A singularity theorem for Einstein-Klein-Gordon theory

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.

arXiv comment19 pages

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• The Return of the Singularities: Applications of the Smeared Null Energy Condition
• A new derivation of singularity theorems with weakened energy hypotheses
• Evaporation of four-dimensional dynamical black holes sourced by the quantum trace anomaly
• Singularity theorems in the effective field theory for quantum gravity at second order in curvature
• Conformal and kinetic couplings as two Jordan frames of the same theory
• How Much NEC Breaking Can the Universe Endure?
• Hawking-type singularity theorems for worldvolume energy inequalities
• Wormhole restrictions from quantum energy inequalities