Talk:Einstein–Cartan theory

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This page contains/contained simple mistake[edit]

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 compatible 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,]

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 translation 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 (talkcontribs) 05:40, 31 March 2008 (UTC)

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

(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 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 08 January 2008 —Preceding unsigned comment added by (talk) 19:09, 8 January 2008 (UTC)

I see some troubling edits creeping in.

(1) In the last section (GR plus matter with spin implies EC), I found this very troubling statement: "Of course, it is possible to get rid of the torsion by fiat if we simply introduce a Lagrange multiplier term, but even that would not be stable under the renormalization group once quantum effects are taken into account."

Using ordinary assumptions of classical physics, you cannot get rid of the torsion in the simple case of a classical spin fluid. (Example of a classical spin fluid: Consider a continuum approximation to a distribution of rotating galaxies as a classical fluid with spin.) That is what [Petti 1986] establishes.

This statement is similar to saying "We simply differentiate and the problem goes away" without saying what is differentiated with respect to what. Whew.

I took the liberty of removing this statement. If anyone believes something remotely resembling the original statement belongs in the article, please contact me at

(2) In the opening section, I found this statement: "Basically, it is just a fancy name to describe general relativity with a nonzero torsion."

We might was well say that "Basically, electromagnetism is just a fancy name for a theory that combines electric and magnetic fields." This statement contributes nothing and should be removed. (Also, the comma in the preceding sentence should be removed.)

Unfortunately, I cannot edit this section, which is under the control the someone appointed by Wikipedia.

(3) In the section "Motivation", I found this statement: "In general relativity, the Einstein curvature tensor models local gravitational forces, and it is equal (up to a gravitational constant) to the stress-energy tensor (also called the energy-momentum-stress tensor since it contains all such)"

I changed this to "In general relativity, the Einstein curvature tensor models local gravitational forces, and it is equal (up to a gravitational constant) to the momentum tensor P[a,b]."

"(The momentum tensor is also called the stress-energy tensor, the energy-momentum tensor, and the energy-momentum-stress tensor. Special relativity shows that energy, momentum, momentum flux, and stress are different spacetime components of the same covariant object, known most compactly as the momentum tensor.)"

(4) The next sentence says, "The symmetry of the Einstein curvature tensor forces the momentum portion of the tensor to be symmetric."

The "momentum portion" certainly should include the momentum density; it might include the momentum flux; it might also include the stress, which are fluxes of momentum. It might also include the energy if you mean 4-momentum. Also, all this 3-D thinking obscures the main point.

I changed this to "The symmetry of the Einstein curvature tensor forces the momentum tensor to be symmetric."

(5) The section title "Geometry and Formulation" is trying to say something but fails.

I changed the title to "Geometric Foundations".

I have not taken the time to review the entire article. This article has been damaged numerous times by irrelevant advanced mathematics. Classical physics and classical Riemann-Cartan geometry are the only ingredients needed to establish the main conclusion.

This article poses a challenge to Wikipedia's presumption that everyone can edit things in a democratic fashion. This policy must be balanced by reliance on qualified experts in areas like this one. This article is worth getting right: Einstein-Cartan theory is the only extension to general relativity to be proven necessary in a master theory of classical (non-quantum) physics since the cosmological constant was added around 1920 and established experimentally in the 1990s with the discovery that the expansion of the universe is accelerating. Finding a relativity physicist who understands EC theory is not easy. I would suggest approaching the editors of the journal "Classical and Quantum Gravity" and asking for help.

Rjpetti1 (talk) 21:52, 2 August 2008 (UTC)

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

Needs clarification[edit]

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,]

Needs less technical introduction[edit]

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,]

Citation format[edit]

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,]

Flaws in non-quantum general relavitity?[edit]

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,]

C Code[edit]

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
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 mention of C code in the article on Einstein-Cartan theory.

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

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]

Work of Myron Evans[edit]

VOLKER204@CS.COM A dispassionate analysis of Myron Evans' work on this is needed. As Einstein showed gravitation is curved spacetime, Evans indicates electromagnetism 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 05:12, 22 December 2006 (UTC)

Reply: Agreed. The editors of the Myron Evans article 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 article. I'm sure some of the editors are the same for both articles. 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 article 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)

Importance Rating[edit]

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 05:12, 22 December 2006 (UTC)

Work still needed?[edit]

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.

  1. The notion of indices associates with flux boxes seems obscure to me. Can it be explained in more detail?
  2. 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] July 8, 2007.

gauge theory section[edit]

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 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 accrued 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 (asymmetric) 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 (talk) 21:29, 13 December 2007 (UTC)

Biased crap[edit]

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. -- 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. -- 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. -- 17:50, 1 November 2007 (UTC)

If the author of the section above will contact me with an open, informed and capable mind, I will explain why general relativity and matter with intrinsic angular momentum together force you to accept Einstein-Cartan extension of general relativity. It has been an amazing journey for me to hear the reactions of people who make such basic assessments based on what other people think, and do not think through the arugments themselves. This is the road to stagnation and belief without reason. Whew!

Richard Petti email:,

Rjpetti1 (talk) 03:53, 28 December 2008 (UTC)

This article is complete nonsense[edit]

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, 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)
I agree that tetrads are mainstream; they were taught to us in 1980/81 (by Michael Duff who is certainly mainstream and non-fringe) - my understanding is that torsion is unavoidable when spin is included in GR. --Michael C. Price talk 09:59, 1 November 2010 (UTC)

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

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.

General suggestions[edit]

  • 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 the 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-authors, 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.
  • 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.

Value of the theory in a broader perspective[edit]

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.)


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)

Not all matter theories with nonzero spin[edit]

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)

See the section of the article on "two types of spin" that has been added since this comment was written. Spacetime spin to be different from the spin that results from just counting the quantum numbers in representations of the symmetry groups of tensor fields. rjpetti Rjpetti (talk) 23:42, 23 December 2010 (UTC)

Moved from Torsion field[edit]

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)

Curvature and Torsion from Local Gauge Invariance[edit]

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 .

