The formation of trapped surfaces in the gravitational collapse of spherically symmetric scalar fields with a positive cosmological constant

João L Costa

doi:10.1088/1361-6382/abb075

Abstract

Given spherically symmetric characteristic initial data for the Einstein-scalar field system with a positive cosmological constant, we provide a criterion, in terms of the dimensionless size and dimensionless renormalized mass content of an annular region of the data, for the formation of a future trapped surface. This corresponds to an extension of Christodoulou’s classical criterion by the inclusion of the cosmological term.

Highlights

The existence of a trapped surface - a compact codimension 2 spacelike submanifold with negative inward and outward null expansions - has profound implication for the global structure of spacetime: most notably, under quite general causal assumptions and fairly weak energy conditions, it guarantees the existence of a non-empty black hole region [11, 12] and, by Penrose’s Singularity/Incompleteness Theorem, it implies future causal geodesic incompleteness
In [7], Christodoulou established a criterion for the dynamical formation of trapped spheres in the context of a characteristic initial value problem for the spherically symmetric Einstein-scalar field system; this was an essential step in his seminal proof of both the Weak and Strong Cosmic censorship conjectures, under appropriate regularity conditions for such system, that was completed in [9]
In celebrated work [10], Christodoulou extended his criterion to the context of the Einstein vacuum equations without symmetry assumptions; in the mean time, important developments have been achieved in that setting [1, 2, 25, 26]

Summary

Introduction

Introduction and main resultThe existence of a (future) trapped surface - a compact codimension 2 spacelike submanifold with negative inward and outward (future) null expansions - has profound implication for the global structure of spacetime: most notably, under quite general causal assumptions and fairly weak energy conditions, it guarantees the existence of a non-empty black hole region [11, 12] and, by Penrose’s Singularity/Incompleteness Theorem, it implies future causal geodesic incompleteness.In [7], Christodoulou established a criterion for the dynamical formation of trapped spheres in the context of a characteristic initial value problem for the spherically symmetric Einstein-scalar field system; this was an essential step in his seminal proof of both the Weak and Strong Cosmic censorship conjectures, under appropriate regularity conditions for such system, that was completed in [9]. In this paper we start this study by revisiting Christodoulou’s original criterion and extending it to the context of solutions to the Einstein-scalar field system with a positive cosmological constant. 4. Condition (iii) above has no parallel in the Λ = 0 case and is related to the fact that, under general conditions [17, Section 3], an apparent cosmological horizon - a causal hypersurface which is the union of marginally anti-trapped spheres, i.e., where ∂ur = 0 and ∂vr > 0 - must intersect the initial cone C0+ in the region 1/Λ < r ≤ 3/Λ.

Results

Conclusion