Dear Lukasz:

Einstein-Cartan theory is almost completely forgotten in the U.S.A. but there are some specialists in Europe still working in this and other theories incorporating torsion as a fundamental feature of gravity. The review by Friedrich Hehl et al.: General relativity with spin and torsion: Foundations and prospects." Rev. Mod. Phys. 48, 393–416 (1976) is the most cited work about this topic.

Recently, a polish researcher, Nikodem Popławski, has also used torsion cosmologies as another way to obtain inflation effects:  "Nonsingular, big-bounce cosmology from spinor-torsion coupling". Phys. Rev. D 85 (10): 107502.

A more recent and good review on "Torsion gravity" was written by Richard Hammond:

Rep. Prog. Phys. 65 (2002).

Max Tegmark and Alan Guth at MIT also think that a phenomenological theory of torsion could be tested in the Solar System (see: Constraining Torsion with Gravity Probe B: http://arxiv.org/abs/arXiv:gr-qc/0608121)

This work has been polemic and it has been dismissed by the advocates of the Poincaré gauge theory of gravity (particularly F. Hehl).

At least, this means that Einstein-Cartan theory is still alive although it is certainly not in the mind of most physicists or even General Relativistics. 

Subjective impression of this Gauge Gravity: "I love when you do that focus pocus to me. The way that you touch, you've got the power to heal. You give me that look, it's almost unreal. It's almost unreal." (ROXETTE's song "Almost Unreal").

Objective impression as quote: "The first such idea after Förster’s, of course, was Hermann Weyl’s gauge approach to gravitation and electromagnetism, unacceptable to Einstein and to Pauli for physical reasons [246, 292]". My comment: 1) The metric tensor fully describes the geometry of spacetime: there is rule, how to measure intervals between points. 2) The geometry sets the way of parallel transport. Conclusion: the metric tensor is sufficient for parallel transport, therefore the inclusion of other fields in Christoffel symbols is wrong.

http://relativity.livingreviews.org/open?pubNo=lrr-2004-2&page=articlesu9.html

Dear Lukasz:

Einstein-Cartan theory is almost completely forgotten in the U.S.A. but there are some specialists in Europe still working in this and other theories incorporating torsion as a fundamental feature of gravity. The review by Friedrich Hehl et al.: General relativity with spin and torsion: Foundations and prospects." Rev. Mod. Phys. 48, 393–416 (1976) is the most cited work about this topic.

Recently, a polish researcher, Nikodem Popławski, has also used torsion cosmologies as another way to obtain inflation effects:  "Nonsingular, big-bounce cosmology from spinor-torsion coupling". Phys. Rev. D 85 (10): 107502.

A more recent and good review on "Torsion gravity" was written by Richard Hammond:

Rep. Prog. Phys. 65 (2002).

Max Tegmark and Alan Guth at MIT also think that a phenomenological theory of torsion could be tested in the Solar System (see: Constraining Torsion with Gravity Probe B: http://arxiv.org/abs/arXiv:gr-qc/0608121)

This work has been polemic and it has been dismissed by the advocates of the Poincaré gauge theory of gravity (particularly F. Hehl).

At least, this means that Einstein-Cartan theory is still alive although it is certainly not in the mind of most physicists or even General Relativistics. 

I agree with Acedo, Einstein-Cartan theory is almost forgotten.

Luca Fabbri also made some interesting comments on this issue in the following discussion: https://www.researchgate.net/post/Laboratory_tests_of_Einstein-Cartan_theory2

Dear friends,

Thank you for your answers.

Luis, your mini review enlightened the issue brilliantly.

Olivier, the link you posted is also very informative.

There is no fundamental reason to ascertain that Torsion has no role in Nature.The statement saying that the theory is forgotten in the US is very comical. There are several alternative approaches to Physical theories and depending upon the available experimental and observational evidences,one is said to be better than the other. Yes,it is true that all the observations done outside the matter distribution indicate that Einstein's theory of GR is the best. One should know that EC theory also agrees with this, and there are no experimental or observational  result  done within the matter distribution,which can decide about EC theory. Mathematically it is a beautiful theory and in a sense the real extension of Special Relativity.Similar questions can be asked about several other approaches including quantising gravity. No theory becomes correct simply by popular opinion or how many are writing papers on it.

Dear Lukasz:

Your question is very important and interesting in the sense that The Einstein-Cartan^* theory can be exactly quantized and then it is special relativized. This leads to a theory of quantum gravity . The resulting theory is also a theory of quantum cosmology. A relation connecting time, temperature and cosmological constant is derived based on this theory, and the value of the cosmological constant is in very good agreement with the value known otherwise. 

