# What are the pros and cons of Einstein-Cartan Theory?

As an alternative to General Relativity, i hear that it can avoid the initial big bang singularity as well as the singularities in black holes, so why does it appear to be talked about so little? If anyone can enlighten me a bit on the successes and shortcomings of it i would be very grateful.

• I'd like to supplement the good technical reasons already given for the lack of coverage of EC Theory with the remark that the theory may be in conflict with the very old notion that the formation of the world was emotionally motivated, as EC provides for cosmologies which are eternal to the past, and consequently incompatible with the "universal creation events" popular in many political constituencies that fund education in physics. Nevertheless, such cosmologies are progressing even in such environments, as seen at arxiv.org/abs/1401.7639 and arxiv.org/abs/1007.0587. – Edouard Sep 14 at 4:10

One main reason why the ECKS theory (Einstein-Cartan-Kibble-Sciama) is so unpopular even today is its extreme mathematical complexity. The connection isn't symetric anymore, so it changes a lot of things in the mathematical formalism (for example: the covariant derivative applied to a simple scalar field doesn't commute anymore): $$$$\tag{1} \Gamma_{\mu \nu}^{\lambda} \ne \Gamma_{\nu \mu}^{\lambda}.$$$$ The non-symetric connection implies the existence of a tensor field called torsion, which is a three indices tensor field: $$$$\tag{2} T_{\mu \nu}^{\lambda} \equiv \Gamma_{\mu \nu}^{\lambda} -\Gamma_{\nu \mu}^{\lambda}.$$$$ This tensor field could be extracted everywhere and added to the Lagrangian as an extra field, so most people prefer to use standard GR (with a symmetric connection and standard formalism), and simply add the new tensor field by hand in the Lagrangian.

You may explore the Wikipedia page, but beware: it has a few subtle mistakes:

https://en.wikipedia.org/wiki/Einstein%E2%80%93Cartan_theory

So PROS: It's a natural extension of General Relativity. GR feels more "complete" with spacetime torsion.

CONS: It's terribly more complicated than GR without torsion. Torsion occurs only inside matter, in cases of extremely high density states, so it's very hard to test it in labs! Propagating torsion outside matter is probably too weak to be observable at all (if it could propagate outside at all!).

• One of the things that Wikipedia used to say about EC was that the theory was "not renormalizable". It doesn't say that anymore (at least, not "in so many words"), and the page (last modified Aug. 5) is currently unchallenged. Renormalizability seems to be especially important in such fractal cosmologies as inflation, so fans of EC might keep an eye on it: Who knows how much progress has been lost thru the resulting disuse of Einstein's work with Cartan? – Edouard Aug 20 at 16:15
• The EC (or ECKS) theory is actually viable, and very interesting in its own right. It may even feel pretty natural as an extension of GR. But sadly, its predictions are only different from GR inside matter, in case of very high density states, where torsion could express itself. About propagating torsion outside matter, well, it's probably too weak to be measurable at all! – Cham Aug 20 at 17:15
• The torsion in the EC-based "cosmology with torsion" to which I'm referring (described in Poplawski's 2010 paper by that name) does occur entirely within what would (in at least some definitions) be considered "matter": In it, the trajectories of newly-materialized fermions (separated by tidal effects, on opposite sides of the outward-propagating event horizon of a star undergoing gravitational collapse, from their former partners in virtual particle-antiparticle pairs, during that horizon's outward propagation) are accelerated and reversed during contact with much larger stellar fermions. – Edouard Aug 20 at 17:33
• For a more complete visualization, I'd like to add that the new local universe (or temporal iteration) is formed when the accelerated particles reach a zone, still within the original star's volume, that's a 3-dimensional analog of the surface of a sphere (i.e., like "the skin of a basketball"). The difference in scale between each "parenting" LU & its "baby" is huge, but maybe not as great as that difference between particles and black holes that's discussed by Susskind in his 2014 "ER=EPR" lecture, available free on YouTube. – Edouard Aug 20 at 17:46

A con i know: The Dirac equation becomes nonlinear and therefore the superposition principle used for the quantisation doesn't work anymore.

But it should be mentioned, that the difference in predictions is so little different from GR that nowadays we aren't able to measure which one is "correct".

• QFT becomes nonlinear whenever interaction is involved. For example, the Dirac equation in QED is nonlinear, as evidenced in Fermi's effective 4-fermion interaction term. So what's wrong/con with nonlinearity? – MadMax Aug 27 at 17:53