In the 1960s Sciama and Kibble were very excited about Einstein-Cartan theory, and tried to persuade people to take it seriously. The root problem was that they did not (to my knowledge) have a derivation of Einstein-Cartan theory from general relativity. Therefore the lack of experimental evidence and its reputation as a failed theory - worse, a nutball theory that tarred anyone who touched it - derailed their efforts. rjpetti Rjpetti (talk) 23:46, 23 December 2010 (UTC)

Problem With Spin Orbit Coupling?[edit]

The article says something along the lines of GR not modeling spin-orbit coupling, and further claims this is the only flaw with GR as a classical theory. I am a graduate student starting my PhD and hence I am not at the level of some other conversations here I admit, but this seems odd to me. I have never encountered spin-orbit coupling in any classical context. I actually just recently came across an exercise that shows that symplectic formalism of classical mechanics can describe spin quite easily. However a mathematical description of something and its being observed in reality are two different things. What does spin angular momentum mean in a classical context?

Classical spin means angular momentum that is contained in some structure that is too small to be explicitly modeled in a classical model you are examining. Consider a cosmological model that includes a distribution of galaxies that have correlated spin. A simple example of spin in classical physics is a fluid model of this configuration. If galaxies exchange orbital angular momentum with the angular momentum of the fluid, then you have classical spin-orbit coupling in this model. rjpetti Rjpetti (talk) 00:00, 24 December 2010 (UTC)

There was also some very poor writing on the page in general as well as some odd (non standard) usage of terms. As an example: the Noether current of the rotation symmetry of space time is referred to as orbital angular momentum in most works on (say) field theory that I've seen. If you look at just the spatial coordinates, then this is the Noether current associated with the rotation of space. This, as any third year physics undergrad will tell you is obviously angular momentum. That's why its usually called the generalized density of (orbital) angular momentum. This Noether current comes from a symmetry in spatial coordinates; spin in quantum mechanics is itself dissociated from the coordinates. It can interact with other wave functions and impact their dependence on coordinates, but in and as of itself the space of spins is separate from real space. So it seems odd to talk about a spin current resulting from something which is coordinate dependent.

A blackbelt in continuum mechanical engineering (not a third year student, and probably not any blackbelt in the field) can tell you that if a spinning gyroscope or distribution of gyroscopes transfers its rotational angular momentum to the surrounding medium, then, on a scale where the gyroscopes are represented as spinning particles, spin orbit coupling has occurred, and the stress tensor will be nonsymmetric during the exchange process. I found the mechanical engineering literature more helpful than the physics literature on this point. rjpetti Rjpetti (talk) 00:11, 24 December 2010 (UTC)

Anyhow I don't claim to be an expert on GR, but this article:

1) Needs to use standard physics terminology and language, 2) Either is saying some things that are untrue or is explaining them very poorly.

PS Telling me that I don't know enough to understand this article is moot, I admit I don't know enough to go through the math or understand every detail. But when a physics graduate student can't even read the opening paragraph of a physics article without statements that seem blatantly false jumping out at him, something is wrong either in veracity or explanation. Nirf (talk) 21:09, 10 July 2008 (UTC)

Torsion, Quantum Entanglment, EC theory[edit]

While not being a geometer - many of the mathematical details are over my head - I think the critics of this article are objecting to its neutrality mainly based on it disagreeing with what they were taught, and not on good foundational arguments. I'll try to argue my point based more on a visual approach than a mathematical symbolic approach because the symbolic approach is short-hand for the visual approach anyway.

Angular momentum is the only way for a object to increase its kinetic energy classically without translational movement. Quantum mechanical "spin" seems to be just a formalization of the classical idea of angular momentum, but described in static, or alternatively, stairstep situations. However, classically if one accelerates an object in space gradually it gains energy gradually and the gravitational field likewise increases. Thus mass change can be seen as a gradual change in angular momentum caused by acceleration. This can be done in a gradual manor without quantum jumps occurring. GR describes this as resulting in a bending of space but an equally good description of it is as photon transfer from quantum vacuum energy to the object in a spherical shell of reduced vacuum energy surrounding the particle. It conserves the total energy of the universe. Otherwise one could have many high kinetic energy objects in close proximity with no consequence to the total energy of the universe. Energy has to be injected into the bodies and it must come from somewhere. Otherwise the universe would not have causality. The vacuum energy provides it and it is experienced as gravity.

However in quantum mechanics entanglement can be seen as a situation where gravitational fields around massive particles that are "rapidly" separated do not increase their gravitational fields. This is a proven phenomenon when photons are split in two when moving through a partially silvered mirror. They then exhibit identical properties, including spin, polarization etc., even though they can be far removed from each other. This is a case where the magnitude of the angular momentum vector does not increase as the two photons separate from each other, or AM does not increase requisite to the acceleration. Contrary to popular opinion entanglement is almost never perfect. A tunneling effect does occur though, which is simply another name for torsion. This is not something exotic - it is involved in creating the original mass of all fundamental particles. It is also why a single photon does not exhibit mass - there is no energy encapsulated in space. A single photon really is a point charge. However the energy in the path between two entangled photons does exhibit all the properties of mass. The energy expended in the "torsion fiber" connecting them is what creates mass. In the case of entangled photons the energy encapsulated in this fiber is so low it will quickly decay and become un-entangled. But that is not what happened in the early universe where it is speculated that the big bang accelerating energy itself created the tunneling effect required to create protons, etc. There is nothing unusual or exotic about torsion. It is what occurs whenever the local quantum vacuum cannot provide sufficient energy for the change in angular momentum needed for the acceleration produced. It creates a skidding effect on the angular momentum of each particle during acceleration and a subsequent perfect tunneling path between the two particles. (In the case of single photons angular momentum would be more akin to changes in angular velocity, or polarization, because each photon does does not really have any mass, just charge energy.)

