GR without Geommetry or coordinates.

[Kevin_spencer2]

Is there a method to describe GR without using coordinates or metrics $$g_{ab} , x_{a}(s)$$, using only Analytic or Algebraic methods?,

the main problem with a geommetryc aspect is that you can't use it when dealing QM since at this level there is no Geommetry.

 

[MeJennifer]

Is there a method to describe GR without using coordinates or metrics $$g_{ab} , x_{a}(s)$$, using only Analytic or Algebraic methods?,

the main problem with a geommetryc aspect is that you can't use it when dealing QM since at this level there is no Geommetry.

The problem is in the metric. In space-time with a Euclidean metric the distance between two points is calculated using the Pythagorean theorem. This is not the case for space-time with a Lorentzian metric. Strictly speaking a Lorentzian metric is not even a proper metric.

While neighborhoods using a Euclidean metric are like hyperspheres they are not so for a Lorentzian metric.
Because of this, two neighborhoods could interact in a rather non-local way.
And there's the rub, but also the irony, while QEM attempts to come to terms with non-locality using a Euclidean metric, the Lorentizian metric already has some non-local aspects. But because of that the path-integral functions don´t converge.
But so-far nobody has been able to put it all together.

 

[Jheriko]

the main problem with a geommetryc aspect is that you can't use it when dealing QM since at this level there is no Geometry.

Really? I always thought that QM is based on Euclidean or Minkowskian geometry depending on the context...

If there was no geometry how can we meaningfully describe a wave function? After all it takes coordinates in some underlying geometry as parameters...

 

[robphy]

Here's one interesting paper:

Robert Geroch, "Einstein algebras"
Comm. Math. Phys. 26, no. 4 (1972), 271–275
http://projecteuclid.org/Dienst/UI/1.0/Summarize/euclid.cmp/1103858122

 

[Demystifier]

You can write
$$g_{\mu\nu}(x)=\eta_{\mu\nu}+h_{\mu\nu}(x)$$
and view gravity as a spin-2 field $$h_{\mu\nu}(x)$$ propagating in flat Minkowski spacetime with the metric $$\eta_{\mu\nu}$$.
For example, such a view is pushed forward by Feynman in his "Feynman Lectures on Gravitation".

 

[Chris Hillman]

Not gtr, and not non-geometric enough?

Hi all,

You can write
$$g_{\mu\nu}(x)=\eta_{\mu\nu}+h_{\mu\nu}(x)$$
and view gravity as a spin-2 field $$h_{\mu\nu}(x)$$ propagating in flat Minkowski spacetime with the metric $$\eta_{\mu\nu}$$.
For example, such a view is pushed forward by Feynman in his "Feynman Lectures on Gravitation".

Maybe I misunderstood something, but what you wrote down here looks like the starting point for weak-field gtr, in which one treats (mathematically speaking) small linear perturbations of Minkowski metric as a tensor field propagating on the (unobservable) flat background. This is not equivalent to fully nonlinear gtr, and it is not entirely self-consistent. However, it can be fixed up (in principle!) to yield something locally equivalent to gtr on neighborhoods with trivial topology.

More to the point, I don't see how this addresses the question asked by the OP. The paper by Wald cited by Robphy does seem to be relevant, however.

 

[lalbatros]

Kevin_spencer2,

Space without space, that's the ultimate program of physics. Isn't it?

Michel

 

[Demystifier]

Maybe I misunderstood something, but what you wrote down here looks like the starting point for weak-field gtr, in which one treats (mathematically speaking) small linear perturbations of Minkowski metric as a tensor field propagating on the (unobservable) flat background. This is not equivalent to fully nonlinear gtr, and it is not entirely self-consistent.

It is consistent and is not merely an approximation. However, the price payed is that the Einstein equation written in terms of eta and h (rather than g) has an infinite number of terms. But the remarkable fact is that the Einstein equation (in terms of eta and h) can be DERIVED in this way, by requiring that a spin-2 field in flat spacetime is coupled consistently to the energy-momentum.

 

[MeJennifer]

It is consistent and is not merely an approximation.

But what about black holes for instance?

 

[Chris Hillman]

Clarification

It is consistent and is not merely an approximation.