1.Book:   Effective Theory of Quantum Gravity : Soluble Sector: S. Mishra and J Christian. Abramis, UK ,2011.

2. Exactly soluble sector of quantum gravity , Joy Christian

Phys. Rev. D 56, 4844 – Published 15 October 1997

3. A Quantum Mechanical Relation Connecting Time, Temperature, and Cosmological Constant of the Universe: Gamow’S Relation Revisited as a Special Case, Subodha Mishra

International Journal of Theoretical Physics 09/2008; 47(10):2655-2662. · 

Note:  * Einstein-Newton-Cartan Theory

My statement about that torsion is forgotten in the US or unpopular is only a personal impression obtained by reading the literature in the subject. Some years ago there was a debate among Steven Weinberg and Friedrich Helh also commented in the blogs: http://www.math.columbia.edu/~woit/wordpress/?p=529

Weinberg does not believe that torsion has any role in physics but it is only an opinion. On the other hand, there is other people at MIT who has published some influential papers on the topic of torsion (see  comment above).

Feynman's saying: "It the theory does not agree with experiment then it is wrong" is the best guide in physics. Of course, popularity does not count on the verification of a theory as well as beauty (as Feynman also quipped).

Really, I am working on this topic (literature surveys are usually not done by historical scholarly interest) so my opinion about ECT and other similar extensions of General Relativity  is that there are very worth studying as alternatives to other approaches.

It is very interesting whether the (some notion of) torsion is really a physical entity and not only "just a tensor".

Subodha, thank you for pointing out the associations with quantized theory of gravity. I will read through your papers since the issue of predicting the correct value of cosmological constant is quite curious. In my personal opinion, this solution can be more viable than for example the one based on anthropic arguments. It is really interesting that ECT can be exactly quantized.

Luis, I too am interested in studying various extensions to the GR. Right now I am going through some review papers concerning Modified Gravity. For me, personally, ECT is very elegant and mathematically aesthetic. Thus it attracted my attention.

Cartan proposed Einstein-Cartan theory (EC) to Einstein in the 1920s because affine torsion is basic differential geometry; neither of them had any idea what it meant physically [Debever Robert, editor, 1979 “Elie Cartan – Albert Einstein, Letters on Absolute Parallelism 1929-1932” p 5-13]. During the twentieth century, the consensus on EC was that it is a speculation that requires an assumption (that torsion is nonzero) beyond general relativity (GR), has no empirical support, and solves no significant problems. Learning EC requires learning Riemann-Cartan geometry, which includes affine torsion along with a Riemannian metric. Worse, EC became socially somewhat toxic, in that people who got involved with EC were in danger of being considered not serious.

To answer your question, let me summarize the strong evidence for EC of which I am aware.

a) EC extends GR to describe exchange of orbital and intrinsic angular momentum, which fixes the most outstanding problem in GR as the master theory of classical physics.

b) EC is the minimal extension of GR that fixes this problem: EC arises either as the mathematical ‘completion’ of GR, or by relaxing the ad-hoc assumption that torsion is zero; also, EC is identical with GR where spin is zero.

c) The main features of EC can be derived from GR with no added assumptions or parameters [Petti, http://arXiv.org/abs/1301.1588].

d) Cosmological models based in EC generate inflationary expansion from classical geometry [Poplawski, http://arXiv.org/abs/1007.0587, http://arXiv.org/abs/1111.4595]. This result suggests that quantized torsion is the inflaton field, and that recent observations by the BICEP2 project of gravitational waves generated by inflation may be the first empirical observations of the effects of torsion.

e) In EC, black holes and the big bang do not have singularities; fermions do not cause ultraviolet divergences in quantum field theory, and they cannot form strings [Poplawski, http://arxiv.org/abs/0910.1181].

In my opinion, items (a) thru (e) qualify EC as a better classical limit for gravitation than GR. In the absence of direct empirical evidence, item (c) is the most compelling. EC should be considered as a classical starting point in any investigation of quantum gravity; it is not.

The key problem is that we do not have direct observable evidence the effects of torsion. This would be possible only where spin densities are very high, for example in the early universe.

I expect three factors will delay acceptance of EC until long after adequate evidence has been assembled, for two reasons:

i) Physicists demand direct empirical evidence before granting some credibility. (Old joke: an intelligence test asks, “What is s2+2?” Mathematician’s answer: “4”. Physicist’s answer: “4.000000000000”.)

ii) Historically, the scientific community is pretty good about accepting innovations that extend the existing ideas new situations. However, innovations that alter the fundamental assumptions in a field (the “paradigm”) often are not accepted until about a generation after the critical evidence has been assembled.

iii) Since EC was at one time quite toxic (and in some quarters it still is), its waiting time will be long.Item (c) was published in 1986, and items (d), (e), and (f) since 2010.

