Is a Theory of Gravity Possible in Flat Minkowski Spacetime?

[MaxwellsDemon]

Have there ever been any attempts to create a theory of gravity within the context of a flat 4-D Minkowski spacetime? Rather than taking the general relativistic route of gravity being a manifestation of curved spacetime geometry, has anyone historically ever attempted to put the more "classical" ideas of gravitational fields and potentials within the context of flat spacetime of special relativity? Certainly it wouldn't be as beautiful conceptually as GR, but I would imagine it could be done. I wonder what sorts of predictions such a theory would make and how they would differ from observed phenomena...

 

[Ich]

I think everybody except AE tried this route. You can find a section in MTW where they discuss these alternative approaches.
IIRC you get either GR in disguise, or end up with the wrong predictions, like no light deflection if you simply try to make Newtonian gravity relativistic.

 

[bcrowell]

MTW is getting extremely out of date, but section 39.2 says that the only theories that were consistent with observations in that era were metric theories, the only exception being Cartan's theory involving torsion. To me, torsion seems just as strange as Riemannian curvature, if not stranger; you still have a manifold with a connection, but you just take a different piece of the connection to be the interesting part. In general, the 1970's were the golden age of the interplay between theory and experiment in relativity, with competing (metric) theories being tested empirically. A good description is given in the book Was Einstein Right? by Clifford Will. Anyway, this is all about theories that don't make all the same predictions as GR. What you want to know about is a reformulation of GR that makes the same predictions as GR.

There is of course linearized gravity, which is only an approximation, but which does describe everything on a preexisting Minkowski background. I think that people who study gravitational waves, etc., sometimes expand things in powers of the gravitational field. It's probably true that all the currently available empirical evidence can be successfully described with such an expansion up to some power n. I don't know what n would be. It's hard for me to imagine how you'd describe, e.g., a black hole singularity on a flat background -- but maybe I'm just not imaginative enough :-)

People like Lee Smolin claim that one of the serious problems with string theory is that it assumes a background spacetime. I think the party line among string theorists is that this is just a mathematical convenience, and actually string theory in background-independent, just not manifestly so.

The answer to the converse of your question is "yes:" one can formulate Newtonian gravity using a curved spacetime metric that is locally Newtonian rather than locally Minkowskian.

 

[bcrowell]

I think everybody except AE tried this route.

I think the dead-end theory that Einstein was working on before he settled on general covariance was one with no clear geometrical interpretation (at least, I've heard it described that way).

IIRC you get either GR in disguise, or end up with the wrong predictions, like no light deflection if you simply try to make Newtonian gravity relativistic.

You can get light deflection in Newtonian gravity, if you simply treat light as a material particle that is initially moving at the speed of light. The deflection in a 1/r2 field ends up being off by a factor of 2 compared to the relativistic result. In this context, I think it makes more sense to talk about linearized gravity, which does give the right deflection, and does assume a Minkowsian background.

 

[MaxwellsDemon]

One of the reasons I ask is because it might be easier to come up with a quantum theory of gravity if you take this sort of an alternative route...maybe its not as conceptually beautiful GR on a big scale, but might lead to a better understanding of quantum phenomena and the relationship between the four forces. I haven't gotten very far in MTW yet, I started with Wald.

 

[atyy]

"The one exception is Nordstrom’s 1913 conformally-flat scalar theory, which can be written purely in terms of the metric; the theory satisfies SEP, but unfortunately violates experiment by predicting no deflection of light."
http://relativity.livingreviews.org/Articles/lrr-2006-3/

"This can now be used to rewrite the Nordstrom gravitational equation as R = 24.pi.G.T"
http://arxiv.org/abs/gr-qc/0405030

See also the mention of Nordstrom's theory and the Ehlers and Rindler reference here:
http://www.einstein-online.info/en/spotlights/equivalence_deflection/index.html

 

[atyy]

One of the reasons I ask is because it might be easier to come up with a quantum theory of gravity if you take this sort of an alternative route...maybe its not as conceptually beautiful GR on a big scale, but might lead to a better understanding of quantum phenomena and the relationship between the four forces. I haven't gotten very far in MTW yet, I started with Wald.

Weinberg takes this route, as does Feynman.

http://arxiv.org/abs/gr-qc/9607039
The Quantum Theory of General Relativity at Low Energies
John F.Donoghue

So there is a quantum theory of gravity at low energies. The only question is whether the UV completion requires introducing new degrees of freedom.

If the answer is no, then this is called Asymptotic Safety.
http://relativity.livingreviews.org/Articles/lrr-2006-5/

If the answer is yes, then string theory is an exploration of this possibility.

 

[meopemuk]

Have there ever been any attempts to create a theory of gravity within the context of a flat 4-D Minkowski spacetime? Rather than taking the general relativistic route of gravity being a manifestation of curved spacetime geometry, has anyone historically ever attempted to put the more "classical" ideas of gravitational fields and potentials within the context of flat spacetime of special relativity? Certainly it wouldn't be as beautiful conceptually as GR, but I would imagine it could be done. I wonder what sorts of predictions such a theory would make and how they would differ from observed phenomena...

