Dispersion relation and surface gravity of universal horizons

In Einstein-aether theory, violating Lorentz invariance permits some super-luminal communications, and the universal horizon can trap excitations traveling at arbitrarily high velocities. To better understand the nature of these universal horizons, we first modify the ray tracing method, and then use it to study their surface gravity in charged Einstein-aether black hole spacetime. Instead of the previous result by Cropp et al., our results show that the surface gravity of the universal horizon is dependent on the specific dispersion relation, kUH = 2(z - 1) k(uh)/z, where z denotes the power of the leading term in the superluminal dispersion relation, characterizing different species of particles. And the associated Hawking temperatures also are different with z. These findings, which coincide with those derived by the tunneling method, provide some full understanding of black hole thermodynamics in Lorentz-violating theories.