Einstein Field Equations and Flat Space Time
Bill Hobba
deduction of the EFE from flat space time and spin 2 particles as was done
by Feynamn in his Lectures on Gravitation. I first read this many years ago
before my math had had improved. I have recently reread it and am still
struck by how the assumption of the principles of QM and flat space-time
inevitably leads to space-time curvature and the EFE ie SR + QM implies GR.
It would seem that somehow QM contains within it the principle of general
invariance upon which GR is based. What this means I have no idea. The
only conclusion I have reached is it is just another thing that points to
the incompleteness of our theories.
Has anyone else any ideas?
Thanks
Bill
Steve Carlip
> [...] I am still struck by how the assumption of the principles of QM
> and flat space-time inevitably leads to space-time curvature and
> the EFE ie SR + QM implies GR. It would seem that somehow QM
> contains within it the principle of general invariance upon which
> GR is based.
There's a bit more to it. You also need, at least, a massless spin two
interaction that couples universally. While this doesn't involve
general covariance in an obvious way, a massless spin two field has
a gauge invariance that's ``as big'' as diffeomorphism invariance
(i.e., that's parametrized by a vector field), and the universality of
the coupling rules out any noninvariant ``background.''
Steve Carlip
Marion Hobba
"Steve Carlip" <sjca...@ucdavis.edu> wrote in message
news:abbmsv$hp4$1...@woodrow.ucdavis.edu...
Thanks Steve.
Do you have a reference that goes into this in greater detail? I am taking
a break from work and have a bit of time to check into some things that have
been on my mind. I will be going through Feynmans Lectures on Gravitation
looking for assumptions and hope to get a better feel for what is happening
here.
Thanks
Bill
Steve Carlip
> "Steve Carlip" <sjca...@ucdavis.edu> wrote in message
> news:abbmsv$hp4$1...@woodrow.ucdavis.edu...
>> Bill Hobba <bho...@bigpond.net.au> wrote:
>> > [...] I am still struck by how the assumption of the principles of QM
>> > and flat space-time inevitably leads to space-time curvature and
>> > the EFE ie SR + QM implies GR.
>> There's a bit more to it. [...]
> Do you have a reference that goes into this in greater detail? I am
> taking a break from work and have a bit of time to check into some
> things that have been on my mind. I will be going through Feynmans
> Lectures on Gravitation
That's a good place to start. Two standard papers are by Deser,
Gen. Rel. Grav. 1 (1970) 9 and Class. Quant. Grav. 4 (1987) L99,
which deal with classical calculations. You might also look at a
paper by Boulware and Deser, Ann. Phys. 89 (1975) 193, for a more
quantum field theoretical argument (based on earlier work by
Weinberg). You might also look at two articles, by Duff and Deser,
in the book _Quantum Gravity: An Oxford Symposium_ (edited
by Isham, Penrose, and Sciama, Clarendon Press, 1975).
Steve Carlip
Jonathan Thornburg [remove -animal to reply]
> Standard interpretation of general relativity says that particles goes
> through geodesics because of space time internal curvature: theory,
> experiments suggest some equations, which looks like being a result of
> internal curvature of spacetime.
> But if we live on some curved manifold, it intuitively should be
> embedded somewhere(?) (and for example black holes are some spikes)
> So why our energy/matter is imprisoned on something infinitely flat?
There's a fascinating analysis due to Deser ["Self-interaction and
gauge invariance", General Relativity & Gravitation 1 (1970), 9-18;
see also his later paper "Gravity from self-interaction in a curved
background", Classical and Quantum Gravity 4 (1997), L99-L105],
summarized in part 5 of box 17.2 of Misner, Thorne, & Wheeler's book.
Quoting from that latter summary:
"The Einstein equations may be derived nongeometrically by
noting that the free, massless, spin-2 field equations
[[for a field $\phi$]]
$$ [[...]] $$
whose source is the matter stress-tensor $T_{\mu\nu}$, must
actually be coupled to the \emph{total} stress-tensor,
including that of the $\phi$-field itself.
[[...]]
Consistency has therefore led us to universal coupling, which
implies the equivalence principle. It is at this point that
the geometric interpretation of general relativity arises,
since \emph{all} matter now moves in an effective Riemann space
of metric $\mathcal{g}^{\mu\nu} = \eta^{\mu\nu} + h^{\mu\nu}$.
... [The] initial flat `background' space is no longer observable."
In other words, if you start off with a spin-2 field which lives on a
flat "background" spacetime, and say that its source term should include
the field energy, you wind up with the original "background" spacetime
being *unobservable in principle*, i.e. no possible observation can
detect it. Rather, *all* observations will now detect the effective
Riemannian space (which is what the usual geometric interpretation of
general relativity posits from the beginning).
