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

LEJ Brouwer <intuitionist1@xxxxxxxxx> wrote:

[...]
We would like to find the metric outside of a static point particle.
Clearly, before we even start, we know that the solution metric, like
the problem itself, must be time-independent. Anyway, we start by
writing the general form for a spherically symmetric metric WITHOUT
imposing the requirement that the solution be static. Despite this, we
are still able to make a change of coordinates in which the solution
metric takes a static form.

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.

[...]

Anyway, we notice that an infalling particle appears to take an
infinite amount of time to reach the 'event horizon' at r = 2m, where
some of the metric components either vanishing or become divergent.
Unfortunately the infalling particle reaches the horizon in finite
proper time, so that it is not clear where it goes after this - so our
solution must be incomplete.

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.

[...]
We take a more rational approach and note that our initial formulation
of the problem was not sufficiently general to take into account all
possible solutions, which could be multivalued in r. Using the method
of Synge, we are able to derive the complete solution, containing both
exterior patches of the 'maximal' (sic) Kruskal extension.

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.

Mr Fuckwit
again jumps up with excitement, and proclaims vociferously that the
entire plane, including both the black and white hole interior
solutions must be included. We tell him once again to calm down, and
that the interior solutions are still non-static and are still not
valid solutions to our original problem.

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.

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)

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.

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

We quickly realize, however, that this observation depended on our choice
of coordinates. Having already realized that the original coordinate
system broke down, we change to any one of an enormous number of known
coordinate systems that are well-behaved at the horizon, and find that
there is nothing at all strange happening to the acceleration of a freely
falling particle. We *do*, on the other hand, find that it would take
infinite acceleration for a particle to remain at rest at the horizon.
We feel gratified that our original reaction that "something unusual must
be happening" was correct, with the added benefit that we now understand
what the unusual something is.

We also perform some calculations showing that the area of the horizon
is 16m^2,

Yes

but that it is at distance zero from the central mass,

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

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. 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!).

Noting that there is no curvature singularity at the horizon, and that
the infalling particle must go somewhere, we realise that the only
physically consistent scenario is that a particle beginning in region
I, which has light cones pointing upwards, must cross to the other side
of the 'wormhole' on reaching the horizon at which point it enters into
region II (which is spatially superimposed upon region I), but now
travelling backwards in time relative to region I, so that forward
light cones in region II point downwards.

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. 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.

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

Steve Carlip
.