Point-support spacetime densities: distributions, continuity equation and endpoint fluxes

If $I=\mathrm{\varnothing }$, the stated functionals and all their derivatives are zero, proving the proper branch. Otherwise assume $I$ nonempty. The graph map $\gamma (t)=(z(t),t)$ is $C^1$ on $I$ and injective, with values in $\mathrm{\Omega }$; for $\psi \in \mathcal{D}(\mathrm{\Omega })$ the pullback $\phi :=\psi \circ \gamma :I\to \mathbb{C}$ is $C^1$, and $\mathrm{supp}\phi \subseteq {\gamma }^{-1}[\mathrm{supp}\psi ]$: if $\phi (t)\mathrm{\ne }0$ then $\gamma (t)\in \mathrm{supp}\psi$. In the proper branch ${\gamma }^{-1}[\mathrm{supp}\psi ]$ is compact, so $\mathrm{supp}\phi$ is a closed subset of a compact set, hence compact in $I$; thus $\phi \in C_c^1(I)$ and ${\phi }^{\mathrm{\prime }}$ is continuous.

In the proper branch, $\rho$ is a distribution. Let $K\subseteq \mathrm{\Omega }$ be compact and $\psi \in {\mathcal{D}}_K$, so that $\mathrm{supp}\phi \subseteq C_K:={\gamma }^{-1}[K]$, a compact subset of $I$; if $C_K=\mathrm{\varnothing }$ then $\phi =0$. Otherwise $u:=\min C_K$ and $v:=\max C_K$ lie in $I$ with $C_K\subseteq [u,v]\subseteq I$, and $\mid \phi \mid \le p_0(\psi )$ on $[u,v]$ because $\psi$ vanishes off $K$ and $p_0(\psi )={\sup }_K\mid \psi \mid$; hence $\mid ⟨\rho ,\psi ⟩\mid \le 2\mid q\mid {\int }_u^v\mid \phi \mid dt\le 2\mid q\mid (v-u)p_0(\psi )$ by [F5], a finite-order estimate of order zero, so $\rho$ is a distribution by [F1].

The automatic properness claims hold. If $\mathrm{\Omega }={\mathbb{R}}^3\times I$ and $K\subseteq \mathrm{\Omega }$ is compact, then the time projection ${\mathrm{p}\mathrm{r}}_t[K]$ is a compact subset of $I$ by [F3] and ${\gamma }^{-1}[K]$ is closed in $I$ (preimage of the closed set $K$ under the continuous map $\gamma$) with ${\gamma }^{-1}[K]\subseteq {\mathrm{p}\mathrm{r}}_t[K]$; being closed in $I$ and contained in ${\mathrm{p}\mathrm{r}}_t[K]$, it is closed in the subspace ${\mathrm{p}\mathrm{r}}_t[K]$, hence compact as a closed subset of a compact metric space by [F3]. If $I=\mathbb{R}$, then ${\gamma }^{-1}[K]$ is closed and contained in the bounded set ${\mathrm{p}\mathrm{r}}_t[K]$, hence compact in $\mathbb{R}$ by [F3].

In the proper branch, $J$ is a distribution: with the notation of step 2.1, $\mid z^{\mathrm{\prime }}\mid$ is continuous on $[u,v]$, hence bounded there by $M_K<\mathrm{\infty }$, and $\mathrm{\mid }⟨J_i,\psi ⟩\mathrm{\mid }\le 2\mid q\mid {\int }_u^v\parallel z^{\mathrm{\prime }}(t){\parallel }_2\mid \phi (t)\mid dt\le 2\mid q\mid M_K(v-u)p_0(\psi )$, again a finite-order estimate of order zero.

Endpoint branch, well-definedness: let $I=(a,b)$, with $z$ extending continuously to $[a,b]$, ${\int }_a^b\parallel z^{\mathrm{\prime }}{\parallel }_2<\mathrm{\infty }$ and both endpoint events in $\mathrm{\Omega }$. Then $\phi =\psi \circ \gamma$ extends continuously to $[a,b]$ with $\mid \phi \mid \le p_0(\psi )$ and $\mathrm{\mid }{z}_i^{\mathrm{\prime }}\phi \mathrm{\mid }\le p_0(\psi )\parallel z^{\mathrm{\prime }}{\parallel }_2$ on $(a,b)$; by the comparison test [F6] the improper integrals ${\int }_a^b\phi$ and ${\int }_a^b{z}^{\mathrm{\prime }}\phi$ converge absolutely (the first against the constant $p_0(\psi )$, the second against $p_0(\psi )\parallel z^{\mathrm{\prime }}{\parallel }_2$), so $⟨\rho ,\psi ⟩:=q{\int }_a^b\phi$ and $⟨J,\psi ⟩:=q{\int }_a^b{z}^{\mathrm{\prime }}\phi$ are well-defined finite values; they are linear in $\psi$ by linearity of the truncated integrals and passage to the limit [F5, F6], and for $\psi \in {\mathcal{D}}_K$ they obey $\mid ⟨\rho ,\psi ⟩\mid \le 2\mid q\mid p_0(\psi )(b-a)$ and $\mathrm{\mid }⟨J_i,\psi ⟩\mathrm{\mid }\le 2\mid q\mid p_0(\psi ){\int }_a^b\parallel z^{\mathrm{\prime }}{\parallel }_2$, so they are distributions by [F1].

