The Stress–Energy Noise Kernel

The semiclassical Einstein equation retains only the mean stress tensor. Its leading measure of discarded matter fluctuations is the noise kernel: the connected, symmetrized two-point function of the renormalized stress tensor. It is not an ordinary function at coincidence, but a positive distribution whose meaning is fixed by smearing with smooth tensor test fields.

Required background. Partial Differential Equations and Green Functions supplies distributions and test-function smearing.

Helpful background. Quantum Fields in Curved Spacetime supplies Hadamard states and stress operators, Einstein Equations supplies covariant conservation, Black-Hole Geometry and Mechanics supplies later horizon applications, and Semiclassical Gravity supplies the mean-field approximation whose fluctuations are being measured.

Connected stress fluctuations

On a fixed background $g_{ab}$ in a specified state $\rho$, define the centered operator

$$\hat t_{ab}(x) =\hat T^{\mathrm{ren}}_{ab}(x) -\langle\hat T^{\mathrm{ren}}_{ab}(x)\rangle_\rho\,\hat I.$$

The noise kernel is the bitensor distribution

$$N_{ab,c'd'}(x,y) =\frac12\left\langle \left\{\hat t_{ab}(x),\hat t_{c'd'}(y)\right\} \right\rangle_\rho .$$

Primed indices live at $y$. Centering removes disconnected fluctuations of the mean, while the anticommutator selects the real symmetric covariance. For a linear field, local c-number counterterms cancel in $\hat t_{ab}$; this does not make the coincidence limit a finite number. The correct statement is distributional, as emphasized in Hu and Verdaguer 2008, § 3.2, Eqs. (3.11)–(3.13).

Smearing, positivity, and singular support

For a real, smooth, compactly supported symmetric test tensor $f^{ab}$, set

$$\hat t[f]=\int dV_x\,f^{ab}(x)\hat t_{ab}(x).$$

Then

$$N[f,f] =\iint dV_xdV_y\, f^{ab}(x)N_{ab,c'd'}(x,y)f^{c'd'}(y) =\left\langle\hat t[f]^2\right\rangle_\rho\ge0.$$

This positive-semidefinite quadratic form permits a Gaussian stochastic tensor $\xi_{ab}$ with $\mathbb E[\xi_{ab}]=0$ and $\mathbb E[\xi_{ab}(x)\xi_{c'd'}(y)]=N_{ab,c'd'}(x,y)$. A point detector corresponds to a singular test function and is not licensed by this construction. Its spatial profile, switching function, and tensor contraction are physical parts of the observable, not disposable regulators.

For a quasifree free-field state, Wick’s theorem expresses the separated-point kernel through derivatives and products of its Wightman function. A non-Gaussian state also contributes a connected four-field function; a two-point function alone then does not determine the noise. Hadamard singularities occur at coincidence and along null geodesic relation for both massive and massless fields. A mass changes propagation inside the light cone, not the null characteristic singularity. Point separation exposes this structure, but a proposed observable must still define an admissible pullback or smearing. The explicit point-separated construction and the warning against treating coincidence as an ordinary value are developed in Hu and Verdaguer 2008, §§ 5.1–5.2.

Conservation and conformal trace

Stress-tensor conservation gives Ward identities in each argument,

$$\nabla_x^aN_{ab,c'd'}(x,y)=0, \qquad \nabla_y^{c'}N_{ab,c'd'}(x,y)=0,$$

in the distributional sense, including any contact terms required by the operator definition. Consequently the stochastic source can be chosen conserved, so it is compatible with the linearized Bianchi identity.

For a conformal free field, the renormalized trace anomaly is a local c-number functional of the metric. Centering therefore removes it from $\hat t^a{}_a$, and the corresponding trace of the noise kernel vanishes. This means that there is no stochastic correction to that anomaly in this setting; it does not mean that the anomaly itself vanishes. Hu and Verdaguer state both conservation and the zero conformal trace fluctuation in Hu and Verdaguer 2008, § 3.2, pp. 10–11.

What the kernel does and does not encode

The noise kernel retains the second cumulant of stress fluctuations. A Gaussian stochastic source reproduces observables controlled by that cumulant, but it discards higher connected stress correlators unless the approximation or state makes them negligible. A small mean stress is no guarantee of small noise: interference can suppress the first moment while leaving a substantial variance.

Nor is $N_{ab,c'd'}$ itself a metric fluctuation. It becomes an induced metric covariance only after convolution with the retarded gravitational Green function and after gauge, boundary, initial-state, and cutoff choices are fixed. The next page performs that propagation with the Einstein–Langevin equation.

Exercise

Let $f^{ab}$ and $q^{ab}$ be real compactly supported test tensors. Prove the Cauchy–Schwarz bound

$$\lvert N[f,q]\rvert^2\le N[f,f]N[q,q],$$

and explain why it can fail to be meaningful for unsmeared point evaluations.

Solution

For any real number $\lambda$, positivity gives

$$0\le N[f+\lambda q,f+\lambda q] =N[f,f]+2\lambda N[f,q]+\lambda^2N[q,q].$$

If $N[q,q]\gt 0$, the discriminant of this quadratic cannot be positive, which yields $N[f,q]^2\le N[f,f]N[q,q]$. If $N[q,q]=0$, positivity for both signs of $\lambda$ forces $N[f,q]=0$, so the same result follows. A delta function at a point is not a smooth test tensor, and the product distributions defining the stress covariance generally have a divergent or undefined coincidence pullback. Thus the formal quantities $N(x,x)$ used in a pointwise version of the inequality need not exist.

References

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