Talk:Einstein–Cartan theory

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[User: R. J. Petti, rjpetti@alum.mit.edu]

[edit] This page contains/contained simple mistake

The original article talked about affine differential geometry. This is not what should have been written. The article is about affine connexions and Euclidean differential geometry. In affine differential geometry we introduce volume forms on a manifold instead of metrics. We look for volume forms which are compatable with certain connexions. In Euclidean differential geometry we introduce metrics on a manifold which are compatible with certain connexions. Euclidean differential geometry and affine differential geometry are VERY different; I should know: my PhD was about the affine setting. I have corrected the author's mistakes. [User: D. Davis, d.davis@liv.ac.uk]

As a geometer, I would like to comment that affine geometry can mean different things. The user D. Davis is studying differential geometry of manifolds where the relevant structure group is SL(n). In Cartan's theory, relevant for GR, the affine group is the semi-direct product of the translation group with the orthogonal group (Lorentz group), although sometimes geometers use the whole GL(n). The word affine connection is also used to mean slightly various things in the text books. In Kobayashi-Nomizu it just means a connection on the tangent bundle with structure group GL(n), which can have torsion, but there is also a notion of an affine Cartan connection, which has structure group the semi-direct product mentioned above. It comes equipped with a soldering form and a preferred reduction to the subgroup GL(n). Parallel translatoin with respect to a Cartan connection is an affine map, not just a linear map of the tangent space (It does not map the zero vector to the zero vector). In fact, it rolls the tangent space along the curve. This is the one needed here for the EC version of GR with torsion. The holonomy of a Cartan connection around a small loop has both a translational part (controlled by the torsion) and a rotational part (controlled by the curvature). The second Bianchi identity which plays a role for conservation laws involves both curvature and torsion. (The Levi-Civita connection is the unique metric preserving one with zero torsion!) I am just saying things that are well known to differential geometers, but perhaps the article is not very clear about the mathematical background. I would leave it to the physicists to decide whether it is a important conceptual improvement on standard GR, although torsion does play a role in supergravity. —Preceding unsigned comment added by Bodomar (talk • contribs) 05:40, 31 March 2008 (UTC)

[edit] I don't think this page needs work.

(Except for removing the table of alternative gravity theories at the end.)

I wrote virtually all of the article on Einstein-Cartan theory. Thanks to the people who improved the formula formatting. I think it is fine as of Nov. 4, 2006.

I had to correct or delete several content edits by other people. Requested actions:

(1) Remove the comment that this page needs work, or tell me what the needed changes are.

(2) Remove the table of alternative gravity theories at the end of the article on Einstein-Cartan theory. The table says nothing about EC theory, and it misrepresents EC theory as one of a large group of speculations about gravitation. The main point of the article is that EC theory is the only extension of non-quantum general relativity to have been been proven necessary since about 1920. Who should remove this table? If no one answers in a reasonable time, I will remove the table from the article on EC theory.

(3) If you want to change this page, please contact me first. [User: R. J. Petti, rjpetti@alum.mit.edu] Rjpetti 04:50, 22 December 2006 (UTC)

I would like to respond to another year of accumulated commentary below.

Until now I have focused my comments on the mathematics and physics. However, few people respond to that, and instead write conjectures about my motivation, extraneous ideas, and anything but the relevant mathematics and physics. So let me recount the human side of this story.

At the 1989 GRG meeting in Colorado, I approached some people with my 1986 proof that EC is a necessary extension of GR. Three people understood the proof (though without all the epsilon-delta convergence arguments worked out in detail).

1. Mauro Francaviglia of Universita di Torino, who was head of the meeting session on alternative gravitation theories, understood the proof after 45 minutes of discussion. At the end of our discussion I asked him, "Can anyone understand GR and this (my 1986) paper and not accept EC?" He responded that it was not possible.

2. Jean Krisch of University of Michigan understood the proof after ten minutes because years of working with torsion had convinced her that EC is correct, but she had not seen a proof. She said this was the most important work in the foundations of relativity she had seen in decades, and other remarks that I shall not repeat.

3. Yuval Ne'man understood it after five minutes of hallway discussion. Later he wrote me that he recognized that EC was no longer a unproven conjecture, and that the work "is of the highest quality and I have often quoted your results", and other comments I shall not repeat.