Try http://arxiv.org/abs/physics/0612019 and Chapter 13 in http://arxiv.org/abs/physics/0504062. All known relativistic gravitational effects can be reproduced.

Eugene.

 

[Bob_for_short]

Have there ever been any attempts to create a theory of gravity within the context of a flat 4-D Minkowski spacetime?

Yes, the Russian academician A.A. Logunov has worked out such a theory - RTG. There are his articles on arXiv on this subject. His theory describes all know experimental data and predicts something interesting in respect of black holes (there is none) and the Universe evolutions.
The attractive features of his RTG are:

1) There are additive conservation laws,

2) The gravitational filed is as physical as the EMF and other fields.

 

[bcrowell]

One of the reasons I ask is because it might be easier to come up with a quantum theory of gravity if you take this sort of an alternative route...maybe its not as conceptually beautiful GR on a big scale, but might lead to a better understanding of quantum phenomena and the relationship between the four forces. I haven't gotten very far in MTW yet, I started with Wald.

I wouldn't recommend starting with Wald. He makes essentially no contact with experiment, and the emphasis is all on making everything look mathematically elegant, without ever explaining what it means. For a first book, you might be better off with Rindler. MTW also does significantly better than Wald at maintaining contact with reality.

 

[bcrowell]

His theory describes all know experimental data and predicts something interesting in respect of black holes (there is none) and the Universe evolutions.

There are actually lots of different theories that predict something other than a Schwarzschild-style black hole, but the chances of distinguishing one from the other observationally seem pretty slim. Is the cosmological stuff anything that could be tested?

The gravitational filed is as physical as the EMF and other fields.

Doesn't this imply that it violates the equivalence principle?

 

[AEM]

MTW is getting extremely out of date, but section 39.2 says that the only theories that were consistent with observations in that era were metric theories, the only exception being Cartan's theory involving torsion.

I find your comment about MTW interesting. Since it was published in 1973, or thereabouts, and was written before that, it is certainly dated. I'm curious, what book(s) would you consider to be up to date in general relativity?

 

[bcrowell]

I find your comment about MTW interesting. Since it was published in 1973, or thereabouts, and was written before that, it is certainly dated. I'm curious, what book(s) would you consider to be up to date in general relativity?

Carroll, Spacetime and Geometry: An Introduction to General Relativity, 2003 (There's a free online version, too.)

Schutz, A First Course in General Relativity, 2e, 2009

Rindler, Relativity: Special, General, and Cosmological, 2006

I've read Carroll and Rindler, but not Schutz.

 

[George Jones]

I find your comment about MTW interesting. Since it was published in 1973, or thereabouts, and was written before that, it is certainly dated. I'm curious, what book(s) would you consider to be up to date in general relativity?

I can't speak for bcrowell (I see that he replied while I was composing this post), but I don't think that there is any recently published, up-to-date general relativity book that is comparable to MTW. MTW, with its two tracks, is both introductory and advanced, and, because of its length, MTW has both breadth and depth. A real masterpiece.

One area where MTW has fallen quite out-of-date is cosmology, so the cosmology section of any recently published general relativity book will be more up-to-date.

At the late undergraduate level, I usually recommend Gravity:An Introduction to Einstein's Relativity by James Hartle. At the beginning graduate level, General Relativity: An Introduction for Physicists by Hobson, Efstathiou, and Lasenby or Spacetime and Geometry: An Introduction to General Relativity by Sean Carroll (widely use as a text, I think). Carroll's book, unusually for an introductory book, has an introductory chapter on quantum field theory in curved spacetime.

Weinberg's latest book is an advanced book about the recent developments in cosmology.

I'll think some more about this.

 

[granpa]

http://arxiv.org/abs/physics/0408077

Henri Poincare and Relativity Theory
Authors: A. A. Logunov
(Submitted on 17 Aug 2004 (v1), last revised 6 Jul 2005 (this version, v4))

Abstract: The book presents ideas by H. Poincare and H. Minkowski according to those the essence and the main content of the relativity theory are the following: the space and time form a unique four-dimensional continuum supplied by the pseudo-Euclidean geometry. All physical processes take place just in this four-dimensional space. Comments to works and quotations related to this subject by L. de Broglie, P.A.M. Dirac, A. Einstein, V.L. Ginzburg, S. Goldberg, P. Langevin, H.A. Lorentz, L.I. Mandel'stam, H. Minkowski, A. Pais, W. Pauli, M. Planck, A. Sommerfeld and H. Weyl are given in the book. It is also shown that the special theory of relativity has been created not by A. Einstein only but even to a greater extent by H. Poincare. The book is designed for scientific workers, post-graduates and upper-year students majoring in theoretical physics

 

[atyy]

http://arxiv.org/abs/astro-ph/0006423
Reflections on Gravity
Norbert Straumann
"Although this field theoretic approach, which has been advocated repeatedly by a number of authors, starts with a spin-2 theory on Minkowski spacetime, it turns out in the end that the flat metric is actually unobservable, and that the physical metric is curved and dynamical. "

 

[hamster143]

Carroll, Spacetime and Geometry: An Introduction to General Relativity, 2003 (There's a free online version, too.)

