Green-hyperbolic free scalar interface

Assume the Axiom of Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)), as used to form the time-oriented Lorentzian geometric data below. At each $p$ in a smooth four-manifold $M$, declare two ordered bases of $T_{p}M$ equivalent exactly when the change-of-basis determinant is positive; the two equivalence classes are the orientations of $T_{p}M$. A smooth orientation of $M$ chooses one of these two classes at every $p$, locally constantly in smooth local frames; an oriented manifold is $M$ equipped with a chosen smooth orientation. Fix an orientation on $M$. Let $(M,g,N)$ be a Hausdorff, second-countable, oriented and time-oriented smooth Lorentzian four-manifold without boundary, with signature $(-,+,+,+)$, and assume it is globally hyperbolic in the convention of Chronology, causality, strong causality, global hyperbolicity and time functions: strongly causal and with compact causal diamonds. Curves and the causal sets $J^{\pm }(K)$ use the locally absolutely continuous causal-curve convention of Causal curves, chronological and causal reachability, and inextendibility.

Write the positive metric density in a chart as ${\mu }_g=\mathrm{\mid }\det (g_{ab}){\mathrm{\mid }}^{1\mathrm{/}2}\mathrm{\mid }d{x}^{0}d{x}^{1}d{x}^{2}d{x}^3\mathrm{\mid }$. This is a coordinate-independent smooth density: if $y=\psi (x)$, then $g_y=(D{\psi }^{-1}{)}^T{g}_{x}D{\psi }^{-1}$, so $\mathrm{\mid }\det (g_y){\mathrm{\mid }}^{1\mathrm{/}2}=\mathrm{\mid }\det D{\psi }^{-1}\mathrm{\mid }\mathrm{\mid }\det (g_x){\mathrm{\mid }}^{1\mathrm{/}2}$, the test-density transformation rule of Scalar distributions on smooth manifolds. Let ${\mathcal{D}}_g(M):=C_c^{\mathrm{\infty }}(M;\mathbb{R})$ have the real test-function topology transported by $f\mapsto f{\mu }_g$ from that supplier's compactly supported smooth test densities. Give $C^{\mathrm{\infty }}(M;\mathbb{R})$ its usual topology of uniform convergence of every coordinate derivative on each compact subset.

For a real $V\in C^{\mathrm{\infty }}(M;\mathbb{R})$ define the scalar Klein–Gordon-type operator $P=-{\mathrm{\square }}_g+V$, where ${\mathrm{\square }}_{g}f:={\mu }_g^{-1}{\mathrm{\partial }}_a({\mu }_g{g}^{ab}{\mathrm{\partial }}_{b}f)$ in a chart. Its principal symbol is $-g^{ab}{\xi }_a{\xi }_b$. No physical units are assigned; this is a mathematical differential operator. The Green-data interface below requires formal self-adjointness with respect to ${\mu }_g$, meaning ${\int }_{M}f(Pu){\mu }_g={\int }_M(Pf)u{\mu }_g$ for every $f\in {\mathcal{D}}_g(M)$ and $u\in C^{\mathrm{\infty }}(M;\mathbb{R})$; these integrals are finite because $P$ is local. The real solution space with spacelike-compact support is

${\mathrm{Sol}}_{\mathrm{s}\mathrm{c}}(M):=\{u\in C^{\mathrm{\infty }}(M;\mathbb{R}):Pu=0\text{ and }\mathrm{supp}u\subseteq J^{+}(K)\cup J^{-}(K)\text{ for some compact }K\subset M\}\mathrm{.}$

A Green-data interface for $(M,g,N,P)$ consists of continuous real-linear maps $E_{+},E_{-}:{\mathcal{D}}_g(M)\to C^{\mathrm{\infty }}(M;\mathbb{R})$, called respectively retarded and advanced Green maps, such that for each $f\in {\mathcal{D}}_g(M)$,

$P{E}_{+}f=E_{+}Pf=f,P{E}_{-}f=E_{-}Pf=f,$

$\mathrm{supp}(E_{+}f)\subseteq J^{+}(\mathrm{supp}f),\mathrm{supp}(E_{-}f)\subseteq J^{-}(\mathrm{supp}f)\mathrm{.}$

The interface also includes uniqueness of the smooth Cauchy problem for $P$. For a smooth spacelike Cauchy hypersurface $\mathrm{\Sigma }$ as defined in Spacelike hypersurfaces, acausal sets, Cauchy sets, domains of dependence and Cauchy horizons, let $\nu$ denote its unique future-directed unit timelike normal, characterized by orthogonality to $T\mathrm{\Sigma }$, $g(\nu ,\nu )=-1$ and $g(\nu ,N)<0$; the Cauchy data of a smooth $u$ are $u{\mathrm{\mid }}_{\mathrm{\Sigma }}$ and the normal derivative $\nu (u){\mathrm{\mid }}_{\mathrm{\Sigma }}$. The uniqueness condition is that a smooth solution of $Pu=0$ with both data zero on $\mathrm{\Sigma }$ is zero.

Scalar distributions on smooth manifolds

Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model

Causal curves, chronological and causal reachability, and inextendibility

Chronology, causality, strong causality, global hyperbolicity and time functions

Spacelike hypersurfaces, acausal sets, Cauchy sets, domains of dependence and Cauchy horizons

The inverse metric, musical isomorphisms and index raising

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)

This is an interface definition, with no existence axiom. Under Countable Choice and Dependent Choice, Existence and uniqueness of continuous scalar Green maps on smooth globally hyperbolic spacetimes constructs and uniquely determines its continuous Green maps for every smooth globally hyperbolic background and arbitrary smooth real potential specified here. Its proof reconstructs Cauchy evolution from the local symmetric-hyperbolic suppliers.