Stress-Energy Must be Singular on the Misner Space Horizon even for Automorphic Fields

Abstract

We use the image sum method to reproduce Sushkov's result that for a massless automorphic field on the initial globally hyperbolic region $IGH$ of Misner space, one can always find a special value of the automorphic parameter $\alpha$ such that the renormalized expectation value $⟨\alpha |T_{ab}|\alpha ⟩$ in the {\it Sushkov state} ``$⟨\alpha |\cdot |\alpha ⟩$'' (i.e. the automorphic generalization of the Hiscock-Konkowski state) vanishes. However, we shall prove by elementary methods that the conclusions of a recent general theorem of Kay-Radzikowski-Wald apply in this case. I.e. for any value of $\alpha$ and any neighbourhood $N$ of any point $b$ on the chronology horizon there exists at least one pair of non-null related points $(x,x^{\prime })\in (N\cap IGH)\times (N\cap IGH)$ such that the renormalized two-point function of an automorphic field $G_{\mathrm{ren}}^{\alpha }(x,x^{\prime })$ in the Sushkov state is singular. In consequence $⟨\alpha |T_{ab}|\alpha ⟩$ (as well as other renormalized expectation values such as $⟨\alpha |{\varphi }^2|\alpha ⟩$) is necessarily singular {\it on} the chronology horizon. We point out that a similar situation (i.e. singularity {\it on} the chronology horizon) holds for states on Gott space and Grant space.

10 pages, LaTeX, 2 postscript figures