Scientists should just get over thinking about torsion as being Einstein's failed attempt at unification. Its really just a form of propaganda propagated by entrenched interests in the science community. Instead they should think about it as the precursor to the ultimate combining of GR and quantum mechanics through the proven concept of quantum entanglement. An open mind is a wonderful thing. You guys should try it. You might like it. (talk) 01:37, 30 July 2008 (UTC)

Actually quantum "spin" is not literally angular momentum at all-it is something completely different and very weird. [1] (talk) 21:04, 12 April 2010 (UTC)

On non-NPOV controversy[edit]

I think the POV controversy for this subject is invalid. The individual(s) that are complaining about Petti's neutrality base it on 2 things.

1.Einstein-Cartan theory is tied to a famously failed unification scheme.

2.Petti has cited several of his own published articles in prestigious journals.

I'll address both things consecutively. Regarding the history of "public" opinion regarding EC there is no doubt that the jury verdict is in. But I regard the verdict as being as tainted as one produced by a jury of racist white southerners judging a black man in the 1920s South. The historical narrative of Einstein wasting the last half of his life on this is well known but is only half the story. Einstein was working on a geometric approach to unification that took its cue from both Kalusa-Klein theory of 5 dimensional space (one for spin) and from Cartan's math. I feel there is nothing in Einstein's approach that bears reproach except that he was just too early.

During the midst of this work the Quantum revolution was in full swing. The mathematical results from that revolution were stunning and revolutionary. But more than that they were totally counter-intuitive. Einstein objected vociferously about this counter-intuitiveness, especially the Copenhagen Interpretation which suggested that there is no reality apart from that which is observed. Einstein was a man on his own following his own path but the air was being sucked out of the room for him by all the energy and excitement going into this quantum revolution.

The problem is this era was the very beginning of a steady march in discovering the encyclopedia of particles we now know. And these sub-atomic particles were all tied to quantum theory. So the very scientists who were making these discoveries, not all of them though, put it out through the grapevine that they were making progress, and because Einstein complained vociferously about the counter-intuitiveness of QM, that Einstein was going nowhere. And the press and then succeeding generations of scientists bought it, hook, line, and sinker. Einstein said he was 75 years ahead of his time regarding this work and I believe he really was.

Today the steady progress of discovering new subatomic particles has ended and we are back to working on the exact same thing Einstein was working on 75 years ago - combining gravity and quantum mechanics. The steady progress from the quantum revolution ended 25 years ago and it brought us no closer to combining these two things. Yet Einstein's ideas from torsion and teleparallism brought the idea of quantum entanglement, which Einstein himself introduced in his EPR paper in 1935. And entanglement seems to be at the heart of both QFT and the idea of what makes particles have mass and how that mass in turn creates gravity. Yet still we seem to have individuals complaining about this article, kissing up to the scientific establishment and complaining about Einstein getting more respect for these ideas than he is due. What is even worse, by misrepresenting the value of these precursor ideas to quantum entanglement they erroneously misrepresent EC theory as being a dead end. It becomes the situation where a very likely suspect in a crime is mistakenly taken off rhe suspect list and the crime goes unsolved for decades. That is what has been happening in physics. Instead of actually investigating the very promising possiblilities of adding just one more dimension - spin - physicists have gone off on a wild goose chase of working on hyperdimensional space in string theory- which has produced nothing valuable in physics. (math might be a different matter)

Regarding point 2 of Petti citing his own papers: Sometimes a person has to take the bull by the horns and use his own expertise and previous work to right a long standing wrong. I'm sure if there were more courageous individuals in the recent past who were also willing to work on this scientifically "unacceptable" subject he wouldn't have found the need. While I don't understand all the math, and cannot say whether there are errors, the complainers about this article will have to do a better job of explaining themselves on a mathematical basis. If they don't, then I will take their previous criticism as being invalid and based on their own prejudices and will remove the criticism hovering above the header of the article. (talk) 05:32, 14 August 2008 (UTC)

Timetable for resolution of POV Controversy[edit]

I have not yet seen any objections to the latest rebuttal of the POV controversy. It may be that opposition to removing the POV banner has just not yet been formulated. If that is the case then I don't want to remove the POV criticism if there are justifiable arguments to retain it, but I don't think it should be left in indefinitely if there are not arguments. So if there are no reasonable and logical arguments on this discussion page by Sept 15 against removing it then I will remove the POV banner from the article.

I would also be interested in hearing from "impartial" physicists on the accuracy of the article, both mathematically and otherwise. An "impartial" viewpoint on this particular subject may be difficult to come by because this area of physics has framed no negatively in popular culture for so long. But we can still try to find those special people who are both impartial and knowledgeable. It is possible that Mr. Petti's mathematical argument is entirely accurate in the article. However, it would be good to also find these rare individuals who could confirm or deny the accuracy of different parts of the article, and possibly contribute to this controversial subject. (talk) 19:43, 20 August 2008 (UTC)

Personally, I think you are being too generous. One week from the time you asked for an explanation should be sufficient. JRSpriggs (talk) 21:25, 20 August 2008 (UTC)
I don't want someone to come back with an excuse that they were in Timbuktu and incommunicado and then have the excuse that my timeline was arbitrary. Just because I think the naysayers' to this article haven't been very fair doesn't mean I should be unfair. Don't worry, I can recognize B.S. counterarguments if I see them and will act accordingly. (talk) 20:53, 21 August 2008 (UTC)

No, really, this is POV[edit]

I'm quite interested in Einstein-Cartan theory: the curvature tensor used in string theory includes torsion (as derived from the 2-form B-field), so I'd like to better understand what that torsion might mean and my impression is that this is what EC-theory is all about. I'm far from an expert on the topic, but I've looked into it a little. But even to my relatively inexpert eye there are many portions of the current article that seem strongly POV. (For the record, I formed that impression well before I saw that a primary author of this article was writing about his own work.)

The first and perhaps most glaring example of this that I noticed was the paragraph where the article claims that the 50 year controversy over the foundations of the theory had been completely resolved by a paper from just three years ago. Is there nobody worth noting who still thinks the controversy is unresolved? Even if not, shouldn't at least some attention be given in the article to what this controversy was and to the (formerly) competing approaches to resolving it? The fact that the very next paragraph here begins by asserting that "The best way to formulate Einstein–Cartan theory is..." (emphasis mine) really brings the POV aspects of the article home.

