# Dynamical obstruction to perpetual motion from Lorentz-violating black holes

## TL;DR

This paper demonstrates that in Lorentz-violating black holes, energy extraction via Penrose processes is impossible under attractive gravity, upholding the second law of thermodynamics in these theories.

## Contribution

It establishes a geometric inequality that prevents perpetual motion energy extraction in Lorentz-violating black holes, applicable to Einstein-ether and Hoava solutions.

## Key findings

- Energy extraction cannot occur in attractive gravity scenarios.
- The geometric inequality holds for all known Lorentz-violating black holes.
- Perpetual motion violates the second law in these theories.

## Abstract

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-\ae{}ther and Ho\v{r}ava black hole solutions.