Universal horizons and black hole spectroscopy in gravitational theories with broken Lorentz symmetry

The violation of Lorentz invariance (LI) in gravitational theories, which allows superluminal propagations, dramatically alters the causal structure of the spacetime and modifies the notion of black holes (BHs). Instead of metric horizons, now universal horizons (UHs) define the boundaries of BHs, within which a particle cannot escape to spatial infinities even with an infinitely large speed. Then, a natural question is how the quasi-normal modes (QNMs) of a BH are modified, if one considers the UH as its causal boundary. In this paper, we study in detail this problem in Einstein-Aether theory, a vector-tensor theory that violates LI but yet is self-consistent and satisfies all observations to date. Technically, this poses several challenges, including singularities of the perturbation equations across metric horizons and proper identifications of ingoing modes at UHs. After overcoming these difficulties, we show that the QNMs of the Schwarzschild BH, also a solution of Einstein-Aether theory, consist of two parts, the metric and aether parts. The QNMs of the metric perturbations are quite similar to those obtained in general relativity and are consistent with current observations of gravitational waves. But the ones from aether perturbations are different, and our numerical studies indicate that they are even not stable. The latter is consistent with our previous studies, which showed that the stealth Schwarzschild BH suffers a Laplacian instability along the angular direction. The method and techniques developed in this paper can be applied to the studies of QNMs in other theories of gravity with broken LI.