Even without such glaring examples, though, the tone of much of the article feels closer to advocacy than encyclopedic. Most of the "Geometric insights" section, for example, feels like it would be more at home in the Conclusions section of an individual published paper than in an encyclopedia article. (I don't know that these claims are not standard parts of the EC-theory literature, but I have some mild familiarity with the topic and I had not heard them before. They are certainly not broadly known and accepted among physicists working in closely related fields.) The third "insight" in that section leaves me feeling particularly uncomfortable: I haven't read the references in order to judge the math behind this claim, but it sounds entirely unlike what I know of standard GR (despite the fact that it presumably ought to reduce to GR in the limit of zero torsion). More broadly, given the somewhat controversial (or perhaps more accurately, disregarded) status of EC-theory in the physics community as a whole, I'm uncomfortable that this article makes no mention of that controversy but rather presents its (recent!) conclusions in the tone of long-established scientific fact.

As noted, all my comments above were based only on my reading of the article itself, without knowledge of its authorship. Discovering on this talk page that the same person whose methods and conclusions are described here as "the best" actually wrote much of the article just strengthened my overall impression. I expect that Rjpetti means well and is making an honest effort to share recent progress in his field of expertise. But this article as it stands is not acceptable, and I think that contributors here should be extremely cautious about making more than cosmetic edits to articles dealing directly with their own outside work.--Steuard (talk) 00:36, 28 May 2009 (UTC)

I appreciate your well reasoned argument. It's a refreshing change from "remove this BS immediately" as in previous comments. There are definitely problems in the article and in the tone of the article. Being the one who has defended Petti in the past I will try to explain my problems with the POV tag in this particular case, which is somewhat different from the usual case.
Previous discussion here for the POV tag centered primarily on vague arguments with a bullying tactic to to just remove certain things. I found it very disrespectful to Petti and also very mean spirited. Most of those attacks piggybacked on EC theory being a discredited theory in the public press, but never analyzed the actual ideas in a coherent way. While the author may have approached the subject in an unconventional way I felt that the need for a serious analyzing of the theory was the most important thing. In other words, the subject is so latent with previous superficial baggage that I wanted to encourage an article in which the subject was taken seriously, if imperfectly. What seemed to be happening otherwise is that the bullying crowd in physics, and believe me there is a group that does that, would just marginalize the theory as "a failed effort at unification".
So basically what I'm saying is Petti was doing a real service just to take the subject seriously and not take the easy route that 90% of physicists take. I thought, and still think, he should get credit for that. So I would like to oppose the idea that this article is just POV. I actually believe that the conventional view of completely writing off EC theory is the real POV that is without merit. So yes, Petti may have a dog in this fight that is not the most desireable situation, but on the whole it is counterbalanced by the need to oppose the very superficial conventional wisdom being propogated about EC theory. I don't want to give some of these bullies who just jump on each physics bandwagon more ways to do damage than they already have. Most physicists just take it as fact that there is nothing of substance that can be taken from EC, because that is what they were taught, sort of like Iraq being linked to Al Qaida. Even physicists are just too gullible. I think most of them don't know what EC really is except that its been discredited. It's similar to string theory being the only game in town just because experts have declared it so. As I said, just like everyone else physicists often are just too credulous. (talk) 23:52, 4 June 2009 (UTC)
But surely you can see that what you're arguing for is that this should be taken seriously as a subject of research, not that this Wikipedia article should present Petti's view as an NPOV description of the subject? I was a bit surprised to see the article claim that matter with spin requires this extension to GR. Given that we know there is matter with spin and GR is still used, this simply can't be accepted as fact by a majority of physicists. (talk) 15:34, 18 July 2010 (UTC)
Yes, the critical point is that the article asserts that EC is "proven". Thank you for drawing attention to this fact. For this reason, I have added a brief section on the appropriate meanings and standards of proof in different parts of mathematics and physics. I also added a section on the two types of spin, to address the comments about Maxwell fields not coupling to torsion. As these comments point out, using veirbeins to make a fundamental distinction between theories is a pedestrian way to talk about fiber bundle structure of theories without getting to the foundations of the theory.
I have added in the discussion of proof a paragraph that says the validity of the claimed proof is considered a matter of controversy by some physicists, and I listed what I think are the main objections.
A quarter century ago (as of 2011), I published a proof of the type described in the section that I just added to the article. I have discussed the proof with numerous people at a GR conference and by mail. Except for comments that there is no experiemntal evidence (which have merit), and some comments leading to refinements (such as making explicit the two types of spin), I have not heard much that directly addresses the key issue. rjpetti Rjpetti (talk) 10:55, 2 December 2010 (UTC)
I noticed that the actor inserting the POV tag is a string theorist. Having myself noted that string theory itself is not rigorous in many ways I will have to say that the POV tag is being placed by someone who has his own POV issues, i.e. this insertion is not coming from a scrupulously unbiased person. Because the insertion of a POV banner in an article is so damning I have concluded that only a person without a dog in the fight should issue it and am removing it. Because one doesn't like an article because it conflicts with your own views does not entitle one to issue such a damning proclamation as the article being POV. The bullying tactic taken up by string theorists to squelch all other opinions cannot be tolerated in Wikipedia. (talk) 20:30, 6 June 2009 (UTC)

Just as a small observation, I am a physics grad student, and have a very hard time taking this article at all seriously because of its bizarre enthusiasm for what is essentially the tetrad formalism, dressed up in such a way as to indicate that Petti thinks he rediscovered it all on his own. For example "The best way to formulate Einstein–Cartan theory is to distinguish between tangents to the spacetime M and tangents to an associated flat affine fiber space, X" is an extremely strange way of saying that it is useful to distinguish between internal Lorentz indices and spacetime indices. And the whole section on "Third geometric insight" is similarly laughable, again espousing an extreme amount of enthusiasm for this "insight" that local frames and the frame bundle can play a role in the formulation of general relativity. The amount of detail Petti goes into, and his extremely unconventional way of framing what is actually a very simple formulation of general relativity, makes me think that he must have no idea what he is talking about. It gives the article (which is basically self-promotion, as far as I can tell) very much the usual crackpot flavor of "I've not looked at what anyone else has done in the area, but guys, look, I solved this huge problem that everyone's stuck on!" So while I don't particularly care to critique this article in-depth or get in an extended war about how to fix it, I thought I'd at least pop onto the talk page and give a perspective on why I (and probably anyone else with background in the field) cannot really see this article as anywhere near reasonable.

