The Einstein–Langevin Equation

The Einstein–Langevin equation propagates quantum stress fluctuations into metric fluctuations without promoting the metric itself to a full quantum operator. Its stochastic source has zero mean, noise-kernel covariance, and covariant conservation. Solving the equation separates uncertainty already present in the initial metric data from fluctuations induced by quantum matter.

Required background. Partial Differential Equations and Green Functions supplies retarded Green functions and inhomogeneous evolution.

Helpful background. Quantum Fields in Curved Spacetime supplies the underlying stress correlators, Einstein Equations supplies the linearized geometric operator, Black-Hole Geometry and Mechanics supplies horizon boundary conditions, and Semiclassical Gravity supplies the background mean equation.

From the mean equation to a stochastic equation

Let $\bar g_{ab}$ be a self-consistent semiclassical solution and let $h_{ab}$ be a perturbation. Write the complete renormalized linear-response operator as

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

where $\delta E_{ab}$ includes the finite local gravitational couplings and $\Pi^{\mathrm{ret}}$ is the causal stress response. Introduce a classical Gaussian random tensor $\xi_{ab}$ satisfying

$$\mathbb E[\xi_{ab}(x)]=0, \qquad \mathbb E[\xi_{ab}(x)\xi_{c'd'}(y)] =N_{ab,c'd'}(x,y).$$

The linear Einstein–Langevin equation is

$$(Lh)_{ab}(x)=8\pi G\,\xi_{ab}(x).$$

Because $N$ is positive semidefinite and conserved, this Gaussian source exists as a generalized random field and obeys $\bar\nabla^a\xi_{ab}=0$ with probability one in the distributional sense. The stochastic equation and these properties are derived in Hu and Verdaguer 2008, § 3.2, Eqs. (3.13)–(3.16).

Retarded solution and two kinds of fluctuation

After gauge fixing or passing to gauge-invariant variables, let $G^{\mathrm{ret}}_{ab}{}^{cd}(x,y)$ invert $L$ with the chosen retarded boundary and initial conditions. The solution is

$$h_{ab}(x)=h^{(0)}_{ab}(x) +8\pi G\int dV_y\, G^{\mathrm{ret}}_{ab}{}^{cd}(x,y)\xi_{cd}(y),$$

where $h^{(0)}$ solves $Lh^{(0)}=0$. Average the initial data with a distribution independent of $\xi$. If both means vanish, the symmetrized stochastic covariance is

$$\begin{aligned} \mathbb E[h_{ab}(x)h_{c'd'}(x')] ={}&C^{\mathrm{intr}}_{ab,c'd'}(x,x')\\ &+(8\pi G)^2\iint dV_y dV_z\, G^{\mathrm{ret}}_{ab}{}^{ef}(x,y) N_{ef,gh}(y,z) G^{\mathrm{ret}}_{c'd'}{}^{gh}(x',z). \end{aligned}$$

The first term is intrinsic: it propagates uncertainty in homogeneous metric data. The second is induced: matter stress fluctuations are filtered through the gravitational response. Stability of the homogeneous equation constrains only the first term. A stable response can still amplify a broad or resonant noise spectrum enough to invalidate the mean-field description.

Relation to quantum metric correlations

For $N$ identical free matter fields, write $G=\bar G/N$ with $\bar G$ fixed and scale the local curvature couplings consistently. Matter loops then organize a $1/N$ expansion. At leading nontrivial order, the stochastic two-point function reproduces the symmetrized quantum metric two-point function once the initial stochastic covariance is chosen to match the initial graviton state and the same response prescription is used. Hu and Verdaguer state the correspondence and its large-$N$ scope in Hu and Verdaguer 2008, §§ 3.2–3.3.1, especially Eq. (3.17).

This is a two-point correspondence, not an identification of a classical random metric with the complete quantum gravitational state. The Gaussian source fixes no higher stress cumulants, and leading $1/N$ does not include graviton loops at the same level as matter loops.

