Hello everybody,

I have given a conceptual and mathematical account of general relativity
on my website. This is an introductory level course, from first
principles, including sections on the required mathematics for those who
either have not done it, or need revision.

Thank you. I am reading through it.

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

In

On Thu, 7 Feb 2008 12:10:20 +0000 (UTC), Oh No
NotI@charlesfrancis.wanadoo.co.uk> wrote:

Hello everybody,

I have given a conceptual and mathematical account of general relativity
on my website. This is an introductory level course, from first
principles, including sections on the required mathematics for those who
either have not done it, or need revision.

Thank you. I am reading through it.


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

In
http://www.teleconnection.info/rqg/images/equivalenceprinciple/Equivale
ncePrinciple-2.gif

the orbits of the objects in the satellite don't look quite right.

I would expect each orbit to be an ellipse.

If the orbits were mirror images of each other (so as to ensure the
same orbital period), then they would intersect at only two points
rather than at four points.

Thanks for your interest. I used ellipses with perpendicular major axes,

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

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).

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

Oh No <NotI@charlesfrancis.wanadoo.co.uk> 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,
..

In article <CKNbQuSauWqHFwWB@charlesfrancis.wanadoo.co.uk>, Charles Francis
wrote:
I have given a conceptual and mathematical account of general relativity
on my website. This is an introductory level course, from first
principles,
[[...]]

In article <62kohfF23emliU1@mid.individual.net> I commented on the
"Black Holes" section of the web site. Unfortunately, I mistakenly
deleted the last section of my comments from the posted version.
Here is the missing section:

Explaining general relativity in a way that doesn't leave significant
conceptual misunderstandings behind is *hard*, and I'm sorry to be
so negative in my comments.

It's very instructive to see how masters
of this craft have tackled it -- besides the article by Price and
the book by Hartle I mentioned above, the books

@book {
Schutz,
author = "Bernard F. Schutz",
title = "A First Course in General Relativity",
publisher = "Cambridge University Press",
address = "Cambridge (UK)",
year = "1985, 1986",
isbn = "0-521-25770-0 (hardcover), 0-521-27703-5 (paperback)",
}

@book {
Kenyon,
author = "Ian R. Kenyon",
title = "General Relativity",
publisher = "Oxford University Press",
address = "Oxford (UK)",
year = "1990, 1991",
isbn = "0-19-851995-8 (hardcover), 0-19-851996-6 (paperback)",
}

Robert Geroch
"General Relativity from A to B"
University of Chicago Press, 1978,
ISBN 0-226-28863-3 (hardcover),
-28864-1 (paperback)

are well worth studying. (The last one manages to describe many of
the ideas of general relativity without using any mathematics at all!)

ciao,

Oh No <NotI@charlesfrancis.wanadoo.co.uk> wrote:
I have given a conceptual and mathematical account of general relativity
on my website. This is an introductory level course, from first
principles,
[[...]]

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.

Hello everybody,

[snip]

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

In article <62kohfF23emliU1@mid.individual.net>,
jonathan@helium.soton.ac.uk says...
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!

I tried to post before on this subject, but it seems my thoughts on the
matter were 'over-speculative'. Unfortunately I have been unable to
figure out exactly the point at which this over-speculation begins, so I
hope the readers of sci.physics.research will be willing to point out
any errors I am making in the following argument:

As I understand it, Stephen Hawking has recently been considering the
evolution of a system in which a black hole - as nearly as we can
determine - forms, perhaps absorbs an observer (although Hawking does
not mention this), and subsequently evaporates. If quantum theory is
correct, the final state of the system encodes somehow the initial state
and subsequent history of all the matter in the system, although the
information might be impractical to decode. (An analogy sometimes made
is the smoke from a burned book, which in principle encodes the text of
the book, but does not differ in any obvious way from the the smoke of a
book with different text.)

