Causality, Singularities, and the Topology of MOTS

Abstract

This Master's Thesis explores the global causal structure of Lorentzian manifolds, utilizing techniques from differential geometry and geometric analysis to study spacetime singularities and apparent horizons. First, we establish the foundational topological properties of causal curves and the critical role of global hyperbolicity. Through the kinematics of timelike congruences and the Raychaudhuri equation, we formally present Hawking's Singularity Theorem. This result proves that, under the strong energy condition, timelike geodesics generically focalize and collapse into conjugate points, rendering the spacetime geodesically incomplete. Subsequently, the framework is adapted to the degenerate regime of null geometry to study the causal boundaries defined by trapped surfaces. This work highlights a profound geometric dichotomy: while the non-degenerate timelike geometry inevitably succumbs to pathological singularities, the degenerate null geometry framing these singularities exhibits remarkable topological rigidity. By analyzing the spectral properties of the stability operator associated with Marginally Outer Trapped Surfaces (MOTS), we prove the Galloway-Schoen theorem. In particular for the four-dimensional spacetime satisfying the dominant energy condition, the topology of a compact apparent horizon is strictly constrained to the sphere $S^2$, or the flat torus $S^1 \times S^1$ in the strictly Ricci-flat limit.