Re: Is general relativity incompatible with the Newtonian limit?
- From: tessel@xxxxxx
- Date: Sat, 24 Sep 2005 16:12:08 +0000 (UTC)
On Fri, 23 Sep 2005, Juan R. wrote:
> In one of his last works Mathematical Foundations of
> Quantum Theory. (Academic Press, Inc., 1978) Dirac claimed:
[delete Dirac's critique of QFT circa 1978]
> I always thought that general relativity had been *completely* verified
> in experiments
Did you perhaps mean that you always thought that "to date, no
widely-accepted experiment or observation has been generally agreed to
clearly -violate- gtr"?
(Which is probably a fair statement of the current situatoin, but see also
the Pioneer Effect thread. This mystery is gaining ever more attention,
but of course "something went wrong with gtr" is just one of many
explanations which have been proposed, and still seems to most of us, I
think, still fairly unlikely to ultimately become the accepted
explanation, although only time will tell.)
> There are several obvious flaws in GR, including discrepancies with data
> especially in the extragalactic regime.
GR does have flaws which are widely recognized, but discrepancies with
data is emphatically -not- one of them! And to be fair, GR also has many
widely recognized -virtues-, including this: it's so darned good that it's
been so intensively studied that scads of -theoretical- flaws have been
identified. None of its competitors have come in for that kind of
scrutinity (not a good thing, but understandable).
What about experimental flaws? As in disagreement with Nature?
Well, there are all kinds of mysteries in cosmology, and the possibility
that our Gold Standard Theory of Gravitation is finally beginning to fail
at large scales, after decades of sturdy use, must always be considered.
Indeed, this has been repeatedly suggested over the decades for various
things which were eventually put down to other causes. Not many would say
that a possible failure of gtr is the leading contender among seriously
discussed possible explanations of any given contemporary cosmological
mystery, however.
> Still my main motivation is if GR is incompatible with NG in the same
> form that RQFT is incompatible with NRQM like claimed by Dirac.
Apples and oranges, surely? Dirac was talking (I guess) about divergent
integrals; nothing like that is happening here; you just misunderstood
what you read in Wald.
> When one ask by the Newtonian limit of GR one heard in Baez page
>
> http://math.ucr.edu/home/baez/RelWWW/wrong.html
>
> that "The theorem stating that gtr does indeed go over to Newtonian
> gravitostatics in the very weak field, very slow motion limit is proven
> in detail in almost every gtr textbook."
Just curious: did you visit that page after reading the Physics Forums
"sticky note:" pointing to it? BTW, see the byline before complaining to
Baez.
> Then one read one of those textbooks and discover like NG is 'obtained'
> from GR, but one finds that derivation is NOT rigorous.
It is well known that "limits" can indeed be tricky. But if you read the
textbooks and understand what statement is being proven, you can usually
see that the alleged proof is good enough for physics :-/
> Then one takes a more rigorous textbook (Wald) and one discovers that,
> effectively, NG is NOT derived from GR in the linear regime. Those
> textbooks are wrong, in the linear regime a = 0.
What do you mean, "derived from"?
It sounds like you might have confused linearized gtr (weak fields; very
roughly speaking first order in the mass parameter, which does indeed kill
off a lot of interesting and important behavior) with the slow motion weak
field limit, which is even more stringent.
This might not be your fault if you are only reading Wald, since in my
experience his discussion in chapter 4 confuses many readers exactly on
this point, because the distinction between weak field and slow motion
weak field is not stated clearly enough. Reading one of the more
elementary textbooks such as Carroll, Schutz, or Stephani might help here.
> Then Wald explains that one may go beyond the linear regime,
Where exactly does he say "beyond the linear regime"? I can't seem to
find this in section 4.4a.
> he does and obtains eq. 4.21 of Wald
(4.4.21) (roughly, equation of motion of test particles, aka force law),
right? And don't forget (4.4.17) (Laplace equation).
> a = - GRAD (Phy)
^^^^^
Phi
> whereas the Newtonian law is
>
> a = - GRAD (Phy)
> A priori one has obtained Newtonian law, but is not. a in GR is not a
> in NG,
I think you might be thinking of relativistic momentum versus Newtonian
momentum. If so, right, these -are- different, but they do agree -up to
first order- in the velocity, as Einstein pointed out. That is, in the
slow motion limit, they do agree:
mv/sqrt(1-v^2) = mv + mv^3/3 + O(v^5)
= mv + O(v^2)
> time in GR is not the time in NG.
No, but for slowly moving test particles, up to first order the proper
time counted off by the test particle does agree with the Newtonian time
approximately:
(Delta t)/sqrt(1-v^2) = (Delta t) + O(v^2)
> Phy in GR is a retarded field.
I have always been careful to point out (as does Wald, on p. 76) that this
approximation also assumes -slowly changing- fields. Or even static
fields, if you are just analyzing the motion of test particle and checking
that in weak static fields, slowly moving test particles to approximately
obey Newtonian laws, as they should. That is what Wald is verifying.
> Phy in NG is an instantaneous potential, The GR above formula contains
> c, Newtonian formula does not contain c, etc.
Wald uses geometric or relativistic units in which c=G=1 by fiat.
> Of course, if one takes limit c --> infinite
Ah. This might be the problem. Wald is thinking of the magnitude of the
velocity of the test particles as being "small" wrt c=1; you want to let c
tend to infinity. Either do it Wald's way or reinsert the c,G factors
(see the appendix) to do it your way.
> the Schwartzild metric
^^^^^^^^^^^
Schwarzschild
> g00 = (1 - 2Phy/c^2) --> 1
>
> and
>
> gRR = -(1 - 2Phy/c^2)^-1 --> -1
Try it again with c=G=1. Try expanding the metric components to first
order in m. Can you pick off the Newtonian potential? Now try the
general case following Wald.
> Then one discovers that the so called 'derivation' given in textbook is
> incorrect.
It looks fine to me; I think you just misunderstood something.
"T. Essel" (hiding somewhere in cyberspace)
.
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