Searching for Spacetime Defects

Defects in graphene.
Image source: IOP.

Whether or not space and time are fundamentally discrete is one of the central questions of quantum gravity. Discretization is a powerful method to tame divergences that plague the quantization of gravity, and it is thus not surprising that many approaches to quantum gravity rely on some discrete structure, may that be condensed matter analogies, triangulations, or approaches based on networks. One expects that discretization explains the occurrence of singularities in general relativity as unphysical, much like singularities in hydrodynamics are merely mathematical artifacts that appear because on short distances the fluid approximation for collections of atoms is no longer applicable.

But finding experimental evidence for space-time discreteness is difficult because this structure is at the Planck scale and thus way beyond what we can directly probe. The best tests for such discrete approaches thus do not rely on the discreteness itself but on the baggage it brings, such as violations or deformations of Lorentz-symmetry that can be very precisely tested. Alas, what if the discrete structure does not violate Lorentz-symmetry? That is the question I have addressed in my two recent papers.

In discrete approaches to quantum gravity, space-time is not, fundamentally, a smooth background. Instead, the smooth background that we use in general relativity – the rubber sheet on which the marbles roll – is only an approximation that becomes useful at long distances. The discrete structure itself may be hard to test, but in any such discrete approach one expects the approximation of the smooth background to be imperfect. The discrete structure will have defects, much like crystals have defects, just because perfection would require additional explanation.

The presence of space-time defects affects how particles travel through the background, and the defects thus become potentially observable, constituting indirect evidence for space-time discreteness.To be able to quantify the effects, one needs a phenomenological model that connects the number and type of defects to observables, and can in return serve to derive constraints on the prevalence and properties of the defects.

In my papers, I distinguished two different types of defects: local defects and non-local defects. The requirement that Lorentz-invariance is maintained (on the average) turned out to be very restrictive on what these defects can possibly do.

The local defects are similar to defects in crystals, except that they are localized both in space and in time. These local defects essentially induce a violation of momentum conservation. This leads to a fairly straight-forward modification of particle interactions whenever a defect is encountered that makes the defects potentially observable even if they are very sparse.

The non-local defects are less intuitive from the particle-physics point of view. They were motivated by what Markopoulou and Smolin called ‘disordered locality’ in spin-networks, just that I did not, try as I might, succeed in constructing a version of disordered locality compatible with Lorentz-invariance. The non-local defects in my paper are thus essentially the dual of the local defects, which renders them Lorentz-invariant (on the average). Non-local defects induce a shift in position space in the same way that the local defects induce a shift in momentum space.

I looked at a bunch of observable effects that the presence of defects of either type would lead to, such as CMB heating (from photon decay induced by scattering on the local defects) or the blurring of distant astrophysical sources (from deviations of photons from the lightcone caused by non-local defects). It turns out that generally the constraints are stronger for low-energetic particles, in constrast to what one finds in deformations of Lorentz-invariance.

Existing data give some pretty good constraints on the density of defects and the parameters that quantify the scattering process. In the case of local defects, the density is roughly speaking less than one per fm⁴. That’s an exponent, not a footnote: It has to be a four-volume, otherwise it wouldn’t be Lorentz-invariant. For the non-local defects the constraints cannot as easily be summarized in a single number because they depend on several parameters, but there are contour plots in my papers.

The constraints so far are interesting, but not overwhelmingly exciting. The reason is that the models are only for flat space and thus not suitable to study cosmological data. To make progress, I'll have to generalize them to curved backgrounds. I also would like to combine both types of defects in a single model. I am presently quite excited about this because there is basically nobody who has previously looked at space-time defects, and there’s thus a real possibility that analyzing the data the right way might reveal something unexpected. And into the other direction, I am looking into a way to connect this phenomenological model to approaches to quantum gravity by extracting the parameters that I have used. So, you see, more work to do...