Causality violation without time-travel: closed lightlike paths in G\"{o}del's universe

TL;DR

This paper demonstrates that causality violations in G"odel's universe can occur through sequences of lightlike paths, not just closed timelike curves, suggesting a new way to violate causality without large accelerations.

Contribution

It introduces the concept of causality violation via relay-like sequences of null geodesics in G"odel's universe, expanding understanding of causal structures.

Findings

Existence of lightlike paths connecting any two events in G"odel's universe.

Minimum of eight geodesic segments needed for such lightlike paths.

Causality can be violated using light signals and mirrors, without large accelerations.

Abstract

We revisit the issue of causality violations in G\"{o}del's universe, restricting to geodesic motions. It is well-known that while there are closed timelike curves in this spacetime, there are no closed causal geodesics. We show further that no observer can communicate directly (i.e.\ using a single causal geodesic) with their own past. However, we show that this type of causality violation can be achieved by a system of relays: we prove that from any event $P$ in G\"{o}del's universe, there is a future-directed lightlike path - a sequence of future-directed null geodesic segments, laid end to end - which has $P$ as its past and future endpoints. By analysing the envelope of the family of future directed null geodesics emanating from a point of the spacetime, we show that this lightlike path must contain a minimum of eight geodesic segments, and show further that this bound is attained. We prove a related general result, that events of a time orientable spacetime are connected by a (closed) timelike curve if and only if they are connected by a (closed) lightlike path. This suggests a means of violating causality in G\"{o}del's universe without the need for unfeasibly large accelerations, using instead a sequence of light signals reflected by a suitably located system of mirrors.