Dynamical obstruction to perpetual motion from Lorentz-violating black holes

Black holes in Lorentz-violating theories have been claimed to violate the second law of thermodynamics by perpetual motion energy extraction. We revisit this question for a Penrose splitting process in a spherically symmetric setting with two species of particles that move on radial geodesics that extend to infinity. We show that energy extraction by this process cannot happen in any theory in which gravity is attractive, in the sense of a geometric inequality that we describe. This inequality is satisfied by all known Einstein-æther and Hořava black hole solutions.

Figure 1

Spacetime diagram of the static, spherically symmetric and asymptotically flat spacetime corresponding to the effective metric $g_{ab}^B$ of the fastest particle, in the case where $g_{ab}^B$ describes a black hole with a single Killing horizon and a spacelike singularity. The solid brown line marks the horizon for the $A$ particles, which can be thought of as an ergosurface. The solid blue line marks the horizon for the $B$ particles, which can be thought of as the event horizon. The shaded part in between is the ergoregion. The dotted red line marks the singularity. The spacetime outside the $A$-horizon will be referred to as the outside region and the spacetime inside the $B$-horizon as the black hole.

Figure 2

The configuration of the two light cones and the Killing vector ${\chi }^a$ in the ergoregion, in the frame $(u^a,s^a)$.