Metric Perturbations and Stress–Tensor Response

A small metric perturbation changes the quantum stress tensor through a local geometric variation and a nonlocal causal polarization kernel. The physical kernel is retarded: it vanishes when the source point is outside the causal past of the observation point. Linearizing the renormalized semiclassical equation with that kernel gives the correct test of whether a self-consistent background amplifies or damps gauge-invariant disturbances below the cutoff.

Required background. Partial Differential Equations and Green Functions supplies retarded kernels, distributions, and boundary-value methods used in the response equation.

Helpful background. Quantum Fields in Curved Spacetime supplies stress operators and states, Einstein Equations supplies the linearized geometry, Black-Hole Geometry and Mechanics supplies later horizon boundary conditions, and Semiclassical Gravity supplies the mean equation being perturbed.

Functional response of the stress tensor

Let $\bar g_{ab}$ and $\rho$ solve the renormalized semiclassical equation, and write $g_{ab}=\bar g_{ab}+h_{ab}$. Keep the initial density operator fixed and vary the subsequent metric history. Define the retarded functional kernel by

$$\delta\langle T_{ab}(x)\rangle_{\mathrm{ren}} =\int dV_y\,\Pi^{\mathrm{ret}}_{ab}{}^{cd}(x,y)h_{cd}(y) +O(h^2).$$

The kernel contains two qualitatively different pieces,

$$\Pi^{\mathrm{ret}}=\Pi^{\mathrm{local}}+\Pi^{\mathrm{nonlocal}}_{\mathrm{ret}}.$$

The local part includes the explicit metric dependence of the stress operator and the contact terms needed to renormalize its variation. Finite local terms assigned to $\langle T_{ab}\rangle$ contribute here; those already assigned to the geometric side belong in $\delta E_{ab}$ below and must not be counted twice.

The normalization follows from the choice of metric variable. Since $h_{cd}=\delta g_{cd}$ is a covariant metric perturbation,

$$\delta S_{\mathrm m}=\frac12\int dV_y\,T^{cd}(y)h_{cd}(y).$$

Linear response to this source gives, away from contact,

$$\Pi^{\mathrm{nonlocal}}_{ab}{}^{cd}(x,y) =\frac{i}{2}\,\theta(x^0-y^0) \left\langle[T_{ab}(x),T^{cd}(y)]\right\rangle.$$

The measure and index placement are those in the defining convolution. Varying $g^{cd}$ instead reverses the coupling sign, so the metric variable and kernel must be translated together. In Hu and Verdaguer 2008, § 4.2–4.3, Eqs. (4.6)–(4.9) and (4.17), the separated-point combination is $H=-i\theta\langle[T,T]\rangle$ and the stress variation is $-\frac12\int Hh$, giving the displayed $+i/2$. The general local/contact and causal structure is also derived in Anderson, Molina-París, and Mottola 2003, § III, Eqs. (3.4)–(3.9), PDF. Microcausality makes the stress commutator vanish at spacelike separation. Since the expectation of a commutator of Hermitian observables is imaginary, the response is real.

An independent perturbation of the initial state adds a term $\delta_\rho\langle T_{ab}\rangle$. It is not contained in this fixed-state kernel and acts as an additional source for the linearized metric equation.

The linearized semiclassical equation

Let $\delta E_{ab}[h]$ denote the linear variation of the renormalized geometric side, including curvature-squared terms. The homogeneous response equation is

$$\delta E_{ab}[h](x) -8\pi G\int dV_y\, \Pi^{\mathrm{ret}}_{ab}{}^{cd}(x,y)h_{cd}(y)=0.$$

Initial data select a retarded solution. In a stationary background, Fourier transformation with time dependence $e^{-i\omega t}$ turns the convolution into multiplication, but the analytic sheet, branch cuts, and boundary conditions remain part of the definition. A pole in the upper half $\omega$-plane corresponds to growth in this convention only if it belongs to a gauge-invariant retarded observable and lies below the cutoff.

The in–out kernel is generally unsuitable here: it is time ordered rather than retarded and can produce a complex equation. Anderson, Molina-París, and Mottola formulate the real closed-time-path variation and gauge-invariant validity criterion in Anderson, Molina-París, and Mottola 2003, §§ II–III, PDF.

Ward identities and gauge transformations

The exact linearized conservation identity is most transparent for a mixed-index stress tensor:

$$\bar\nabla_a\delta\langle T^a{}_b\rangle +\delta\Gamma^a{}_{ac}\langle\bar T^c{}_b\rangle -\delta\Gamma^c{}_{ab}\langle\bar T^a{}_c\rangle=0.$$

The mixed-index variation includes the variation of the inverse metric used to raise the first index. When the background mean stress vanishes, this reduces to a transverse distributional response. Otherwise the connection terms require local contact contributions. They ensure that a pure gauge perturbation $h_{ab}=2\bar\nabla_{(a}\xi_{b)}$, with state data transformed consistently, changes the stress expectation by its Lie derivative rather than creating a physical response. Dropping them can make a formally transverse calculation gauge dependent.

Three checks are therefore independent. The kernel must have retarded support; its Ward identity must match that of $\delta E_{ab}$; and a pure gauge perturbation must reduce the complete equation to the Lie derivative of the background equation, which vanishes on shell.

Local limits and nonlocal memory

If all frequencies and curvature scales are well below a heavy mass $m$, the kernel admits a derivative expansion. Its analytic low-frequency terms renormalize local operators, while thresholds and logarithms retain nonlocal information. In the limit $m\to\infty$ at fixed low frequency, the nonlocal response decouples after matching the local couplings. In a massless theory or near a particle-production threshold, no finite local series reproduces the entire retarded kernel.

Similarly, a static susceptibility probes $\Pi^{\mathrm{ret}}(\omega=0)$ but does not determine dissipative behavior at finite frequency. A slowly varying source can accumulate memory over long times even if each instantaneous derivative is small. The relevant expansion includes both frequency-to-cutoff ratios and a secular-time bound.

Scope and limitations

Linear response tests perturbations about a mean solution accurate to the declared order. A first-order correction may consistently use a quantum source on the leading classical background when the omitted source response is second order; that ordering must be checked rather than assumed. Bounded homogeneous response does not bound matter-induced fluctuations. The next pages add the noise kernel and stochastic source.

Fixed-background QFT supplies the commutator and state; this chapter calculates their metric response. Fine-grained information recovery and microscopic graviton interactions remain outside the linear matter-loop approximation. When a physical response becomes nonlinear or reaches the cutoff, the linear equation no longer supports the claimed prediction.

Exercise

Assume a stationary scalar response $K^{\mathrm{ret}}(t)$ vanishes for $t\lt 0$ and is real. Show the reality relation obeyed by its Fourier transform and explain why a pole at $\omega=i\gamma$ with $\gamma\gt 0$ signals growth for the site’s time convention.

Solution

Reality gives $K^{\mathrm{ret}}(\omega)^*=K^{\mathrm{ret}}(-\omega^*)$. With fields proportional to $e^{-i\omega t}$, a pole at $\omega=i\gamma$ contributes $e^{\gamma t}$ and therefore grows. Retarded support alone does not decide whether the pole is physical: it must survive gauge projection, lie on the retarded physical sheet, and have $\gamma$ below the effective cutoff. A cutoff-scale pole may instead be an artifact of a truncated higher-derivative response.

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.