Schutz, A First Course in General Relativity, 2e, 2009

Rindler, Relativity: Special, General, and Cosmological, 2006

I've read Carroll and Rindler, but not Schutz.

Schutz is an elementary book, there's nothing in it that would've been "too new" for MTW.

Wald is a fairly modern book and it's a must-read, even if it appears a bit dated (published in 1984), there weren't many significant advances in GR since then. I would've liked a book that expounds some more on 3+1 decomposition and Hamiltonian approach (since those are the basic tools at the foundation of LQG), and talks about work by Hawking, Penrose & such since 1984, and perhaps a word or two about numerical GR, but I'm not aware of any such books.

I'm not familiar with Carroll, I've just looked through Rindler's table of contents and it does not look like it's any more "modern" or even as complete as Wald.

 

[Bob_for_short]

http://arxiv.org/abs/physics/0408077

Henri Poincare and Relativity Theory
Authors: A. A. Logunov
(Submitted on 17 Aug 2004 (v1), last revised 6 Jul 2005 (this version, v4))

Yes, but it is not RTG. It is a tribute to the academician H. Poincaré whose continuous work in this filed resulted in formulation of all SR stuff: principle of relativity as the experimental fact (Saint Louis, 1904, USA), mechanics and electrodynamics, including all 4-invariants, etc.

 

[George Jones]

I would've liked a book that expounds some more on 3+1 decomposition and Hamiltonian approach.

Take a look at chapter 4 in Eric Poisson's excellent notes,

http://www.physics.uoguelph.ca/poisson/research/agr.pdf,

but this might be too elementary. Also, Chapter 3 on hypersurfaces and junction conditions is not treated in very many elementary books.

These notes evolved into the excellent book A Relativist's Toolkit: The Mathematics of Black Hole Mechanics.

 

[MaxwellsDemon]

Doesn't this imply that it violates the equivalence principle?

I'm guessing that it would violate the equivalence principle...in this sort of "flat spacetime with a gravitational field" theory I would imagine that the equivalence of gravitational and inertial mass is still viewed as a happy little experimental coincidence.

 

[Bob_for_short]

There are actually lots of different theories that predict something other than a Schwarzschild-style black hole, but the chances of distinguishing one from the other observationally seem pretty slim. Is the cosmological stuff anything that could be tested?

Eventually, maybe.

I answered the OP question as "yes, it is possible and done." Comparison of different theories is another question.

Doesn't this imply that it violates the equivalence principle?

It "violates" the total geometrisation of acceleration and gravity. The gravity in RTG is just an effective Riemann space-time for matter but the true space-time is the Minkowski's one. Its metric is non-trivially decoupled from the graviational field in the field equations.

 

[Bob_for_short]

http://arxiv.org/abs/astro-ph/0006423
Reflections on Gravity
Norbert Straumann
"Although this field theoretic approach, which has been advocated repeatedly by a number of authors, starts with a spin-2 theory on Minkowski spacetime, it turns out in the end that the flat metric is actually unobservable, and that the physical metric is curved and dynamical. "

The flat Minkwski space-time has a group of symmetry leading to ten additive conservation laws for matter and gravity field. It is not so in any curved space-time, especially in dynamical where the gravity is not a field at all.

 

[hamster143]

Take a look at chapter 4 in Eric Poisson's excellent notes,

http://www.physics.uoguelph.ca/poisson/research/agr.pdf,

but this might be too elementary. Also, Chapter 3 on hypersurfaces and junction conditions is not treated in very many elementary books.

These notes evolved into the excellent book A Relativist's Toolkit: The Mathematics of Black Hole Mechanics.

Straumann (2004) looks interesting. Newer and more comprehensive than Wald. Not sure if it's worth the price, though (MSRP $109, vs. Wald's $48).

 

[heldervelez]

http://arxiv.org/abs/gr-qc/0210005"
Logunov, 256 pages

In the framework of the special theory of relativity, the relativistic theory of gravitation (RTG) is constructed. The energy-momentum tensor density of all the matter fields (including gravitational one) is treated as a source of the gravitational field. The energy-momentum and the angular momentum conservation laws are fulfilled in this theory. Such an approach permits us to unambiguously construct the gravitational field theory as a gauge theory. According to the RTG, the homogeneous and isotropic Universe is to be ``flat''. It evolves cyclewise from some maximal density to the minimal one, etc.
The book is designed for scientific workers, post-graduates and upper-year students majoring in theoretical physics.