I agree with Richard about the "toxic" status of EC theory in the scientific community. Poincaré gauge theory could be elegant, in the sense that it follows other achievements in field theory during the XXth century. However, it is a haunting task to find any unquestionable evidence of its validity. 

In EC theory, torsion is confined to the bodies and it requires high spin densities for their effects to be of any importance. Obviously, it is not simple to perform experiments inside high density objects and the reference to the early Universe it is not very helpful in most cases. 

A possible explanation of inflation can arise also from other ideas, including the plethora of models for f(R) gravity: See the wikipedia entry on this topic and references therein.

Moreover, EC is not the only consistent theory of gravity including torsion as a fundamental tensor. In the late seventies, Hayashi and Shirafuji proposed a theory whose basic ingredients where a tetrad reference frame and torsion and they were succesful to explain the basic tests of GR using only torsion in a zero curvature Weitzenböck spacetime. 

See: http://journals.aps.org/prd/abstract/10.1103/PhysRevD.19.3524

Other theories with propagating torsion (including physical effects outside the bodies themselves) are also possible and the papers on torsion phenomenology by the MIT people and also this one: 

http://arxiv.org/abs/1101.2791

are in this line. So, perhaps the path is to find a role for torsion in physics (experimentally) and then proceed to pinpoint the correct theory among the neverending forest of hypotheses.

As far as I know, torsion and curvature are tensors that are commonly used to model continuum with translational and rotational dislocations. I agree, this comment is not directly related to the EC theory (present discussion). However, this is to illustrate that torsion (and curvature) is not just a mathematical tensor field, at least in the domain of classical continuum mechanics … a limit case of continuum relativistic mechanics … as far as I know.

Well, indeed, the present treatment of (the) Einstein-Cartan theory seemingly preserves nothing from the first author (that is, from Einstein’s absolute parallelism – which is not so good a term but quite an interesting possibility for equations with high symmetry group uniting symmetries of both Special and General relativities). For more details one can look here: 

http://fqxi.org/community/forum/topic/2459

We can derive Einstein-Cartan theory (EC) from general relativity (GR) with no additional assumptions. Here is an outline of the derivation and further thinking.

1) GR cannot describe exchange of classical spin and macroscopic orbit angular momentum.

Most relativity physicists with an opinion on this believe that GR can model all aspects of classical spin. First, they seem to believe (without explicitly saying so, as far as I can tell) that that there is no such thing as classical spin-orbit coupling; every time I mention classical spin-orbit coupling to one such physicist, he reverts to discussion of quantum spin. However the essence of turbulence in classical statistical fluid mechanics is that orbital angular momentum is transferred to smaller scales with no classical lower limit on scale [Kolmogorov 1941]. Secondly, it is well known in classical continuum mechanics that exchange of orbital and intrinsic angular momentum requires that the momentum tensor be non-symmetric during the exchange, which GR cannot manage since Riemannian geometry dictates that the Einstein tensor is symmetric. Third, they seem to believe that, since GR can describe classical spin and classical orbital angular momentum, it can give a full account of spin in classical physics. Clearly this is not enough, given that statistical fluid mechanics identifies classical spin-orbit coupling to be the essence of turbulence. 

Conclusion of step 1: GR is not the ultimate a master theory of classical spacetime physics. Read on to see why EC is better. (It is risky to claim that any theory is the ultimate theory of anything.)

2) We can derive EC from GR with no additional assumptions or parameters. Here is how.

2-a) Compute the translational holonomy for closed loops in all orientations around a single Kerr rotating mass. Compute the holonomy/area for the loop, where the area of the loop is that seen by an observer at "infinity".

Note: Curvatures of all kinds (Riemann, Yang-MIlls, and torsion) IS limits of holonomy/area as the loops gets smaller (though it seems most physicists think this relationship of holonomy/area and curvature is just an intuitive heuristic). This definition of curvature is the contravariant version of the definition of curvature in terms of Cartan differential forms. Holonomy/area without a limit is an integral surrogate for torsion. 

2-b) Form a regular array of Kerr masses and take the limit as the number of Kerr masses increases to form a continuum with constant densities of mass, angular momentum and charge. The result is torsion, and the spin-torsion field equation of EC.

