Lorentz violation and perpetual motion

We show that any Lorentz-violating theory with two or more propagation speeds is in conflict with the generalized second law of black hole thermodynamics. We do this by identifying a classical energy-extraction method, analogous to the Penrose process, which would decrease the black hole entropy. Although the usual definitions of black hole entropy are ambiguous in this context, we require only very mild assumptions about its dependence on the mass. This extends the result found by Dubovsky and Sibiryakov, which uses the Hawking effect and applies only if the fields with different propagation speeds interact just through gravity. We also point out instabilities that could interfere with their black hole perpetuum mobile, but argue that these can be neglected if the black hole mass is sufficiently large.

Figure 1

The radial 4-momentum covector space at the point $x$ in the ergoregion ($p_a{m}^a=0$ for all vectors $m^a$ tangent to the symmetry sphere through $x$). The momenta with vanishing Killing energy $\mathcal{E}=p_a{\xi }^a$ lie along the dashed line. $\mathcal{E}$ is negative for $p_A$, since $x$ lies in the $A$ ergoregion, while it is positive for all momenta on the $B$ null cone. The total 4-momentum $P$ has positive $\mathcal{E}$, hence it points above the dashed line.