On First Singularities in the Spherically Symmetric Einstein--Maxwell--Scalar Field Equations

Abstract

We investigate the behavior of "first singularities" in the spherically symmetric Einstein--Maxwell--(real) Scalar field system. We prove that the area-radius function must necessarily extend to zero on all first singularities. This improves on previous characterizations of first singularities, which could not exclude the possibility that the area-radius function diverges or that a first singularity is preceded by a region of infinite spacetime volume. Key to this proof are controls over the area radius function obtained through monotonicity, a previously known bound on the scalar field, and a difference in scaling in a term in one of Einstein's equations. The result has several applications. First, it allows us to strengthen the $C^2$ formulation of the strong cosmic censorship conjecture established by Luk and Oh. It also suggests the existence of a Cauchy hypersurface of maximal area in the maximal future globally hyperbolic development of two-ended asymptotically flat initial data, which may be of possible interest in high energy physics.