Conservation Consistency of the Semiclassical Equation

The semiclassical Einstein equation is consistent only if its renormalized quantum source is covariantly conserved with respect to the same metric that appears on the geometric side. Conservation is not an optional check imposed after solving: it is the Ward identity of diffeomorphism invariance and a constraint on regulators, counterterms, state evolution, and approximations. A finite-looking stress tensor that violates this identity cannot source the unmodified Einstein tensor.

Required background. Einstein Equations supplies the contracted Bianchi identity, and Gravitational Actions and Boundary Terms supplies the diffeomorphism-invariant variational problem.

Helpful background. Quantum Fields in Curved Spacetime supplies covariant stress-tensor renormalization, while Black-Hole Geometry and Mechanics supplies the flux and horizon balances used later.

The diffeomorphism Ward identity

Let $W_{\mathrm{ren}}[g;\rho]$ be the matter effective action, including a state specified on an initial surface or by invariant boundary conditions. Under an infinitesimal diffeomorphism generated by a compactly supported vector $\xi^\mu$,

$$\delta_\xi g^{\mu\nu}=-2\nabla^{(\mu}\xi^{\nu)}.$$

Using $\langle T_{\mu\nu}\rangle_{\mathrm{ren}}=-2(\sqrt{-g})^{-1}\delta W_{\mathrm{ren}}/\delta g^{\mu\nu}$ gives

$$0=\delta_\xi W_{\mathrm{ren}} =\int d^4x\sqrt{-g}\, \langle T_{\mu\nu}\rangle_{\mathrm{ren}}\nabla^\mu\xi^\nu =-\int d^4x\sqrt{-g}\, \xi^\nu\nabla^\mu\langle T_{\mu\nu}\rangle_{\mathrm{ren}}.$$

Arbitrariness of $\xi^\nu$ yields

$$\nabla^\mu\langle T_{\mu\nu}\rangle_{\mathrm{ren}}=0.$$

Boundary terms vanish here because $\xi^\mu$ has compact support; with a physical boundary, the Ward identity includes the declared boundary stress and flux. The geometric derivation and its classical consequences belong to Bianchi Identity and Stress-Energy Conservation; the present page applies it to a renormalized quantum source.

Matching the geometric side

For the four-derivative truncation,

$$E_{\mu\nu} =G_{\mu\nu}+\Lambda g_{\mu\nu} +16\pi G\left(\alpha H^{(R^2)}_{\mu\nu}+\beta H^{(C^2)}_{\mu\nu}\right).$$

Each tensor is a variation of a diffeomorphism-invariant local action, so

$$\nabla^\mu E_{\mu\nu}=0$$

identically. Therefore $E_{\mu\nu}=8\pi G\langle T_{\mu\nu}\rangle_{\mathrm{ren}}$ propagates the gravitational constraints only when the source obeys the same identity. Hu and Verdaguer display the conserved curvature tensors and renormalized equation in Hu and Verdaguer 2008, § 3.1, Eqs. (3.7)–(3.10).

A trace anomaly does not contradict this statement. It makes $g^{\mu\nu}\langle T_{\mu\nu}\rangle_{\mathrm{ren}}$ nonzero for a classically conformal field while leaving the covariant divergence zero. Confusing trace conservation with spacetime conservation produces a false inconsistency.

Choices that preserve conservation

A covariant point-splitting, dimensional, or locally covariant Hadamard prescription preserves the Ward identity after the required local counterterms are included. Finite scheme changes also preserve it because their ambiguity is a conserved local curvature tensor. Shapiro derives the counterterm structure from a covariant effective action in Shapiro 2008, §§ 2–3, PDF.

By contrast, the following operations require repair or an explicit exchange term:

• a coordinate momentum cutoff that treats time and space differently;
• subtracting only selected components or adiabatic orders;
• evolving a state with one metric while inserting its stress tensor into another;
• switching an external source without including the work done by that source; or
• truncating mode sums in a way that violates the stress Ward identity.

If matter is an open subsystem, $\nabla^\mu T^{\mathrm{matter}}_{\mu\nu}$ need not vanish separately; the environment or prescribed source carries the compensating flux. The total source appearing in the geometric equation must still be conserved.

Approximation and numerical checks

An approximation accurate through $O(\epsilon)$ should satisfy

$$\nabla^\mu\langle T_{\mu\nu}\rangle_{\mathrm{approx}} =O(\epsilon^2),$$

not necessarily zero beyond its retained order. The residual must be no larger than the first omitted term. Enforcing exact conservation by an arbitrary projection can hide rather than fix a wrong local tensor, so one should derive the correction from the same effective action or Ward identity.

In a homogeneous cosmology, the temporal component reduces to $\dot\rho+3H(\rho+p)=0$ for an isolated isotropic source. In a stationary black-hole exterior it becomes equality between boundary fluxes and the change of the conserved charge. These limits give independent checks that are easier to evaluate than the full tensor divergence.

The initial-value interpretation is equally important. Projecting the field equation twice along the hypersurface normal gives the Hamiltonian constraint, while one normal and one spatial projection gives the momentum constraint. The Bianchi identity converts the time derivative of these constraints into a combination of the evolution equations and $\nabla^\mu T_{\mu\nu}$. Thus conserved quantum stress data propagate initially satisfied constraints; a source residual generically drives constraint violation rather than merely changing a gauge choice.

Fixed-background QFT supplies the renormalized source; self-consistent backreaction requires that source to be conserved on the solved geometry. Fine-grained information recovery and microscopic quantum gravity are separate issues and cannot be inferred from satisfying this local identity.

Exercise

In a spatially flat FLRW geometry, assume a diagonal source $T^\mu{}_\nu=\operatorname{diag}(-\rho,p,p,p)$. Derive the conservation equation and explain what a nonzero residual would mean in a semiclassical calculation.

Solution

Evaluating $\nabla_\mu T^\mu{}_0=0$ with $\Gamma^i{}_{j0}=H\delta^i_j$ gives $-\dot\rho-3H\rho-3Hp=0$, hence $\dot\rho+3H(\rho+p)=0$. A residual larger than the first omitted order signals inconsistent subtraction, state evolution, or truncation, unless an explicitly modeled external sector carries the opposite energy transfer. It cannot be absorbed by changing only the initial Friedmann constraint because the Bianchi identity propagates that constraint using the conservation law.

References

• Hu, B. L., and Verdaguer, E. 2008. “Stochastic Gravity: Theory and Applications.” Living Reviews in Relativity 11, 3. DOI.
• Shapiro, I. L. 2008. “Effective Action of Vacuum: Semiclassical Approach.” Classical and Quantum Gravity 25, 103001. Open PDF.