Charged Liouville transport preserves the future massive shell

Fix $(x_0,P_0)$ with $C(x_0,P_0)=0$. Since $M>0$, $P_0\mathrm{\ne }0$; by nondegeneracy of $g_{x_0}$ there is a tangent vector $V$ with $g_{x_0}(P_0,V)\mathrm{\ne }0$. In any bundle chart around $(x_0,P_0)$, choose a fiber coordinate direction $P^j$ on which this nonzero covector has a nonzero component. The map from the eight chart coordinates to the four base coordinates, $C$, and the other three fiber coordinates has Jacobian determinant equal, up to sign, to $\mathrm{\partial }C\mathrm{/}\mathrm{\partial }{P}^j=2g_{jb}{P}^b\mathrm{\ne }0$. By [F2] it is a local diffeomorphism, so $C=0$ is locally a smooth codimension-one coordinate slice. The future timelike sheet is an open component of this level set. The supplied tangent-bundle structure carries [A2].

The differential of $C$ along ${\mathcal{L}}_{F,q}$ is computed using the covariant horizontal derivative and $\mathrm{\nabla }g=0$: $dC({\mathcal{L}}_{F,q})=2g(P,cq{F}^{\mathrm{♯}}(P))=2cq{P}^a{F}_{ab}{P}^b=0.$ The last contraction vanishes because $P^a{P}^b$ is symmetric in $a,b$ while $F_{ab}$ is antisymmetric. Thus ${\mathcal{L}}_{F,q}$ lies in the kernel of $dC$, which is the tangent space to the regular level set from step 1.1; the future sheet is open in that set, so the vector field is tangent there as well. Its restriction is therefore well defined.