Hadamard subtraction and semiclassical model contracts

Let $(M,g)$ be a smooth Lorentzian four-manifold of signature $(-,+,+,+)$. Use its unique torsion-free, metric-compatible connection $\mathrm{\nabla }$ and the curvature convention $R(X,Y)Z={\mathrm{\nabla }}_X{\mathrm{\nabla }}_{Y}Z-{\mathrm{\nabla }}_Y{\mathrm{\nabla }}_{X}Z-{\mathrm{\nabla }}_{[X,Y]}Z$. Define $R^a{}_{bcd}$ in each chart by $R({\mathrm{\partial }}_c,{\mathrm{\partial }}_d){\mathrm{\partial }}_b=R^a{}_{bcd}{\mathrm{\partial }}_a$, and sum repeated indices; $g^{ab}$ denotes the inverse matrix to $g_{ab}$. Then set $R_{abcd}=g_{ae}{R}^e{}_{bcd}$, ${\mathrm{Ric}}_{bd}=g^{ac}{R}_{abcd}$, $S=g^{bd}{\mathrm{Ric}}_{bd}$, and $G_{ab}={\mathrm{Ric}}_{ab}-\frac{1}{2}S{g}_{ab}$ using this convention. In a chart whose coordinates have units of length, use the positive metric volume density $d{\mathrm{v}\mathrm{o}\mathrm{l}}_g=\mathrm{\mid }\det (g_{ab}){\mathrm{\mid }}^{1\mathrm{/}2}\mathrm{\mid }d{x}^{0}d{x}^{1}d{x}^{2}d{x}^3\mathrm{\mid }\mathrm{.}$

A compactly supported inverse-metric variation is a smooth family of smooth Lorentzian metrics $g_{ϵ}$ for $ϵ$ in an open interval about $0$, with $g_0=g$, for which $h^{ab}:={{\mathrm{\partial }}_{ϵ}{g}_{ϵ}^{ab}\mid }_{ϵ=0}$ is a compactly supported smooth symmetric contravariant two-tensor. Every such $h^{ab}$ is admissible for a sufficiently small parameter interval because Lorentzian signature is open and $h^{ab}$ has compact support. For a smooth metric $q$, write the curvature-squared densities as ${\mathcal{L}}_{R^2}(q):=S[q{]}^{2}d{\mathrm{v}\mathrm{o}\mathrm{l}}_q$ and ${\mathcal{L}}_{{\mathrm{Ric}}^2}(q):={\mathrm{Ric}}_{cd}[q]{\mathrm{Ric}}^{cd}[q]d{\mathrm{v}\mathrm{o}\mathrm{l}}_q$. Define the first variations by integrating their pointwise density derivatives at $ϵ=0$, not by subtracting possibly infinite total actions: $\delta I_{R^2}[g;h]:={\int }_M{{\mathrm{\partial }}_{ϵ}{\mathcal{L}}_{R^2}(g_{ϵ})\mid }_0,\delta I_{{\mathrm{Ric}}^2}[g;h]:={\int }_M{{\mathrm{\partial }}_{ϵ}{\mathcal{L}}_{{\mathrm{Ric}}^2}(g_{ϵ})\mid }_0\mathrm{.}$ These derivatives are smooth compactly supported densities determined only by $g$ and $h$, even if $g_{ϵ}-g$ is not compactly supported for nonzero $ϵ$. Indeed, in a chart each density coefficient is a smooth function of the inverse metric and its coordinate derivatives through order two on the open set of nondegenerate metrics. The parameter chain rule therefore makes its derivative at zero a finite linear combination, with smooth background coefficients, of $h^{ab}$ and its coordinate derivatives through order two. Every such term vanishes outside $\mathrm{supp}h$. Smooth joint dependence allows the parameter and coordinate derivatives to commute, so no higher-order parameter terms of the family enter. This proves both compact support and independence of the chosen family with the same first inverse-metric derivative; integration of these compact densities is finite. In particular $h=0$ gives zero first variations. No finiteness of the actions at nonzero parameter, or compact support of their differences, is assumed or concluded.

For a particular model instance, supply smooth symmetric covariant two-tensors $H_{ab}^{(R^2)}$ and $H_{ab}^{({\mathrm{Ric}}^2)}$ that satisfy, for every compactly supported inverse-metric variation, $\delta I_{R^2}[g;h]={\int }_M{{\mathrm{\partial }}_{ϵ}{\mathcal{L}}_{R^2}(g_{ϵ})\mid }_0={\int }_M{H}_{ab}^{(R^2)}{h}^{ab}d{\mathrm{v}\mathrm{o}\mathrm{l}}_g,$ $\delta I_{{\mathrm{Ric}}^2}[g;h]={\int }_M{{\mathrm{\partial }}_{ϵ}{\mathcal{L}}_{{\mathrm{Ric}}^2}(g_{ϵ})\mid }_0={\int }_M{H}_{ab}^{({\mathrm{Ric}}^2)}{h}^{ab}d{\mathrm{v}\mathrm{o}\mathrm{l}}_g\mathrm{.}$ The displayed positive signs, inverse-metric variation, and curvature convention fix the counterterm sign and normalization. In length coordinates both tensors have units ${\mathrm{m}}^{-4}$. This Definition specifies the selected tensors as model inputs and does not assert their existence for every possible background or any solution of a gravitational field equation.

