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

LEJ Brouwer wrote:
JanPB wrote:
No. It should read: "Despite this, we are still able to make a change
of coordinates in which the solution metric takes either of the two
forms:

ds^2 = -A^2 dt^2 + B^2 dr^2 + r^2 dOmega^2
or:
ds^2 = +A^2 dt^2 - B^2 dr^2 + r^2 dOmega^2".

For some mysterious reason (laziness?) most texts insist on the first
metric only (which is the wrong thing to do) and only later pull the
second metric like a rabbit out of hat which naturally looks very fishy
to the reader - exactly as you describe below.

It's probably more likely to do with the fact that there should not be
an explicit time-dependence in the solution of a manifestly
time-independent problem.

We solve the Einstein field equations, and discover the usual exterior
Schwarzschild solution. Now, Mr Fuckwit, our resident self-proclaimed
GR 'expert', notices that the if we let r < 2m, we get another solution
of the field equations,

Actually, if we start off correctly by considering both forms of the
metric then both ranges (r>2m and r<2m) follow from solving the
equations. The interior is not obtained by "oh, look, I can plug in
r<2m and it's still a solution" kind of thing. Some time ago the thread
you initiated made me rederive this solution with some care. I TeX'd it
and put it at http://www.mastersofcinema.org/jan/t.pdf
I hope it addresses some of these issues. I used Cartan's moving frames
as I find Christoffel symbols way too tedious.

I am sure there is nothing wrong with your derivation - the problematic
issue for me is the second of the two equations you set out to solve.

OK, you can't have both. If there is nothing wrong with my derivation
then automatically you cannot complain about the interior portion
because it's no longer pulled out of thin air. Right from the start we
know the metric can be of either of the two forms.

even though it is non-static

We've only assumed spherical symmetry (and signature 2, etc.).

Yes, but coordinate transformations can be made to bring the general
solution to a static form.

I'm losing you again - the interior is not static and it cannot be
"brought" to a static form.

[...]
In order not to upset Mr Fuckwit
too greatly, he might mention that Mr Fuckwit's metric may turn out to
be the solution to some, non-static problem, but certainly not the one
we are trying to solve. But also that Mr Fuckwit should take note that
his metric has a rather nasty singularity right in the middle of it, so
that the chances of it being the solution to any physically reasonable
problem are rather slim.

That's a problem of a sort of different scientific magnitude. It is
thought that this is a consequence of GR being a classical theory and
incorporating QM in it will probably resolve this.

I don't think QM is relevant if we are considering GR on its own.

Right.

I am
slightly confused that Steve Carlip and others worry (understandably)
about the original Schwarzschild solution having an edge, and therefore
disappearing off into nothingness, but in the same breath advocate an
extension to the original solution which contains a singularity into
which all infalling particles disappear into who-knows where. Why is
the same objection not raised about the presence of this singularity,
and why is the extension any better than the original?

Probably because the "edge" is avoidable by extending across the
horizon while the central singularity is unavoidable.

--
Jan Bielawski

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