Re: More on the controversy about the Schwarzschild radius and black holes.
- From: carlip-nospam@xxxxxxxxxxxxxxxxxxx
- Date: Thu, 10 May 2007 23:45:43 +0000 (UTC)
LEJ Brouwer <intuitionist1@xxxxxxxxx> wrote:
On Apr 24, 11:41 pm, carlip-nos...@xxxxxxxxxxxxxxxxxxx wrote:
LEJ Brouwer <intuitioni...@xxxxxxxxx> wrote:
[...]
In Crother et al's view,
coordinates can be given a physical significance a priori - e.g. if
the direction of t is defined to be timelike, and the direction of r
is defined to be spacelike when a problem is stated, then it must
retain these properties in the solution.
How do you "define [a coordinate] to be timelike"? What does that mean?
You can *say* "t is timelike," but why should that affect anything?
Hopefully I have clarified what I mean in my response to Jan.
Sorry, I still don't understand.
[...]
The point to note at this point is that r and t are _not_ merely
labels - they have a definite physical attribute determined by their
respective spacelike and timelike natures.
What physical attributes are those? How do you determine them? Or is
this more magical thinking: you chant, "t shall be time" and that makes
it time?
If we choose spacetime index (+,---), then if in a given coordinate
system the line element looks like this:
ds^2 = A dt^2 - r^2 dOmega^2 - B dr^2
where A and B are positive, then t parametrises time and r
parametrises radius.
But that's not a statement about the coordinates, it's a statement about
the coordinates *and the metric.* Your argument seems to be
(1) t parametrizes time when A is positive;
(2) therefore t has a "timelike nature";
(3) therefore A can't be negative, because that would violate (2).
Is it not clear that this is circular?
[...]
The conclusion to be reached then is that analytic continuation of a
solution is only valid if it respects the physical constraints of the
original problem. If the original constraints are ignored, well, a
different problem is being solved, with the possibly that the
resulting solutions are unphysical. [If Carlip et al disagree with
this conclusion, then I would very much like to see a proof that the
interior solution does indeed solve the original problem and
corresponds to a physically realisable vacuum].
I've answered this in the past. If you want to ask whether the solution
is physically reasonable, set up physically reasonable initial data and
see how they evolve. In this case, for example, you can write down an
exact solution for a collapsing spherically symmetric shell or sphere of
matter, with a time coordinate that has the direct physical meaning as
the proper time as measured by an observer standing on the surface of the
collapsing matter. In the resulting exact solution the matter passes
through the horizon with nothing strange happening, and the solution
evolves into an ordinary Schwarzschild black hole -- both exterior and
interior -- typically described in Painleve-Gullstrand coordinates. You
can find two simple examples in Adler, Bjorken, Chen, and Liu, "Simple
analytical models of gravitational collapse," American Journal of Physics
73 (December 2005) 1148-1159.
Yes, it is clear that such coordinates give a nice smooth evolution
from exterior to interior.
Specifically, they give nice smooth evolution of a real physical system.
Furthermore, the Painleve-Gullstrand version is described in coordinates
that have clear physical meanings in terms of measurements by observers
in the system.
I just have doubts that that evolution
corresponds to physical reality - and partly because of the
discontinuity in the transformation to Schwarzschild (or similar)
coordinates at the event horizon.
I understand that you like Schwarzschild coordinates. But I don't
understand how 'I can't describe this in my favorite coordinates'
translates to 'this doesn't correspond to physical reality.'
Why are there no coordinate choices
in which the metric remains diagonal at every point on the path of a
radially infalling particle and which does not suffer from some kind
of discontinuity at the horizon?
There are. The Kruskal form of the metric is usually usually written
in terms of two null coordinates U and V, but you can define T=U+V
and R=V-U. Then the Kruskal form has "a metric [that] remains diagonal
at every point on the path of a radially infalling particle and which
does not suffer from some kind of discontinuity at the horizon."
On the other hand, I also don't understand your aversion to off-diagonal
terms in the metric. If you want to describe a rotating system in
ordinary, flat spacetime, you will find that corotating coordinates
have an off-diagonal term that goes as d\phi dt. So why should it be
such a surprise that comoving coordinates for radially collapsing matter
have a dr dt term in the metric?
Steve Carlip
.
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