I don't know what you mean by "it", but you should probably look at say MTW section 18.1, where you will see that what you wrote down is the starting point of the theory of linearized perturbations of Minkowski spacetime, aka "weak-field gtr", in which the field equation is the "linearized EFE". Next, look at Box 17.2, which summarizes the results I was referring to when I stated (correctly) that "linearized gtr" is not self-consistent, and that by trying to fix this deficiency, one is inexorably led, following Deser 1970, to gtr.

However, the price payed is that the Einstein equation written in terms of eta and h (rather than g) has an infinite number of terms. But the remarkable fact is that the Einstein equation (in terms of eta and h) can be DERIVED in this way, by requiring that a spin-2 field in flat spacetime is coupled consistently to the energy-momentum.

It sounds like you are in fact talking about Deser 1970, but I think you missed another crucial point: Deser's formulation is only locally equivalent to gtr (as in "local neighborhood"). When you study spacetime models in classical gtr with nontrivial topology (e.g. the Kerr vacuum), this distinction is essential. This should be obvious now that I have pointed it out to you!

If not, I quote a length from a post by Steve Carlip to sci.physics.relativity from February 2000, in which the key sentence is

And as a physicist, not a mathematician, I consider this the key issue. GR a la Deser, Feynman, Weinberg, et al. is certainly not globally equivalent to standard GR.

Here is the complete post:

From carlip@***.*** Sun Feb 27 12:48:34 2000
Date: 24 Feb 2000 19:36:17 GMT
From: Steve Carlip <carlip@***.***>
Newsgroups: sci.physics.relativity, sci.physics
Subject: Re: Black "Holes"? [Attn: Steve Carlip!]

In sci.physics Chris Hillman <hillman@***.***> wrote:
 
> Now, I'd like to try to offer a lengthy and thoughtful response  

Too lengthy, I'm afraid.  I have time only for a very short reply.

> First, there is an issue regarding the nature of gtr as a theory in
> mathematical physics, namely the question of whether Deser's 
> scheme yields "a reinterpretation of gtr in terms of a spin-two 
> self-interacting field on an (unobservable) Minkowski vacuum 
> background".  I think it would be correct to say that it yields a 
> -local- interpretation of gtr, in the  sense that -any local 
> neighborhood- in -any- exact solution to the EFE should 
> arise from Deser's scheme.

Agreed.

> However, it certainly is not possible to recover, in particular, 
> the Kerr vacuum without gluing together local neighborhoods 
> each of which are homeomorphic to R^4 

Agreed.

> Second, there is an issue regarding which exact solutions in gtr 
> are "physically reasonable" [...]

> It is not clear to me that all "physically reasonable" solutions in
> gtr are homeomorphic to R^4.

Agreed.  

> Third, there is an issue regarding which exact solutions in 
> gtr are "physically realistic" [...]

> I don't think it given the current "state of the art", that
> there is any reason to expect that all "physically realistic 
> spacetimes" in gtr must be homeomorphic to R^4.

Agreed.

> Fourth, there is an issue regarding the body of available 
> observational and experimental evidence regarding 
> gravitational phenomena, namely whether any evidence 
> forces us to accept that the universe in which we live  [...]
> is not homeomorphic to R^4.  You say that you don't 
> know of any, and off the top of my head, I guess I do not
> either.

Right.  And as a physicist, not a mathematician, I consider this
the key issue.  GR a la Deser, Feynman, Weinberg, et al. is
certainly not globally equivalent to standard GR.  But at the
moment, there is no direct evidence that allows us to choose
between them.  I personally find the geometric approach much
more appealing, and I think it has historically been more
fruitful, but I also recognize that for now this is a matter of
taste rather than ``truth.''

> Fifth, there is the question of whether "real but unobservable 
> background spacetimes" have a legitimate role in physics.  

This is an old argument, and I don't expect anyone to resolve
it in a newsgroup.  If you read some of the discussions of non-
Euclidean geometry from 100 years ago, you'll find exactly
the same issues coming up.  The Poincare disk model of H^2,
for example, was invented to precisely to show the difficulty
of distinguishing a real non-Euclidean geometry from a
Euclidean geometry with funny measuring instruments.  
There's a nice book by Sklar about this sort of question.

