  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.