Since the original background spacetime is now completely unobservable
(insofar as Deser's analysis is valid & applicable), it seems to me
that discussing its hypothesised properties is no longer physics.
It's also worth pointing out that we discussed "backgrounds versus the
Einstein equations" in this newsgroup back in May 2002, in a thread
beginning at
http://www.lns.cornell.edu/spr/2002-05/msg0041432.html
Later in that thread Steve Carlip gave some references to
quantum-field-theory extensions of Deser's argument:
http://www.lns.cornell.edu/spr/2002-05/msg0041725.html
--
-- From: "Jonathan Thornburg [remove -animal to reply]" <jth...@astro.indiana-zebra.edu>
Dept of Astronomy, Indiana University, Bloomington, Indiana, USA
"Washing one's hands of the conflict between the powerful and the
powerless means to side with the powerful, not to be neutral."
-- quote by Freire / poster by Oxfam
Tom Roberts
> [referencing Deser's papers]
> In other words, if you start off with a spin-2 field which lives on a
> flat "background" spacetime, and say that its source term should include
> the field energy, you wind up with the original "background" spacetime
> being *unobservable in principle*, i.e. no possible observation can
> detect it. Rather, *all* observations will now detect the effective
> Riemannian space (which is what the usual geometric interpretation of
> general relativity posits from the beginning).
This seems self-inconsistent to me: the topology of the initial (flat)
spacetime is, in general, incompatible with the topology of the (curved)
Riemannnian spacetime. So how can they both be applied? What are the
ramifications of topological consistency being a selection rule for
allowed manifolds? -- it seems to me that all of the FRW manifolds of
cosmology would be ruled out. Indeed, I suspect that all non-trivial
manifolds would be ruled out. Does this imply that the geometric
approach of GR is more basic, and the field-theory with its flat
"background" is merely a local limit (so the global topology of the flat
"background" is irrelevant)?
Or alternatively, does this imply that these models (GR and QFT) are
being stretched beyond their limits, and we need a better, more
comprehensive theory before this will make sense?
Tom Roberts
Jonathan Thornburg [remove -animal to reply]
| [referencing Deser's papers]
| In other words, if you start off with a spin-2 field which lives on a
| flat "background" spacetime, and say that its source term should include
| the field energy, you wind up with the original "background" spacetime
| being *unobservable in principle*, i.e. no possible observation can
| detect it. Rather, *all* observations will now detect the effective
| Riemannian space (which is what the usual geometric interpretation of
| general relativity posits from the beginning).
Tom Roberts <tjrobe...@sbcglobal.net> wrote:
> This seems self-inconsistent to me: the topology of the initial (flat)
> spacetime is, in general, incompatible with the topology of the (curved)
> Riemannnian spacetime. So how can they both be applied? What are the
> ramifications of topological consistency being a selection rule for
> allowed manifolds?
As I understand it (based on the summary in Misner, Thorne, & Wheeler),
Deser's analysis is a purely local one, i.e. the phrase
"*all* observations will now detect the effective Riemannian space"
means (only) that these observations will be consistent with the
Einstein field equations, which are purely a statement about the
*local* differential geometry of spacetime.
So Deser's analysis basically says that the flat background is
unobservable by any local experiment.
The Einstein equations (& Deser's argument) don't say anything
about the global topology of spacetime -- that has to be worked
out separately. One could imagine (gedanken) experiments that
could measure such things, e.g. "seeing all the way around a closed
universe".
--
-- "Jonathan Thornburg [remove -animal to reply]" <jth...@astro.indiana-zebra.edu>
Jonathan Thornburg [remove -animal to reply]
| [referencing Deser's papers]
| In other words, if you start off with a spin-2 field which lives on a
| flat "background" spacetime, and say that its source term should include
| the field energy, you wind up with the original "background" spacetime
| being *unobservable in principle*, i.e. no possible observation can
| detect it. Rather, *all* observations will now detect the effective
| Riemannian space (which is what the usual geometric interpretation of
| general relativity posits from the beginning).
Tom Roberts <tjrobe...@sbcglobal.net> wrote:
> This seems self-inconsistent to me: the topology of the initial (flat)
> spacetime is, in general, incompatible with the topology of the (curved)
> Riemannnian spacetime. So how can they both be applied? What are the
> ramifications of topological consistency being a selection rule for
> allowed manifolds?
As I understand it (based on the summary in Misner, Thorne, & Wheeler),
Deser's analysis is a purely local one, i.e. the phrase
"*all* observations will now detect the effective Riemannian space"
means (only) that these observations will be consistent with the
Einstein field equations, which are purely a statement about the
*local* differential geometry of spacetime.