If someone more dedicated to Wikipedia than I am cares to reform this article, a good place to start would be Hammond's 2002 review paper on torsion gravity. Domenic Denicola (talk) 05:53, 31 July 2009 (UTC)

Horrible article[edit]

Honestly, this is the worst article I've ever seen at Wikipedia. This does not seem like an encyclopedia article, but rather a bunch of aimless, incoherent rambling on heterodox ideas.

It does not help that the grammar is atrocious. For example, from the very first sentence, we learn "Einstein–Cartan theory ... extends general relativity to by handle spin angular momentum". But what does it mean to extend something "to by"? If this page had legs, it would try and kick the reader in the nuts. Doubledork (talk) 20:45, 4 August 2010 (UTC)

Yes, horribledly written. But EC / tetrads is mainstream and deserves a thorough treatment. It was taught to us in 1980/81 (by Michael Duff who is certainly mainstream and non-fringe) - my understanding is that torsion is unavoidable when spin is included in GR. --Michael C. Price talk 09:47, 1 November 2010 (UTC)

Yes, the introduction is quite terrible (and I wote most of the article) - it is "a bunch of aimless, incoherent rambling on heterodox ideas". One of serious problems is the truly incompetent edits that continue to accumulate. I have ignored this for about two years, but I will straighten out the introduction again...

I have just re-edited the introduction to focus on the context-setting information that is appropriate for an introduction. I moved the information that is irrrelevant to EC to a new section at the end entitled "Other basic physical theories that employ affine torsion". This article continues to be a learning experience for me in the kind of fierce resistance that new ideas can bring, even when accompanied by a mathematical proof published 24 years ago. Very interesting. rjpetti Rjpetti (talk) 19:25, 28 November 2010 (UTC)

Removed section on hints of discreteness in microphysics[edit]

The section on discreteness in microphysics contains insights for futher research. It is not central to an article on EC theory and it seems to be causing controversy. So I chose to eliminate it. With this deletion, I would like to know in what areas the article needs attention from experts, other than the common assumption that EC is a failed theory. Rjpetti (talk) 03:41, 15 March 2011 (UTC)


This article is tainted. Even now, Rjpetti appears to be the most prolific editor. Five years ago he wrote: "I don't think this page needs work. [...] I wrote virtually all of the article [...] If you want to change this page, please contact me first." which demonstrates clear misunderstanding of the mission here (no page is above further improvements, and no individual may interpose themselves as gatekeeping authority). But the more serious problem is that Rjpetti's enthusiasm for his or her own original work has undermined the credibility of this article. That is an unfortunate shame because the topic still sounds potentially interesting.

  • The article (especially the lead) needs to talk far more about how the topic is perceived by experts in related fields. (Non-bias has to be seen to be done. If this really is uncontentiously the true successor of GR, quote the recent textbooks that say so. No, didn't think so, so explain the mainstream viewpoint, what would their rationalisations be for holding back?)
  • The whole section defining the word "proof" needs to go. It might be relevent to an article titled proof, but it makes this article look crackpotty.
  • If there is some historic controversy, then add a section to explain and detail it, don't try to ignore the elephant in the room. If the most widely noted use of this title and concept was in a since-falsified model, describe it and how it was similar or different, when did this happen, what part was falsified, etc. Clarify which should be understood by the term when it is encountered in the literature (as used by authors other than rjpetti obviously).

Cesiumfrog (talk) 00:59, 23 May 2011 (UTC)

(Comment on the opening paragraph:) Five years ago this article attracted many irrelevant and nonsensical edits, plus a few valid points that required clarifications now included in the article. This problem ended only when the editor entered the discussion on August 14, 2008 with a masterful summary of historical and sociological issues surrounding EC in the scientific community, and a request that at least some criticisms of the article address the mathematics and physics.

I understand that Wikipedia prefers multiple authorship, and authors that are not writing about their own work, to filter out low-quality work and to get an objective point of view. As I read it, the editor's comment August 14, 2008 took the point of view that Wikipedia has grown in stature to the point where this general filter about authoring should not be decisive in cases like this.

The key scientific issue is whether EC is one of many speculations surrounding GR, or whether it is proven that GR plus spinning matter imply torsion, and in the most important cases, torsion enters the field equations exactly as in EC. The proof uses only methods of classical differential geometry, and it can be summarized in three sentences.