Hawking analyses the evolution of the wave function of such a system,
summing over the histories on all possible geometric backgrounds, and
concludes - if I understand matters correctly - that the contribution to
the final state of the wave function from topologically non-trivial
geometries is zero. That is to say, the final state of the wave
function is equal to the sum over the histories in which a classical
black hole did not actually form.

Now this final state contains the encoded history of the infalling
observer, albeit in hard-to-read form. The history of the infalling
observer, whatever it is, must therefore be a history compatible with an
existence that throughout took place in a spacetime of trivial topology.

I conclude, therefore, that if Hawking is correct, whatever happens to
the infalling observer must differ from the canonical description given
in typical literature about general relativity, i.e, the
"spaghettification" and eventual arrival near a mysterious central
singularity at r=0, where it is conceded by almost almost all theorists
that general relativity must break down. Because the part of such a
journey below the Schwarzschild radius r=2m takes place in a
topologically non-trivial spacetime. Correct?

Abd finally, since Hawking has not been shown to be incorrect, and
nobody has provided any better solution to the information loss problem,
then
it would be rash to assert that the canonical description of what
happens the infalling observer is true, especially in a site intended
for the education of the public - especially when there are so many old
sites that assert this.

That concludes my argument. If there's an error in it, I'd love to know
where.

Note that the r=0 singularity is not involved
here, and (for a sufficiently massive black hole) all happens in
*weak* gravitational fields.

The above seems to assume that a breakdown of general relativity must
necessarily be associated with large local gravitational stresses.
However I see no reason why this must be so. Can someone point it out?

On 7 Lut, 14:10, Oh No <N...@charlesfrancis.wanadoo.co.uk> wrote:
Hello everybody,

[snip]

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

Hi Charles,

I found such a fragment on your page:

"It was widely thought that the universe should endure from
everlasting to everlasting. In order to make this possible Einstein
made a modification to the field equation, by including a repulsive
force to balance gravitational attraction."

You must know that the "gravitational attraction" is an idea from the
Newtonian gravitation, not subscribed to even by Newton himelf, who
similarly like Einstein didn't believe in (spooky) "action at
distance". Einstein happened to eliminate this "force" from
gravitation repalcing it by the inertial force due to curvature of
spacetime. So why are you talking about some "repulsive force" and the
"gravitational attraction" which is dead for almost a century?

And BTW, the remark that "the cosmological term was the biggest
blunder of his life" was made by Einstein not because of the math
(which is obviously flawless) but since after this discovery he was
botherd by every cosmologist and his brother until Enstein has
forbidden his secretary to let in anybody who wanted to talk with him
about the universe (soure: Roy Glauber, my teacher and Einstein's
coworker). Apparently George Gamow didn't have Einstein's sense of
humor and didn't get the joke (or didn't want to). How anyone could
think that the discovery of the cosmological constant could be a
blunder?

I do not think that either the inclusion or non-inclusion of the

The remark [the bigest blunder] was supposedly made
at a time when the cosmological constant was regarded as unnecessary. It
was not removed because of a fault in the maths, but because it lacks an
underlying mechanism - something which is still true today.

On 1 Kwi, 17:23, Oh No <N...@charlesfrancis.wanadoo.co.uk> wrote:

[snip]
The remark [the bigest blunder] was supposedly made
at a time when the cosmological constant was regarded as unnecessary. It
was not removed because of a fault in the maths, but because it lacks an
underlying mechanism - something which is still true today.

Einsein's cosmological constant was made to counter the curvature that
would force the collapse of the unierse, which was not observed. So it
was something just giving the equations their badly needed physical
sense, since otherwise (without the cosmological constant) they
wouldn't describe the universe as it is.

What do you mean by "underlying mechanism"? Mechanism of what?

It is
just the math describing a stable universe as Einstein have seen it.

And it turned out that the equations can't do without the cosmological
constant.

So why don't we accept that Einstein was clearly joking
calling it a biggest blunder?

At the time when Einstein supposedly made the remark, no one believed in

Thus spake Gerry Quinn <gerryq@indigo.ie
In article <62kohfF23emliU1@mid.individual.net>,
jonathan@helium.soton.ac.uk says...
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!

