Vacuum polarization of scalar and spinor fields near closed null geodesics

Hawking has recently proposed a ‘‘chronology protection conjecture,’’ which states that closed timelike curves cannot form in the real Universe. The most likely mechanism for enforcing this conjecture, if it is correct, is a divergent vacuum polarization at the Cauchy horizon (‘‘chronology horizon’’) where closed timelike curves first try to form. Hawking has proved that, if the chronology horizon is compactly generated, then it contains one or more smoothly closed null geodesics. Because it seems likely that all the horizon’s generators emerge from these closed null geodesics, a sufficiently strongly divergent vacuum polarization at the closed null geodesics is likely to destroy the chronology horizon completely and thereby prevent closed timelike curves from forming. In this paper we compute the details of the divergence near the closed null geodesics, in a generic spacetime with a compactly generated chronology horizon—thereby generalizing earlier computations, in special spacetimes, by Hiscock and Konkowski, Kim and Thorne, and Frolov. We carry out the computation for both a conformal scalar field and a two-component spinor field. We show that, for an observer who will pass through a point on the closed null geodesic after a small interval of proper time δt, the leading-order divergence is always proportional to (δt${)}^{\mathrm{-}3}$ and has the same tensorial structure as the stress-energy of a null fluid moving along the closed null geodesic. We also show that, by contrast with flat spacetime, there is in general no cancellation between the divergent vacuum energies of a combination of fields that, in flat spacetime, would be related by supersymmetry: two conformal scalar fields and one two-component spinor field. We discuss the implications of these results for Hawking’s chronology protection conjecture.