3) EC, GR, and teleparallel gravity are the only gauge theories that pass all experimental tests so far [Hel 2013].

3-a) EC is the minimal extension of GR that fixes the problem with spin-orbit exchange. It agrees with GR wherever spin density is negligible. If you relax GR’s ad-hoc assumption that torsion=0, then the resulting theory is EC.

3-b) Teleparallel gravity uses torsion to model gravitation and sets Riemannian curvature to zero. This is a radical deviation from GR that is unnecessary.

4) EC is relevant to at least three important fields of gravitational research. 

3-a) EC generates explosive cosmic expansion in closed Friedman cosmologies [Poplawski 2010a, 2012]. Quantized torsion may be the long-sought inflation field.

3-b) EC with semi-classical-quantum models of matter eliminate eliminates singularities in the big bang (which becomes the big bounce) and in black holes [Poplawski 2010b].

3-c) Since spin is so important in quantum phenomena, EC is a better classical limit for a successful quantum theory of gravitation. 

4) The only missing factor is direct empirical evidence of torsion. Mathematical physics is much more sophisticated than even 25 years ago, so that it is not possible to add well-justified terms to field equations like GR. The best mathematical models -- GR, the Standard Model -- are now so restrictive that they sometimes get ahead of experimental physics in areas like cosmology, inflation theory, and the Higgs boson. 

We have little choice but to adopt EC where it makes a difference - in the early universe, in singular (near-singular) situations, and possibly in neutron stars and other settings. 

At the April APS meeting, I will make a presentation and a poster session on the derivation of EC from GR. I have a paper on Arxiv on this that I will update in the next week or two with a much improved version 15.

Hi,

I've just completed one master degree in fundamental physics and from a few week i read E. Cartan work.

Spinnor fields in fundamental forces play big roles.

Then, I found remarkably his vision to integrate a torsion tensor which its components are different and independant from the degrees of freedom used in the so-called curvature tensor to draw the geometry of complex systems in the universe.

But something's still perturbing me..

Is someone could take few minutes to explain me why the equivalence principle should be violated ?

Sincerely,

Aurelien

Hi Aurelien, 

What do you think violates the equivalence principle? Spinors and torsion are only present where intrinsic angular momentum is present -- more precisely, torsion exists where angular momentum exists on a smaller scale than the natural scale of a (classical or quantum) physical model.  

For example, turbulence consists of transport of orbital angular momentum to smaller distance scales until the cycles are smaller than the scale on which classical models describe fluids. This is the essence the analysis of turbulence using statistical fluid mechanics. The angular momentum must go somewhere when it is transformed in this way, and that is an example of classical intrinsic (non-orbital) angular momentum. The absence of torsion makes it "impossible" to construct the correct classical interpretation of turbulence. 

 Sincerely,

Richard

Torsion suffers from lack of empirical evidence: "Final results of the GP-B experiment were announced at NASA HQ in Washington DC on 4 May 2011.

The experimental results are in agreement with Einstein's theoretical predictions of the geodetic effect (0.28% margin of error) and the frame-dragging effect (19% margin of error)." If the null result is due to the fact that "torsion (if it exists) couples only to the intrinsic spin of elementary particles" how else could one test the theory?    

If the particles in Einstein-Cartan spacetime move along autoparallels instead of geodesics there exist a possibility that spinless particles could sense torsion. See, for example, the papers of Riccardo March et al. :

http://arxiv.org/abs/1101.2789 and  http://arxiv.org/abs/1101.2791

However, this is not accepted among most  torsion "experts".

@Aurelien, as stated by Misner, Thorne and Wheeler the EEP is that in the locality of each event (point of spacetime) the nongravitational physics is that of SR. since SR is verified to a 10^-20 level of accuracy people feel free of strong constraints to question the mass only source of gravitation -- universality --  as long as it converges to SR. And since spin as mass is an invariant of the Poincare' group why not include it as a source of of gravity? This of course could break universality in the sense that it may couple with another constant than G, involve other mechanisms etc. Maybe Wei-Tou Ni (see https://www.researchgate.net/profile/Wei_Tou_Ni) would be kind enough to take time to explain the theoretical work in view of the current empirical constraints in a thorough way.    

I agree completely with the summary in the excellent presentation of the Einstein -- Cartan Theory by A. Trautman, gr-qc/0606062.

I have been working in EC for many years and most recently I apply torsion gravity to the alternative theories models to get dynamo equations [ phys lett B, 2012] AND TO INVESTIGATE anti dynamo theorem in other curvature settings , more recently yet we have noticed that models with gravity can be used in magnetogenesis and galactic dynamo seeds 9see pkease my recent papers on JCAP 2014 and Class and quantum gravity....

