In Einstein's theory of general relativity, Riemann tensor is assumed to be symmetric, due to great simplification of the theory compared to the one with non-symmetric Riemann tensor, and an absolute lack of experimental evidence to the contrary for 100 years since it was discoveried. Torsion, which is the cause of asymmetry of the Riemann tensor is set to zero. I do not know why this author is so excited about torsion tensor, the idea that torsion could be non-zero has been around for at least 90 years, but got zero traction. Disclaimer : I have a phd in string theory.
I don't have a PhD in string theory, but would like to make the point that many times in the history of science ideas that were originally regarded as uninteresting lie dormant for a long time only to usefully surface much later in the context of new evidence.
In this case, the researcher seems to be excited specifically because of the potential of torsion to explain dark energy -- a recently discovered phenomena (although of course, oddly presaged by Einstein's cosmological constant hack).
The classical theory of gravity with torsion certainly does not explain dark matter. The author is trying to build some quantum theory of gravity, where torsion is required, speculating that it would explain dark matter. I should add here that the attempts to build a quantum theory of gravity without torsion failed in the last 100 years (string theory is the best candidate), and the author is trying to add even more complexity by adding torsion tensor.
I'm trying to follow you as a (very interested) layman, so excuse me if I'm being stupid. Couldn't the failure to build quantum gravity be BECAUSE torsion has been excluded (which is I believe what the author suggests)? Maybe this is something that has to be looked into.
I'm thinking that a marriage of physics, mathematics and CS might be necessary to overcome our limits in understanding these structures. Something like an IBM Watson for physicists, where a computer is fed with all informations we have and solves an optimisation problem to come up with a unified theory explaining all the phenomenons with the least complex solution (i.e. the least universal constants). Another requirement would be to have 42 as an error code for all possible failures in the calculation ;).
The author paraphrases "spacetime tells matter how to move, and matter tells spacetime how to curve". The torsion axiom extends this to "...and matter tells spacetime how to curve and twist". The way he explains it makes it sound like an elegant way to complete the General Relativity hypothesis, which Einstein described as "the happiest thought of his life". If an axiom has a certain elegance about it, then perhaps a quantum theory of gravity without torsion is adding the complexity, not the other way around.
Mind you, string theory itself includes torsion: the Riemann tensor that arises in (e.g.) curved backgrounds includes contributions from the NS-NS B-field that correspond precisely to the effects of torsion. I don't think that non-zero torsion in itself is an outlandish idea, but I completely agree that it's fair to be skeptical of claims that torsion has significant observable effects. Disclaimer: I also have a PhD in string theory.
I'm not sure I understand. In Riemannian geometry it might be convenient to use a connection with non-vanishing torsion, but it is not required. What do you mean by 'includes'?
Well, to give an explicit example (sorry, non-specialists!), consider the nonlinear sigma model action for superstring theory in an arbitrary background geometry (that is, arbitrary spacetime metric G and NS-NS B-field; we'll ignore the dilaton). The coefficient of the 4-fermion term is the Riemann tensor. But if you construct that Riemann tensor just from the metric (as one does in general relativity without torsion), you'll omit essential pieces: you also need to include terms in the Riemann tensor resulting from a non-zero torsion tensor T=-dB (in differential form notation). A similar situation holds for the covariant derivatives in the fermion kinetic terms.
Now, it's entirely up to you whether you write that 4-fermion term coefficient as "Riemann tensor (including torsion)" or as "Riemann tensor (torsion free) + (lots of weird, arbitrary-looking interaction terms involving derivatives of B)". So on some level, there's no reason that you must use a connection with non-vanishing torsion. But I would claim that the equations are much more elegant (and give deeper insight) when expressed with torsion "built-in". [Fun fact: So does Polchinski, but he's not talkative about it. If you look up "torsion" in the index of Vol. 2, the first reference is to the page with the equations I've referenced above... but the word "torsion" doesn't appear anywhere in the text of the page!]
Aside: If anyone out there is interested in how this torsion stuff fits into the mathematics of Relativity, you might have a look at my notes on how it would be incorporated into Bob Wald's textbook: http://www.slimy.com/~steuard/teaching/tutorials/GRtorsion.p...
