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 ?



Singularity theorems for semiclassical gravity
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](http://arxiv.org/abs/1507.06299) and subsequent [paper](https://arxiv.org/abs/1507.00297) 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](http://dx.doi.org/10.1088/0264-9381/28/12/125009) and references therein.




---


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](http://dx.doi.org/10.1088/0264-9381/28/21/215008); the basic mutual nonimplication between superluminal propagation and energy conditions are discussed already in the introduction. 