A similar issue is that Gerry is considering his background
space and time as ``absolute elements,'' elements that affect
matter but are unaffected by it.  Certainly the direction of
physics has been away from such elements, and towards the
idea that anything that has an effect can itself be affected.
But again, this is not a question of physics, but more one
of philosophy (in the sense of the word that has a faint
overtone of disapproval when spoken by a physicist).

Steve Carlip

By the way, the "nice book by Sklar" is indeed very nice. The full citation is at http://www.math.ucr.edu/home/baez/RelWWW/reading.html#phil

 

[Chris Hillman]

Clarification

But what about black holes for instance?

Demystifier apparently misunderstood what I was referring to (the points I actually had in mind are widely appreciated in the field). Regarding the fact that standard gtr is not globally equivalent to the reformulation by Deser et al., I stress that this reformulation cannot capture nontrivial topology, and standard black hole models are not topologically trivial. It is still possible (although this requires increasing strain!) to claim that it has not, perhaps, yet been "decisively proved" to everyone's satisfaction that the universe in which we live has nontrivial topology, and standard black holes are thought be misleading in the interior region (search on "Poisson-Israel model", "mass inflation" and so on), but this does obviate the point that there is no good reason to expect that nontrivial topology is somehow physically forbidden; quite the contrary, in fact.

 

[pervect]

Demystifier apparently misunderstood what I was referring to (the points I actually had in mind are widely appreciated in the field). Regarding the fact that standard gtr is not globally equivalent to the reformulation by Deser et al., I stress that this reformulation cannot capture nontrivial topology, and standard black hole models are not topologically trivial. It is still possible (although this requires increasing strain!) to claim that it has not, perhaps, yet been "decisively proved" to everyone's satisfaction that the universe in which we live has nontrivial topology, and standard black holes are thought be misleading in the interior region (search on "Poisson-Israel model", "mass inflation" and so on), but this does obviate the point that there is no good reason to expect that nontrivial topology is somehow physically forbidden; quite the contrary, in fact.

I haven't had time to look at the issues as closely as I would like, but I still think that the folks who propose that the universe might have a trivial topology could be onto something, so I don't necessarily share Chris's views on this matter of the unlikelyhood of this state of affairs. On the other hand, I suspect he's looked at it more than I have. On the other other hand (the gripping hand :-), an obscure science fiction reference), I'd really like to see if I could find the expressed opinions of (for example), Baez, Bunn, and Carlip on this issue (they being frequent posters to spr, at least at one time, so that it might be possible to divine their opinion on these matters by digging through some usent archives).

[add]
Why is this relevant at all? Well, the point is that if you consider the Desser theories being discussed (spin 2 theories on a not-directly observable underlying flat Minkowski geometry) that you wind up with a globally trivial topology. So this invites us to consider the question of "what do we really know about the topology"?

If we ever find "circles in the sky" (see http://arxiv.org/abs/gr-qc/9602039) or a wormhole, that would be a "smoking gun" that would convince me that the universe did in fact have a non-trivial topology.

Topology changing transitions in quantum gravity would also be a strong hint that the universe didn't have a trivial topology. While trying to find papers by Carlip on the topic of the toplogy of the universe, I stumbled across http://xxx.lanl.gov/abs/gr-qc/9406006 which seems to suggest that such transitions are infinitely suppressed.

 

[Chris Hillman]

I haven't had time to look at the issues as closely as I would like, but I still think that the folks who propose that the universe might have a trivial topology could be onto something, so I don't necessarily share Chris's views on this matter of the unlikelyhood of this state of affairs.

Don't forget about all those astrophysical objects out there which look an awful lot like black holes, to mention just one issue.

If we ever find "circles in the sky" (see http://arxiv.org/abs/gr-qc/9602039) or a wormhole, that would be a "smoking gun" that would convince me that the universe did in fact have a non-trivial topology.

This refers to the suggestion by Cornish and Weeks that spatial hyperslices in "topologically realistic" FRW models might have nontrivial topology. IOW, the "best fit" FRW models might be none of the ones discussed in classic textbooks like Hawking and Ellis, but quotient manifolds of these. Also an important point, but not the most important one here.

Topology changing transitions in quantum gravity would also be a strong hint that the universe didn't have a trivial topology. While trying to find papers by Carlip on the topic of the toplogy of the universe, I stumbled across http://xxx.lanl.gov/abs/gr-qc/9406006 which seems to suggest that such transitions are infinitely suppressed.