Francaviglia gave me strong individual introductions to about a half dozen other people at the meeting. Not a single one of them understood my explanation. I tried to approach other people who might be interested; not one of them gave me any indication that he understood it.

Lee Smolin expressed the following ideas about string theory research, as recounted in the article "Unstrung" by Jim Holt, in The New Yorker, October 2, 2006: "The initiators of the dual revolution a century ago-Einstein, Bohr, Schrödinger, Heisenberg-were deep thinkers, or 'seers.' They confronted questions about space, time, and matter in a philosophical way. The new theories they created were essentially correct. But, Smolin writes, 'the development of these theories required a lot of hard technical work, and so for several generations physics was 'normal science' and was dominated by master craftspeople.' Today, the challenge of unifying those theories will require another revolution, one that mere virtuoso calculators are ill-equipped to carry out."

The comments by Smolin about string theory are relevant to EC as a change in the foundations.

[User: R. J. Petti, rjpetti@alum.mit.edu] Rjpetti 08 January 2008 —Preceding unsigned comment added by 207.172.130.220 (talk) 19:09, 8 January 2008 (UTC)

[edit] Record of previous claims that this article needs work

[edit] Needs clarification

Needs clarification, elaboration (maybe split since already long), links to related articles.

Reply: Many links are inserted. If you want more, please add them or tell me what you want. [User: R. J. Petti, rjpetti@alum.mit.edu]

[edit] Needs less technical introduction

Reply: It would be nice, but I think it is optional. This article is of interest only to people who know general relativity fairly well. Please suggest changes. [User: R. J. Petti, rjpetti@alum.mit.edu]

[edit] Citation format

Citations should conform with WikiProject GTR.---CH (talk) 01:39, 15 September 2005 (UTC)

Reply: I don't know your conventions and I could not easily follow your link to find out, so I stopped pursuing this. [User: R. J. Petti, jpetti@alum.mit.edu]

[edit] Flaws in non-quantum general relavitity?

And I just noticed "one known flaw"--- I wish! See objections to general relativity and especially its talk page (the article itself is currently terribly imbalanced and rather incoherent).---CH (talk) 01:41, 15 September 2005 (UTC)

Reply: Please tell me what are the other known flaws in non-quantum general relativity. I know only of the inability of GR to handle spin-orbit coupling, which is fixed by EC theory. [User: R. J. Petti, rjpetti@alum.mit.edu]

[edit] C Code

I have found that the equation can be represented by a C code with

INCLUDE (objects), Main (Kernel, singularity) and a brace ( for C code molecules).

(unsigned comment by 71.131.226.30)

I hope you realize why this statement doesn't make any sense as it stands. But maybe you already knew that when you wrote it... ---CH (talk) 22:24, 9 November 2005 (UTC)

Reply: These comments seem misplaced. There is no mentione of C code in the article on Einstein-Cartan theory.

[edit] O(p,q) or SO(p,q) or neither?

An anon changed SO(p,q) to O(p,q) on the grounds that the latter is the orthogonal group, while the former is usually called special orthogonal group. In fact, even without reading the article one has to suspect that the connected component of the orthogonal group might be the group intended. Someone with more time and energy please figure it out and fix the problem! ---CH 16:50, 14 March 2006 (UTC)

Reply: O(p,q) is fine. The point is minor. [User: R. J. Petti rjpetti@alum.mit.edu]

[edit] Work of Myron Evans

VOLKER204@CS.COM A dispassionate analysis of Myron Evans' work on this is needed. As Einstein showed gravitation is curved spacetime, Evans indictes electomagnetism is spinning spacetime. Geometrically, it makes good sense. Which equations will prove to be absolutely true remains to be seen.

Reply: What is the work of Myron Evans? This article in EC theory has nothing to do with electromagnetism, except that electromagnetism fits into EC theory as well as it fits into general relativity. [User: R. J. Petti rjpetti@alum.mit.edu] Rjpetti 05:12, 22 December 2006 (UTC)

Reply: Agreed. The editors of the Myron Evans arcticle make him out to be some kind of a quack, yet his work is similar (at least to a layman like myself) to the subject matter in this arcticle. I'm sure some of the editors are the same for both arcticles. This is a place for free information exchange and debate. This isn't a scientific journal, and careers aren't on the line. This can be an anonymous forum. The Myron Evans arcticle is link page to dogmatic scientific rhetoric written by people who either don't understand his work or think they have something to prove. They do not believe in true scientific advancement let alone healthy debate.

