Finslerian lightconvex boundaries: applications to causal simplicity and the space of cone geodesics $\mathcal{N}$

Abstract: Our outcome is structured in the following sequence: (1) a general result for indefinite Finslerian manifolds with boundary $(M,L)$ showing the equivalence between local and infinitesimal (time, light or space) convexities for the boundary $\partial M$, (2) for any cone structure $(M,\mathcal{C})$ which is globally hyperbolic with timelike boundary, the equivalence among: (a) the boundary $\partial M$ is lightconvex, (b) the interior $\mathring{M}$ is causally simple and (c) the space of the cone (null) geodesics $\mathcal{N}$ of $(\mathring{M},\mathcal{C})$ is Hausdorff, (3) in this case, the manifold structure of $\mathcal{N}$ is obtained explicitly in terms of elements in $\partial M$ and a smooth Cauchy hypersurface $S$, (4) the known results and examples about Hausdorfness of $\mathcal{N}$ are revisited and extended, leading to the notion of {\em causally simple spacetime with $T_2$-lightspace} as a step in the causal ladder below global hyperbolicity. The results are significant for relativistic (Lorentz) spacetimes and the writing allows one either to be introduced in Finslerian technicalities or to skip them. In particular, asymptotically AdS spacetimes become examples where the $C^{1,1}$ conformal extensions at infinity yield totally lightgeodesic boundaries, and all the results above apply.