Semiclassical equation with fixed counterterms and state

Use SI units, a smooth globally hyperbolic Lorentzian four-manifold $(M,g)$ without boundary of signature $(-,+,+,+)$, and length coordinates $x^0=ct$. Let Newton's gravitational constant $G>0$ have units ${\mathrm{m}}^3\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{s}}^{-2}$ and let $c>0$ have units $\mathrm{m}{\mathrm{s}}^{-1}$. Use the curvature convention $R(X,Y)Z={\mathrm{\nabla }}_X{\mathrm{\nabla }}_{Y}Z-{\mathrm{\nabla }}_Y{\mathrm{\nabla }}_{X}Z-{\mathrm{\nabla }}_{[X,Y]}Z$ and ${\mathrm{Ric}}_{bd}=g^{ac}{R}_{abcd}$, $S=g^{bd}{\mathrm{Ric}}_{bd}$, and $G_{ab}={\mathrm{Ric}}_{ab}-\frac{1}{2}S{g}_{ab}$; thus $S$ is the scalar curvature and $G_{ab}$ is the Einstein tensor in that convention. Use the positive Lorentzian volume density $d{\mathrm{v}\mathrm{o}\mathrm{l}}_g=\mathrm{\mid }\det (g_{ab}){\mathrm{\mid }}^{1\mathrm{/}2}\mathrm{\mid }d{x}^{0}d{x}^{1}d{x}^{2}d{x}^3\mathrm{\mid }$. The framework uses the same SI meanings of $G$ and $c$ as its selected fixed-background free scalar-field model; no collapse geometry or scattering premise from that model is imposed here.

Select finite constants ${\mathrm{\Lambda }}_{\mathrm{c}\mathrm{c}}$, ${\alpha }_{\mathrm{c}\mathrm{t}}$, and ${\beta }_{\mathrm{c}\mathrm{t}}$, with units ${\mathrm{m}}^{-2}$, ${\mathrm{m}}^2$, and ${\mathrm{m}}^2$ respectively; ${\mathrm{\Lambda }}_{\mathrm{c}\mathrm{c}}$ is the cosmological constant. Fix a positive renormalization scale $\mu$ in ${\mathrm{m}}^{-1}$, finite renormalization conditions $\mathbf{c}$, and a positive normalized state $\omega$ on the selected free scalar-field algebra, with Hadamard two-point distribution. Use the prescribed local stress tensor $T_{ab}^{\mathrm{r}\mathrm{e}\mathrm{n}}[g,\omega ]$, obtained by subtracting one state-independent Hadamard singular kernel from the state's two-point distribution, applying a finite-order bidifferential operator and adding the fixed local curvature correction. The tensors $H_{ab}^{(R^2)}$ and $H_{ab}^{({\mathrm{Ric}}^2)}$ are the selected smooth symmetric tensors whose pairings with compactly supported inverse-metric variations give, with positive sign, the first variations obtained by integrating the pointwise derivatives of the scalar-curvature-squared and Ricci-squared densities at the reference metric. These are the compactly supported density derivatives specified by the variation-tensor definition; no finite total curvature action or finite-parameter action difference is required. Also fix a retarded boundary rule for selecting the state/source data on the modeled domain. A retarded Green operator for the selected Klein-Gordon operator $P$ means a supplied linear map $E_R:C_0^{\mathrm{\infty }}(M)\to C^{\mathrm{\infty }}(M)$ satisfying both Green identities $P{E}_{R}f=f=E_{R}Pf$ for every $f\in C_0^{\mathrm{\infty }}(M)$ and $\mathrm{supp}(E_{R}f)\subseteq J^{+}(\mathrm{supp}f)$, where $J^{+}(K)$ is the set of points reachable from $K$ by future-directed causal curves. The retarded rule specifies the chosen $E_R$ and initial state data; no future boundary data are inferred from the equation. Let $E_{\mathrm{c}\mathrm{u}\mathrm{t}}>0$ joules be the chosen effective-model cutoff; its numerical value and admissible input domain are supplied with each model instance.

Adopt the equation $G_{ab}+{\mathrm{\Lambda }}_{\mathrm{c}\mathrm{c}}{g}_{ab}+{\alpha }_{\mathrm{c}\mathrm{t}}{H}_{ab}^{(R^2)}+{\beta }_{\mathrm{c}\mathrm{t}}{H}_{ab}^{({\mathrm{Ric}}^2)}=\frac{8\pi G}{c^4}{T}_{ab}^{\mathrm{r}\mathrm{e}\mathrm{n}}[g,\omega ]$ on that domain, in the stated regime below $E_{\mathrm{c}\mathrm{u}\mathrm{t}}$. With $R_{abcd}$, $S$ and $G_{ab}$ of units ${\mathrm{m}}^{-2}$ and the two $H$ tensors of units ${\mathrm{m}}^{-4}$, every term has units ${\mathrm{m}}^{-2}$ in length coordinates; $T_{ab}^{\mathrm{r}\mathrm{e}\mathrm{n}}$ has SI stress-energy units $\mathrm{J}{\mathrm{m}}^{-3}$. This is an adopted effective model equation. It does not assert that any solution exists, is unique or stable, or that it describes a given physical system.

Hadamard and renormalized stress contract

Specified late-time free-field collapse scattering model

The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors

Curvature-squared counterterm variation tensors