(1) Compute the translational holonomy for an equatorial path around a rotating black hole.
(2) Take the limit of very many very small rotating black holes to get a continuum limit.
(3) The result is that GR yields torsion and the field equations of EC.
  • (Response to comment on "perception by experts":) The article states "For decades, Einstein–Cartan theory was considered one of many speculative (and largely ignored) extensions of general relativity..." I believe the article deals with all the objections to the proof that GR and classical spin imply torsion and EC. These objections are:
  • EC is a known failed theory.
  • There is no empirical evidence for EC. The discussion of the different standards of proof in mathematical physics, directly addresses this issue.
  • The section "Two types of spin" addresses another objection, due to ambiguity in the meaning of spacetime spin.
  • Spin and spin-orbit coupling are quantum phenomena, not classical phenomena, so EC is not relevant to classical GR. However, the relativity literature and classical continuum mechanics include much work on classical spin.
  • (Response to comment on the section on proof:) I changed the section header and first line to "Standards of proof in mathematical physics", which I hope better conveys the key idea.
The section referred to briefly introduces the standards of proof in mathematical physics that apply in this situation. It does not define the word "proof," nor is it complete enough for an article about standards of proof. It states which standard this proof meets (derivation by symbolic computation) and which standards it does not meet (empirical evidence, and rigorous functional analysis). This section is necessary to address the objection regarding empirical evidence.
  • (Response to comments on historical controversy:) The article plainly states that this theory was considered a speculation for decades, and it discusses all objections to the proof that I know of. Perhaps to some people this sounds like an elephant in the room, but it sounds quite blunt to me.
It may be possible to rationalize that EC is not included in basic texts on GR. However, it is not possible to rationalize that review articles about EC do not mention the proof that EC is a necessary extension of GR in the presence of classical spin. See for example the article in The Encyclopedia of Physics by A. Trautman (Oxford: Elsevier, 2006, vol 2, pp 189-195, The final paragraph of that article says, "The Einstein-Cartan theory is a viable theory of gravitation ... It is possible that the Einstein-Cartan theory will prove to be a better classical limit of a future quantum theory of gravitation than the theory without torsion." This review article does not mention the classical derivation of EC from GR with spin. If there is a scientific reason for this omission, I am sure all of us would like to know what it is.
I have discussed the material in the EC article with numerous gravitational researchers, and I get four types of response.
  • The proof is valid and it is a very important result. (4 including Yuval Ne'eman in writing)
  • The proof is valid and EC is not important. (1)
  • A brush-off without discussing the proof. (1)
  • A plea of ignorance of EC and non-Riemannian geometry.
I am unaware of any substantial controversy about EC in the open literature since Sciama and Kibble wrote about it more than 40 years ago. The only controversy of sorts is that the proof that GR plus spin implies EC has been met with silence in review articles of EC, and I do not know why. While I don't regard this as an acceptable situation, the fault does not lie with this article or the refereed publications on which it is based. I have previously suggested that the editors of Wikipedia approach the editors of Classical and Quantum Gravity for their opinion of the EC article, specifically the proof that GR plus classical spin implies torsion and EC. We might speculate that omission of the proof in reviews of EC may be due to sociological issues in the scientific community. For example, in August 2008 the editor commented on the need for courage to do original research on EC; and other speculations are possible. If the omission is due to a valid scientific issue, then the EC article should include it. If there is no valid reason for the omission, then the EC article should include this in the historical discussion. The Wikipedia article on EC covers the relevant scientific issues. I do not feel it is appropriate to speculate on sociological issues in that article. Rjpetti (talk) 04:18, 3 July 2011 (UTC)

Merge articles on Einstein-Cartan theory and Einstein-Cartan-Sciama-Kibble theory[edit]

I agree with the suggestion to merge the Wikipedia articles on Einstein-Cartan theory (EC) and Einstein-Cartan-Sciama-Kibble theory (ECSK).

I believe the term that has been adopted in the literature is "Einstein-Cartan theory". For example the article on EC in the Encyclopedia of Physics (2006) uses only the term "Einstein-Cartan theory".

Two references in the ECSK article were not already in the EC article. So I added these references to the EC article under "further reading." Rjpetti (talk) 04:18, 3 July 2011 (UTC)

Thanks for handling my proposed merge. The article needs extensive work. So far I've given a stab at a more direct introduction with in-line citations. I definitely agree that the "standards of proof" section needs to go. At best, it's preachy and off topic. Teply (talk) 19:57, 27 August 2011 (UTC)

Thanks for rewriting the lead, Teply! Now, it appears (confusingly) that there may be two distinct strands of EC theory discussed in this article: the widely-known historic version, and the (less widely known) new version that rjpetti champions; would that be a fair distinction to make in the organisation of the article (to compartmentalise all of the material about rjpetti's new version into one section)? Cesiumfrog (talk) 02:38, 28 August 2011 (UTC)
Wow, after rewriting a whole bunch of this article, I have found that it is even more messed up then it looks at first glance. I feel like blindly deleting half of it, but I am worried about slighting others' contributions just in case they happen to be better at explaining this really difficult subject to a general audience. Maybe I'll back off for a while now that I've set things up better for the next editor. Cesiumfrog, I'm not entirely sure what you mean by rjpetti's version. The citations I placed in the intro belong to what is usually meant by Einstein-Cartan theory. As for the rest of the article's wild ramblings, which without looking closely I presume are mostly rjpetti's contributions, I'm going to have to assume good faith. Most likely it is a semi-lay person's politicized "spin" on an otherwise perfectly legit theory well known to the gr-qc community. Teply (talk) 04:02, 28 August 2011 (UTC)

I conjecture that a correct translation of the first sentence of the previous paragraph would be: "Wow, after rewriting a whole bunch of this article, I have found that it is even more messed up than it was before."

The comments above about "wild ramblings" are typical of the ad-hominem remarks that have been so prominent in this discussion for the past five years, in places where more comprehension is needed.

The judgment of what is "semi-amateur" should be used with some care. I believe in this context it turns on two issues.

  1. The roles of computational tools and geometric insights. As Lee Smolin has pointed out, in the twentieth century the physics community focused so much on computations, that it is now weak on the deep insights and concepts needed to make fundamental advances. This phenomenon is consistent with the inclination of some professionals to replace the deep geometric insights with formulas from tensor calculus.
  2. What is accepted as conventional wisdom. As I recall, as late as 1920, even special relativity was considered so semi-amateur that the physics community could not bring themselves to mention the term in Einstein's Nobel Prize award. As Thomas Kuhn pointed out (in The Structure of Scientific Revolutions), sometimes the only solution for conventional wisdom is death.

If you are interested in the mathematics and physics, please read the following section "Progress summary" on the state of the article. Rjpetti (talk) 04:22, 3 September 2011 (UTC)

Progress summary[edit]

The edits in July and August remove many of the insights that make EC comprehensible from a geometric point of view, and that lie behind the proof that GR plus spin imply EC based on classical differential geometry.

