Update: Some speculations on the topological structure of elementary particles

More from Sabbir:

I just wanted to provide a little update on progress on this subject since last time I wrote.

I mentioned in earlier posts that I suspected that the discretised charge of the elementary fermions might be explained in terms of a ‘defect angle’ which is experienced by neutrinos rotating around the Kerr ring-singularity defining the particle. Because the direction of time changes sign every time the neutrinos pass through the ring singularity, they effectively become bounded standing waves ‘trapped’ in time as they wind around a toroidal surface surrounding the singularity an integral number of times.

Now, after a little research, I discovered that the existence of defect angles in classical gravity corresponding to the intrinsic spin of elementary particles is actually nothing new. Indeed there exists a generalisation of Einstein’s general theory of relativity (GTR) which allows for the existence of ‘torsion’ – in particular, the connection coefficients associated with gravitational curvature on Riemannian manifolds, which are usually taken to be symmetric, are allowed to have an asymmetric component which is associated with particle spin. This particle spin, in the current context, would be associated with the speed-of-light rotation of the Kerr singularity.

It happens that torsion results precisely in the kind of ‘spacetime defect’ (namely the defect angle) that I am looking for to explain the charge of electrons and quarks.

This generalised theory of gravity is called the Einstein-Cartan-Kibble-Sciama (ECKS, or simply EC) theory, as the basic ideas were first introduced by Einstein and Cartan in the 1920s and 30s (I think), and then later fleshed out in much greater detail in the 1960s by Kibble (currently still at Imperial College) and Sciama. The theory is a very pleasing one in the sense that it has been proven that is precisely the local gauge theory associated with the full Poincare invariance group of relativistic spacetime, which consists of both Lorentz transformations and spatial translations – something which cannot be said of GTR.

It has been known since its development in the 1960s that there was a very strong link between Einstein-Cartan theory and the usual theory of defects and dislocations in crystals in condensed matter theory which had been beautifully elucidated by Kondo and others in the 1940s, and that they shared an almost identical mathematical structure. More recently, Petti showed more directly the equivalence between the theory of spacetime defects and Einstein-Cartan theory.

All of this is of course very encouraging in the context of my proposed model. The defect angle required by the model becomes a necessary consequence of the rotation of the ring singularity in the presence of torsion, and furthermore, the most elementary scalar particles (which need not be black holes) can be associated with localised conical singularities, which will cause matter in the exterior to be “attracted” to them (think of straight lines trajectories around a cone that actually look circular or elliptical).

I had claimed that the neutrinos themselves were formed by the gravitational collapse of either ‘gravitational waves’ or ‘gravitons’ (or ‘dilatons’ or ‘axions’ or some other proposed massive scalar particle), without being able to specify precisely which or why. If the analogy between Einstein-Cartan theory and the theory of dislocations is actually more than that, and rather reflects the true nature of spactime, then these ‘gravitons’ may in fact be none other than (geometrical) defects in spacetime, i.e. the conical ‘pinches’ referred to above which give them the appearance of having mass. Furthermore, spacetime itself is then simply a (possibly continuous limit of) some kind of Riemannian crystal lattice, the defects in which give rise to the physical universe that we see.

This would be quite fascinating as it would suggest that there is some kind of ‘substructure’ to the fabric of spacetime – that perhaps we are living on some kind of regular crystalline structure formed within the context of an even deeper physical reality, and that GTR and – if my hypotheses are indeed correct – all of the physical laws that we observe in nature, emerge from this more basic and fundamental physical reality. Indeed it has hard to imagine otherwise, as it would be difficult to explain the existence of defects in a perfectly smooth continuum.

Wassalam,
Sabbir.

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