Observable form of the induced variance

A local component $h_{00}(x)$ is coordinate dependent and is usually too singular to serve as the final comparison. Let $F^{ab}$ be a smooth compactly supported detector tensor that defines the linear observable

$$H[F]=\int dV_x\,F^{ab}(x)h_{ab}(x).$$

A sufficient condition for gauge invariance is $\bar\nabla_aF^{ab}=0$: integration by parts gives $\delta_\xi H[F]=-2\int dV\,(\bar\nabla_aF^{ab})\xi_b=0$ for an allowed gauge transformation, with no boundary contribution. Alternatively one can construct $H$ directly from a relational or gauge-invariant perturbation variable. Smoothness alone does not remove gauge dependence.

Propagate the detector backward with the retarded kernel,

$$Q^{cd}(y)=\int dV_x\,F^{ab}(x)G^{\mathrm{ret}}_{ab}{}^{cd}(x,y).$$

The induced variance then takes the compact form

$$\operatorname{Var}_{\mathrm{ind}}H[F] =(8\pi G)^2N[Q,Q]\ge0.$$

This pairing must exist: finite-time initial data, switching, boundary conditions, and any EFT cutoff have to make the propagated profile $Q$ an admissible argument of $N$. A retarded convolution does not automatically regularize every distribution. With that qualification, smearing belongs to the physical observable; the retarded operator determines which parts of the noise spectrum reach the detector; and positivity follows from the noise kernel rather than from any one tensor component. After both integrations, the result must have the square of the dimension of $H[F]$.

In a stationary problem, the same statement becomes a spectral filter: schematically, the induced power is $|G^{\mathrm{ret}}(\omega)|^2N(\omega)$. Narrow response peaks can therefore magnify a modest noise spectrum. A cutoff must be imposed on the full convolution, not by deleting an unstable homogeneous root while retaining noise at the same frequency.

Causality, gauge, and renormalization checks

The construction passes four checks. First, $G^{\mathrm{ret}}$ has support only when $y$ lies in the causal past of $x$. Second, conservation of $\xi_{ab}$ matches the linearized Bianchi identity. Third, the observable covariance is evaluated only after pure-gauge directions have been removed or contracted with gauge-invariant probes. Fourth, the same renormalized local couplings and stress prescription used to define the background must also occur in $L$.

Changing the homogeneous initial ensemble changes $C^{\mathrm{intr}}$ without changing $N$. Changing the matter state generally changes both $N$ and $\Pi^{\mathrm{ret}}$. A calculation that imports the noise from one state and the response from another need not satisfy the fluctuation–dissipation relation developed on the next page.

Exercise

Consider the scalar analogue $\dot h+\gamma h=\xi(t)$ for $t\ge0$, with $\gamma\gt 0$, deterministic $h(0)=0$, and white noise $\mathbb E[\xi(t)\xi(t')]=2D\delta(t-t')$. Derive $\mathbb E[h(t)^2]$ and identify the retarded stability and late-time fluctuation tests.

Solution

The retarded Green function is $G^{\mathrm{ret}}(t,t')=\theta(t-t')e^{-\gamma(t-t')}$. Hence

$$h(t)=\int_0^t dt'\,e^{-\gamma(t-t')}\xi(t')$$

and

$$\mathbb E[h(t)^2] =2D\int_0^t dt'\,e^{-2\gamma(t-t')} =\frac{D}{\gamma}\left(1-e^{-2\gamma t}\right).$$

The homogeneous solution $e^{-\gamma t}$ decays, so the retarded response is stable. Nevertheless the induced variance approaches $D/\gamma$; whether mean-field theory is valid depends on this value relative to the squared geometric scale that defines a small perturbation. Thus homogeneous stability is necessary but not sufficient.

References

• Hu, B. L., and Verdaguer, E. 2008. “Stochastic Gravity: Theory and Applications.” Living Reviews in Relativity 11, 3. DOI.