  1. The current section on "Geometric insight" makes a very nice contribution about how torsion twists pencils of geodesics and how this relates to spin. However, this contribution is offered as a replacement for all the geometric insights listed below that have been removed.
  2. The edits eliminate the geometric insight about inhomogeneous linear symmetry groups; the interpretation of torsion as translational curvature, and Riemannian curvature as rotational curvature. Instead it treats torsion as a combination of connection coefficients that happens to be antisymmetric, though that is the way many authors treat it. (By the way, the term Christoffel symbol is used incorrectly. This term is used for the connection coefficients only in the case of a torsion-free metric connection.)
  3. The edits eliminate the distinction between base space and fiber tensor factors (tensor indices in the vernacular), the flux box meaning of all antisymmetric combinations of base space indices; the conserved current meaning of all fiber indices; the meaning of all antisymmetric differential operators and divergences as boundary-flux-per-unit-volume in the limit of small volumes. Worst of all, it eliminates the insight that base space tensor factors should never be covariant differentiated. In a deep sense it is just wrong to do this, though it is possible to do a ham-fisted computation that takes covariant derivatives of base space indices, then cancels out all the nonsensical extra terms to get the answer from vastly over-complicated and nonsensical computations.
  4. The edits eliminate the section on two types of spin (representation spin and spacetime spin), which is necessary to distinguish tensor indices that couple to torsion and those that do not. This distinction is yet another corollary of the difference between base space and fiber tensor indices.
  5. the edits eliminate the discussion of spin (including classical intrinsic angular momentum) as dislocations, which is the first successful extension of Einstein's program to (differential-) geometrize more of classical physics beyond gravity since GR itself. Also the interpretation of GR itself in terms of a continuum version of disclinations.
  6. The edits eliminate mentioning that EC can derive momentum as the variational current (Noether current) of translational symmetry; that GR has no satisfactory derivation of this fact because it lacks the translational connection coefficients to vary in the first place.
  7. The section "Field Equations" makes no mention of variation with respect to (translational and rotational) connection coefficients to get the equations of EC. This is vastly superior to varying with respect to the metric. Also, the momentum tensor derived from varying connection coefficients (called P) is the wise one, and the one derived from varying metric tensor is an unwise choice. Here are some reasons why: (a) The deep structure of EC is an affine theory, not a theory based on a metric tensor and torsion that falls out of some equation. (b) The connection with Ashtekar variables is lost; the Asktekar variables are just the translational and rotational connection coefficients rediscovered by GR researchers.
  8. The deleted discussion of the kinds of proof in mathematical physics is important to the proof that GR + spin implies EC because the most common objection is that there is no experimental evidence.
  9. The edits eliminate all discussion of fiber bundles, which is the central advance in differential geometry between 1920 and 1970. The entire GR community has very few people who even passively understand fiber bundles, and improving this situation would in itself be a substantial contribution to the GR community. But all of this is erased, at least in Wikipedia.
  10. The section "Mathematical background" is the usual way that people with little knowledge of post-1940 differential geometry approach connections and covariant derivatives. I get the feeling that no one who is editing this article has read and understood either of the great treatises on differential geometry by Bishop and Crittenden or by Kobayashi and Nomizu, or any other works, that base the entire field on fiber bundles.
  11. the edits eliminate the interpretation of covariant differentiation in terms of parallel translation, and the interpretation of curvature as holonomy-per-unit area. This is the deep idea that makes the proof possible by starting with discrete holonomy and ending with torsion and EC.
  12. Collectively, the deleted ideas and methods listed above made the computations in the proof simple enough to be done manually without computer algebra systems. I agree with the earlier comment that computations in EC are "too hard" (and too senseless I would add) – if you ignore the ideas and methods described above. (Since 1998, the defunct commercial version of Macsyma can do the key computations in the proof in 3 seconds on my vintage 2006 laptop).

This has been a fascinating journey. The signature fact is that, a quarter century after EC was proven a necessary extension of GR, the article on EC in The Encyclopedia of Physics does not mention this fact. Also, the geometric insights that make the proof discoverable and comprehensive are usually displaced by computational tensor calculus without the insights. Rjpetti (talk) 04:22, 3 September 2011 (UTC)

The role Wikipedia has played[edit]

I would like to close with a comment on the role played by Wikipedia in the area of EC. Wikipedia is not generally recognized as an authority by experts. However, in this case I believe Wikipedia has outperformed the top refereed journals in some respects. Let me explain. The top journals will accept important ideas like these because they are committed to academic freedom; and I can buttonhole people at a conference (which I did once) and convince, even excite, some experts at the top of the game. However, the proof of the correctness and necessity of EC, and the insights behind the proof, were dying from the "silent treatment." No one refers to the proof, and sometimes my work finds its way into the GR journals without attribution (such as the idea that rotational curvature is a continuum model of affine defects called disclinations, which I published in 1976, and which I think was the first mention of this in the GR literature, but I haven't done a full search of the early literature). I think this happens because I am not a member of the club, which is extremely competitive and virtually closed to important contributions from outside. In Wikipedia the club cannot kill an idea with the silent treatment. The Wikipedia article exposed many people to EC, the proof, and the ideas behind the proof while the article was still rich in geometric insight. That is better than the top journals have managed. Congratulations. Rjpetti (talk)

For history of science: removal of mention of derivation of Einsetin-Cartan theory from general relativity[edit]

I am sad to see the article no longer mentions the derivation of EC from GR, which is the most important fact about EC. EC is the only way for treatment of spin in a future theory in the long-sought theory of quantum gravity to match up with classical gravitational theory. I say the "only" way because, while other constructions that do this may be possible, EC is the least invasive way to alter GR to accomplish this goal, and we have a derviation of EC from GR. This rejection reflects either collective incompetence to evaluate the derivation, or worse: that a mathematical derivation is not enough proof to change the socially accepted opinion of the majority of the physics community. While the vast majority of results that are ignored are nonsense, the derivation of EC from GR is not. I have tried to publish updates on the derviation with rebuttals and better explanations in CQG and GRG; the editors don't even provide a good rationale for rejection, just shifting excuses. Yes, this kind of behavior still survives in physics. I excuse from this indictment the handful of physicists who acknowledge the correctness and importance of the proof, including the late Yuval Ne'eman, Jean Krisch, Mauro Francaviglia, and others I have heard of indirectly who accept it. Let this episode stand as a sign post for future historians of science. At least Wikipedia has a discussion section of record, whereas elsewhere the entire controversy is erased. Rjpetti (talk) 15:01, 12 December 2012 (UTC)

I would not have tried to publish my ideas in Wikipedia, except that the scientific community has more difficulty than it can admit in dealing with ideas that change existing thought paradigms, and there are few places to go. I find it quite significant that among the Wikepedia commentators the entire controversy comes down to POV and quoting the existing consensus, with virtually no competent discussion of the math or physics.

