Evolution and spherical collapse in Einstein-Æther theory and Hořava gravity

Abstract

We compare the initial value formulation of the low-energy limit of (nonprojectable) Ho\ifmmode \check{r}\else \v{r}\fi{}ava gravity to that of Einstein-\ae{}ther theory when the \ae{}ther is assumed to be hypersurface orthogonal at the level of the field equations. This comparison clearly highlights a crucial difference in the causal structure of the two theories at the nonperturbative level: in Ho\ifmmode \check{r}\else \v{r}\fi{}ava gravity evolution equations include an elliptic equation that is not a constraint relating initial data but needs to be imposed on each slice of the foliation. This feature is absent in Einstein-\ae{}ther theory. We discuss its physical significance in Ho\ifmmode \check{r}\else \v{r}\fi{}ava gravity. We also focus on spherical symmetry, and we revisit existing collapse simulations in Einstein-\ae{}ther theory. We argue that they have likely already uncovered the dynamical formation of a universal horizon and that they can act as evidence that this horizon is indeed a Cauchy horizon in Ho\ifmmode \check{r}\else \v{r}\fi{}ava gravity.