Stress Bi-Tensors and Noise-Kernel Input

The matter input for stochastic gravity is not $\langle T_{\mu\nu}\rangle^2$ and not an unsmeared coincident variance. It is the connected, symmetrized stress bi-distribution with indices at two spacetime points, defined in a state and renormalization prescription and tested for symmetry, conservation, covariance positivity, and admissible smearing.

Required background. Wick Polynomials and Hadamard Point Splitting supplies the composite fields; Renormalized Stress Tensor: Axioms and Curvature Ambiguities supplies the one-point tensor; Quasifree States and Two-Point Functions supplies Gaussian factorization.

Helpful background. Green Functions and Causal Propagators separates symmetric and causal kernels; Hadamard Admissibility: The Two-Point Wavefront Criterion supplies the distributional domain.

Connected symmetrized stress bi-distribution

Use the page-local stress convention

$$\delta\Gamma_{\mathrm m} = \frac12\int\mathrm d^4x\,\sqrt{-g}\, \langle T_{\mu\nu}\rangle\,\delta g^{\mu\nu}.$$

Define the centered stress operator

$$t_{\mu\nu}(x) = T_{\mu\nu}(x)-\langle T_{\mu\nu}(x)\rangle_\omega\mathbf 1$$

and the bi-distribution

$$N_{\mu\nu\rho'\sigma'}(x,y) = \frac12 \left\langle \left\{ t_{\mu\nu}(x),t_{\rho'\sigma'}(y) \right\} \right\rangle_\omega .$$

Primed indices live in $T_y^*M$ and unprimed indices in $T_x^*M$. No parallel transport is needed to define the bitensor. A propagator $g^\mu{}_{\rho'}(x,y)$ is required only when indices are compared or contracted at one point, and its geodesic domain must then be stated.

The subtraction of $\langle T\rangle\langle T\rangle$ is essential. Under a finite c-number shift

$$T_{\mu\nu}\mapsto T_{\mu\nu}+C_{\mu\nu}[g]\mathbf 1,$$

$t_{\mu\nu}$ and hence the separated-point $N$ are unchanged. This cancellation does not remove the finite freedom in extending singular products onto the diagonal $x=y$.

For real compactly supported symmetric test tensors $F^{\mu\nu}$,

$$N[F,F] = \int\mathrm dV_x\,\mathrm dV_y\, F^{\mu\nu}(x) N_{\mu\nu\rho'\sigma'}(x,y) F^{\rho'\sigma'}(y) = \langle t[F]^2\rangle_\omega\ge0.$$

Positivity is therefore a smeared covariance statement. Individual coordinate components of $N(x,y)$ need not be positive.

First application: quasifree scalar state

Write the point-split scalar stress as

$$T_{\mu\nu}(x) = \lim_{x'\to x} \mathcal D_{\mu\nu}^{xx'} \left[ \phi(x)\phi(x')-H_\ell(x,x')\mathbf 1 \right] +C_{\mu\nu}(x)\mathbf 1 .$$

For a quasifree Hadamard state, Wick’s theorem reduces the connected four-point function to the two cross-contractions. At separated points,

$$\begin{aligned} N_{\mu\nu\rho'\sigma'}(x,y) =\operatorname{Re}\lim_{\substack{x'\to x\\y'\to y}} \mathcal D_{\mu\nu}^{xx'}\mathcal D_{\rho'\sigma'}^{yy'} \Big[ &W_\omega(x,y)W_\omega(x',y')\\ &+W_\omega(x,y')W_\omega(x',y) \Big]. \end{aligned}$$

This formula is interpreted distributionally on $M\times M$. It is pointwise smooth only away from the singular support, which includes null-related pairs. The local Hadamard subtractions and finite c-number tensors cancel from the connected separated expression, but they were needed to define each stress operator consistently.

