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Consider the two principles:

  • A QFT of spin-2 particles (gravitons) cannot transmit information faster than the speed of light by special relativity. (Let's make an assumption that such a theory can exist).
  • In GR we can have a metric of space-time in which shortens distances between two points to arbitrarily small distances. (e.g. the value of $g_{xx}$ can be arbitrarily big or small.

In my mind there seems to be a contradiction, that a theory of spin-2 particles on a Minkowski background should not allow information to go outside a lightcone. But the limit is supposed to be GR in which a metric can alter that lightcone to be wider.

My own thoughts are that perhaps the theory of spin-2 particles would only converge when the metric makes the distances the same or larger. This is what happens when we have negative spacial curvature resulting from massive objects. Then there doesn't seem to be a contradiction. Hence a theory of spin-2 particles on a flat background space does seem to be consistent with GR in the limit (and SR in terms of the gravitons) at the same time.

But the spin-2 graviton theory should theoretically be consistent on all starting background spaces not just flat backgrounds.

So the inconsistency seems to be that QFT shouldn't alter the causal structure but metrics do alter the causal structure. (Perhaps the graviton model only works if the new causal structure is a subset of the Minkowski causal structure?)

Is there a resolution to this seeming paradox? This is the main confusion I have with theories like string theory which say that they approximate to general relativity.

Secondly, in the quantum case if GR is the approximation and spin-2 particles on a flat background is the "reality", we should observe particles going faster than the speed of light in respect to the GR space, which also seems wrong. Unless all these processes are magically cancelled.

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    $\begingroup$ I'll get back to you once we have a theoretically- and observationally-consistent theory of quantum gravity. $\endgroup$
    – controlgroup
    Commented Nov 3 at 23:27
  • $\begingroup$ @controlgroup but this is what string theory pertains to be. Also see Feynman's Lectures on Gravity in which he discusses thinking of GR as a spin-2 field. $\endgroup$
    – bob
    Commented Nov 4 at 0:15
  • $\begingroup$ Well, you can consider that four-velocity, governed by the geodesic acceleration equations, is thus governed by the Christoffel symbols, which corresponds to the field tensor in a spin-2 QFT. The Christoffel symbols, governed by the metric, thus also are governed by the metric. $\endgroup$
    – controlgroup
    Commented Nov 4 at 0:20
  • $\begingroup$ Yes, indeed. But how does that answer the question if g_xx = 0.01 then the distance between points seems 10x times as near, so the particle would go 10x the speed of light between the points (in terms of the original Minkowski space-time in which the QFT exists). $\endgroup$
    – bob
    Commented Nov 4 at 0:21
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    $\begingroup$ In essence: information and matter can go faster than $c$, just not along spacelike curves (which in flat space are $v>c$ and in general usually faster than null curves). $\endgroup$
    – controlgroup
    Commented Nov 4 at 0:31

2 Answers 2

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QFT doesn't explicitly require that nothing travels faster than light; quantum theory is completely nonlocal anyway, so the statement that nothing is traveling faster than light is just a statement about commutation relations of observables. You get the result that observables commute outside light cones if you start with a Lorentz-invariant Lagrangian and then canonically quantize by performing the appropriate equal-time commutation relations. The addition of a spin-2 field changes the equal-time commutation relations so that observables which would have commuted without the spin-2 field no longer commute. So effectively, the spin-2 fields can in fact make things go faster than the background spacetime would imply that they can.

