On the thermodynamics of universal horizons in Einstein-{\AE}ther theory

The theories of gravity which violate local Lorentz invariance do not admit a universal maximum speed of signal-propagation. Different field excitations see a different effective metric and hence a different light cone. In these theories, although one can define the Killing horizon in a conventional way, this definition does not capture the notion of a black hole. This is so because there exist modes which see a wider light cone than the one defined by the Killing Horizon and therefore can escape to infinity. However, there exist solutions of these theories which admit a special spacelike hypersurface which acts as a one-way membrane. Signals from beyond this hypersurface can never escape to infinity and are destined to hit the singularity. In this sense this hypersurface acts like a black-hole horizon and is called the Universal Horizon because it traps modes travelling with arbitrarily high velocities. We use the Noether charge method {\it \`a la} Wald to show that a first law, which resembles the first law of thermodynamics, can be formulated for universal horizons in the Einstein-{\AE}ther theory. This seems to suggest that in Lorentz violating theories one should ascribe the thermodynamical properties to the universal horizon and not to the Killing horizon.