G\"odel's universe and the chronology protection conjecture

TL;DR

This paper provides a solution demonstrating that G"odel's universe inherently prevents closed timelike curves, supporting Hawking's chronology protection conjecture through geometric and group-theoretic analysis.

Contribution

It introduces a specific geodesic solution in G"odel's universe that proves the absence of closed timelike curves, confirming the conjecture in three-dimensional gravity.

Findings

No closed timelike curves in the solution

Presence of a closed Cauchy-Riemann surface for protection

Equivalent to a cylindrical gravitational wave front

Abstract

We present a solution for the geodesic motion in G\"odel's universe that provides a particular proof of Hawking's chronology protection conjecture in three-dimensional gravity theory. The solution is based upon the fact that the group of the automorphisms of the Heisenberg motion group H1\timesU(1), modulo discrete sub-group Z, act isometrically on the boundary of the hyperbolic three-dimensional manifold. Closed timelike curves do not exist due to the presence of a closed Cauchy-Riemann surface for chronology protection, with two mirror symmetric sets of helicoidal self-similar modules inside. The present solution is isometrically equivalent to a cylindrical gravitational monochromatic wave front.