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. These results remain true in the presence of additional matter obeying both the strong and weak energy conditions.
Highlights
The conditions under which cosmological models either originate or terminate in a singularity provided an active subject of research in the decades prior to the breakthroughsDedicated to the memory of S.W
There, we first establish a Hawking-type singularity theorem using methods taken from [10] and apply it to the Einstein–Klein–Gordon theory with or without additional matter using our worldline bounds. This provides an analogue to the Penrose-type singularity theorem for the nonminimally coupled scalar field discussed in [10]
General properties of globally hyperbolic spacetimes with compact Cauchy surfaces guarantee the existence of a future-directed S-ray γ —that is, γ is a unit-speed geodesic, issuing orthogonally from S, so that the Lorentzian distance from each γ (τ ) to S is precisely τ, for all τ ∈ [0, ∞)—γ is complete by assumption
Summary
5, we return to worldline bounds, adapted to the special case of solutions to the Einstein–Klein–Gordon system and obtaining a slightly refined bound which holds in the presence of additional matter provided that it obeys the weak and the strong energy condition This bound is used in our discussion of singularity theorems in Sect. There, we first establish a Hawking-type singularity theorem using methods taken from [10] and apply it to the Einstein–Klein–Gordon theory with or without additional matter using our worldline bounds This provides an analogue to the Penrose-type singularity theorem for the nonminimally coupled scalar field discussed in [10].
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
