Singularity theorems for semiclassical gravity

Question

The semi-classical Einstein equations (without a cosmological constant) are $G^{\mu \nu} = 8\pi \langle T^{\mu \nu} \rangle$. I am told that there are serious objections as to why these equations cannot be a completely correct description of nature. (If they are correct, then there is no need for quantum gravity.) From what I have heard, the objections are

1) Superluminal propagation is possible in some cases.

2) If one takes $\frac{1}{\sqrt{2}} (\vert M \ at \ A \rangle + \vert M \ at \ B \rangle)$, where $M$ is large and $A$ and $B$ are widely separated, one arrives at counter-intuitive conclusions.

My question is : Do the Hawking-Penrose singularity theorems still hold in semi-classical gravity ? If so, I shall be happy if pointed out to a rigorous mathematical reference.

I have some physics questions too but perhaps another forum is a better venue for these :

1) Is objection 1) true? Are there are any other objections ?

2) So what if superluminal propagation is possible in extreme situations ? What evidence do we have against it anyway (in such extreme cases that is) ? As for objection 2), does the wave function of the universe allow such pathologies ?

Answer 1

In terms of singularity theorems:

The Hawking-Penrose singularity theorems require certain energy conditions be satisfied; the theorems are in particular not-sensitive to the underlying fields being classical or quantum. So the trivial answer to your question is:

As stated, the singularity theorems hold for semi-classical relativity.

The problem, however, is that even for classical fields the energy conditions are assumptions. For classical fields they are deemed physically reasonable, and most model fields we study do verify them. The main objections when it come to the semiclassical picture is that even though the dominant and null energy conditions may be satisfied by physically reasonable classical fields, they may fail "pointwise" for their quantum analogues. Thus the singularity theorems may fail to apply as the hypotheses used by the theorems may fail to hold. So a better question to ask is:

Are there versions of the singularity theorems that hold for semiclassically reasonable matter models?

The proposed solution is to consider "averaged" versions of the energy conditions.

A great deal of work has been done concerning the "averaged null energy condition". You can look in Kontou's PhD thesis and subsequent paper for more information and references. Some quantum models have been shown to obey the averaged energy conditions.

The next question then is whether these weakened versions of energy conditions are sufficient to give classical phenomena such as Hawking/Penrose singularity theorems. The answer is "yes", with some caveats. For details you can look at the work of Fewster and Galloway and references therein.

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As to the physic questions: I don't know about (2). But (1) for most fields is incorrect. The failure of the dominant energy condition is, to some extent, an independent phenomenon to the failure of finite speed of propagation of fields (and in particular to the presence of superluminal propagation). (Also, superluminal is really the wrong word, the barrier is the speed of gravity, not the speed of light.)

For details see my CQG article; the basic mutual nonimplication between superluminal propagation and energy conditions are discussed already in the introduction.

Answer 2

The semi-classical Einstein equations (without a cosmological constant) are $G^{\mu \nu} = 8\pi \langle T^{\mu \nu} \rangle$. I am told that there are serious objections as to why these equations cannot be a completely correct description of nature. (If they are correct, then there is no need for quantum gravity.) From what I have heard, the objections are [...]

There are many, many issues with semiclassical gravity. Examples of problems in addition to the two you mention: questions about stability of flat spacetime, problems with the Born rule and Copenhagen interpretation due to nonlinearity. The issue is not whether semiclassical gravity is fundamental but whether it's even a reliable approximation, and, if so, which formulation of semiclassical gravity to use and how to put bounds on the errors.

To my mind, there is an overriding physical reason why semiclassical gravity clearly can't be a fundamentally correct theory, which is that it couples a classical field to a quantized field. Physicists realized almost a century ago that this just generically doesn't work. Bohr and his followers wanted to keep the electromagnetic field classical while quantizing the atom. This resulted in the Bohr-Kramers-Slater theory, which lived for only a few years before being disproved experimentally by Bothe. The basic issue is that if you want energy and momentum to be conserved in a quantum context, you need long-range correlations, and those correlations don't exist in a classical field. For example, in the photoelectric effect, you would like the absorbed photon to give its energy to exactly one electron. If it gives its energy to both electron A and electron B, then you've doubled the amount of energy that's present. What prevents this from happening is that the electromagnetic field is quantized, and therefore it can have correlations between what it does at A and what it does at B. Without these quantum correlations, energy and momentum can at best be conserved on a statistical basis. Nonconservation of energy-momentum has never been observed, and if it did happen (locally), it would render the Einstein field equations inconsistent.

Do the Hawking-Penrose singularity theorems still hold in semi-classical gravity ?

Willie Wong has answered this, but I would like to provide some perspective in addition. Semiclassical gravity, if it works at all as an approximation scheme, is an approximation scheme that applies to weak gravitational fields. Therefore if it told us something about singularities, we wouldn't believe it.

Generically, theories of quantum gravity tend to provide mechanisms that prevent the formation of singularities once you get down to the Planck scale.

So what if superluminal propagation is possible in extreme situations ? What evidence do we have against it anyway (in such extreme cases that is)

I think Willie Wong's answer clarifies why this is not really the right way to frame the issue. However, there is a generic objection to superluminal propagation, which is that if it happens, it typically means we don't have existence and uniqueness of solutions to Cauchy problems, so that physics loses its predictive value.