Is space-time a prism?

Tl;dr: A new paper demonstrates that quantum gravity can split light into spectral colors. Gravitational rainbows are almost certainly undetectable on cosmological scales, but the idea might become useful for Earth-based experiments.

Einstein’s theory of general relativity still stands apart from the other known forces by its refusal to be quantized. Progress in finding a theory of quantum gravity has stalled because of the complete lack of data – a challenging situation that physicists have never encountered before.

The main problem in measuring quantum gravitational effects is the weakness of gravity. Estimates show that testing its quantum effects would require detectors the size of planet Jupiter or particle accelerators the size of the Milky-way. Thus, experiments to guide theory development are unfeasible. Or so we’ve been told.

But gravity is not a weak force – its strength depends on the masses between which it acts. (Indeed, that is the very reason gravity is so difficult to quantize.) Saying that gravity is weak makes sense only when referring to a specific mass, like that of the proton for example. We can then compare the strength of gravity to the strength of the other interactions, demonstrating its relative weakness – a puzzling fact known as the “strong hierarchy problem.” But that the strength of gravity depends on the particles’ masses also means that quantum gravitational effects are not generally weak: their magnitude too depends on the gravitating masses.

To be more precise one should thus say that quantum gravity is hard to detect because if an object is massive enough to have large gravitational effects then its quantum properties are negligible and don’t cause quantum behavior of space-time. General relativity however acts in two ways: Matter affects space-time and space-time affects matter. And so the reverse is also true: If the dynamical background of general relativity for some reason has an intrinsic quantum uncertainty, then this will affect the matter moving in this space-time – in a potentially observable way.

Rainbow gravity, proposed in 2003 by Magueijo and Smolin, is based on this idea, that the quantum properties of space-time could noticeably affect particles propagating in it. In rainbow gravity, space-time itself depends on the particle’s energy. In particular, light of different energies travels with different speeds, splitting up different colors, hence the name. It’s a nice idea but unfortunately it’s is an internally inconsistent theory and so far nobody has managed to make much sense of it.

First, let us note that already in general relativity the background depends of course on the energy of the particle, and this certainly should carry over also into quantum gravity. More precisely though, space-time depends not on the energy but on the energy-density of matter in it. So this cannot give rise to rainbow gravity. Worse even, because of this, general relativity is in outright conflict with rainbow gravity.

Second, an energy-dependent metric can be given meaning to in the framework of asymptotically safe gravity, but this is not what rainbow gravity is about either. Asymptotically safe gravity is an approach to quantum gravity in which space-time depends on the energy by which it is probed. The energy in rainbow gravity is however not that by which space-time is probed (which is observer-independent), but is supposedly the energy of a single particle (which is observer-dependent).

Third, the whole idea crumbles to dust once you start wondering how the particles in rainbow gravity are supposed to interact. You need space-time to define “where” and “when”. If each particle has its own notion of where and when, the requirement that an interaction be local rather than “spookily” on a distance can no longer be fulfilled.

In a paper which recently appeared in PLB (arXiv version here), three researchers from the University of Warsaw have made a new attempt to give meaning to rainbow gravity. While it doesn’t really solve all problems, it makes considerably more sense than the previous attempts.

In their paper, the authors look a small (scalar) perturbations over a cosmological background, that are modes with different energies. They assume that there is some theory for quantum gravity which dictates what the background does but do not specify this theory. They then ask what happens to the perturbations which travel in the background and derive equations for each mode of the perturbation. Finally, they demonstrate that these equations can be reformulated so that, effectively, the perturbation travels in a space-time which depends on the perturbation’s own energy – it is a variant of rainbow gravity.

The unknown theory of quantum gravity only enters into the equations by an average over the quantum states of the background’s dynamical variables. That is, if the background is classical and in one specific quantum state, gravity doesn’t cause any rainbows, which is the usual state of affairs in general relativity. It is the quantum uncertainty of the space-time background that gives rise to rainbows.

This type of effective metric makes somewhat more sense to me than the previously considered scenarios. In this new approach, it is not the perturbation itself that causes the quantum effect (which would be highly non-local and extremely suspicious). Instead the particle merely acts as a probe for the background (a quite common approximation that neglects backreaction).

Unfortunately, one must expect the quantum uncertainty of space-time to be extremely tiny and undetectable. A long time has passed since quantum gravitational effects were strong in the very early universe and since then they have long decohered. Of course we don’t really know this with certainty, so looking for such effects is generally a good idea. But I don’t think it’s likely we’d find something here.

The situation looks somewhat better though for a case not discussed in the paper, which is a quantum uncertainty in space-time caused by massive particles with a large position uncertainty. I discussed this possibility in this earlier post, and it might be that the effect considered in the new paper can serve as a way to probe it. This would require though to know what happens not to background perturbations but other particles traveling in this background, requiring a different approach than the one used in this paper.

I am not really satisfied with this version of rainbow gravity because I still don’t understand how particles would know where to interact, or which effective background to travel in if several of them are superposed, which seems somewhat of a shortcoming for a quantum theory. But this version isn’t quite as nonsensical as the previous one, so let me say I am cautiously hopeful that this idea might one day become useful.

In summary, the new paper demonstrates that gravitational rainbows might appear in quantum gravity under quite general circumstances. It might be an interesting contribution that, with further work, could become useful in the search for experimental evidence of quantum gravity.

Note added: The paper deals with a FRW background and thus trivially violates Lorentz-invariance.