Re: The Trouble with Physic(ist)s is that they are Not Even Wrong


carlip-nospam@xxxxxxxxxxxxxxxxxxx wrote:
in a region of spacetime; we know enough elementary differential
geometry to realize that a change of coordinates typically does not
cover the whole manifold.

This is true, but your statement on its own is a bit too vague if you
expect me to read this and then immediately come to the realisation
that the interior must also a correct solution. Clearly there are a few
steps missing in the chain of logic and it would help if you could be a
bit more explicit.

Exactly. Since we are not flat-earthers, we reject the possibility
that the infalling particle falls off the edge of spacetime. (This
would actually be worse than the usual flat earth stereotype -- there
would be no place for the particle to fall to, so it would have to
just poof out of existence.) We also note that the divergence of some
metric components is an indication that our coordinate system is not
valid at r=2m, much as the divergence of the component g^{\theta\theta}
of the metric in ordinary polar coordinates on the plane indicates
that the coordinate system breaks down at the origin. We check this
by noting that the coordinate-independent quantities that we can build
-- the curvature invariants -- are perfectly well-behaved at the
horizon.

Why is falling into a singularity of spacetime any better than falling
off the edge of spacetime?

We then remember that our coordinate system broke down at r=2m, so we
switch to a set of coordinates that is well-behaved there. We immediately
find that the solution is then the whole Kruskal extension, including
the interior regions.

Since we were (apparently) worried about the adjective "maximal," we do
a bit more research, and find that the Kruskal solution is the unique
maximal analytic extension of the solution we originally found for r>2m.

As I recall, when someone asked earlier, you were not able to give a
reference to the original proof of this statement. Are you taking this
on blind faith? I don't doubt that it's true, but it's just curious
that physicists have a habit of taking for granted statements made by
other physicists merely on the basis of their authority rather than on
the basis of a mathematical proof.

Since we are not flat-earthers, and are not prepared to accept a solution
in which test particles fall off the edge of space, we conclude that there
must have been something wrong with our original formulation -- that the
assumption of a static *point* particle must have been inconsistent.

I don't see why the incompleteness of the original solution at all
implies that the assumption of a static point particle is inconsistent.
Sure, the solution you propose is non-static, but that does not
necessarily imply that static solutions do not exist. Unless there is a
uniqueness theorem which states otherwise, like "only _maximal_
extensions" are valid. I prefer "only singularity-free solutions are
valid".

Or, if we are the original poster, we decide that:

Indeed the only two valid
solutions are the two (static) exterior solutions, labelled I and II:
_
\ / /.\
\ / / . \
I \/ II ---> I( x )II
/\ \ . /
/ \ \./
/ \ -
Two (spatially superimposed) An infinite cone with regions I and
II
quadrants with light cones ---> patched along the EH (dotted line)
with
pointing upwards in region I lightcones rotating clockwise around
and downwards in region II. the cone. (We are looking down into
the
cone here - note that regions I and
II
are still spatially superimposed)

Well, at least it's interesting and original.

We then note that the spacetime still has an "edge," and that an infalling
particle can still fall out of space and vanish. Oops! So we realize that
we must have been wrong -- we really should have included regions III and IV.

What edge? It is an infinite cone. There is a pointy bit in the middle,
but I can't say that bothers me at all.

We note that the acceleration on a particle at the event horizon
diverges, so that something unusual must be happening at the event
horizon.

[snip]

Apparently "we" have made a rather bad miscalculation here, since the
actual calculation shows no such thing.

That depends on which solution you are using for the calculation,
doesn't it?

which
must therefore be at (or, rather, just inside) the event horizon.

Having gotten this far, we realize that our original assumption of a
static point mass was, indeed, inconsistent, as we had suspected earlier,
since a point can't have a finite surface area.

Again, this appears to be a logical non-sequitur.

To investigate further,
we try replacing our point mass by a static sphere of fluid, with an
arbitrary equation of state. Sure enough, we find that if we try to
shrink the sphere to one that has a surface area of less than 16 pi m^2,
no static solution exists -- as long as we have an equation of state
for which the speed of sound is less than the speed of light, such a
sphere of fluid inevitably collapses (a rather nonstatic process!).

Unless you have very large hands, you can't physically shrink the
sphere to such a small size. The Helmholtz-Klein mechanism prevents
matter from collapsing to form a black hole:

http://arxiv.org/abs/gr-qc/0605066

Having made this guess, we go back and compute the geodesics that
describe the particle's motion, and discover that it does no such
thing.

Depends what you think the solution manifold looks like, doesn't it?

Oops! So we go back to our earlier observation that the
omission of regions III and IV left an "edge" at which particles
could vanish, and we find that the geodesics do, in fact, reach this
"edge." We thus confirm our earlier suspicion that we really needed
regions III and IV after all.

We realize that we have been rather impolite to a number of people who,
as it turns out, know quite a bit more physics and mathematics than we
do. We politely ask their pardon.

Sorry sir, we won't do it again.

[Or, if we are the original poster, we stick our fingers in our ears
and yell, "Nyah, nyah, I'm smarter than you!"]

I think you may be confusing me with Chris Hillman, John Baez or Lubos
Motl here. I am not like that AT ALL.

Steve Carlip

- Sabbir.

.

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