Wavefront and positive-frequency Hadamard data

Let $M$ first be any Hausdorff, second-countable smooth manifold without boundary and $u\in {\mathcal{D}}^{\mathrm{\prime }}(M)$. No metric, orientation or dimension restriction is used in the following wavefront definition. For each smooth chart $x:U\to \mathrm{\Omega }\subseteq {\mathbb{R}}^n$, let $u_x\in {\mathcal{D}}^{\mathrm{\prime }}(\mathrm{\Omega })$ be the coordinate representative of $u$ in the test-density convention. Map a local covector $(x(p),\xi )\in T^{\ast }\mathrm{\Omega }$ to $(p,d{x}_p^T\xi )\in T^{\ast }M$. Define the global wavefront set by the union over all charts, $\mathrm{WF}(u):={\bigcup }_{(U,x)}\{(p,d{x}_p^T\xi ):(x(p),\xi )\in {\mathrm{WF}}_{x(U)}(u_x)\}\subset T^{\ast }M\setminus 0,$ where the local sets are the local Fourier wavefront sets. Thus the global set is defined using all charts; equality of the chartwise descriptions on overlaps is a separate theorem. The Fourier convention is the displayed $2\pi$ convention.

Let $(M,g)$ be a Hausdorff, second-countable, oriented, time-oriented, globally hyperbolic smooth Lorentzian $n$-manifold without boundary, $n\ge 2$. Here $g$ is a smooth symmetric nondegenerate bilinear form on each tangent space with one negative and $n-1$ positive directions; in charts its signature is $(-,+,\dots ,+)$. An orientation is a smooth choice of orientation in each tangent space. A time orientation selects one of the two components of $\{v\mathrm{\ne }0:g(v,v)<0\}$ as future. A tangent vector is causal when $g(v,v)\le 0$, and is future or past causal when it belongs to the closure of the selected future or past timelike cone. A future causal curve is a continuous map $c:I\to M$, where $I\subset \mathbb{R}$ is an interval, which is piecewise $C^1$ on each compact subinterval, with finitely many pieces there and nonzero future causal tangent on each smooth piece. Past curves are defined with the opposite cone. Endpoint values, when included in $I$, are taken by continuity. Define $J^{+}(p)$ by including $p$ and all $q$ for which a future causal curve $c:[a,b]\to M$, $a<b$, has $c(a)=p$, $c(b)=q$; put $J^{-}(p)=\{q:p\in J^{+}(q)\}$. Here globally hyperbolic means strongly causal and every causal diamond $J^{+}(p)\cap J^{-}(q)$ is compact; strongly causal means every point has arbitrarily small neighborhoods which each causal curve meets in a connected parameter interval. A nonzero cotangent vector is future or past causal, or future or past null, when its metric dual under $g^{-1}$ has the corresponding tangent property.

Here is an explicit construction of the connection used below. A smooth vector field $A$ is a smooth section of $TM$, acting on smooth scalars by $Af=df(A)$; its bracket is the vector field characterized by $[A,B]f=A(Bf)-B(Af)$. A connection is an $\mathbb{R}$-bilinear operation $(A,B)\mapsto {\mathrm{\nabla }}_{A}B$ on smooth local vector fields, commuting with restriction, with ${\mathrm{\nabla }}_{fA}B=f{\mathrm{\nabla }}_{A}B$ and ${\mathrm{\nabla }}_A(fB)=(Af)B+f{\mathrm{\nabla }}_{A}B$ for smooth real $f$. Define it by $2g({\mathrm{\nabla }}_{A}B,C)=Ag(B,C)+Bg(A,C)-Cg(A,B)-g(A,[B,C])+g(B,[C,A])+g(C,[A,B])\mathrm{.}$ Indeed the right side is $C^{\mathrm{\infty }}$-linear in $C$: on substituting $fC$, derivative terms cancel using $[B,fC]=f[B,C]+(Bf)C$ and $[fC,A]=f[C,A]-(Af)C$. On a chart, writing the right side as $K(A,B,C)$ and taking $C={\mathrm{\partial }}_c$, put ${\mathrm{\nabla }}_{A}B=\frac{1}{2}{g}^{ac}K(A,B,{\mathrm{\partial }}_c){\mathrm{\partial }}_a$. Its coefficients are smooth: $g^{-1}=\mathrm{adj}(g)\mathrm{/}\det g$. The $C^{\mathrm{\infty }}$-linearity just checked shows it satisfies the formula for every local $C$. Nondegeneracy gives uniqueness on overlaps, so these local fields agree and define a global connection. Repeated coordinate indices are summed from $1$ to $n$. Expansion with $fA$ and $fB$ gives the two connection rules above. Subtracting the formula with $A,B$ interchanged gives ${\mathrm{\nabla }}_{A}B-{\mathrm{\nabla }}_{B}A=[A,B]$ (torsion zero), and adding the formulas with $B,C$ interchanged gives $Ag(B,C)=g({\mathrm{\nabla }}_{A}B,C)+g(B,{\mathrm{\nabla }}_{A}C)$ (metric compatibility). Conversely these last two identities and the connection rules give the displayed formula by taking the cyclic metric identities and replacing ${\mathrm{\nabla }}_{A}B-{\mathrm{\nabla }}_{B}A$ by $[A,B]$; hence the connection is unique. This dimension-independent construction defines the Levi–Civita connection $\mathrm{\nabla }$ without an unprovided existence premise. On scalars ${\mathrm{\nabla }}_{A}f=Af$; on covectors $({\mathrm{\nabla }}_A\beta )(B)=A(\beta (B))-\beta ({\mathrm{\nabla }}_{A}B)$. Define the smooth metric density ${\mu }_g=\mathrm{\mid }\det (g_{ab}){\mathrm{\mid }}^{1\mathrm{/}2}\mathrm{\mid }d{x}^1\cdots d{x}^n\mathrm{\mid }$ in a chart. It is coordinate-independent: under $y=\psi (x)$, $g_y=(D{\psi }^{-1}{)}^T{g}_{x}D{\psi }^{-1}$, 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 density transformation law. Functional-kernel pairings and contractions on this page use ${\mu }_g$ in each integrated variable; no physical units are assigned.

