  We compare the initial value formulation of the low-energy limit of
(non-projectable) Horava gravity to that of Einstein-aether theory when the
aether 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 non-perturbative level: in Horava
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-aether theory. We discuss its
physical significance in Horava gravity. We also focus on spherical symmetry
and we revisit existing collapse simulations in Einstein-aether 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 Horava gravity.