 @Andrade, could you please point out what is the energy scale at which the torsion becomes a differentiating phenomenon from curvature only and is there an experiment that would be able to test it? 

EC can be derived from GR using classical differential geometry. The derivation takes the continuum limit of a distribution of exterior Kerr masses, as the density of Kerr objects increases and the mass and angular momentum of the bodies decreases so the densities of mass and angular momentum approach constants. The limit is EC, with torsion and the spin-torsion relationship - not GR. This derivation requires with no assumptions beyond GR, no added parameters, no spinors, no quantum theory.

Why does EC matter?

1) GR cannot describe exchange of classical intrinsic angular momentum and orbital angular momentum. The reason is simple: exchange of orbital and intrinsic angular momentum requires that the momentum tensor be nonsymmetric during the exchange, as has been well known in classical continuum mechanics (for at least a century I believe). GR cannot accommodate a nonsymmetric momentum tensor because the Ricci tensor is symmetric in Riemannian geometry. Einstein–Cartan theory fixes this problem in the least invasive way.

2) Poplawski (Arxiv, 2010) has shown that Schwarzschild black holes have no singularity and have a minimum size. The result for black holes affects most or all arguments regarding the so-called Black hole information paradox, which assume that information disappears into the singularity, and/or that black holes gradually evaporate.

3) Poplawski (Arxiv, 2010, 2012) has shown that Friedman cosmologies with positive curvature have no central singularity. Torsion causes a "big bounce" without a singularity in these models. The inflation gurus have been postulating an inflaton field, which is most likely the torsion field.

 4) Turbulent fluids are common on cosmology. The essence of turbulence is conversion of orbital angular momentum to smaller distance scales until the scale of the angular momentum is smaller than the scale of the classical model (known since Kolmogorov's statistical model of fluid mechanics, including turbulence, published in 1941). We don't need quantum theory or spinors to observe the deficiencies of GR in this area.

 5) The definition of intrinsic angular momentum in classical continuum theory (and it is becoming more obvious that all classical theories are continuum limits of deeper structures) is angular momentum that exists on a scale smaller than is modeled in the relevant classical continuum model. I think this is the most sensible definition of intrinsic angular momentum.

In my correspondence with an unnamed expert in gauge gravity, he appeared to maintain that GR perfectly accommodates intrinsic angular momentum because it models conservation of orbital-plus-intrinsic angular momentum, and that current knowledge cannot resolve whether GR or EC is correct. He seems to regard exchange of orbital and intrinsic angular momentum (and perhaps intrinsic angular momentum itself) as strictly a quantum phenomenon. Turbulence is an important factor in the discussion because it involves classical exchange of intrinsic and orbital angular momentum, using the definition of intrinsic angular momentum above.

How can we have a master theory of classical spacetime that cannot at its root allow exchange of orbital and intrinsic angular momentum? 

I published the basic results in 1986 (in GRG), and a major update in 2015 on Arxiv (http://arxiv.org/abs/1301.1588).

I think EC is by now a clear example of a well-known problem in physics: a theory that alters the foundations of a field of physics often requires an extra generation to gain acceptance after it is proven on the merits. This is the central motivation for the distinction between "revolutionary science" and "normal science" made by Thomas Kuhn in his 1962 book "The Structure of Scientific Revolutions." As Planck wrote in his scientific autobiography: "A new scientific truth does not triumph by convincing its opponents and making them see the light, but because its opponents eventually die, and a new generation eventually grows up that is familiar with it." (Quoted in Kuhn, 4th edition, 2012, p 150)

regarding "1) GR cannot describe exchange of classical intrinsic angular momentum and orbital angular momentum." 

a) what is classical intrinsic angular momentum?

b) the EM energy momentum tensor is made symmetric by intent while making sure that no conservation laws are violated -- as far as i know GR doesn't violate any such laws.

c) i read a paper by Andrade showing that for spinless particles one can obtain without torsion the Lorentz force analog. since there is no empirical evidence for torsion the question of spin-spin interaction becomes in my opinion energy scale dependent. 

d) GR and quantum physics are not compatible but LQG for example doesn't start from EC theory and can cope with spin and so do other frameworks as well

a) Classical intrinsic angular momentum is angular momentum on a smaller scale than it included in relevant classical continuum models.

