The Semiclassical Expansion

Semiclassical gravity is not defined merely by putting angle brackets around a stress tensor. It is an approximation in which a mean metric responds to quantum matter while quantum fluctuations of that metric remain parametrically smaller than the retained matter backreaction. The controlling parameter may be a curvature-to-cutoff ratio, a loop-counting parameter, a large number of matter species, or a state-dependent occupation scale; a trustworthy calculation states which one it uses and what correction is omitted first.

Required background. Einstein Equations supplies the geometric field equation and the contracted Bianchi identity used to organize the expansion.

Helpful background. Quantum Fields in Curved Spacetime supplies Hadamard states and renormalized stress tensors, while Black-Hole Geometry and Mechanics supplies the horizon distinctions needed in later applications.

The three semiclassical countings

Let $L$ be the shortest geometric length scale being resolved and let $\ell_{\mathrm P}^2=G$. A low-curvature expansion begins with

$$\varepsilon_{\mathrm{curv}}=\frac{\ell_{\mathrm P}^2}{L^2}\ll1.$$

For a massive field, a local derivative expansion instead uses $\varepsilon_m=(mL)^{-2}\ll1$; for a slowly changing mode, an adiabatic expansion uses $\varepsilon_{\mathrm{ad}}=(\omega\tau_{\mathrm{evol}})^{-1}\ll1$. These parameters answer different questions. Small curvature suppresses local gravitational higher-dimension operators when their coefficients and derivative scales are also controlled, whereas adiabaticity controls time derivatives and does not make a high-frequency state harmless. With many species or large finite curvature couplings, the effective cutoff can be lower than $\ell_{\mathrm P}^{-1}$; $G/L^2\ll1$ alone is then insufficient.

A loop expansion starts from a renormalized closed-time-path effective action,

$$\Gamma[g^+,g^-] =S_{\mathrm{grav}}[g^+]-S_{\mathrm{grav}}[g^-] +W_{\mathrm m}[g^+,g^-]+\Gamma_{\mathrm{grav\,loops}}+\cdots .$$

Varying with respect to the difference metric and then setting $g^+=g^-=g$ gives the causal mean-field equation. At leading matter-loop order it has the schematic form

$$G_{\mu\nu}+\Lambda g_{\mu\nu} +H^{\mathrm{local}}_{\mu\nu} =8\pi G\,\langle T_{\mu\nu}\rangle_{\mathrm{ren}},$$

where $H^{\mathrm{local}}_{\mu\nu}$ contains the variations of renormalized curvature-squared couplings. The state, subtraction prescription, boundary data, and retarded contour are part of the input. This effective-action route and its renormalized equation are developed explicitly in Hu and Verdaguer 2008, § 3.1, Eqs. (3.7)–(3.10).

Why a mean metric can be classical

Consider $N$ identical, independent matter fields and scale $G=\bar G/N$ with $\bar G$ fixed. Their mean stress tensor is extensive,

$$\langle T_{\mu\nu}^{\mathrm{tot}}\rangle =N\langle T_{\mu\nu}^{(1)}\rangle,$$

so $G\langle T^{\mathrm{tot}}\rangle=O(1)$ and matter backreaction survives as $N\to\infty$. Connected fluctuations add rather than square:

$$\langle t_{\mu\nu}^{\mathrm{tot}}t_{\rho\sigma}^{\mathrm{tot}}\rangle_c =N\langle t_{\mu\nu}^{(1)}t_{\rho\sigma}^{(1)}\rangle_c.$$

Consequently the rms stochastic metric source scales as $G\sqrt N=O(N^{-1/2})$, while its variance is $O(N^{-1})$. A metric fluctuation inherits that suppression only if its retarded response stays bounded over the time interval being studied. Matter polarization insertions in a graviton propagator survive at leading order; graviton self-interaction loops and further subleading corrections do not. Hu and Verdaguer 2008, § 3.3.1 gives the diagrammatic counting and explains why this limit yields a classical mean geometry without turning the matter fields classical. The large-$N$ page also scales the local gravitational couplings consistently.

Large $N$ is sufficient bookkeeping, not a claim that nature supplies arbitrarily many species. With $N=1$, a coherent or highly occupied state can also have a sharply peaked stress observable in a specified smearing region, but that conclusion is state- and observable-dependent. Conversely, $N\gg1$ does not rescue modes whose proper frequency crosses the effective cutoff.

Checks on the retained equation

There are three quick independent checks. First, dimensions agree: both $G_{\mu\nu}$ and $G\langle T_{\mu\nu}\rangle$ scale as $L^{-2}$. Second, diffeomorphism invariance gives a Ward identity, so every retained local term and the renormalized expectation value must be covariantly conserved. Third, at fixed finite local couplings the limit $G\to0$ decouples matter backreaction. Setting the mean stress to zero gives a vacuum equation, but it reduces to the two-derivative Einstein vacuum equation only if the retained higher-curvature contributions also vanish or are negligible.

Linear response supplies a stronger test. If a gauge-invariant perturbation below the cutoff grows without bound because of the retarded stress-tensor kernel, neglected metric fluctuations cannot remain small. Anderson, Molina-París, and Mottola formulate this criterion using the causal effective action and emphasize that stability must be assessed only after renormalization and gauge projection Anderson, Molina-París, and Mottola 2003, §§ II–III.

Where the expansion stops

No single number is a universal validity certificate. Control is lost if resolved curvature or frequencies approach the effective cutoff, induced or intrinsic metric correlators become order one, secular terms erase the nominal small parameter, or uncertainty in unfixed higher-order couplings changes a prediction by as much as the retained effect. A compensated renormalization-scheme change is not a physical uncertainty: the complete prediction must remain unchanged through the retained order. A small mean stress tensor alone is therefore inconclusive.

This volume begins after the fixed-background construction of states and observables, which remains with Quantum Fields in Curved Spacetime. It follows mean backreaction and its fluctuations, but it does not derive microscopic spacetime degrees of freedom or fine-grained information recovery. Those questions require separate quantum-gravity or black-hole-information input once the stated expansion ceases to be controlled.

Exercise

For $N$ identical uncorrelated fields with $G=\bar G/N$, determine the $N$-scaling of the mean metric source, its rms fluctuation, and the relative fluctuation. What changes if all fields are perfectly correlated?

Solution

The mean source is $G\langle T^{\mathrm{tot}}\rangle=(\bar G/N)(N\langle T^{(1)}\rangle)=O(1)$. Independence gives a connected variance proportional to $N$, hence an rms metric source $(\bar G/N)\sqrt N=O(N^{-1/2})$ and a relative fluctuation of the same order when the mean is nonzero. Perfect correlation instead makes the connected variance scale as $N^2$; the rms metric source is then $O(1)$. Thus species counting suppresses fluctuations only with an independence or sufficiently weak-correlation hypothesis.

References

• Anderson, P. R., Molina-París, C., and Mottola, E. 2003. “Linear Response, Validity of Semi-Classical Gravity, and the Stability of Flat Space.” Physical Review D 67, 024026. Open PDF.
• Hu, B. L., and Verdaguer, E. 2008. “Stochastic Gravity: Theory and Applications.” Living Reviews in Relativity 11, 3. DOI.