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 C2 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.