Spherically symmetric static black holes in Einstein-aether theory

In this paper, we systematically study spherically symmetric static spacetimes in the framework of Einstein-aether theory and pay particular attention to the existence of black holes (BHs). In the theory, two additional gravitational modes (one scalar and one vector) appear, due to the presence of a timelike aether field. To avoid the vacuum gravi-\ifmmode \check{C}\else \v{C}\fi{}erenkov radiation, they must all propagate with speeds greater than or at least equal to the speed of light. In the spherical case, only the scalar mode is relevant, so BH horizons are defined by this mode, which are always inside or at most coincide with the metric (Killing) horizons. In the present studies, we first clarify several subtle issues. In particular, we find that, out of the five nontrivial field equations, only three are independent, so the problem is well posed, as now generically there are only three unknown functions, $F(r)$, $B(r)$, $A(r)$, where $F$ and $B$ are metric coefficients, and $A$ describes the aether field. In addition, the two second-order differential equations for $A$ and $F$ are independent of $B$, and once they are found, $B$ is given simply by an algebraic expression of $F$, $A$ and their derivatives. To simplify the problem further, we explore the symmetry of field redefinitions, and work first with the redefined metric and aether field, and then obtain the physical ones by the inverse transformations. These clarifications significantly simplify the computational labor, which is important, as the problem is highly involved mathematically. In fact, it is exactly because of these, we find various numerical BH solutions with an accuracy that is at least two orders higher than previous ones. More important, these BH solutions are the only ones that satisfy the self-consistent conditions and meantime are consistent with all the observational constraints obtained so far. The locations of universal horizons are also identified, together with several other observationally interesting quantities, such as the innermost stable circular orbits (ISCO), the ISCO frequency, and the maximum redshift ${z}_{\mathrm{max}}$ of a photon emitted by a source orbiting the ISCO. All of these quantities are found to be quite close to their relativistic limits.

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