I tried to post before on this subject, but it seems my thoughts on the
matter were 'over-speculative'. Unfortunately I have been unable to
figure out exactly the point at which this over-speculation begins, so I
hope the readers of sci.physics.research will be willing to point out
any errors I am making in the following argument:

Looking at what you wrote, I am not sure that it is not Steven Hawking
who was not exploring an overspeculative argument!

As I understand it, Stephen Hawking has recently been considering the
evolution of a system in which a black hole - as nearly as we can
determine - forms, perhaps absorbs an observer (although Hawking does
not mention this), and subsequently evaporates. If quantum theory is
correct, the final state of the system encodes somehow the initial state
and subsequent history of all the matter in the system, although the
information might be impractical to decode. (An analogy sometimes made
is the smoke from a burned book, which in principle encodes the text of
the book, but does not differ in any obvious way from the the smoke of a
book with different text.)

Hawking analyses the evolution of the wave function of such a system,
summing over the histories on all possible geometric backgrounds, and
concludes - if I understand matters correctly - that the contribution to
the final state of the wave function from topologically non-trivial
geometries is zero. That is to say, the final state of the wave
function is equal to the sum over the histories in which a classical
black hole did not actually form.

Now this final state contains the encoded history of the infalling
observer, albeit in hard-to-read form. The history of the infalling
observer, whatever it is, must therefore be a history compatible with an
existence that throughout took place in a spacetime of trivial topology.

I conclude, therefore, that if Hawking is correct, whatever happens to
the infalling observer must differ from the canonical description given
in typical literature about general relativity, i.e, the
"spaghettification" and eventual arrival near a mysterious central
singularity at r=0, where it is conceded by almost almost all theorists
that general relativity must break down. Because the part of such a
journey below the Schwarzschild radius r=2m takes place in a
topologically non-trivial spacetime. Correct?

It is not clear to me that this treatment, if it is Hawking's, is very
useful. Even if we correctly track the wave function, in the collapse
which takes place in measurement, the wave function, and the detailed
information in it is lost.

Abd finally, since Hawking has not been shown to be incorrect, and
nobody has provided any better solution to the information loss problem,
then
it would be rash to assert that the canonical description of what
happens the infalling observer is true, especially in a site intended
for the education of the public - especially when there are so many old
sites that assert this.

Not on account of Hawking's argument, I agree that it is rash to assert
too strongly what happens in the environs of a black hole in the absence
of a true unification between quantum theory and general relativity, and
have sought to avoid doing so in the brief remarks given on the website.

That concludes my argument. If there's an error in it, I'd love to know
where.

Note that the r=0 singularity is not involved
here, and (for a sufficiently massive black hole) all happens in
*weak* gravitational fields.

The above seems to assume that a breakdown of general relativity must
necessarily be associated with large local gravitational stresses.
However I see no reason why this must be so. Can someone point it out?

I agree. I think that general relativity should break down in a vicinity
of the singularity, not only at the singularity. I have an argument
using quantum theory that the radius of the vicinity depends on the mass
of the hole, not on the strength of the classical field which would be
calculated at that radius. My argument suggests that the point at which
classical concepts of space-time break down is actually at the
Schwarzschild radius, notwithstanding the possibility of extending the
mathematical description of a classical manifold beyond that point.

At that point Einstein withdrew
the cosmological constant,

Right.

and it was widely thought that the observed
expansion would lead to eventual contraction.

More or less.

Later again, in the late
1990's, observations lead to accelerating expansion.

Right.

The cosmological
constant was reintroduced, but with opposite sign to that originally
proposed by Einstein.

Completely wrong. Both in the Einstein static universe and in the
currently favoured model, the cosmological constant is positive.

At the time when Einstein supposedly made the remark, no one believed in
the cosmological constant.

Actually, no-one had ever heard of it until then. One could have
something similar in Newtonian theory, but I don't think anyone
discussed it.