I just took a cursory look at Myron Evans' webpage. He's definitely a crackpot who knows how to copy equations from articles and textbooks and use jargon but doesn't know what he's really doing. It's true that he copied equations from articles on EC theory, but he clearly doesn't know what they mean. AnonyScientist (talk) 12:31, 27 December 2007 (UTC)

[edit] Importance Rating

To my knowledge, EC theory is now (2006) the only change to classical (i.e. non-quantum) general relativity to be proven necessary since about 1920. GR must be extended to EC theory to correctly describe the currents of translational and rotational spacetime symmetry, that is, momentum and spin currents. GR is not a closed theory without extending it to EC theory -- you can obtain EC theory solutions as a limiting case of GR solutions for rotating black holes, without any further assumptions. (See my paper of 1986 in the references.)

We are in a period in the history of physics when basic research results are so driven by aesthetic considerations that there are few testable hypotheses to evaluate theories, in particular string theory. EC theory is one of the few recent cases in which one of the foundational theories of physics has been shown to require an extension or amendment.

Proposed direct empirical tests of EC theory involve factors on the order of 10^-40. Empirical evidence may require a venue with very high spin densities. I don't think the problem of finding empirical indirect indications of EC theory has gotten much attention because so few physicists know EC theory is proven, or know even the mathematics of Riemann-Cartan geometry, that is, Riemannian geometry extended to include affine torsion.

I think these factors should be taken into account in setting an importance rating for EC theory. I don't know what rating to give it because I am not calibrated on how you make those decisions.

[User: R. J. Petti rjpetti@alum.mit.edu] Rjpetti 05:12, 22 December 2006 (UTC)

[edit] Work still needed?

As a geometer with interests in math-phys, I like very much EC theory, and the idea to couple spin to gravity via torsion. This is a nice article advocating the merits of this point of view. However, I think it should remain tagged to encourage input from other experts in relativity to add context, background, and a more neutral point of view. Let me answer the specific questions above.

(1) This article needs a more neutral tone, and more explanation. Wikipedia is an encyclopedia. Articles should describe and explain, rather than promote a particular point of view. The question above "Is there really only one classical flaw in GR?" is valid because other opinions to the one of this article seem fairly reasonable. For example, "spin is not a classical concept anyway", or "GR predicts singularities at which it breaks down". The wikipedia advice would be: "Let the theory speak for itself". There is no need to make bold assertions like "general relativity must be extended to Einstein–Cartan theory" or "The best way to formulate...". Additionally, better explanation is not "optional". Admittedly, the interested layman may not get much from such a specialized topic, but some paragraphs of motivation would help. Also, a wide range of physicists and mathematicians could be interested.

(2) I agree the table of alternative gravity theories should be removed, but I suspect whoever placed it there had a similar point to make about the tone/neutrality of this article. I suggest that somewhere near the beginning, this article should state that EC theory is one of many extensions/modifications of GR and link to alternatives to general relativity. It may be the nicest, most natural such classical extension, but again, let the theory speak for itself.

(3) No, this is not how wikipedia works. When you work on an article here, it is never set in stone, and may be modified by anyone at any time, or even replaced entirely. That is one of wikipedia's strengths. There is no editor-in-chief of an article. If you want to post your own definitive summary of EC theory, put it on a preprint archive, or some other forum.

I have a couple of technical questions/comments too.

The notion of indices associates with flux boxes seems obscure to me. Can it be explained in more detail?
I came to this article from Cartan connections, and indeed they are not far beneath the surface in phrases like "The translational part of the affine connection acts like an (inverse) frame field" and "General relativity without translational connection coefficients (which would introduced affine torsion into the theory)...", but these ideas are far from completely explained, and the article on affine connections is no help from this point of view.

Geometry guy 21:12, 14 February 2007 (UTC)

I shall attempt to address the two questions posed by Geometry Guy in the language of differential geometry.

