from regions
B and C once the horizon has formed. As I have said before, my guess
is that at the singularity, particles are reflected back in space and
time towards the event horizon and back into the region B, so x acts
like a spacetime reflecting mirror. At a microscopic level, the
interior region A corresponds to an elementary particle, and it
appears to an external observer (i.e. in region B) that particle-
antiparticle annihilations are taking place at the event horizon,
though physically what is happening is that a single particle is being
reflected off of the singularity at 'x' which is acting as a spacetime
mirror.

Let me know if you agree with the above picture. I am not a betting
man, but I will bet that you do not. :)

- Sabbir.

.

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