Areas and Volumes for Null Cones

Abstract

Abstract. Motivated by recent work of Chruściel,, we prove monotonicity properties and comparison results for the area of slices of the null cone of a point in a Lorentzian manifold. We also prove volume comparison results for subsets of the null cone analogous to the Bishop-Gromov relative volume monotonicity theorem and Günther's volume comparison theorem. We briefly discuss how these estimates may be used to control the null second fundamental form of slices of the null cone in Ricci-flat Lorentzian four-manifolds with null curvature bounded above.

Vacuum Fluctuations and the Small Scale Structure of Spacetime

“…(2) If the stress-energy tensor is bounded below, the expansion in four dimensions has bounds exactly analogous to those of (8), even taking the same functional form [19].…”

The Hawking-Penrose singularity theorem for $C^{1,1}$-Lorentzian metrics

“…Consequently, γ ε k converges in C 1 to the (future) inextendible g-geodesic γ w : [− T , b 0 ) → M with γ w (− T ) = γ(− T ) and γw (− T ) = w. Since our spacetime is non-totally imprisoning (which follows from global hyperbolicity by the same proof as for smooth metrics, [24, Lem. 14.13]), this geodesic must leave the compact set J − (γ( T )) ∩ J + (γ(− T )), hence b 0 > b 9 Recall that a timelike geodesic is globally maximising if it maximises between any two of its points. and in particular b = ∞ and γ w (b) = γ( T ).…”

“…Now, one can look at the level sets S t := exp p (t Ũ ), where Ũ := {v ∈ U : v null, g(T, v) = g( γ(0), T )} for some fixed timelike vector T ∈ T p M , and their shape operators S γ(t) (t) : T γ(t) S t → T γ(t) S t derived from the normal γ(t). Proceeding as in [9,Prop. 3.4] one gets that this shape operator satisfies a Riccati equation along γ and lim tց0 t S γ(t) (t) = id.…”