Stochastic Gravity and Metric Fluctuations

Stochastic gravity adds stress-tensor fluctuations to a semiclassical mean background. It predicts a specified class of metric correlations through a causal response problem, rather than a complete quantum state of geometry.

Enter this chapter

Begin with The Stress-Tensor Noise Kernel to construct a smeared covariance of stress fluctuations.

Helpful background. Stress Bi-Tensors and Noise-Kernel Input and System–Environment Splits and Influence Functionals develop the definitions used by the chapter’s opening pages.

Topics

1. The Stress-Tensor Noise Kernel — Defines the centered symmetrized covariance and its positivity after smearing.
2. Renormalizing Stress Fluctuations and Coincident Limits — Separates a well-defined bi-distribution from an inadmissible pointwise variance.
3. Influence Functionals, Dissipation, and Noise — Derives the causal and stochastic kernels from the same closed-time-path functional.
4. Fluctuation–Dissipation Relations in Stationary States — States the stationarity and KMS hypotheses required for a spectral relation.
5. Einstein–Langevin Dynamics — Solves the linear stochastic response with the explicit $8\pi G$ normalization.
6. Intrinsic and Induced Metric Fluctuations — Separates propagated initial metric uncertainty from matter-induced covariance.
7. Gauge-Invariant Stochastic Observables — Constructs quantities that survive a change of metric gauge.
8. Large-N Quantum–Stochastic Correspondence — Identifies the controlled symmetrized quantum correlator reproduced at leading nontrivial order.
9. Higher Cumulants and Non-Gaussian Noise — Tests Gaussian closure against the connected stress hierarchy.
10. Validity, Decoherence, and What Stochastic Gravity Does Not Capture — Limits claims about classicality, information, and quantum geometry.

How the ideas fit together

The noise kernel is the symmetrized covariance of fluctuations about the mean stress. Together with a retarded response kernel, it sources the Einstein–Langevin equation and generates induced metric fluctuations Hu and Verdaguer 2008, §§ 3.2 and 4.1.

Homogeneous solutions retain intrinsic initial fluctuations, so the induced part alone is not the total covariance. Smearing, gauge-invariant observables, and perturbative order must be fixed before comparing with quantum calculations. Matching symmetrized two-point functions under a large-species expansion does not reproduce all operator orderings or higher quantum correlations.

The comparison below separates the assumptions and tests needed for these different conclusions.

Domain and failure conditions

Layer	Matter input	Metric output and ordering	Gauge and distributional domain	Licensed conclusion	Failure or further method needed
Semiclassical mean	Renormalized $\langle T_{\mu\nu}\rangle$	Mean background $g_{\mu\nu}$	Covariant mean equation; local finite terms fixed	Self-consistent mean geometry within the stated EFT	Stress fluctuations, unstable response, or large metric variance require a stronger layer
Linear response	Retarded stress commutator plus contact terms	Causal susceptibility of $h_{\mu\nu}$	Constraints imposed; pure gauge removed	Stability and response to a prescribed perturbation	Growing physical homogeneous modes invalidate perturbation about the background
Gaussian stochastic	$N=\frac12\langle\{t,t\}\rangle$	Symmetrized induced covariance	Doubly smeared or otherwise distributionally extended; observable gauge invariant	Second moments generated by the Einstein–Langevin equation	Coincident unsmeared products, nonpositive proposed covariance, or important higher cumulants
Leading large-N quantum	$N$ identical matter fields with $NG$ fixed	Leading nontrivial symmetrized metric two-point function	One saddle, controlled initial state, gauge-invariant linear observables	Equality with the corresponding quantum anticommutator at the controlled order	Metric commutators, out-of-time-order observables, graviton loops, or nonuniform $1/N$ limit
Non-Gaussian stochastic	Connected stress cumulants retained to declared order	Selected higher symmetrized metric statistics	Smeared cumulants; existence of a probability representation checked	Corrections to skewness, tails, or other declared statistics	A truncated characteristic functional that is not positive, or uncontrolled cumulant hierarchy
Quantum-gravity description	Quantum matter and metric degrees of freedom	Full operator algebra or quantum state	Diffeomorphism-invariant observable and UV completion specified	Claims about unsymmetrized observables, entanglement, or nonlinear geometry	Use quantum gravity when no low-energy, large-N, weak-field, or open-system reduction controls the requested observable

Where to go next

Continue to Generalized Entropy and Quantum Extremal Surfaces to separate horizon temperature, flux, and entropy as physical quantities.

References

• Hu, B. L., and E. Verdaguer. “Stochastic Gravity: Theory and Applications.” Living Reviews in Relativity 11, 3 (2008). doi:10.12942/lrr-2008-3. Open PDF