For $\psi \in \mathcal{D}(\mathrm{\Omega })$ in the proper branch, the pairing of the divergence is $⟨{\mathrm{\partial }}_t\rho +\mathrm{div}J,\psi ⟩=-⟨\rho ,{\mathrm{\partial }}_t\psi ⟩-⟨J,\mathrm{\nabla }\psi ⟩=-q{\int }_I({\mathrm{\partial }}_t\psi (\gamma (t))+⟨z^{\mathrm{\prime }}(t),\mathrm{\nabla }\psi (\gamma (t))⟩)dt$, the components of $\mathrm{\nabla }\psi$ being the spatial partial derivatives and $\mathrm{div}J$ the sum of the coordinate distributional derivatives of the components of $J$ by [F2].

By the chain rule [F4], $\frac{d}{dt}\phi (t)=D\psi (\gamma (t)){\gamma }^{\mathrm{\prime }}(t)={\mathrm{\partial }}_t\psi (\gamma (t))+⟨\mathrm{\nabla }\psi (\gamma (t)),z^{\mathrm{\prime }}(t)⟩$, since ${\gamma }^{\mathrm{\prime }}=(z^{\mathrm{\prime }},1)$; hence the integrand of step 4.1 is exactly ${\phi }^{\mathrm{\prime }}$.

Since $\phi \in C_c^1(I)$ by step 1.1, there are $u<v$ in $I$ with $\mathrm{supp}\phi \subseteq [u,v]$ and $\phi (u)=\phi (v)=0$; by [F5] applied to the subinterval and by additivity of the integral over subintervals, ${\int }_I{\phi }^{\mathrm{\prime }}={\int }_u^v{\phi }^{\mathrm{\prime }}=\phi (v)-\phi (u)=0$. Therefore $⟨{\mathrm{\partial }}_t\rho +\mathrm{div}J,\psi ⟩=0$ for every $\psi \in \mathcal{D}(\mathrm{\Omega })$, which is the identity ${\mathrm{\partial }}_t\rho +\mathrm{div}J=0$ of distributions.

Endpoint branch, identity: applying [F2] and the chain rule as in steps 4.1 and 5.1 gives $⟨{\mathrm{\partial }}_t\rho +\mathrm{div}J,\psi ⟩=-q{\int }_a^b{\phi }^{\mathrm{\prime }}(t)dt$, where ${\phi }^{\mathrm{\prime }}$ is continuous on $(a,b)$ and $\mid {\phi }^{\mathrm{\prime }}\mid \le C(1+\parallel z^{\mathrm{\prime }}{\parallel }_2)$ for a constant $C$ depending on the fixed test, so the improper integral converges absolutely by the comparison test [F6]; for $0<\epsilon <(b-a)\mathrm{/}2$ the proper fundamental theorem [F5] gives ${\int }_{a+\epsilon }^{b-\epsilon }{\phi }^{\mathrm{\prime }}=\phi (b-\epsilon )-\phi (a+\epsilon )$, and the mixed improper integral is the limit of these truncations as $\epsilon \downarrow 0$ [F6]; continuity of $\phi$ at the endpoints gives $\phi (b-\epsilon )-\phi (a+\epsilon )\to \phi (b)-\phi (a)=\psi (z(b),b)-\psi (z(a),a)$. Hence $⟨{\mathrm{\partial }}_t\rho +\mathrm{div}J,\psi ⟩=q\psi (z(a),a)-q\psi (z(b),b)=⟨q{\delta }_{(z(a),a)}-q{\delta }_{(z(b),b)},\psi ⟩$ for every $\psi \in \mathcal{D}(\mathrm{\Omega })$ by [F2], which is the endpoint identity. If $q=0$, each endpoint contribution and their sum vanish for every test; if $q\mathrm{\ne }0$, division by $q$ shows that an individual contribution vanishes exactly when its endpoint test value is zero, and the sum vanishes exactly when the two endpoint test values are equal.