Energy conservation in semiclassical gravity

Question

I'd like to know whether semiclassical gravity models contain an energy conservation law with the following (heuristic) form:

$$``\text{Energy of classical spacetime + expected energy of quantum fields = constant.}"$$

In semiclassical discussions of black hole evaporation, it is argued that if the quantum field gains energy (in Hawking radiation), then the spacetime must lose energy to compensate, so the black holes shrinks. So it seems that such a conservation law is assumed to hold semiclassically. Is there a precise statement of this semiclassical conservation law? Can it be derived (perhaps via Noether's theorem) from symmetry considerations?

(Note: this question is closely related to a previous question I asked about the Hawking effect. The questions are not duplicates, though, since the current question is more general, and is not specifically about black holes).

Answer 1

Semiclassical gravity is defined by the following procedure:

1. define a background metric ${}^{0}g_{ab}$
2. Use the background metric as the background for some quantum field theory (usually, this means promoting the integral measure of the action to $\sqrt{|g|}d^{4}x$ and replacing partial dertivatives by covariant derivatives,a nd rederiving relevant rules, treating the background metric as exact
3. Figure out the $T_{ab}$ generated by this field
4. Solve the new einstein equation for some ${}^{1}g_{ab}$
5. repeat.

Now, the thing is, unless a metric $g_{ab}$ has a timelike killing vector, or has some closed timelike surface with a timelike killing vector (for instance, "conformal timelike infinity" for asymptotically flat spacetimes), then it does not have a conserved energy, so any "energy conservation" falls down at step (1), before you've even done anything semiclassical. If either of these conditions is accurate at for ${}^{0}g_{ab}$, then you can use that time coordinate to go and find a conserved energy on the spacetime (say, the ADM mass), and you can even look at the difference between what is calculated for point 1. and point 4. and attribute the correction to "energy of quantum fields"

Answer 2

This answer is similar to the excellent one by Jerry Schirmer, but tries to employ a bit less of a perturbative point of view, as I think this might make the concepts a bit clearer.

One can think of Semiclassical Gravity in terms of the semiclassical Einstein Equations, $$G_{ab} = 8 \pi \langle \hat{T}_{ab} \rangle_{\omega}, \tag{1}$$ where $\omega$ is the state of the fields. This equation is discussed, e.g., on Wald's Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics and can be motivated either by the fact that it seems natural or, if you prefer, because it can be derived on the one-loop approximation to perturbative quantum General Relativity (there is a bit more detail and discussion on Wald p. 98). Of course, we can't actually solve Eq. (1) exactly as it is insanely difficult (see Wald pp. 98–100), which pretty much justifies the point of view taken in Jerry Schirmer's answer. The reason I'm introducing Eq. (1) is because I believe it makes the concepts clearer.

More specifically, from Eq. (1), we see there is really nothing much different from what we would have in classical GR as long as we are considering the expectation values of the fields. If spacetime has a timelike Killing field and we can assume the expectation values to be small close to infinity, we can define the spacetime mass in the usual manners and just proceed as expected.

Notice also that if the spacetime possesses classical matter in addition to quantum matter, the equation would simply be modified to $$G_{ab} = 8 \pi \langle \hat{T}_{ab} \rangle_{\omega} + 8 \pi T_{ab},$$ where $T_{ab}$ is the stress tensor for the classical matter, and the remaining remarks persist.

In summary, semiclassical gravity contains such a conservation law pretty much in the same sense classical GR does.

Answer 3

I am not sure how helpful this would be, but maybe I could get the discussion started. So, instead of using Noether's theorem in basic, or classical GR, you can find the conservation of energy equations by solving Killing's equation and finding Killing's fields. For example, if your metric is just a sphere of $r=1$, $ds^2 = d\theta^2 + sin^2(\theta)d\phi^2$, then by using Killing's equation, $\xi_{\alpha;\beta} + \xi_{\beta;\alpha} = 0$, you would find the conserved quantities of the metric/system, which turn out to be the angular momentum operators you may be familiar with from quantum mechanics (but they are not quantum mechanical objects...) This would get you conserved quantities but not an overall conservation equation. For that, you would have to use $\nabla_\mu T^{\mu\nu} = 0$

But, there is a very special case which I believe would help with your question in that if $\nabla_\mu T^{\mu\nu} = 0$ and $R_{\alpha\beta\gamma\delta} = 0$ are true (flat spacetime), then you could find 10 global conservation of energy equations and thus 10 conserved quantities.

Other then that, I am not sure if there is a distinct statement on conservation of energy in semi-classical GR, or even in GR in the first place since it only occurs in special scenarios, such as for the Schwarzschild black hole. And if you want a true semi-classical system, then you would have to pick your favorite version of quantum gravity, and test it in the semi-classical regime (usually easiest if you find the partition function).

Hope this at least points you in the right direction, or draws attention for more answers (preferably better then mine)!