Gödel-type universes and chronology protection in Hořava–Lifshitz gravity

Abstract

In the attempts toward a quantum gravity theory, general relativity faces a serious difficulty since it is non-renormalizable theory. Hořava–Lifshitz gravity offers a framework to circumvent this difficulty, by sacrificing the local Lorentz invariance at ultra-high energy scales in exchange of power-counting renormalizability. The Lorentz symmetry is expected to be recovered at low and medium energy scales. If gravitation is to be described by a Hořava–Lifshitz gravity theory there are a number of issues that ought to be reexamined in its context, including the question as to whether this gravity incorporates a chronology protection, or particularly if it allows Gödel-type solutions with violation of causality. We show that Hořava–Lifshitz gravity only allows hyperbolic Gödel-type space-times whose essential parameters m and ω are in the chronology respecting intervals, excluding therefore any noncausal Gödel-type space-times in the hyperbolic class. There emerges from our results that the famous noncausal Gödel model is not allowed in Hořava–Lifshitz gravity. The question as to whether this quantum gravity theory permits hyperbolic Gödel-type solutions in the chronology preserving interval of the essential parameters is also examined. We show that Hořava–Lifshitz gravity not only excludes the noncausal Gödel universe, but also rules out any hyperbolic Gödel-type solutions for physically well-motivated perfect-fluid matter content.