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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.
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