Hadamard and renormalized stress contract

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