Nonorientable spacetime tunneling

TL;DR

This paper explores a nonorientable spacetime topology modeled as a Klein bottle, revealing how such a structure can form closed timelike curves, exhibit unique lensing effects, and face quantum instabilities that can be mitigated by a time-dependent spatial period.

Contribution

It introduces a novel Klein bottle topology for Misner space, analyzes the formation of closed timelike curves, and investigates quantum stability and potential resolutions.

Findings

Closed timelike curves form in Klein bottle topology.

Exotic matter distribution causes converging lensing at wormhole mouths.

Quantum fluctuations lead to divergences at chronology horizons, but can be mitigated.

Abstract

Misner space is generalized to have the nonorientable topology of a Klein bottle, and it is shown that in a classical spacetime with multiply connected space slices having such a topology, closed timelike curves are formed. Different regions on the Klein bottle surface can be distinguished which are separated by apparent horizons fixed at particular values of the two angular variables that eneter the metric. Around the throat of this tunnel (which we denote a Klein bottlehole), the position of these horizons dictates an ordinary and exotic matter distribution such that, in addition to the known diverging lensing action of wormholes, a converging lensing action is also present at the mouths. Associated with this matter distribution, the accelerating version of this Klein bottlehole shows four distinct chronology horizons, each with its own nonchronal region. A calculation of the quantum vacuum fluctuations performed by using the regularized two-point Hadamard function shows that each chronology horizon nests a set of polarized hypersurfaces where the renormalized momentum-energy tensor diverges. This quantum instability can be prevented if we take the accelerating Klein bottlehole to be a generalization of a modified Misner space in which the period of the closed spatial direction is time-dependent. In this case, the nonchronal regions and closed timelike curves cannot exceed a minimum size of the order the Planck scale.