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

TL;DR

The paper develops a comprehensive Lorentz–Finsler framework for manifolds with timelike boundary, establishing that boundary lightconvexity, interior causal simplicity, and Hausdorffness of the lightspace $ N$ are equivalent in globally hyperbolic cone structures. It provides an explicit global model for $ N$ using a spacelike Cauchy hypersurface $S$ and boundary data, and shows how these results extend to AdS and asymptotically AdS spacetimes through conformal extensions that render the boundary totally lightgeodesic. A new causal ladder step, causally simple spacetimes with $T_2$-lightspace, is proposed and illustrated with both proper examples and counterexamples, highlighting boundary regularity and limit behaviors. The results fuse convexity notions, anisotropic connections, and the geometry of cone geodesics to illuminate causal structure in spacetimes with boundary, with concrete applications to AdS-like geometries and potential extensions beyond global hyperbolicity.

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.

Figures (5)

• Figure 1: The dotted circle represents the boundary of two identified disks in the upper and lower surfaces. The depicted vector is tangent to both surfaces, so that it can be approximated by vectors tangent to both the upper and the lower surface. Along the circle, the glued surface is either non-Hausdorff (if the gluing only affects the open disks but excludes the corresponding circles) or not a manifold (if the two circles are also glued).
• Figure 2: The case of ${M}= I\times [-1,1]$, $I\subset \mathbb{R}$ an interval, $0\in I$ (a subset of $\mathbb{L}^2$). On the left, $M$ and some points of $\mathcal{N}$ (depicted as unfilled points). Here all the depicted curves are lightlike geodesics intersecting either ${S}= \{0\} \times [-1,1]$ or $\partial M= I \times \left\{\pm 1 \right\}$. The dotted lines represent $\pm \mathcal{C}_{}$ at each point. Observe that $[q]^+\in \ell^+_{\partial S }$ and $[r]^-\in \ell^-_{\partial S }$ represent a class of (unparametrized) lightlike geodesics starting from $q$ and ending at $r$ respectively. In this case, one has only one class for each point in $\partial M\setminus {S}$. Each $p\in S$ is crossed by two lightlike geodesics $\gamma_{p,i}$, $i=1,2$. Then, $[p]^{+}_i\in \ell^+_{S}$ and $[p]^-_i\in \ell^-_S$ are $\sim$ related, according to \ref{['eq:8']}. Hence, we have two points in $\mathcal{N}$ (represented at the right figure by $[p]^{+}_{1}$ and $[p]^{+}_{ 2 }$). Finally, for a point $s\in \partial S$ we also have two points $[s]^-$ and $[s]^+$ in $\mathcal{N}$, each one belonging to $\ell^+_{\partial M}\cap \ell^+_S$ and $\ell_{\partial M}^- \cap \ell^-_{S}$ respectively. On the right, a representation of the space $\mathcal{N}$ (which has two connected components), including the points at the left figure.
• Figure 3: In the left-hand figure, a single geodesic $\gamma$ determines a point in $\ell^+_{\partial M}$ (which can be also seen as a point in $\ell^-_{\partial S}$ for red Cauchy $S$), one in $\ell^{-}_{\partial M}$ (which can be also seen as a point in $\ell^+_{\partial S}$ for blue Cauchy $S$) and one in $\ell^{\pm}_S/\sim$ for black Cauchy $S$. Accordingly, these points are represented by red and blue vectors (oriented directions), respectively. Also depicted are the subsets of $\mathcal{N}$ corresponding to the pre-geodesics that end at $p$ (shown as a red arc), start at $r$ (a blue arc), and pass through $q$ close to $\gamma$ (represented by two arcs blue and red, whose points in opposite directions with respect $q$ should be identified). On the right-hand side, the horizontal rectangle represents the cylindrical boundary of the figure on the left cutting it by de vertical line at $p$. Above it, a pictorial representation of the subset of $\mathcal{N}$ projecting onto the vertical gray line through $r$. The blue arc from the left figure is also included in this representation. This arc has been rotated, allowing it to be visualized as a fibered space over the gray line (but all the blue arc projects on $r$).
• Figure 4: A pictorial representation of the Riemannian manifold $(S,g)$ described at the beginning of § \ref{['simpleflatlorentzian']}. The identification of the arrowed radii yields a convex incomplete conic surface. The vertical segments $\sigma_k$ are geodesics converging to both vertical limits $\sigma_-$ and $\sigma_+$. The product $\mathbb{L}\times S$ is causally simple with no $T_2$ lightspace.
• Figure 5: On the left, the convex Riemannian manifold $(S,g_{S})$. The rectangles $R_k$ (in gray) are removed. Some of the geodesics $\sigma_k$ plus $\sigma_{\pm}$ are depicted for visualization. On the right, the corresponding Lorentzian product $M=\mathbb{L}\times S$ (the vertical products on $R_k$ are not depicted for a better visualization). The geodesics $\sigma_k$ are lifted into lightlike pregeodesics $\gamma_k$ converging to both, $\gamma_{\pm}$ (also depicted in the figure).

Theorems & Definitions (86)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Corollary 1.4
• Definition 2.1
• Remark 2.2
• Remark 2.3
• Definition 2.4
• Remark 2.5
• Definition 2.6