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

TL;DR

This paper demonstrates that the stress-energy tensor for automorphic fields in Misner space inevitably becomes singular at the chronology horizon, confirming a general theorem and extending the understanding of quantum field behavior in such spacetimes.

Contribution

It proves that the stress-energy tensor must be singular on the Misner space horizon for automorphic fields, confirming a general theorem and extending previous results.

Findings

Stress-energy tensor is necessarily singular on the Misner space horizon.

Singularity persists for all automorphic parameters and states.

Similar singularity results hold for Gott and Grant spaces.

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 $\langle\alpha|T_{ab}|\alpha\rangle$ in the {\it Sushkov state} ``$\langle\alpha|\cdot|\alpha\rangle$'' (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 $(x,x') \in (N\cap IGH)\times (N\cap IGH)$ such that the renormalized two-point function of an automorphic field $G^\alpha_{\rm ren}(x,x')$ in the Sushkov state is singular. In consequence $\langle\alpha|T_{ab}|\alpha\rangle$ (as well as other renormalized expectation values such as $\langle\alpha|\phi^2|\alpha\rangle$) 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.