The Taming of Closed Time-like Curves

TL;DR

The paper analyzes the time-dependent orbifold $R^{1,d}/\mathbb{Z}_2$ (elliptic de Sitter embedding) to address time-nonorientability and potential closed time-like curves. It advocates a doubled-field, $\mathbb{Z}_2$-invariant QFT consistent with string theory, yielding Minkowski-like backreaction at one loop while revealing an initial-time divergence associated with the orbifold singularity. String calculations support the Minkowski-backreaction picture and motivate the invariant reformulation of QFT, which permits a locally well-defined S-matrix away from the initial slice. Overall, the work provides a framework for quantizing fields on globally time-nonorientable spacetimes and offers insights for cosmological models with elliptic de Sitter geometry.

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.

Figures (11)

• Figure 1: The orbifold $\mathbb{R}^{1,1}/\mathbb{Z}_2$. Also depicted are some identified points and resulting closed timelike curves.
• Figure 2: Three possible time-arrows on the quotient $\mathbb{R}^{1,1}/\mathbb{Z}_2$.
• Figure 3: A view of the quotient spacetime (for 1+1 dimensions). Note the absence of the $x=0$ axis for $t<0$.
• Figure 4: Another view of the quotient spacetime (for 1+1 dimensions). Note the absence of the $t=0$ axis for $x<0$. The $t=0$ axis represents a "big bang" singularity--the beginning of the spacetime.
• Figure 5: The CTC's of Fig. \ref{['fig:newfig1.eps']} are not forward oriented in the quotient.