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[Image] When A falls into the black hole B
gets thermally distributed headache. If Blogger had space for a subtitle it would be “A paper I can’t make up my mind about”. A few months ago a paper appeared on the arxiv that proposed to test quantum gravitational effects with neutrino oscillations.

Quantum Gravity effect on neutrino oscillations in a strong gravitational field
Jonathan Miller, Roman Pasechnik
arXiv:1305.4430 [hep-ph] Models in quantum gravity phenomenology span a spectrum that reaches from conservative but boring to interesting but flaky. This craziness factor is of course somewhat subjective, but the paper in question at first sight seemed to fall somewhere in the middle. In a nutshell, the authors are arguing that neutrino oscillation would be affected in the vicinity of black holes by interaction with gravitons and that this may cause a potentially observable phase distortion. For this they made the assumption that it’s the neutrino mass eigenstates (not flavor eigenstates) that couple to the graviton, and then they had some rather vague explanation that the type of this coupling would depend on the fundamental theory of gravity and thus could be used as a test.

Since it wasn’t originally really clear to me what assumption they made on top of perturbatively quantized gravity and why, I had a longer exchange with the authors in which they patiently answered my dumb questions. They updated the paper two months later and version two is a remarkable improvement over the first version. Alas, I’m still not sure the effect is real. But neither can I find a reason why it’s not real. Let me explain.

First, forget about the neutrino oscillation. That really isn’t so relevant, it’s just that neutrinos can deliver a particularly clean signal because they interact weakly with other stuff. Second, calling the gravitational field that they are concerned with a “strong” field is somewhat misleading. The term is commonly used to mean in the Planckian regime, but the field they talk about is that of a solar mass black hole close by the horizon. That’s strong compared to the field you just sit in, but still far off the Planckian regime. Also forget the stuff about collapse in the abstract, it doesn’t make much sense to me.

But then, note that while it’s often said that gravity is a weak interaction that’s a sloppy statement. Yes, that little fridge magnet and its electromagnetic interaction can overcome the gravitational pull of the whole planet Earth. But if you slam the door the magnet falls down, meaning the forces are quite comparable. How strong gravity is depends on how much mass you accumulate. In the paper the authors make the point that the cross-section for gravitational bremsstrahlung (that’s exchange of a virtual graviton and emission of a real graviton) is tiny for masses of elementary particles all right. But if you put in the mass of a solar mass black hole as one of the interacting ‘particles,’ the cross-section becomes comparable to that of other cross-sections in the standard model.

The original calculation of this cross-section goes back to a paper in the 60s. This is just perturbatively quantized gravity and besides the coupling constants, indices on the propagators, and polarization tensors very similar to the respective qed effect. Having said that, there’s no particular reason bremsstrahlung should be coherent or at least I don’t see one. This would mean then that a particle that passes by the black hole experiences a phase blurring, essentially because the background field is not in fact classical but because the interaction with the gravitational source is mediated by virtual gravitons. Or so the idea. Then they claim that this effect is large enough as to be potentially observable.

Having pulled out the origin of their proposed effect however, the paper suddenly moved to the very conservative end of the spectrum. On that conservative end, you typically find lots of effects that are almost certainly there, but way too small to be observable. If it was possible to find evidence for the quantization of the gravitational background field, evidence that virtual gravitons have been exchanged, this would be amazing.

However, my headache with the paper, which prevails in its revised version, is the following. Treating the black hole as a point particle is almost certainly a bad approximation. In some sense one might say a black hole is as close to a perfect point particle as we’ll ever get. But the distance in which the particle passes by the ‘point’ that is at the center of the black hole is large, much larger than the wavelength of the particle. It takes some time for the particle to pass by the horizon. It shouldn’t exchange one graviton at a fairly high energy (comparable to that of the neutrino in the black hole restframe with non-negligible probability) but it should exchange a lot of very low energy gravitons. This must be so simply because the equivalence principle prevents you from noticing anything on distance scales below the curvature radius. If this passage by the black hole was treated correctly, the effect would almost certainly get smaller. The question is how much smaller. It seems implausible it would vanish completely.

For me this also raises the question whether the cross-section would depend on what you believe is inside the black hole or at its horizon respectively. Eg if you’re a fuzzball fan, the coupling might look quite different than when you believe in baby universes. And let me not even get started on firewalls.

So I’m quite convinced that the effect is actually much smaller than they say, but this only raises the question just how small it is. I’m also not sure whether such an effect, if it exists, would be truly a sign for the quantization of the gravitational field. I mean, to first approximation the graviton exchange just has the effect that the particle moves on a geodesis. If you take into account that the particle itself is quantum and not a point particle, you should also notice some dispersion in a non-homogeneous background. But that’s not a signal for the quantization of the background, just for the quantum nature of the particle. Ie, it is conceivably possible that even if the effect is real, it’s not a signal for quantum gravity.

Having said that, if the particle acquires a phase-blurring as a correction from the quantum nature of the background field, the same effect should exist for charged particles passing by large charged objects in conceivable distance. Let me know if you have a useful reference.

The paper is now on the pile with unsettled cases...