Re: Misinterpretation of the radial parameter in the Schwarzschild solution - a response from Stephen Crothers.
- From: carlip-nospam@xxxxxxxxxxxxxxxxxxx
- Date: Tue, 17 Oct 2006 21:36:49 +0000 (UTC)
LEJ Brouwer <intuitionist1@xxxxxxxxx> wrote:
[...]
In this regard I am forwarding, with the author's permission, the
following message which I received from Stephen Crothers in response to
a number of critical remarks made against him and his work by S. Carlip
and T. Essel on s.p.research, to which he had been denied the right to
respond.
[...]
Carlip says, including Abrams and myself,
"These papers are complete nonsense. In particular, the authors seem
not to understand the basic fact that physics does not depend on what
coordinates one chooses to use."
[...]
These remarks are bereft of scientific merit and indeed of sanity.
First, he assumes, a priori, that the erroneously named Schwarzschild
solution due to Hilbert is a solution,
This is not hard -- take the metric and compute the Ricci tensor,
and see that it vanishes.
and transforms that line element
with its invalid range on the parameter r by his proposal to r-2m,
making his nonsensical horizon at r = 0.
As I said, it helps to understand that physics does not depend on what
coordinates one chooses. If r is a good coordinate, with some range,
then r'=r-2m is also a good coordinate, with a shifted range. This is
elementary differential geometry.
He has however incorrectly assumed:
That there is an "interior" region on Hilbert's metric;
As I said, it helps to understand that physics does not depend on what
coordinates one chooses. If one transforms to coordinates that are
well-behaved at r=2m -- for example, Kruskal-Szekeres coordinates, or
Painleve-Gullstrand coordinates -- it is trivial to see that there is
an interior region corresponding to r<2m. It is also a completely
straightforward computation to see that a freely falling observer who
starts at r>2m will cross to this interior, with nothing strange
happening, in a finite proper time.
That r=2m is a 2-sphere on the Hilbert metric;
Some elementary differential geometry: to find the area of a surface,
compute the induced metric on that surface, and integrate the square
root of its determinant over the surface.
That General Relativity requires necessarily that a singularity must
only occur where the Riemann tensor scalar curvature invariant (the
Kretschmann scalar) is unbounded;
General relativity requires no such thing. There are all sorts of
different kinds of singularities. Taub-NUT space has a singularity
in which the topology becomes non-Hausdorff. There are gravitational
plane wave spacetimes in which all scalars constructed from the
curvature are finite (in fact, zero), but parallel propagated
curvature components in an orthonormal basis blow up; there are
manifolds with conical singularities, in which invariants remain
finite everywhere away from the singularity. There are also manifolds
that are not singular, but that look singular if we make a poor choice
of coordinates.
I have already made the following recommendation to Crothers, though
he as apparently not taken me up on it: if he wants to understand
singularities in general relativity, he should start with Geroch and
Horowitz, "Global structure of spacetimes," chapter 5 of _General
Relativity: An Einstein Centenary Survey_ (edited by Hawking and Israel).
Then read chapter 8 of Hawking and Ellis, _The Large Scale Structure of
Space-time_. After that, try Clarke's book, _The Analysis of Space-Time
Singularities_. The look at Scott and Szekeres on the "abstract boundary":
J.Geom.Phys. 13 (1994) 223, gr-qc/9405063.
That "points" as they relate to "point masses" have the same
properties in Einstein's gravitational field as they do in Minkowski
space, notwithstanding that the latter is pseudo-Euclidean and the
former pseudo-Riemannian. (There is in fact no a priori reason why this
must be so.)
Well, I do assume that a point is a point, and not a sphere with a
surface of finite area. This is not a matter of physics or math, but
of English -- we call a sphere of finite area a "sphere."
That there is one radius in Einstein's geometry.
I don't know what this means. In a spherically symmetric spacetime,
there are an infinite number of coordinates one might call "r", which
all agree asymptotically. For example, the "radius" in isotropic
coordinates differs from the "radius" in Schwarzschild coordinates,
and both differ from the "radius" in harmonic coordinates. If, on
the other hand, "radius" means "physically measured radius on a given
slice of constant time and in a given angular direction," then it is
unique, and has to be calculated. Again, elementary differential
geometry: to find the proper length of a curve, compute the induced
(one-dimensional) metric on the curve and integrate its square root.
I'm afraid this reply has confirmed my original impression.
Steve Carlip
.
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