G\"odel-type universes and chronology protection in Horava-Lifshitz gravity

In the attempts toward a quantum gravity theory, general relativity faces a serious difficulty since it is non-renormalizable theory. Ho\v{r}ava-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 Ho\v{r}ava-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 G\"odel-type solutions with violation of causality. We show that Ho\v{r}ava-Lifshitz gravity only allows hyperbolic G\"odel-type space-times whose essential parameters $m$ and $\omega$ are in the chronology respecting intervals, excluding therefore any noncausal G\"odel-type space-times in the hyperbolic class. There emerges from our results that the famous noncausal G\"odel model is not allowed in Ho\v{r}ava-Lifshitz gravity. The question as to whether this quantum gravity theory permits hyperbolic G\"odel-type solutions in the chronology preserving interval of the essential parameters is also examined. We show that Ho\v{r}ava-Lifshitz gravity not only excludes the noncausal G\"odel universe, but also rules out any hyperbolic G\"odel-type solutions for physically well-motivated perfect-fluid matter content.