On an ultrastatic background, insert the positive-frequency mode expansion of $W_\omega$ and smear both arguments with compact test tensors before summing. Then verify:

$$N_{\mu\nu\rho'\sigma'}(x,y) = N_{\rho\sigma\mu'\nu'}(y,x)$$

under simultaneous point and index-pair exchange, and

$$\nabla^\mu_xN_{\mu\nu\rho'\sigma'}=0, \qquad \nabla^{\rho'}_yN_{\mu\nu\rho'\sigma'}=0$$

as distributional identities away from declared contact, background-force, anomaly, and boundary terms. Phillips and Hu give the point-separated Gaussian construction and its coincidence analysis Phillips and Hu 2001, §§II–III, Eqs. (2.1)–(3.29).

Numerically, form a covariance matrix $N[F_i,F_j]$ for a finite set of real test tensors. It must be symmetric positive semidefinite within error. A negative eigenvalue larger than the numerical uncertainty signals inconsistent subtraction, quadrature, or state data.

What is not yet a stochastic source

$N$ contains the symmetrized connected fluctuations. The causal response instead involves the stress commutator, retarded support, and metric-variation contact terms. Neither kernel can be reconstructed from the other without additional stationary/KMS hypotheses, and even then the relation is a fluctuation–dissipation statement with a specified frequency convention Hu and Verdaguer 2020, §§3.1–3.3.

This page does not postulate a classical random stress. Chapter 10 may introduce a Gaussian source $\xi_{\mu\nu}$ whose covariance equals $N$ within a controlled approximation. Higher connected stress cumulants and the full quantum metric state are not encoded in that replacement.

Adversarial coincidence test

Take $y\to x$ in the displayed $N$ before smearing. Products of two stress tensors are more singular than the one-point stress, so the result generally diverges or requires new contact-term extensions. The finiteness of $\langle T_{\mu\nu}(x)\rangle$ proves nothing about $\langle T_{\mu\nu}(x)T_{\rho\sigma}(x)\rangle$.

The strongest result without a diagonal extension is the separated bi-distribution and every admissibly smeared covariance it defines. A local “noise amplitude” requires its own smearing or renormalization prescription and cannot be imported as a universal scalar.

Bitensor and failure maps

The structure map ends this chapter with a typed matter output: a connected symmetric covariance, not yet a metric correlator.

The exported object retains two spacetime arguments, primed indices, state data, and smearing domain; the map is schematic and not to scale.

The failure map blocks the common shortcuts from a finite mean stress to a pointwise variance or full stochastic metric prediction.

Mean stress, connected symmetric covariance, causal commutator, and metric fluctuations are distinct objects; the map is schematic and not to scale.

Use Domain and failure conditions. Record field and state, stress prescription, connected subtraction, index bundles and transport if used, separated-point domain, contact extensions, boundary terms, test tensors, covariance positivity, and conservation residuals.

Check your understanding

Why does a finite shift $C_{\mu\nu}\mathbf 1$ cancel from $N$?

Solution

The same shift appears in the operator and its expectation, so $t_{\mu\nu}=T_{\mu\nu}-\langle T_{\mu\nu}\rangle\mathbf1$ is unchanged. This argument does not constrain contact terms created when two insertions coincide.

Chapter 10 owns the noise-kernel name, influence functional, Einstein–Langevin equation, and stochastic metric response. The handoff from this page is only $N_{\mu\nu\rho'\sigma'}$ with its state, subtraction, distributional domain, smearing, and checks. The literature basis and this claim boundary were checked through 10 August 2026.

References

• Bei-Lok Hu and Enric Verdaguer, “Stochastic Gravity: Theory and Applications,” Living Reviews in Relativity 23 (2020), article 3, DOI, arXiv:0802.0658.
• Nicholas G. Phillips and Bei-Lok Hu, “Noise Kernel in Stochastic Gravity and Stress Energy Bi-Tensor of Quantum Fields in Curved Spacetimes,” Physical Review D 63 (2001), 104001, DOI, arXiv:gr-qc/0010019.