Example 1: Cosmological models with continuum models of a dust of galaxies do not include a small enough scale to model the orbital angular momentum of galaxies. Nevertheless, the angular momentum of the galaxies is still there in the physical world. This is classical intrinsic angular momentum.  GR can't handle it, but EC can.

Example 2: Turbulent fluids transform orbital angular momentum to smaller scales without limit in classical physics. When the angular momentum is transformed to a smaller scale that the relevant classical fluid model can handle, this is classical intrinsic angular momentum.  GR can't handle it, but EC can.

b) GR cannot model exchange of classical orbital and classical intrinsic angular momentum, because this requires the momentum tensor to be nonsymmetric during the exchange, as I discussed in my previous post.

c) The empirical evidence for torsion and EC is the presence of galactic turbulence, and (probably) cosmic inflation.The Lorentz force is due to covariant derivatives that use a connection with curvature an a complex 1-dimensional line bundle (the EM field). I don't see the connection of the Lorentz force with this discussion. Also, I am treating GR as a classical continuum theory because I can derive EC from that without getting into the complexities of quantum mechanics, which doesn't fit well with GR anyway.

d) The classical limit of LQG is EC, even though many researchers in LQG think the classical limit is GR.  Intrinsic angular momentum is extremely important in quantum theory; GR can't handle exchange of orbital-intrinsic a.m., which is a fundamental interaction in quantum theory. I propose (without proof) that ignoring torsion and EC has been a substantial hurdle in making gravitational theory and quantum theory fit.

I am sorry if my first post was not clear; I thought it addressed all these issues except the one about LQG. I should have added that to my list of arguments for EC.

on "d) The classical limit of LQG is EC, even though many researchers in LQG think the classical limit is GR. " -- can you please show evidence for this claim? as far as i understand this is the teleparallel theory with Weizenbock connection which doesn't have torsion. 

Asher, I request a clarification of what you wrote above. Do you mean 

"The classical limit of LQG is the teleparallel theory with Weizenbock connection which doesn't have torsion."

@Petti, I was completely wrong the Weizenbock doesn't have curvature. thanks.   

Asher,

In April 2015 I read on the web that EC is the classical limit of LQG. By fall 2015, I could no longer find that statement anywhere on the web, using the same searches I used before. This assertion evidently was removed at some point in 2015.

The Wikipedia article on LQG asserts "Any candidate theory of quantum gravity must be able to reproduce Einstein's theory of general relativity as a classical limit of a quantum theory." It also states that researchers have not been able to prove this.I am totally unsurprised. EC is critical to correct modeling of intrinsic angular momentum, and spin is much more central to quantum theory than is intrinsic a.m. in classical physics. 

Asher,

In April 2015 I read on the web that EC is the classical limit of LQG. By fall 2015, I could no longer find that statement anywhere on the web, using the same searches I used before. This assertion evidently was removed at some point in 2015.

The Wikipedia article on LQG asserts "Any candidate theory of quantum gravity must be able to reproduce Einstein's theory of general relativity as a classical limit of a quantum theory." It also states that researchers have not been able to prove this.I am totally unsurprised. EC is critical to correct modeling of intrinsic angular momentum, and spin is much more central to quantum theory than is intrinsic a.m. in classical physics. 

for what it is worth I think that there is something in it

Einstein-Cartan theory (EC) is a modest extension of general relativity (GR) – not a competitor to GR, as its detractors like to claim – that has numerous benefits.  First I will outline the case for EC, then I will comment on the degree of acceptance at the end of this post.

PART 1: THE CASE FOR EINSTEIN-CARTAN THEORY

1) EC can model exchange or orbital and sub-scale angular moment; GR cannot.

• “Subscale a.m.” means a.m. on a smaller scale than the classical model. This includes classical intrinsic a.m., as in turbulence, and quantum intrinsic a.m. known as quantum spin.
• It has been known for a century in classical continuum mechanics that for a medium to cause a body to rotate, the momentum tensor must be non-symmetric during the exchange. You can see this by considering the surface tractions needed to cause a small cube of matter to rotate: the antisymmetric part of the momentum tensor is the source of a.m. transferred to the cube.
• The inability of GR to model a.m. exchange is fundamental: Riemannian geometry requires that the Einstein curvature is symmetric, so in GR the momentum tensor is symmetry.
• EC is the minimally invasive change in GR that can do this. EC is identical to GR when orbital and intrinsic a.m. are not being exchanged; in particular, when intrinsic a.m. = 0. Intrinsic a.m. is generally negligible in classical physics, except in turbulent fluids in which the inertial region covers several orders of magnitude.

 2) EC can be derived from GR with no additional assumptions, at two levels.