I have published an updated derivation of EC from GR with no additional assumptions at . The conclusions of that article are shown below.

The case for a revised consensus on Einstein-Cartan theory[edit]

Einstein–Cartan theory currently satisfies a substantial array of criteria for adoption: (a) it can be derived from general relativity with no added assumptions or parameters; (b) it extends general relativity to describe exchange of orbital and intrinsic angular momentum, which fixes the most outstanding problem in general relativity as the master theory of classical physics; (c) it is the minimal extension of general relativity that can fix this problem, because it arises merely by relaxing the ad-hoc assumption that torsion is zero, and because Einstein–Cartan theory is identical with general relativity where spin density is zero; (d) cosmological models based in Einstein–Cartan theory provide an explanation for inflation theory based on classical geometry, without relying on speculations about quantum fields; (e) Einstein–Cartan theory generates new predictions that can in principle validate or falsify the theory, but it cannot be validated by empirical results due to current limitations in technology.

Changing the old consensus on Einstein-Cartan theory[edit]

Since the original publication of this derivation over a quarter century ago, another generation of physicists has been trained in the 90-year-old consensus that Einstein–Cartan theory is but one among many speculations surrounding general relativity. An exception in the older generation was the late Yuval Ne'eman, who wrote in a private communication to me: "Your work on GR and the EC 'theory' [in quotes to indicate it is no longer a speculation] are of the highest quality and I have often quoted your results."

Thomas Kuhn analyzed such transitions in his landmark book The Structure of Scientific Revolutions [Kuhn 1962, 2012]. The history of science contains numerous cases in which a new theory was seriously considered or adopted only a generation or more after the new "paradigm" – the essential experiments, assumptions and models – was sufficiently developed. Kuhn distinguishes “normal science,” which extends the current paradigm to new situations, from "revolutionary science," which changes the basic paradigm to improve or to extend the scope of the science. If the new paradigm challenges the old paradigm before its superiority is completely established, it is appropriate that more conservative and more innovative scientists differ in the credence they give to the two paradigms. Kuhn emphasized that a paradigm shift is based on the reality or promise of better analysis of observations, and is part social, and part "enthusiasm" to make the change.

The paradigm shifts from the Ptolemaic system to Newtonian mechanics to special relativity to general relativity each are much larger than the shift to Einstein–Cartan theory. The latter shift requires relatively modest changes:

  • Drop the ad-hoc assumption in general relativity that torsion is zero.
  • The new theory must provide a better and simpler solution to a problem, namely a complete classical treatment of conservation of angular momentum (and a classical explanation of the origin of cosmic inflation, and the treatment of spin in the classical limit of a future theory of quantum gravity).
  • New experimental results are needed to validate the relationship between spin and torsion. This we do not yet have.

The problem with the comments in Appendix E, section E-3, is that they focus on technical criticisms of the derivation of Einstein–Cartan theory that either are relatively easy to address, or they insist on a level of rigor that perhaps should be met, but that is not the norm for such complex nonlinear new theories. Kuhn points out that when the Copernican model of the solar system was adopted by the leaders in physics, it was not yet able to provide better predictions of the motions of the planets than could the Ptolemaic system it displaced.

In 1996, David Goodstein, then vice provost of the California Institute of Technology, published an article in which he states that the current peer review system is becoming counterproductive due to the strong competitive pressures in science. Goodstein asserts that the peer review system is widely abused in many fields of science to delay or prevent dissemination and adoption of new results from competitors [2]. Regardless of the intentions of the participants, such competitive forces can raise the standards of rigor required of new paradigms to inappropriate levels, especially in fields that are moving relatively slowly such as basic physics (and less so in fast-moving fields such as genetics and cell biology).

In the author’s opinion, the norms of science suggest two solutions to this situation: either a suitably substantial objection to the claims herein surfaces and is openly discussed, or the next generation of physicists should be informed that Einstein–Cartan theory is derivable from general relativity with no additional assumptions, and that the only substantial weakness in the case for the theory is absence of direct observational evidence. Rjpetti (talk) 06:27, 11 January 2014 (UTC)

Response to monologue[edit]

Hi Rjpetti,

I would not have tried to publish my ideas in Wikipedia, except that the scientific community has more difficulty than it can admit in dealing with ideas that change existing thought paradigms, and there are few places to go. I find it quite significant that among the Wikepedia commentators the entire controversy comes down to POV and quoting the existing consensus, with virtually no competent discussion of the math or physics.

As you say, you have tried to publish your original ideas on wikipedia. It is explicitly not this encyclopedia's purpose to publish fringe work such as this, and those policies (of quoting mainstream consensus) are deliberately intended to ward off technical arguments (which lets us resolve editing disputes quickly without determining the prerequisite competence, sufficient background knowledge, and technical skill of our contributors).

If you aren't being taken seriously in the academic literature, and the textbooks taught at renowned universities still do not yet promulgate your theories, I suggest you try discussing them in a place like physicsforums. You might find it easier to make converts on semi-amateur forums such as that, which may in turn gather and help sway the tide of academic opinion. Or, at least, you will be quite likely to find qualified professionals there who are willing give up their time to hear you out, to explain any of their objections, and to discuss the topic further with you. Cesiumfrog (talk) 01:40, 12 January 2014 (UTC)
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