Careful, 2+1 gtr is very different from 3+1 gtr (as Carlip knows).

 

[Demystifier]

OK Chriss and MeJenifer, you are right that the equivalence is only local. Both theories lead to the Einstein equation, which is a local equation. Coordinate singularities (like those on the horizon of a black hole) are true singularities in the Deser at al nongeometrical formulation of gravity. This is why the Einstein geometrical interpretation is much more appealing. Still, I find the idea that gravity could be just another field in spacetime quite interesting. Perhaps there is a way to do it consistently. I do not know yet how, but this is something I think about.

 

[Chris Hillman]

Citatation plus caveat

you are right that the equivalence is only local. Both theories lead to the Einstein equation, which is a local equation. Coordinate singularities (like those on the horizon of a black hole) are true singularities in the Deser at al nongeometrical formulation of gravity. This is why the Einstein geometrical interpretation is much more appealing. Still, I find the idea that gravity could be just another field in spacetime quite interesting. Perhaps there is a way to do it consistently. I do not know yet how, but this is something I think about.

Agreed.

I'd go a bit further and stress that while Einstein's Big Idea (geometrizing gravitation to express its universality in the clearest possible way) has many virtues, it carries a price:

1. by its very nature (mathematically speaking), in gtr it is hard to keep track of energy transfer between the gravitational field and Other Things,

2. by its very nature, reasonable boundary values can be hard to write down in a given situation, solutions found locally are often hard to interpret globally, and so on.

3. etc.

If you don't know about "teleparallel gravity" http://arxiv.org/find/gr-qc/1/all:+teleparallel/0/1/0/all/0/1 this might interest you. But be careful, "teleparallel equivalent of gtr" appears to be another misnomer. A number of sci.physics.research denizens discussed this back in the day, and concluded that apparently Itin claims rather than you can formulate all the theories "near" gtr, but not gtr itself, using a theory featuring a Weitzenboeck connection (vanishing curvature, so "flat" in a sense, but nonvanishing torsion).

 

[Kevin_spencer2]

The question is, Why (2+1) Quantum gravity, which is completely renormalizable and can be quantizied in several manners, is completely different from (3+1) gravity?, from the geommetrical aspect it shouldn't be but why?, or if we could treat the fact that in 3-D we have a one more component -"z"- as a perturbation.

Isn't Semi-classical Quantum gravity plus corrections in order of $$\hbar$$ a good approach to the theory?.

 

[robphy]

Here are some classical examples to suggest that physics in (2+1)-spacetimes is different from (3+1)-spacetimes:
- vanishing of Weyl in (2+1)-spacetimes
- Huygens principle
- etc... (I'm sure others can add to this list...)

So, one might expect success of a particular approach [whether classical or quantum] in one choice of dimension might not carry over to other choices.

 

[coalquay404]

The question is, Why (2+1) Quantum gravity, which is completely renormalizable and can be quantizied in several manners, is completely different from (3+1) gravity?, from the geommetrical aspect it shouldn't be but why?, or if we could treat the fact that in 3-D we have a one more component -"z"- as a perturbation.

Isn't Semi-classical Quantum gravity plus corrections in order of $$\hbar$$ a good approach to the theory?.

An excellent discussion of the solvability and quantization of (classical) gravity in 2+1 dimensions and its relation to its 3+1 dimensional big brother is available http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=JMAPAQ000031000012002978000001&idtype=cvips&gifs=yes :

Title: HOW SOLVABLE IS (2+1)-DIMENSIONAL EINSTEIN GRAVITY
Author(s): MONCRIEF V
Source: JOURNAL OF MATHEMATICAL PHYSICS 31 (12): 2978-2982 DEC 1990
Document Type: Article
Language: English
Cited References: 7
Addresses: MONCRIEF V (reprint author), YALE UNIV, DEPT PHYS, 217 PROSPECT ST, NEW HAVEN, CT 06511 USA
YALE UNIV, DEPT MATH, NEW HAVEN, CT 06511 USA
Publisher: AMER INST PHYSICS, CIRCULATION FULFILLMENT DIV, 500 SUNNYSIDE BLVD, WOODBURY, NY 11797-2999
Subject Category: PHYSICS, MATHEMATICAL
IDS Number: EK824
ISSN: 0022-2488