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

Covariant field families and variations

The time orientation in The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors can be supplied locally; no global time-orientability assumption is needed here. At each point choose a timelike tangent vector and extend its chart components constantly near that point. Smoothness of $g$ makes this vector field timelike on a sufficiently small open neighborhood, where its cone selects a time orientation. Apply the supplier on each such neighborhood. Its unique torsion-free, metric-compatible connections agree on overlaps: each restricted connection satisfies those same two conditions for the restricted metric, regardless of which timelike cone was selected. They therefore define one global smooth connection. The displayed curvature operator, lowering and contractions then agree on overlaps as well, since they use only that connection and $g$, not the local cone selections. Thus the supplier's smooth tensor definitions apply to every background in this Definition, including backgrounds with no global time orientation. This local argument makes no simultaneous choice of a cone at every point.

Let $(M,g)$ be a Hausdorff, second-countable, oriented, time-oriented, globally hyperbolic smooth Lorentzian four-manifold without boundary, with signature $(-,+,+,+)$. An orientation is a smooth choice of orientation in each tangent space, and a time orientation selects the future timelike cone. Use the causal-curve and $J^{\pm }$ conventions of Wavefront and positive-frequency Hadamard data: a future causal curve is a continuous map $c:I\to M$ on a real interval $I$, piecewise $C^1$ on each compact subinterval with finitely many pieces there, and with nonzero future-directed causal tangent on each smooth piece; a past causal curve uses the opposite cone. If an endpoint belongs to $I$, its value is the continuous endpoint value. Then $J^{+}(p)$ consists of $p$ and every $q$ reached by a future causal curve $c:[a,b]\to M$, $a<b$, with $c(a)=p$ and $c(b)=q$, and $J^{-}(p)=\{q:p\in J^{+}(q)\}$. Strong causality means every point has arbitrarily small neighborhoods that each causal curve meets in a connected parameter interval. Global hyperbolicity means strong causality and compact causal diamonds $J^{+}(p)\cap J^{-}(q)$. Fix a real scalar Klein-Gordon operator $P=-g^{ab}{\mathrm{\nabla }}_a{\mathrm{\nabla }}_b+V$, where $\mathrm{\nabla }$ is the Levi-Civita connection and $V$ is a prescribed smooth real scalar of dimension ${\mathrm{m}}^{-2}$ in length coordinates. Use the canonical field length degree ${\mathrm{m}}^{-1}$ used for the framework's free scalar field, so its two-point distribution and subtraction kernel have length degree ${\mathrm{m}}^{-2}$. Supply the free-field $\ast$-algebra ${\mathcal{A}}_P$ and its real test-function smearing map $\mathrm{\Phi }:C_c^{\mathrm{\infty }}(M;\mathbb{R})\to {\mathcal{A}}_P$; smearing uses $d{\mathrm{v}\mathrm{o}\mathrm{l}}_g$. This is model input, not a field-algebra construction claim. In the massless minimally coupled collapse specialization, Specified late-time free-field collapse scattering model specifies bosonic Fock quantization, the smeared field operators, and the positive normalized state convention. A state $\omega$ is a positive normalized linear functional on ${\mathcal{A}}_P$, so $\omega (A^{\ast }A)\ge 0$ and $\omega (1)=1$. Its two-point distribution $W_{\omega }\in {\mathcal{D}}^{\mathrm{\prime }}(M\times M)$, in the test-density convention, is defined by the pairings $W_{\omega }(f,h)=\omega (\mathrm{\Phi }(f)\mathrm{\Phi }(h))$ for compactly supported smooth real tests $f,h$, extended complex linearly. Require $P_x{W}_{\omega }=P_y{W}_{\omega }=0$ and $\mathrm{WF}(W_{\omega })={\mathcal{C}}_g^{+}$, where ${\mathcal{C}}_g^{+}$ consists of pairs $(x,\xi ;y,-\zeta )$ with nonzero future-null covectors $\xi ,\zeta$ that are equal at a coincident point or are the metric-dual tangents, with a common positive scale, along one nonconstant future-directed null geodesic; the two base points need not be ordered along that geodesic. This is the positive-frequency Hadamard convention in Wavefront and positive-frequency Hadamard data, whose definition fixes this relation without a base-point order. $\mathrm{WF}$ records nonzero cotangent directions in which localized Fourier transforms fail rapid decay. The framework's massless, minimally coupled collapse scalar is one specialization; its Schwarzschild and scattering assumptions are not imposed on this general contract.