•  First, consider a rotating Kerr mass (which need not be a black hole). You can compute the integral version of affine torsion, called translational holonomy. It is equal to the a.m. of the black hole. This is the integral version of the spin-torsion field equation that is the main feature of EC.
• Second, consider the continuum limit of a distribution of Kerr masses with correlated rotations. The limit yields affine torsion and the spin-torsion relationship of EC, not GR.

See my paper and poster at https://www.researchgate.net/profile/Richard_Petti/contributions or http://www.arxiv.org/abs/1301.1588.

 3) EC is extremely well-motivated from a mathematical point of view.

• EC is not derived by adding yet another ad-hoc term to GR. Affine torsion (translational curvature) is the partner of Riemannian curvature (rotational curvature) in the general curvature tensor in the deepest theory of affine differential geometry, based on Cartan differential forms on fiber bundles.

 4) EC has many implications for gravitational research

•  Black holes do not contain singularities in EC. (Poplawski, 2010)
• Closed Friedmann cosmologies in EC have no singularity. Torsion force dominates at super-high densities and causes “big bounce.” The “big bang” really the “big bounce” (Poplawski 2010)
• Is torsion the “inflaton field”? In closed Friedmann EC cosmology, torsion drives inflation-like expansion (Poplawski 2010a, 2012). Does torsion meet all requirements?
• Is EC the classical limit of quantum gravity? Intrinsic a.m. is far more important in quantum physics than in classical physics. A good theory of quantum gravity should have torsion or a quantum precursor of torsion.
• Does torsion trace occur in virtual fields joining ends of fermion lines? Analogy with dislocations: ends of screw dislocations are joined by edge dislocations that contain in-plane torsion.

 PART 2: ACCEPTANCE OF EINSTEIN-CARTAN THEORY

The reaction of the relativity community to EC will not be a bright spot in the history of science. The problem is a generic one in science.

• Thomas Kuhn segments science into two types. “Normal science” consists of extending applications of accepted theories to new problems. Science is probably better than any other human activity at doing this quickly and effectively. “Revolutionary science” is science that alters one of the basic ‘paradigms’ (concepts, math methods, empirical methods) in a field of science. This second kind of science often takes an extra 1-2 generations to gain acceptance after the decisive evidence is in place. This kind of science often catalyzes bitter conflicts and scientific purges. See Kuhn’s landmark book “The Structure of Scientific Revolutions” 1962, 4th ed 2012.
• Max Planck wrote, “A new scientific truth does not triumph by persuading its opponents and making them see the light, but rather because its opponents eventually die, and a new generation grow up that is familiar with it.” (Max Planck, “Scientific Autobiography and Other Papers,” 1949, p 33-34. Quoted in Kuhn’s book, 4th ed, p 150.)
• There are exceptions. Yuval Ne’eman (independent discoverer of SU(3) symmetry and chief scientist on the Israeli bomb) wrote me that the first half of my derivation of EC was of the highest quality, that he had often quoted the work.
• To oversimplify, we need the curators and extenders of normal science, and the creators of revolutionary science. Their thought processes are often quite different, as Einstein eloquently described in http://www.neurohackers.com/index.php/fr/menu-top-neurotheque/68-cat-nh-spirituality/99-principles-of-research-by-albert-einstein

Today, most of the resistance to EC takes the form of insistence on excessive rigor, excessive emphasis on empirical evidence as the standard of credibility, omission of EC from surveys of the field (though this seems to be improving), failure to grant research positions to EC researchers (Germany and Poland are notable exceptions), and ignorance of what EC is and how it fits.

Dear Lukasz,

I personally think that the theory in itself is elegant because it does not involve requiring torsion to vanish and its place in physics is the one for which Einstein gravity gets completed in the sense that it becomes possible to couple also the spin beside the energy of matter fields: the fact that spinors exist is itself the single most fundamental reason to work with torsion. In addition, the idea for which torsion could only be seen at very high spin densities was due to the fact that for long torsion-gravity was based on the simple and not the most general Einstein-Hilbert type of action, in which case the constant with which torsion couples to the spin need not be the Newton constant and the spin does not need to be as dense as people believed thirty years ago (see http://arxiv.org/pdf/1201.5498.pdf for details).

As torsion couples to the spin, torsion induces in the effective regime spinorial non-linear interactions that can mimic very closely the structure of the weak forces among leptons (http://arxiv.org/pdf/1011.2373.pdf), and in the case of massive neutrinos there doesn't even need to be mass non-degeneracy to have neutrino oscillations (http://arxiv.org/pdf/1504.03545.pdf).

