Wormholes and naked singularities in the complete Hořava theory

Abstract

We find the static spherically symmetric solutions (with vanishing shift function) of the complete nonprojectable Ho\ifmmode \check{r}\else \v{r}\fi{}ava theory explicitly, writing the space-time metrics as explicit tensors in local coordinate systems. This completes previous work of other authors who have studied the same configurations. The solutions depend on the coupling constant $\ensuremath{\alpha}$ of the $({\ensuremath{\partial}}_{i}\mathrm{ln}N{)}^{2}$ term. The $\ensuremath{\lambda}=1/3$ case of the theory does not possess any extra mode; hence, the range of $\ensuremath{\alpha}$ is in principle not limited by the linear stability of any extra mode. We study the full range of $\ensuremath{\alpha}$, both in the positive and negative sectors. We find the same wormhole solutions and naked singularities that were found for the Einstein-aether theory in a sector of the space of $\ensuremath{\alpha}$. There also arise wormholes in other sectors of $\ensuremath{\alpha}$. Our coordinate systems are valid at the throats of the wormholes. We also find the perturbative solutions for small $\ensuremath{\alpha}$. We give this version of the solutions directly on the original radial coordinate $r$, which is particularly suitable for representing the exterior region of solutions with localized sources.