A singularity theorem for Einstein-Klein-Gordon theory

TL;DR

This paper extends Hawking's singularity theorem to Einstein-Klein-Gordon models by deriving bounds on the effective energy density, demonstrating that solutions with initial contraction are geodesically incomplete.

Contribution

It introduces new lower bounds on local averages of the effective energy density for Klein-Gordon fields, enabling a Hawking-type singularity theorem beyond the strong energy condition.

Findings

Lower bounds on local averages of EED for Klein-Gordon solutions

Proven singularity theorem for Einstein-Klein-Gordon models

Applicability to various coupling constants including minimal and conformal

Abstract

Hawking's singularity theorem concerns matter obeying the strong energy condition (SEC), which means that all observers experience a nonnegative effective energy density (EED), thereby guaranteeing the timelike convergence property. However, there are models that do not satisfy the SEC and therefore lie outside the scope of Hawking's hypotheses, an important example being the massive Klein-Gordon field. Here we derive lower bounds on local averages of the EED for solutions to the Klein-Gordon equation, allowing nonzero mass and nonminimal coupling to the scalar curvature. The averages are taken along timelike geodesics or over spacetime volumes, and our bounds are valid for a range of coupling constants including both minimal and conformal coupling. Using methods developed by Fewster and Galloway, these lower bounds are applied to prove a Hawking-type singularity theorem for solutions to the Einstein-Klein-Gordon theory, asserting that solutions with sufficient initial contraction at a compact Cauchy surface will be future timelike geodesically incomplete.