With respect to Dark Matter, torsionally-induced non-linear potentials entail a gravitational behaviour which, in the case of galaxies, could describe the flattening of rotation curves (http://arxiv.org/pdf/1211.3837.pdf), and because these are normally interpolated in terms of the Navarro-Frenk-White profile, then it comes as a consequence that torsion is the reason for the appearance of the D8 operator in effective field theories of DM.

So, as it should be clear, there are theoretical as well as phenomenological reasons to work with torsion.

Please feel free to contact me, if you need more information.

By the way, almost all my works are about torsion, so if you want to have an overview you can have a look at https://inspirehep.net/author/profile/L.Fabbri.2 which also has the advantage that I can reply to any comment you might have.

Dear All,

Thank you very much for such an informative input!

Luca, your works are very interesting and I will certainly go through them. Thank you for willing to answer my question which will surely appear.

All the best,

Lukasz

... any time

Quoted from Wheeler's A Journey into Gravity and Spacetime, 1990, p. 128.  "This single simple expression---the Einstein-Cartan equation--- gives us the most vivid image that mankind has ever won of the living heart of gravity."  This is the focus of his chapter 7, and the core of his geometrodynamic modeling of general relativity.

The EC theory represents an ab-initio approach of dealing with dynamics of geometric field associating with nongeometric complement.manifested from a single cause like inflation as per today's knowledge. The nongeometric part with its torsional feature can be stipulated in principle to have compatibility with the metricity of space-time manifold as well as with the geometric curvature properties for it to corroborate to the stress balance between the geometric and nongemetric parts as expressed in Einstein's field equation. With the involvement of torsion as the nnongeometric feature, the inflationary state, dark matter and dark energy and interplay of gauge fields in dynamical evolution can be realized. The quantization of the dynamic process involving the geometric field of course calls for innovative way of axiomatic translation of the canonical variables in EC formulations where renormalizibility  is an open challenge.  

Dasarathi Das:

Why do you say that torsion is a non-geometric concept? If so, S. Kobayashi and K. Nomizu (for example) would not have included this repeatedly in their classic text "Foundations of Differential Geometry", Vol.I and II, Interscience Publishers 1963. (See, for example Vol.I, Sect. III.2.)

Norbert:

You are absolutely correct about the tomb by Kobayashi and Nomizu. Let me be more specific about the ways that affine torsion is part of geometry.

a) The basic Cartan structure group of affine differential geometry is a semi-direct product of GL(n,R) and the translations R^n. If there is a metric, this becomes O(n,R) and R^n. The significance of the "semidirect product" is that the the group is canonically split into a rotation group and a translation group; equavalently, the origin of flat affine associated fibers is fixed, and not a dynamic varaible (I am not sure the semidirect product condition will survive as we get more advanced geometric field theories. This would make the origin into a dynamic variable, as in my 2006 paper in CQG.)

b) This structure group implies that Cartan structure equations include both (rotational) curvature and torsion (which is translational curvature).

c) The discrete analogs of translational and rotational curvature are well-known to metallurgists and crystallographers as dislocation and disclination densities respectively. See my 2001 paper in GRG for more explanation with pictures.

Cartan tried to persuade Einstein to extend GR with torsion because, to Cartan, torsion is an integral part of affine differential geometry. Einstein assumed torsion=0 because he had a closed theory of gravitation without it, and he did not know what torsion was good for. Einstein told Cartan the did not understand what Cartan was trying to tell him. (See Debever Robert, editor, 1979, "Elie Cartan – Albert Einstein, Letters on Absolute Parallelism 1929-1932", Princeton University Press, p 5-13.) 

We now know that torsion enables inclusion of coupling of intrinsic and orbital a.m. in gravitational theories. This makes the exclusion of torsion an ungeometric assumption, while the inclusion of torsion is extremely geometric.

Torsion also causes a repulsion of spins that causes an inflation-like expansion at very high densities, and that becomes immeasureably small in the expanded universe with very low spin density. So torsion is probably the "inflaton field" that the inflation researchers seek. My impression is that the specialists in inflation do not know enough geometry to be persuaded to do the work needed to test whether torsion meets all known conditions to be the cause of cosmic inflation.

--Richard Petti

In the brief write up, I distinguished the purely geometric space time curvature as used by Einstein from the potential based curvatures generally used in designing the torsion field.

Hi Dasarathi,

Please read my paper at http://www.arxiv.org/abs/1301.1588 . It should persuade you that EC is more geometrical then any other theory, even GR which is a proper subset of EC.

--Richard