Area Theorem and Smoothness of Compact Cauchy Horizons

Abstract

We obtain an improved version of the area theorem for not necessarily differentiable horizons which, in conjunction with a recent result on the completeness of generators, allows us to prove that under the null energy condition every compactly generated Cauchy horizon is smooth and compact. We explore the consequences of this result for time machines, topology change, black holes and cosmic censorship. For instance, it is shown that compact Cauchy horizons cannot form in a non-empty spacetime which satisfies the stable dominant energy condition wherever there is some source content.

The boundary of the chronology violating set

“…Proof. The proof coincides with that of [24,Theor. 18] where this time the completeness of the future hypersurface must be assumed while there it was a consequence of it being a Cauchy horizon.…”

“…18]. We stress once again [24] that physically speaking it is incorrect to demand the validity of the null genericity condition on a compact set as done by some authors [31], thus its violation does not imply that the spacetime is unphysical.…”

“…Do the same on the slice t = 1 but let the new ellipse be rotated of π/2 radians with respect to the former. A point belonging to both The differentiability of topological hypersurfaces generated by past inextendible lightlike geodesics has been studied in [2][3][4]24]. This analysis was carried out having in mind Cauchy horizons but, as it is clarified with [3, Theor.…”

“…If it is know that if B f = H − (S) for some partial Cauchy surface S (e.g. see next section) then the condition on the geodesic completeness of B f can be dropped, for in this case one can use directly [24,Theor. 18].…”

The vacuum conservation theorem

“…This result will allow us to show that under the weakened stable dominant energy condition compact Cauchy horizons cannot form [10], a result which seems significative for a final resolution of the weak/strong cosmic censorship conjecture.…”