# Singularity theorems The singularity theorem is stated as follows: Let $(M,g)$ be globally hyperbolic with a *non-compact* Cauchy surface $\Sigma$. Assume that the *Einstein equation* and *[[0480 Null energy condition|NEC]]* are satisfied and that $M$ contains a trapped surface $T$. Let $\theta_0<0$ be the maximum value of $\theta$ on $T$ for both sets of null geodesics orthogonal to $T$. Then at least one of these geodesics is future-inextendible and has affine length no greater than $2/\theta_0$. \[From Reall's lecture notes on Black Holes.\] ## Proof 1. [[0480 Null energy condition|NEC]] & trapped surface => $d\theta/d\lambda\le -\frac{1}{2}\theta^2$ (using [[0408 Raychaudhuri equation|Raychaudhuri equation]]) & $\theta_0<0$ => reach $\theta=-\infty$ (conjugate point) within $2/|\theta_0|$ 2. $\dot{J}^+(T)$ compact: - need to show it is bounded and closed - closed: automatic because it is the boundary of topological space - bounded: $\dot{J}^+(T)$ cannot have a conjugate point to $T$ because having a conjugate point implies we can deform smoothly to timelike curves connecting $p$ and $T$, meaning $p$ would be in the interior of ${J}^+(T)$ not boundary 3. contradiction: construct a homeomorphism between $\dot{J}^+(T)$ and a Cauchy surface $\Sigma$, but $\Sigma$ is non-compact but $\dot{J}^+(T)$ is compact! 1. global hyperbolicity => one-to-one map from $\dot{J}^+(T)$ to $\Sigma$ 2. former is closed => latter (the image) is closed 3. former is a 3-manifold => latter is open 4. latter both open and closed (and connected) => the image is the whole set $\Sigma$ => bijective map constructed ## Refs - original: [[1965#Penrose]] - [[1970#Hawking, Penrose]]: relaxes globally hyperbolic condition - Bousso bound [[1999#Bousso (May)]] - [[2010#Wall (Oct)]] - removes [[0480 Null energy condition|NEC]] from assumptions but requires [[0082 Generalised second law|GSL]] - smeared NEC (for finite regions) - [[FreivogelKontouKrommydas2020]] - Bousso's new singularity theorem - [[2022#Bousso, Shahbazi-Moghaddam (Jan)]] and [[2022#Bousso, Shahbazi-Moghaddam (Jun)]] - [[Rsc0036 Banff Gravitational Emergence in AdS-CFT]] - focusing on general null surfaces - [[2023#Ciambelli, Freidel, Leigh]] - rigidity and MOTS - [[2024#Galloway, Ling]] ## Related - [[0408 Raychaudhuri equation]] - [[0243 Quantum focusing conjecture]]