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

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.