Given a differential k-form (that is, an antisymmetric covariant k-tensor), is exterior derivative (which is a k+1 form) can be defined at a point x as follows. Compute The integral of the k-form over the boundary of a small k+1 coordinate box at x, divided by the k+1 volume of the box, in the limit as the k+1 box gets arbitrarily small. This limit defines the exterior derivative in the k+1 form applied to a tiny k+1 surface element (antisymmetric contravariant k+1-tensor) that lies in the same hyperplane as the family of small k+1 boxes. The k+1 indices on the exterior derivative tell you in what k+1 hyperplane the k+1 boxes sits. This construction is a limit of Stokes' theorem as the volume of the box of integration goes to zero. In words, the exterior derivative of a k-form (where a k-form is viewed as a flux through k-surfaces) is the surface-flux-per-unit-volume of a tiny k+1 box small.

Cartan connections provide the "right" way to view spacetime translations in general relativity, but especially in EC theory, where translational symmetry is handled better than in general relativity. See my paper that explains this as the conclusion of 50 years of somewhat confused discussion in the research literature about how to include translational spacetime symmetries in general relativity (by solving the problem in EC theory). Many published research papers about translational symmetry violate a theorem in Kobayashi and Nomizu's "Foundations of Differential Geometry." Reference: Petti, R. J. (2006): "Translational Spacetime Symmetries in Gravitational Theories," Class. Quantum Grav., 23, 737-751. [User: rjpetti rjpetti@alum.mit.edu] July 8, 2007.

[edit] gauge theory section

besides being badly formatted and hard to understand, is it written by (a follower of) Jack Sarfatti? I'd vote for deleting the whole section 2. - Saibod 19:45, 1 July 2007 (UTC)

Not only is the section on gauge theory badly written, but the topic is irrelevant to the foundations of EC theory to the first (and second) order. A key point about EC theory is that the need to include torsion arises in classical GR without spinors, variational principles, etc. So I removed this section. - Richard Petti rjpetti@alum.mit.edu 7 July 2007

Einstein-Cartan does not arise to specifically resolve various problems in classical GR so much as to provide a gravitational theory on a more complete foundation -- namely one that incorporates both a metric and connection as independent objects. The ability to resolve the problems it does (e.g. providing a curved-space continuum version of the spin-orbit decoupling) are just added bonuses.

Both the Sarfatti material and the original article need cleaning up on a crucial point being obscured or dealt with in a roundabout way: namely, that Einstein-Cartan (or any metric-affine gravity theory) serves to complete the analogy (???):Riemannian = Affine Space:Vector Space = General Affine Group:General Linear Group.

Metric-affine gravity treats the manifold as one whose tangent spaces are affine spaces, rather than just vector spaces (i.e., what's called an affine bundle). This is the point that's being obscured. The connection + frame is therefore just a connection for the affine group; particularly, it's a Cartan connection. Both of these concepts (Cartan connection, affine group) are described on other Wikipedia pages, and need linking to.

Sarfatti's point about linking the kinematics of this field to a gauge field are dead-on, and are part of the standard material that's acrued around Einstein-Cartan (e.g. Hehl). The two Cartan structure equations are just the equations relating the potentials to the field strengths, for the Cartan connection. For pure gravity, the spin tensor and (assymmetric) Einstein tensor arise naturally as the derivative of the Lagrangian with respect to the two parts of the Cartan connection.

If nothing more, the Trautman article (arXiv:gr-qc/0606062 v1) really ought to be linked to. It also contains indirect reference to the "equation of state" (section 3.1) which relates the stress tensor to the Einstein-Cartan stress and spin tensors; as well as the related issue of describing how Einstein-Cartan decomposes into Riemannian gravity + extra theory for torsion (and/or contorsion) as well as what's non-trivial about the decomposition (i.e., energy positivity in EC can lead to non-energy-positive sources in the Riemannian part of the decomposition). Trautman also brings into focus what the empirical issues are: the spin-spin coupling term and the energy non-positivity issue, and resultant loopholes in the singularity theorems. —Preceding unsigned comment added by 64.24.187.111 (talk) 21:29, 13 December 2007 (UTC)

[edit] Biased crap

Am I the only person who sees something wrong with the fact that the author of this article, who is controlling it with an iron fist, is the most-cited person in the article? This is a classic example of someone ringing their own bell, to the potential exclusion of reality. This article is F-level at best, and needs to be totally rewritten. As it exists now, it's massively POV, though it would take someone pretty well-versed in Einstein's field equations to see it. --75.58.54.17 19:42, 27 October 2007 (UTC)

On top of being POV, the article is also OR. This article practically defines the phrase "synthesis of published material designed to advance a position" and calling EC theory proven and necessary is undeniably a novel narrative, since nobody else seems to find any problem with GR. --75.58.54.17 20:00, 27 October 2007 (UTC)

Wikipedia is not a publisher

Wikipedia is supposed to compile human knowledge. It is not a vehicle to make personal opinions become part of human knowledge. In the unusual situation where the opinions of an individual are important enough to discuss, it is preferable to let other people write about them.

