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The violation of Lorentz invariance (LI) in gravitational theories, whichallows superluminal propagations, dramatically alters the causal structure ofthe spacetime and modifies the notion of black holes (BHs). Instead of metrichorizons, now universal horizons (UHs) define the boundaries of BHs, withinwhich a particle cannot escape to spatial infinities even with an infinitelylarge speed. Then, a natural question is how the quasi-normal modes (QNMs) of aBH 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-tensortheory that violates LI but yet is self-consistent and satisfies allobservations to date. Technically, this poses several challenges, includingsingularities of the perturbation equations across metric horizons and properidentifications of ingoing modes at UHs. After overcoming these difficulties,we show that the QNMs of the Schwarzschild BH, also a solution ofEinstein-aether theory, consist of two parts, the metric and aether parts. TheQNMs of the metric perturbations are quite similar to those obtained in generalrelativity and are consistent with current observations of gravitational waves.But the ones from aether perturbations are different, and our numerical studiesindicate that they are even not stable. The latter is consistent with ourprevious studies, which showed that the stealth Schwarzschild BH suffers aLaplacian instability along the angular direction. The method and techniquesdeveloped in this paper can be applied to the studies of QNMs in other theoriesof gravity with broken LI.

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