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    $\begingroup$ Can you give some mathematics to this. I was under the impression that one of the rules of QFT is that observable commute outside the light cone even for a spin-2 theory. Otherwise is that still QFT? I'm not sure how a theory in which says that observable commute outside a lightcone gives rise to something in which things don't commute outside the lightcone. BTW, I'm not talking about QFT on curved space, I'm talking about a theory of graviton interactions on flat space which is supposed to approximate to curved space. $\endgroup$
    – bob
    Commented Nov 4 at 3:53
  • $\begingroup$ To give more clarity, the Feynman propagators for gravitons are just like those for bosons but with more indices. $\endgroup$
    – bob
    Commented Nov 4 at 4:01
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    $\begingroup$ Propagators are defined using the vacuum. In a vacuum, there are no gravitons and so everything should commute outside the light cone like normal. If the quantum state is not the vacuum, and in particular contains gravitons, then the commutation properties of observables can change. $\endgroup$
    – Travis
    Commented Nov 4 at 4:20
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    $\begingroup$ Ah... that's interesting. But would the region in a gravity well of a star be called a vacuum or a place filled with gravitons. It feels like we are solving GR to get a vacuum and then suggesting gravitons move about on this. This feels like semi-classical quantum gravity. $\endgroup$
    – bob
    Commented Nov 4 at 4:40
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    $\begingroup$ Yes, the gravitational field is composed of gravitons, just like the electromagnetic field is composed of photons. Just because we only detect one or two of them at a time doesn't mean there aren't much more. Bosons are not conserved in number like Fermions are. And also, the uncertainty principle applies as always: so if a photon/graviton is truly traveling in only one specific direction, and therefore has a definite momentum, then it is a wave that exists over all of space. $\endgroup$
    – Travis
    Commented Nov 4 at 12:06
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Your problem seems to stem from a misunderstanding of gravitons. Firstly, you need General Relativity and QFT at the same time. Then, expand on a background metric (Minkowski, or (anti) de Sitter if you have a cosmological constant) and do all the QFT stuff around that curved or not background.

In this sense, gravitons are just perturbations to the background metric, so of course they shorten lengths and dilate time. Your problem essentially comes from the fact that you did the reasoning backward.

We have working theories of quantum gravity, like string theories as you point out in the comments, or the less high intellectual cost of the EFT approach to quantum gravity, see https://arxiv.org/abs/1702.00319. I could make an Ad for my own because it is now published in CQG, but I won't.

In a sense, this is just QFT in curved space, where your space is almost flat (or almost (anti) de Sitter)! For finding the "gravitons" inside a star (like you said in a comment), well just do a coherent state of gravitons... For QFT in curved space, a good introduction is Birrell & Davies.

Finally, concerning your argument about lightcones, the resolution essentially boils down to my first paragraph: gravitons make them change because gravitons perturb the background metric, that's as simple as that.

I honestly don't understand your last paragraph. Everything goes at most as the speed of light if it carries information (like particles), even in curved space.

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  • $\begingroup$ I don't think I did misunderstand. Expanding on a background metric means you only consider small perturbations away from a particular solution. This is fine for an approximation in some circumstances but will obviously break down at high enough energies when the perturbations become too large. You are answering a different question to the one I posed. What you are talking about is QFT on curved space time, not deriving curved space from graviton interactions on flat space-time. $\endgroup$
    – bob
    Commented Nov 5 at 1:07
  • $\begingroup$ "Gravitons peturb the background metric" is meaningless to me unless you provide some mathematical equation to back it up. $\endgroup$
    – bob
    Commented Nov 5 at 1:12
  • $\begingroup$ @bob Your question is based on a misunderstanding of gravitons, as I said, and as your first comment shows (I never talked about small perturbations). Your "paradox" comes from this misunderstanding. Rather than answering this wrong paradox, I fixed the misunderstanding --- not for you alone, but for anyone having the same line of reasoning as you. For your second comment, just plug $g_{\mu \nu} = g^{(0)}_{\mu \nu}+\kappa h_{\mu \nu}$, where $h_{\mu \nu}$ is the perturbation to $g^{(0)}_{\mu \nu}$ into the Einstein-Hilbert action and see what happens. $\endgroup$
    – Jeanbaptiste Roux
    Commented Nov 5 at 5:56
  • $\begingroup$ Yes that works when kappa is small but not large. $\endgroup$
    – bob
    Commented Nov 5 at 19:21
  • $\begingroup$ @bob The point of $\kappa$ being small is to perform an asymptotic power series. This differs entirely from the assumption $|h_{\mu \nu}| \ll 1$. Of course, when the energetic scale of any process is close to the Planck energy, quantum gravity as a QFT is plagued by divergences. But this problem is far from your question. $\endgroup$
    – Jeanbaptiste Roux
    Commented Nov 6 at 6:17

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