Wikipedia is not a soapbox

It can be tempting to write about yourself or projects you have a strong personal involvement in. However, do remember that the standards for encyclopedic articles apply to such pages just like any other, including the requirement to maintain a neutral point of view, which is difficult when writing about yourself. Creating overly abundant links and references to autobiographical articles is unacceptable.

Wikipedia is not a textbook

The purpose of Wikipedia is to present facts, not to teach subject matter.

"There is a qualitative theoretical proof showing that general relativity must be extended to Einstein-Cartan theory when matter with spin is present."

This is POV, and opinion. "Qualitative theoretical proof" is a triple oxymoron. Mathematical consistency does not imply physical reality. I'm also unaware of what someone is attempting to refer to as qualitative mathematics.

"Therefore general relativity cannot properly model spin-orbit coupling."

This article is framed based on how its well-accepted alternative theory (allegedly) fails, rather than how this theory succeeds. This is an attack article, and is therefore politics.

This article, as written, should not exist. Delete it, delete it, delete it. Alternatives to general relativity do not deserve equal treatment on Wikipedia. Wikipedia is not a democracy. This is minority opinion fringe theoretical physics, and it is not noteworthy until or unless it has been accepted by the scientific community. This has not. --75.58.54.17 17:50, 1 November 2007 (UTC)

[edit] This article is complete nonsense

People had many confusions around 1920. But all those confusions have gone away. Statements that general relativity suffers from a "flaw" because it cannot describe some couplings and therefore must be extended by adding new fields are absolutely absurd. No serious physics paper in the last 70 years used anything like this theory. This theory has been falsified for 70 years and only plays a historical role. All tests of gravity, general relativity etc. ever made only refer to the normal gravity and general relativity without torsion etc. Just search for einstein-cartan-theory at scholar.google.com, for example, to see that there are also no well-known articles that would talk about this bizarre topic. The article must be rewritten as a history-of-physics article. Best wishes, Lubos Motl --Lumidek 07:55, 6 November 2007 (UTC)

Basically, whenever you add a term like $|e|\overline{\Psi}e^{\mu}_a\gamma^a\left( \partial_\mu - \frac{i}{4}\omega_{ab\mu}\sigma^{ab} \right){\Psi}$ to the GR Lagrangian density, we get EC theory. So, you've been working with EC all along!!! It's just that you don't realize it. Here's an easy one-minute exercise for you; take the GR action with a Dirac term and vary with respect to the spin connection and see what you get as the Euler-Lagrange equation. AnonyScientist (talk) 12:24, 27 December 2007 (UTC)

[edit] Don't dismiss but put in proper perspective instead

As I see it, this article does hold value, and should not be dismissed too lightly. At the same time, it will benefit from some further enhancements.

[edit] General suggestions

Primarily present Riemann-Cartan theory as a bona fide mathematical theory of geometry, more general than (pseudo-)Riemannian geometry (which it is).

Present it subsequently as a candidate for generalization of General Relativity (see next section for some hints); the references indicated below already go in this direction. (Appropriate formulation to be found, so as to avoid counterproductive controversy.)

In order to mitigate the justified suspicion of self-indulgence on te part of the principal author, I suggest at least to add some further references, which might include:

- Ne'eman, Y.; Hehl, F.W.: Test Matter in a Space-time with Nonmetricity. Classical and Quantum Gravity, 14 (1997), A251-A259.
- Hehl, F.W.; Mielke, E.W.; Tresguerres, R.: Skaleninvarianz und Raumzeitstruktur. In Geyer, B.; Herwig, H., Rechenberg, H. (Eds.): Werner Heisenberg. Physiker und Philosoph, p 299-306. Spektrum Akademie Verlag, Heidelberg, 1993.
- Hehl, F.W.; McCrea, J.D., Mielke, E.W.; Ne'eman, Y.: Progress in Metric-affine Gauge Theories of Gravity: Field Equations, Noether Identities, World Spinors and Breaking of Dilation Invariance. Physics Reports, 258 (1995), 1-171.

