The Taming of Closed Time-like Curves

TL;DR

This paper investigates the properties of a time-reversal orbifold related to elliptic de Sitter space, showing that closed causal curves can be eliminated and proposing a reformulation of quantum field theory to handle divergences.

Contribution

It demonstrates the removal of closed causal curves through a proper time function and introduces a reformulated QFT approach to address divergences in this non-orientable spacetime.

Findings

Closed causal curves disappear with a proper time function.

Naive QFT yields divergences, but string theory suggests a zero stress tensor.

A reformulated QFT reduces divergences, except at a potential initial singularity.

Abstract

We consider a $R^{1,d}/Z_2$ orbifold, where $Z_2$ acts by time and space reversal, also known as the embedding space of the elliptic de Sitter space. The background has two potentially dangerous problems: time-nonorientability and the existence of closed time-like curves. We first show that closed causal curves disappear after a proper definition of the time function. We then consider the one-loop vacuum expectation value of the stress tensor. A naive QFT analysis yields a divergent result. We then analyze the stress tensor in bosonic string theory, and find the same result as if the target space would be just the Minkowski space $R^{1,d}$, suggesting a zero result for the superstring. This leads us to propose a proper reformulation of QFT, and recalculate the stress tensor. We find almost the same result as in Minkowski space, except for a potential divergence at the initial time slice of the orbifold, analogous to a spacelike Big Bang singularity. Finally, we argue that it is possible to define local S-matrices, even if the spacetime is globally time-nonorientable.