No special relativity is not effected. This local splitting is equivalent to the existence of a Lorentz structure at a manifold (a globally non-vanishing vector field).
But more is true in the exotic context: exotic smoothness admit locally hyperbolic structures and the isometry of the 3D...
Now I found some rest to answer this question: how does this space looks like?
For simplicity I will concentrate on exotic versions of S3×ℝ. For the exotic space, there is no splitting into space and time globally. Therefore, there is no global, smooth S3 for a fixed time. If one wants to...
I will start only with a little overview what was done in the last 6 or 7 years. The informations above are a little bit outdated...
But at first let me state that nearly all 4-dimensional manifolds admit exotic differential structures. For compact 4-manifolds there are countable infinite many...
In my opinion, twisted geometries will give a better semi-classical limit. The problem is viewable in perturbative quantum gravity. Every graviton gives only a (in principle) beglectable contribution and you need uncountable infinite many gravitons to get measurable contribution. Regge calculus...
I agree with marcus: I can only describe my own view. A real evaluation is not possible, even experimental verifications are missed.
But back to twisted geometries: I want to compare it with the two approaches to geometry: Riemann or Cartan.
The discrete version of Riemannian geometry is Regge...
As far as I know the motivations are different: Witten found twistor-like relations between string amplitudes. With the help of these ideas, some people were able to express loop amplitudes in some QFT (mostly supersymmetric) by tree amplitudes.
As I explained above, the LQG use of twistor...
Twistor theory was originally designed to obtain an effective spinor representation of the conformal group SO(4,2) of the Minkowski space, i.e. the representation of SU(2,2). All constructions work only for the flat Minkowski space and Penrose (around 1976) tried to extend it to the curved...
The usual process to generate gravitational waves is part of classical GRT. Currently there is no contribution of quantum gravity to this part. Therefore sring theory is also not able to produce any contribution to this topic. As far as I know scalar-tensor theories do not produce a different...
Because of this article: http://physics.aps.org/articles/v8/108
it is for me more likely that the vacuum tends to the stable region. But one needs more data especially to determine the Higgs potential.
See also http://arxiv.org/abs/1507.08833
One possible research program is the Causal set theory of Rafael Sorkin and co-workers. But it seems you look for another representation of quantum theory?
In that case check out the approaches using model (or topos) theory especially forcing.
Interesting for you is maybe also my own approach...
http://arxiv.org/abs/1601.06436
Smooth quantum gravity: Exotic smoothness and Quantum gravity
Torsten Asselmeyer-Maluga
(Submitted on 24 Jan 2016)
Over the last two decades, many unexpected relations between exotic smoothness, e.g. exotic R4, and quantum field theory were found. Some of these...
Because of this discussion, I had a chance to look into this paper.
In principle, kodama above is right: it is a combination of NCG and LQG to get better control about the limit. But it is also much more.
Originally, the work started with the construction of the spectral triple some years ago...
In principle I agree with: the discrete spektra is the most important point, see the smooth Schrödinger equation.
I also agree that the point is not fundamental in GR. If I remember correctly, it is the famous hole argument of Einstein
(see http://plato.stanford.edu/entries/spacetime-holearg/)...
For me, it is a dogma that the spacetime in quantum gravity has to be discrete. As far as I know, there is no experiment showing the discreteness. It is interesting that a smooth manifold has much to do with discrete structures.
I will only mention a few one: (for more details, see my essay at...