+ others. In fact, there exists a substantial body of work on this by Friedrich Hehl, with various co-authers, in the period 1970 up to this day.

And also:

- Debeyer, R.: Elie Cartan - Albert Einstein. Lettres sur le Parallelisme Absolu 1929 - 1932. Académie Royale de Belgique et Princeton University Press, Brussels, 1979.
- Vargas, J.G.; Torr, D.G.: The Cornerstone Role of the Torsion in Finslerian Physical Worlds. General Relativity and Gravitation, 27 (1995), 629-644.
- Trautman, A.: Foundations and Current Problems of General Relativity. In: Trautman, A.; Pirani, F.A.E.; Bondi, H. (Eds.): Brandeis Summer Institute: Lectures on General Relativity, p 1-248. Prentice Hall, 1964.
- Trautman, A.: On the Einstein-Cartan Equations, I - III. Bulletin de l' Académie Polonaise des Sciences. Série des Sciences math., astr. et phys., 20 (1972), 185-190, 503-506, 895-896.
- Hehl, F.W.; Heinicke, Ch.: Über die Riemann-Einstein-Struktur der Raumzeit und ihre möglichen Gültigkeitsgrenzen, Philosophia naturalis, Band 37, Heft 2 (2000)
- Hehl, F.W.; Obukhov, Y. N.: Foundations of Classical Electrodynamics: Charge, Flux, and Metric. Birkhauser, Boston, 2003.
- Hehl, F.W.; Obukhov, Y. N.: Elie Cartan's torsion in geometry and in field theory, an essay; arXiv:0711.1535v1.

etc.

This said, the article should also make clear that while (as argued below) a worthwhile area of complementary research etc. etc., Riemann-Cartan geometry has not superseded Riemannian General Relativity.

[edit] Value of the theory in a broader perspective

Ideally, this topic should be presented as part of a more general overview of "space-time geometries" (including Weyl and Finsler geometries), indicating motivations, merits and failures of such attempts. Failure typically being lack of experimental evidence or indeed evidence to the contrary.

A scheme of such general geometries is given by Proberii (Metric-affine Scale-covariant Gravity. General Relativity and Gravitation, 26 (1994), 1011-1054) as: general Affine / Weyl-Cartan ("geodesic lightcone") / Weyl (zero torsion) / Riemann-Cartan (zero Weyl vectorfield) / Riemann (no torsion, no Weyl) / Minkowski (zero curvature).

Space-time geometries with torsion (as is the case with Riemann-Cartan geometry) have been investigated not only with the aim of including a spin contribution to energy-momentum, but also to check for space-time models suitable at the nano-scale, which is dominated by quantum effects, and even as an approach to the quantization of space-time itself.

Further theoretical interest in this model resides in the fact that a (Poincaré) gauge field formulation is possible, which necessarily leads to torsion (the gauge potentials may be interpreted as curvature resp. torsion of a Riemann-Cartan space. These models are sometimes referred to as Einstein-Cartan-Sciama-Kibble theories.

One series of variants of space-time geometries with torsion, has been initiated by Trautman. A spin density is introduced, with the explicit aim of avoiding singularities. If not a "flaw", these are at least a nuisance in standard Riemannian spacetimes, where "physics breaks down".

Based on correspondence between Einstein and Cartan, Torr/Vargas have also pointed out the fact that Finsler geometries generally exhibit non-zero torsion.

Apart from all this, there is an important foundational aspect to considering more general space-time geometries: the choice of a 4-dimensional pseudo-Riemannian manifold as "the" model for space-time leaves many physical questions unanswered.

There have been many attempts to give a more explicit, physically motivated axiomatics of space-time, in which the Riemannian choice is then derived (instead of postulated at the outset). Famous contributions have been made by Reichenbach, Synge, Ehlers-Pirani-Schild and many others.