So Deser's analysis basically says that the flat background is
unobservable by any local experiment.
The Einstein equations (& Deser's argument) don't say anything
about the global topology of spacetime -- that has to be worked
out separately. One could imagine (gedanken) experiments that
could measure such things, e.g. "seeing all the way around a closed
universe".
--
-- "Jonathan Thornburg [remove -animal to reply]" <jth...@astro.indiana-zebra.edu>
Gerry Quinn
tjrobe...@sbcglobal.net says...
> Jonathan Thornburg [remove -animal to reply] wrote:
> > [referencing Deser's papers]
> > In other words, if you start off with a spin-2 field which lives on a
> > flat "background" spacetime, and say that its source term should include
> > the field energy, you wind up with the original "background" spacetime
> > being *unobservable in principle*, i.e. no possible observation can
> > detect it. Rather, *all* observations will now detect the effective
> > Riemannian space (which is what the usual geometric interpretation of
> > general relativity posits from the beginning).
The above argument is valid only if the spin-2 (graviton) field is
considered fundamental. But a theory with fundamental gravitons doesn't
actually work anyway; it blows up at sufficiently high energies and is
not fixable by renormalisation etc.
So a theory based on gravitons must instead be an effective field theory
that breaks down at a sufficiently high energy scale. And where it
breaks down, observations will, directly or indirectly, be affected by
the underlying nature of spacetime. Thus it is not true that *all*
observations will detect the effective Riemannian space.
> This seems self-inconsistent to me: the topology of the initial (flat)
> spacetime is, in general, incompatible with the topology of the (curved)
> Riemannnian spacetime. So how can they both be applied? What are the
> ramifications of topological consistency being a selection rule for
> allowed manifolds? -- it seems to me that all of the FRW manifolds of
> cosmology would be ruled out. Indeed, I suspect that all non-trivial
> manifolds would be ruled out. Does this imply that the geometric
> approach of GR is more basic, and the field-theory with its flat
> "background" is merely a local limit (so the global topology of the flat
> "background" is irrelevant)?
I agree that it is inconsistent, but my conclusion would be that
(assuming that gravity is explained based on a non-fundamental graviton
field) non-trivial topologies would be unphysical.
- Gerry Quinn
Tom Roberts
> [about spin-2 gravitons on flat background spacetime]
> my conclusion would be that
> (assuming that gravity is explained based on a non-fundamental graviton
> field) non-trivial topologies would be unphysical.
I would state this quite differently: Assuming that gravity is explained
based on a non-fundamental graviton field, it is an experimental issue
to determine the underlying manifold, including its curvature and
topology. Current attempts to formulate such theories use Minkowski
spacetime for simplicity, but there's no reason to expect that to hold
in the real world.
Many/most current attempts to explore quantum gravity are abandoning the
whole notion of manifolds, so this may well be moot.
Tom Roberts
Gerry Quinn
when saving it from my inbox into my s.p.r articles-to-be-moderated queue.
I apologise to the author and other readers for this, and for the ensuing
loss of threading in many newsreaders.
-- jt]]
In article <ivr7l.10266$D32....@flpi146.ffdc.sbc.com>, tjroberts137
@sbcglobal.net says...
> Gerry Quinn wrote:
> > [about spin-2 gravitons on flat background spacetime]
> > my conclusion would be that
> > (assuming that gravity is explained based on a non-fundamental graviton
> > field) non-trivial topologies would be unphysical.
>
> I would state this quite differently: Assuming that gravity is explained
> based on a non-fundamental graviton field, it is an experimental issue
> to determine the underlying manifold, including its curvature and
> topology.
Agreed... but have you any reason at all to suppose it is anything but
approximately flat on the scale of the observable universe, and of some
pretty simple topology? In that case any small-scale non-trivial
topology - or any topology at all that differs from the background -
would be unphysical. [Unless spacetime can somehow have multiple
topologies at once and still be stable and consistent.]
> Current attempts to formulate such theories use Minkowski
> spacetime for simplicity, but there's no reason to expect that to hold
> in the real world.
But there's no reason either for any background curvature corresponding
to gravity. The whole point of the graviton theory is to explain
gravity as just another field, like electromagnetism. Sure, the
background might in principle be curved or have an exotic topology for
some other reason, but any such curvature would be unrelated to any
curvature or topology predicted by GR.
> Many/most current attempts to explore quantum gravity are abandoning the
> whole notion of manifolds, so this may well be moot.
They can do what they like at the Planck scale and beyond, but at lower
energy scales it is obvious that spacetime is best described in terms of
manifolds (effective manifolds, if you prefer). It is consistency
between effective manifolds of this kind that is relevant to this
thread.
- Gerry Quinn