Outer trapped surfaces are dense near MOTSs

Piotr T Chruściel,Gregory J Galloway

doi:10.1088/0264-9381/31/4/045013

Piotr T Chruściel, Gregory J Galloway

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https://doi.org/10.1088/0264-9381/31/4/045013

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Abstract

We show that any vacuum initial data set containing a marginally outer trapped surface S and satisfying a ‘no KIDs’ condition can be perturbed near S so that S becomes strictly outer trapped in the new vacuum initial data set. This, together with the results in Eichmair et al (2012), gives a precise sense in which generic initial data containing marginally outer trapped surfaces lead to geodesically incomplete spacetimes.

Highlights

In [8] results were obtained concerning the topology of three-dimensional asymptotically flat initial data sets (S, g, K)
An immersed marginally outer trapped surface (MOTS) is the image under a finite covering map p : S → S of a MOTS Sin the initial data set (S, g, K ), where gand Kare the pullbacks via p of g and K, respectively
The condition appears in the classical Hawking–Penrose singularity theorem [10, section 8.2, theorem 2], where it is required to hold along all inextendible causal geodesics

Summary

Introduction

In [8] results were obtained concerning the topology of three-dimensional asymptotically flat initial data sets (S , g, K). Penrose singularity theorem [10, section 8.2, theorem 2], where it is required to hold along all inextendible causal geodesics While this condition may seem physically reasonable, mathematically it is rather unsatisfactory, especially from an initial data point of view. As should be clear from the proof of proposition 1.1, theorem 1.2 (together with a covering argument from the proof of [8, theorem 3.2] when the MOTS is non-separating) implies the following: let (S , g, K) be a vacuum initial data set, with S noncompact, and suppose S is a MOTS in (S , g, K), without local KIDs near S. There exists an arbitrarily small smooth local perturbation of the initial data to a new vacuum initial data set (S , g , K ) whose maximal globally hyperbolic development is null geodesically incomplete. When S is an interior submanifold, a subsequent deformation-and-smoothing argument in the other direction provides the desired initial data

Definitions

Conformal deformations

Integral identities and uniqueness