For each open convex normal neighborhood $U\subset M$ (an open set such that for every $p\in U$ the exponential map at $p$ restricts to a diffeomorphism from a star-shaped open neighborhood of $0\in T_{p}M$ onto $U$), supply a locally covariant Hadamard singular bidistribution $H_{\mu }$ for $P$. The scale $\mu >0$ is fixed and has units ${\mathrm{m}}^{-1}$. “Hadamard singular bidistribution” here means the prescribed local, state-independent singular part determined by $(g,P,\mu )$, with $W_{\omega }-H_{\mu }\in C^{\mathrm{\infty }}(U\times U)$. The parametrix pieces, the bidifferential operators below, and their resulting coincident values must be compatible on overlaps. This is supplied data: the definition proves no parametrix construction or general existence theorem.

Supply a finite-order bidifferential operator $D_{ab}(x,y)$ which maps a smooth biscalar on $U\times U$ to a smooth covariant symmetric two-tensor at $x$; any coefficient bitensors and parallel transports needed to define its indices are part of the supplied operator. Also supply a smooth, symmetric, state-independent local curvature tensor $C_{ab}[g;\mu ,\mathbf{c}]$, where $\mathbf{c}$ is the fixed list of finite renormalization conditions. “Local” means determined by the values of $g$, $P$, and their derivatives at the point, through finitely many curvature contractions; it has no state or distant-geometry dependence. Require local covariance: whenever $\psi :(M,g)\to (M^{\mathrm{\prime }},g^{\mathrm{\prime }})$ is a smooth embedding with ${\psi }^{\ast }{g}^{\mathrm{\prime }}=g$, preserves the time orientation, has causally convex image (every causal curve segment in $M^{\mathrm{\prime }}$ with endpoints in the image stays in the image), and preserves the prescribed scalar potential, $V={\psi }^{\ast }{V}^{\mathrm{\prime }}$, compare prescriptions with the same scale $\mu$ and finite conditions $\mathbf{c}$. Potential preservation is equivalently the operator identity $P({\psi }^{\ast }f)={\psi }^{\ast }(P^{\mathrm{\prime }}f)$ for every $f\in C^{\mathrm{\infty }}(M^{\mathrm{\prime }})$; the metric connection terms intertwine because $\psi$ is an isometry. Require the assigned local kernels, bidifferential operators and curvature corrections on $(M,g,P)$ to be the pullbacks of those on $(M^{\mathrm{\prime }},g^{\mathrm{\prime }},P^{\mathrm{\prime }})$. This requirement supplies no preferred state assignment. For stress expectations, compare supplied states only when their two-point distributions obey $W_{\omega }=(\psi \times \psi {)}^{\ast }{W}_{{\omega }^{\mathrm{\prime }}}$, where this pullback is restriction to the open image followed by transport through $\psi \times \psi$. Then the smooth-remainder formula below requires $T^{\mathrm{r}\mathrm{e}\mathrm{n}}[g,\omega ]={\psi }^{\ast }{T}^{\mathrm{r}\mathrm{e}\mathrm{n}}[g^{\mathrm{\prime }},{\omega }^{\mathrm{\prime }}]$. No covariance equality between unrelated states or different potentials is imposed. Require conservation, ${\mathrm{\nabla }}^a{T}_{ab}^{\mathrm{r}\mathrm{e}\mathrm{n}}=0$, with the index raised using $g^{ab}$.

Define the renormalized stress tensor by $T_{ab}^{\mathrm{r}\mathrm{e}\mathrm{n}}[g,\omega ](x):={\lim }_{y\to x}{D}_{ab}(x,y)(W_{\omega }-H_{\mu })(x,y)+C_{ab}[g;\mu ,\mathbf{c}](x)\mathrm{.}$ The limit is taken inside a convex normal neighborhood. It exists because $W_{\omega }-H_{\mu }$ is smooth and $D_{ab}$ has finite order. In SI units and length coordinates $x^0=ct$, the normalization of $D_{ab}$ is part of the chosen scalar-field prescription and its output, as well as $C_{ab}$, has stress-energy units $\mathrm{J}{\mathrm{m}}^{-3}$. No pointwise product of the singular distributions $W_{\omega }$ and $H_{\mu }$ is taken.

Wavefront and positive-frequency Hadamard data

Specified late-time free-field collapse scattering model

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

Let $g$ be one fixed smooth globally hyperbolic Lorentzian four-metric and let ${\omega }_1,{\omega }_2$ be two positive Hadamard states for the same real scalar Klein-Gordon operator. Use one identical prescription: the same local parametrix $H_{\mu }$, bidifferential operator $D_{ab}$, scale $\mu$, overlap convention, and state-independent curvature correction $C_{ab}$. Then, on each convex normal neighborhood, $T_{ab}^{\mathrm{r}\mathrm{e}\mathrm{n}}[g,{\omega }_1]-T_{ab}^{\mathrm{r}\mathrm{e}\mathrm{n}}[g,{\omega }_2]={\lim }_{y\to x}{D}_{ab}(x,y)(W_{{\omega }_1}-W_{{\omega }_2})(x,y),$ where the difference on the right is smooth and its coincidence limit exists. Adding the same state-independent local curvature tensor to both stress prescriptions leaves this difference unchanged.

Hadamard and renormalized stress contract