Time Separation and Scattering Rigidity for Analytic Lorentzian Manifolds

TL;DR

The paper proves three analytic-rigidity results for Lorentzian manifolds with timelike boundary: (i) the exterior time separation data determines an unknown compact region up to analytic isometry when no lightlike cut points occur; (ii) the boundary time separation data determines the whole manifold up to analytic isometry under a non-cut-point and non-trapping hypothesis near the boundary; and (iii) the interior and complete scattering data near the light cone determine the manifold up to analytic isometry under non-trapping lightlike geodesics, without requiring causality. The approach combines reconstructing lightlike travel-time data from exterior data, constructing a global bijection that preserves travel times, and extending to an analytic isometry via boundary-jet determination and collar extensions. The work also clarifies how interior and complete scattering information relate in both Riemannian and Lorentzian settings and provides boundary-determination results and lens–scattering connections. These results advance rigidity theory in the analytic Lorentzian setting, offering constructive procedures and broad applicability beyond strictly convex boundaries or globally causal spacetimes.

Abstract

In this work, we prove the following three rigidity results: (i) in a real-analytic globally hyperbolic spacetime $(M,g)$ without boundary, the time separation function restricted to a thin exterior layer of a unknown compact subset $K \subset M$ determines $K$ up to an analytic isometry, assuming no lightlike cut points in $K$; (ii) in a real-analytic globally hyperbolic spacetime $(M,g)$ with timelike boundary, the boundary time separation function determines $M$ up to an analytic isometry, assuming no lightlike cut points near $M$ and lightlike geodesics are non-trapping; (iii) in a real-analytic Lorentzian manifold $(M,g)$ with timelike boundary, the interior and complete scattering relations near the light cone, each determines $M$ up to an analytic isometry, assuming that lightlike geodesics are non-trapping. We emphasize in all of these three cases we do not assume the convexity of the boundary of the subset or the manifold. Moreover, in (iii) we do not assume causality of the Lorentzian manifold, and allow the existence of cut points. Along the way, we also prove some boundary determination results, the connections between the interior and complete scattering relations, and the connections between the lens data and the scattering relation, for Riemannian manifolds and Lorentzian manifolds with boundaries.

Figures (7)

• Figure 1: For $(x, v) \in \partial_-TM$, denote the corresponding geodesic by $\gamma$, then $(y, w)$ is the point and direction at which $\gamma$ leaves $M^\circ$ for the first time, or equivalently its first time reaching $\partial M$; and $(z, u)$ is where the geodesic fully leaves $M$. The scattering relation $(x,v) \to (z, u)$ and $(y, w) \to (z, u)$ will be included in the complete scattering relation, but interior scattering relation will only record $(x, v) \to (y, w)$. The $(y, w) \to (z, u)$ part will not be recorded in the interior scattering relation as $(y, w)$ is tangential to the boundary.
• Figure 2: The first step goes from $t_1$ to $t_2'$, but there are two cases based on whether $\gamma(s_1)$ is in $K$ (left graph) or $K^c$ (right graph). The second step goes to $t_3'$, which can be shown is strictly larger than $s_1$ in both cases. Since $s_1 - t_1$ has a uniform lower bound, this shows for every two steps we gain a uniform lower bound for the recovery.
• Figure 3: Locally consider Fermi coordinates around a fixed geodesic, and construct the analytic Riemannian metric $g^R$ there. Around $z$ and $\varphi_0(z)$, the metrics agree (preserved by $\varphi_0$).
• Figure 4: The inductive step: we choose $x_{j+1}'$ sufficiently close to $x_{j+1}$ such that it is still contained in the convex neighborhood $U_{j+1}$. There exists a unique timelike geodesic segment from $x_{j+1}'$ to $x_{j+1}$.
• Figure 5: In the left picture, if a causal curve intersects with the boundary infinitely many times, then it can be replaced by a piecewise geodesic curve via a finite cover with geodesically convex neighborhoods. By analyticity, the piecewise geodesic curve switches between $M$ and $M^c$ only finitely many times. In the right picture, for each timelike curve $\beta_1 \circ \cdots \circ \beta_N$, we keep the same $\beta_j$ as $\gamma_j$ if it is in the exterior region, and find a $\gamma_j$ that is at least almost the same length as $\beta_j$ if it is in the interior. This is possible since the boundary time separation functions agree.

Theorems & Definitions (90)

• Theorem 1.1
• Theorem 1.2
• Definition 1.1
• Theorem 1.3
• Theorem 1.4
• Definition 2.1
• Lemma 2.1
• proof
• Definition 2.2: BEE96
• Definition 2.3: BEE96