Write ${\mathrm{\nabla }}_{{\mathrm{\partial }}_a}{\mathrm{\partial }}_b={\mathrm{\Gamma }}_{ab}^c{\mathrm{\partial }}_c$; then ${\mathrm{\Gamma }}_{ab}^c=\frac{1}{2}{g}^{cd}({\mathrm{\partial }}_a{g}_{bd}+{\mathrm{\partial }}_b{g}_{ad}-{\mathrm{\partial }}_d{g}_{ab})$ by the defining formula. Along a smooth curve, ${\mathrm{\nabla }}_{\dot{\gamma }}\dot{\gamma }$ has components ${\ddot{\gamma }}^c+{\mathrm{\Gamma }}_{ab}^c(\gamma ){\dot{\gamma }}^a{\dot{\gamma }}^b$. An affinely parametrized null geodesic is a smooth curve $\gamma :I\to M$ on an open interval $I\subset \mathbb{R}$, with nonzero tangent, and $g(\dot{\gamma },\dot{\gamma })=0$ and ${\mathrm{\nabla }}_{\dot{\gamma }}\dot{\gamma }=0$; it is future-directed when $\dot{\gamma }$ lies in the chosen future causal cone. Two nonzero future-null cotangent vectors $(p,\xi )$ and $(q,\zeta )$ are related, written $(p,\xi )\sim (q,\zeta )$, if $p=q$ and $\zeta =\xi$, or if there are a nonconstant future-directed affinely parametrized null geodesic $\gamma :I\to M$, parameters $s,t\in I$ in either order, and a common scale $\lambda >0$ such that $p=\gamma (s),q=\gamma (t),\xi =\lambda g_p(\dot{\gamma }(s),\cdot ),\zeta =\lambda g_q(\dot{\gamma }(t),\cdot )\mathrm{.}$ These covectors are the metric-dual tangents to that geodesic and are related by induced covector parallel transport. There is no requirement $s\le t$: the future-frequency choice concerns the covectors, not an ordering of the two base points. The coincident case includes every nonzero future-null covector with $\zeta =\xi$. Define the positive-frequency null relation ${\mathcal{C}}_g^{+}:=\{(p,\xi ;q,-\zeta ):(p,\xi )\sim (q,\zeta ),\xi \text{ future null}\}\subset T^{\ast }(M\times M)\setminus 0.$ For a fixed scalar second-order operator $P:C^{\mathrm{\infty }}(M;\mathbb{C})\to C^{\mathrm{\infty }}(M;\mathbb{C})$ given by $P=-g^{ab}{\mathrm{\nabla }}_a{\mathrm{\nabla }}_b+X^a{\mathrm{\nabla }}_a+V$, with smooth vector field $X$, smooth scalar $V$, inverse metric $g^{ab}$, and Levi-Civita derivative $\mathrm{\nabla }$ (extended complex-linearly), define its action on distributions as follows. If in a chart $P={\sum }_{\mathrm{\mid }\alpha \mathrm{\mid }\le 2}{a}_{\alpha }{\mathrm{\partial }}^{\alpha }$, its transpose on compactly supported test densities is $P^{\mathrm{t}}(h\mathrm{\mid }dx\mathrm{\mid })={\sum }_{\mathrm{\mid }\alpha \mathrm{\mid }\le 2}(-1{)}^{\mathrm{\mid }\alpha \mathrm{\mid }}{\mathrm{\partial }}^{\alpha }(a_{\alpha }h)\mathrm{\mid }dx\mathrm{\mid }\mathrm{.}$ There is no complex conjugation. Integration by parts characterizes it intrinsically by $\int (Pf)\omega =\int f(P^{\mathrm{t}}\omega )$ for every smooth scalar $f$ and test density $\omega$; hence coordinate expressions agree. It preserves compact support and is continuous on the test-density space, since each output derivative seminorm is bounded by finitely many input derivative seminorms on the same compact support. Put $⟨Pu,\omega ⟩=⟨u,P^{\mathrm{t}}\omega ⟩$. On $M\times M$, $P_x$ and $P_y$ differentiate only the first or second coordinate block, respectively, with those coefficients and the same transpose convention on product test densities; this also specifies their action on nonfactorized test densities. A scalar Hadamard bisolution is a distribution $W\in {\mathcal{D}}^{\mathrm{\prime }}(M\times M)$ satisfying $P_{x}W=P_{y}W=0$ and $\mathrm{WF}(W)={\mathcal{C}}_g^{+}$. Write $W^{\mathsf{T}}={\tau }^{\ast }W$ for the pullback by $\tau (x,y)=(y,x)$. A fixed antisymmetric part means the equality $W-W^{\mathsf{T}}=iE$ for one specified distribution $E$. This definition asserts no existence of $W$ and fixes no field-operator or commutator sign beyond the displayed convention.

Scalar distributions on smooth manifolds

Local Fourier definition of a wavefront set

Smooth manifolds and their smooth charts

Cotangent space and cotangent bundle as a disjoint union