As "physical" postulates are added, the generality of the geometry is restricted further and further. It appeared especially difficult to motivate why a Weyl structure should ultimately "collapse" to the Riemannian one, without recourse to quantum mechanical considerations. This was finally achieved in the period 1995-2000 by Schröter and Schelb. (Schröter, J.; Schelb, U.: Remarks concerning the Notion of Free Fall in Axiomatic Space-Time Theory. General Relativity and Gravitation, 27, 1995, 605 + further publications by these authors.)

[edit] Conclusion

From the above, it should at least be clear that the topic is meaningful also in mathematical and foundational physics, and a part of the ongoing research on the nature and modeling of space-time. So definitely not nonsense.

And let's not forget that several (often more radical) generalizations / modifications of the classical space-time model are currently being investigated actively and in earnest by many renowned scientists. They do not shun such things as complexifications ("imaginary time"), change of metric signature, change of topology, twistors etc. - --Marc Goossens 14:23, 2 December 2007 (UTC)

[edit] Not all matter theories with nonzero spin

The main author has claimed that EC is needed for all GR theories coupled to matter fields with a nonzero spin. Technically, this is not correct. It's only needed for GR theories with matter fields that couple to the spin connection. This does not include Maxwell and Proca fields, for instance.

AnonyScientist (talk) 11:54, 27 December 2007 (UTC)

[edit] Moved from Torsion field

The following recent addition to Torsion field seems more suited to inclusion here. However, it needs wikification, so I have added it here to the talk page instead of the article. silly rabbit (talk) 21:32, 18 March 2008 (UTC)

[edit] Curvature and Torsion from Local Gauge Invariance

T.W. Kibble in 1961 ("Lorentz invariance and the gravitational field", J. Math. Phys. 2 (1961) 212-21) showed that both the curvature and torsion fields are compensating gauge potentials from locally gauging the 10-parameter universal Poincare space-time symmetry group of Einstein's 1905 Special Relativity for the global actions of all matter fields. The equivalence principle is embodied in the universal tetrad coupling to the matter fields (classical and quantum). For example, given a 1905 SR first-rank tensor Aa in globally flat Minkowski space-time, where a = 0,1,2,3 the corresponding 1916 GR tensor in locally curved space-time is Au = eu^aAa where eu^a are the 16 tetrad components. The 4 tetrad Cartan 1-forms are then e^a = e^aue^u where eu is a local basis in curved spacetime, e.g. e^u -> dx^u is a coordinate basis of 1-forms. These General Coordinate Invariant (GCI) tetrads split as e^a = (Minkowski)^a + B^a, where B^a is the compensating inertial field tetrad corresponding to a Local Non-Inertial Frame (LNIF - Misner,Thorne, Wheeler "Gravitation") from the local gauging. Indeed, B^I is analogous to a Yang-Mills vector field for a non-Abelian internal symmetry group. Einstein's 1916 GR has the zero torsion field constraint imposed ad-hoc. This means that the 6 spin-connection 1-forms S^a^b = - S^b^a are entirely determined from the 4 tetrad 1-forms as shown in eq. (2.89) of Rovelli's "Quantum Gravity." Although the curvature corresponds to a rotational disclination defect (see Hagen Kleinert's webpage from Free University of Berlin) in parallel transport of vectors around an infinitesimal parallelogram of one infinitesimal displacement parallel transported by another, and torsion corresponds to a 2nd order translational gap in that parallelogram, nevertheless, the curvature in Einstein's 1916 GR comes only from locally gauging the 4-parameter translational subgroup of the 10-parameter Poincare group. One gets the dynamically independent torsion field in the spin-connection only by locally gauging the 6-parameter homogeneous Lorentz subgroup of the Poincare group. This was done by Utiyama ("Invariant Theoretical Interpretation of Interaction" (1956) Phys. Rev 101, p.1597) without locally gauging the 4-parameter translation group. That is, the GCT transformations of Einstein's 1916 GR are put in by hand ad-hoc. The dynamically independent spin connection in this case of course still induces a curvature. That is, the curvature field 2-form is R^a^b = dS^a^b + S^ac/\S^cb, and the torsion field 2-form is T^a = de^a + S^ac/\e^c, where the Einstein-Hilbert action density is proportional to R^a^b/\e^c/\e^d contracted with the completely antisymmetric tensor {a,b,c,d}. Note also that Einstein's fundamental invariant is ds^2 = guvdx^udx^v = (Minkowski)abe^e^b .

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