Geometric Analysis of Particular Compactly Constructed Time Machine Spacetimes

TL;DR

This paper analyzes the geometric structure of specific compact time machine spacetimes, focusing on the pseudo Schwarzschild and pseudo Kerr metrics, revealing their geodesic properties and limitations.

Contribution

It introduces the pseudo Kerr spacetime as a rotating generalization and establishes its time machine structure and global properties.

Findings

Pseudo Schwarzschild spacetime is geodesically incomplete.

Pseudo Kerr spacetime has a well-defined time machine structure.

The analysis highlights limitations in extending these spacetimes to complete ones.

Abstract

We formulate the concept of time machine structure for spacetimes exhibiting a compactely constructed region with closed timelike curves. After reviewing essential properties of the pseudo Schwarzschild spacetime introduced by A. Ori, we present an analysis of its geodesics analogous to the one conducted in the case of the Schwarzschild spacetime. We conclude that the pseudo Schwarzschild spacetime is geodesically incomplete and not extendible to a complete spacetime. We then introduce a rotating generalization of the pseudo Schwarzschild metric, which we call the the pseudo Kerr spacetime. We establish its time machine structure and analyze its global properties.