sci.physics.research

Oh No wrote:
> I have given a conceptual and mathematical account of general relativity
> on my website. This is an introductory level course, from first
> principles,
[[...]]
> Certainly it is a lot shorter, and I hope more
> approachable than any textbook that I know of, as I have found a number
> of treatments which simplify, and I hope clarify, those from the books
> (though I don't treat every topic one might find in a textbook).

You might find it instructive to compare your treatment with those of
@article{
author = "Richard H. Price",
title = "General Relativity Primer",
journal = "The American Journal of Physics",
volume = 50, number = 4,
year = 1982, month = "April",
pages = "300--329",
note = "errata in 52(4), April 1984, p. 366--367",
snote = "++good introduction to general relativity, assuming only
that the reader has ``a familiarity ... with partial
differential equations and their application in physics,
as would certainly result from, say, a junior- or
senior-level course in electrodynamics''"
}
or James B. Hartle's book "Gravity: an Introduction to Einstein's General
Relativity" (Addison-Wesley, New York, 2002, ISBN-10: 0-8053-8662-9,
ISBN-13: 9780805386622).


> Any comments on anything inadequate or not clear will be gratefully
> received.

I took a quick look at part of your treatment of black holes. I am
sorry to say that it is likely to leave the unwary with significant
conceptual misunderstandings:

You write the Schwarzschild metric in Schwarzschild coordinates,
and say "The Schwarzshild metric becomes singular at the Schwarzschild
radius." But you don't say that this is only a coordinate singularity,
nor do you make it clear that the metric (as an abstract geometric
object independent of any particular coordinate system) is non-singular
there. Indeed, you don't distinguish in your wording between the
metric (an abstract geometric object independent of any particular
coordinate system) and the coordinate components of the metric.
For an "introductory level course" you might reasonable omit writing
out the metric in non-singular-at-the-horizon coordinates (eg
Kruskal-Szekeres, Eddington-Finkelstein, Painleve-Gulstrand, ...),
but I think it's very important to point out that non-singular
coordinate systems do exist.

You then write
"For a normal star or planet, this [[the coordinate singularity
of the Schwarzschild metric at r=2GM]] is not important because
most of the mass is outside this value of r,"
No, that's not the reason. The Schwarzschild metric is only a
solution of the Einstein equations for a *vacuum* region of spacetime,
so it's not valid (., it's not a solution of the Einstein equations)
inside a star or planet (or anywhere else where there's a non-vanishing
stress-energy tensor). The point is not that the r=2GM coordinate
singularity of the Schwarzschil metric is _unimportant_ there, it's
that the actual spacetime metric there is not the Schwarzschild metric!

Your statement "there is a theoretical possibility that a body
could exist which is so dense that its mass is contained within
its Schwarzschild radius. Such a body is called a black hole."
will likely lead the unwary to think that a body must be very
dense in order to be a black hole. This is of course false --
a black hole can have arbitrarily low density (if it's big enough).
(For example, a black hole of mass 10^{12} solar masses (about
that of our galaxy) has a mean density less than that of
Earth-sea-level air.)

You write
"From the pespective of an external observer, the redshift factor
k = (g00)^{1/2} becomes zero at the Schwarzschild radius, showing
that time, and all physical processes, slow down as matter approaches
the black hole. The external observer would calculate that matter
does not actually fall through the Schwarzschild radius, but stops
at it."
Again, this will mislead the unwary. k --> 0 (and the other properties
of the Schwarzschild metric) doesn't mean that "time slows down as
matter approaches the black hole", it means that if an observer falling
into the black hole sends regular signals out to an observer far from
the black hole, they will arrive slower and slower, and eventually
stop arriving just as the falling-in observer crosses the event
horizon.

You go on to say
"This raised an issue as to whether a singularity could
actually form according to the equations of relativity. This was
resolved by Roger Penrose who showed that one can."
Eeek -- that's not what Penrose showed! It was known long before
Penrose that a singularity *could* form (Oppenheimer & Snyder
published the first calculation of gravitational collapse to a
singularity in 1939!). The question was whether this result was
generic, . whether a singularity *must* form in *any* gravitational
collapse, . in *any* spacetime containing a horizon. Penrose &
Hawking showed that the answer to this latter question is "yes".

Your next paragraph attempts to resolve some of these issues, but
is again likely to mislead the unwary. You write
"If instead of using coordinates defined by an external observer,
stationary with respect to the hole, we use coordinates determined
by an observer falling into it, it can be shown that no singularity
arises in coordinates at the Schwarzschild radius."
This is ok so far. But then you write
"Apparently the observer simply falls through empty space at the
Schwarzschild radius, into a region from which he can no longer
communicate with the external observer, ..."
The word "Apparently" is simply wrong. The observer *does* fall
through the Schwarzschild radius into a region from which she can
no longer communicate with the external observer. There's no
"apparently" about it! Note that the r=0 singularity is not involved
here, and (for a sufficiently massive black hole) all happens in
*weak* gravitational fields.

You then proceed to write
[[about the observer who has just fallen in through the
Schwarzschild radius]]
"... eventually meeting a singularity at r = 0. That is the
solution according to Einstein's field equation, but the
question arises as to whether it is a real physical solution.
The meaning of a singularity is that known laws of physics
break down. We cannot say, from classical generality,
precisely at what point the laws of physics break down in
the vicinity of a singularity. Relational quantum gravity
will reexamine this issue in the light of a unification
with quantum theory."
This part looks ok... except that your wording is likely to leave
some readers thinking that this also applies to (the coordinate
singularity at) the Schwarzschild radius.