It has previously been shown that the Einstein equation can be derived from the requirement that the Clausius relation dS = δQ/T hold for all local acceleration horizons through each spacetime point, where dS is one quarter the horizon area change in Planck units, and δQ and T are the energy flux across the horizon and Unruh temperature seen by an accelerating observer just inside the horizon. Here we show that a curvature correction to the entropy that is polynomial in the Ricci scalar requires a non-equilibrium treatment. The corresponding field equation is derived from the entropy balance relation dS = δQ/T + diS, where diS is a bulk viscosity entropy production term that we determine by imposing energy-momentum conservation. Entropy production can also be included in pure Einstein theory by allowing for shear viscosity of the horizon.

We study black hole solutions in general relativity coupled to a unit timelike vector field dubbed the "aether". To be causally isolated a black hole interior must trap matter fields as well as all aether and metric modes. The theory possesses spin-0, spin-1, and spin-2 modes whose speeds depend on four coupling coefficients. We find that the full three-parameter family of local spherically symmetric static solutions is always regular at a metric horizon, but only a two-parameter subset is regular at a spin-0 horizon. Asymptotic flatness imposes another condition, leaving a one-parameter family of regular black holes. These solutions are compared to the Schwarzschild solution using numerical integration for a special class of coupling coefficients. They are very close to Schwarzschild outside the horizon for a wide range of couplings, and have a spacelike singularity inside, but differ inside quantitatively. Some quantities constructed from the metric and aether oscillate in the interior as the singularity is approached. The aether is at rest at spatial infinity and flows into the black hole, but differs significantly from the the 4-velocity of freely-falling geodesics.

We review the status of "Einstein-AEther theory", a generally covariant theory of gravity coupled to a dynamical, unit timelike vector field that breaks local Lorentz symmetry. Aspects of waves, stars, black holes, and cosmology are discussed, together with theoretical and observational constraints. Open questions are stressed. * Based on a talk given by T. Jacobson at the Deserfest.

The time independent spherically symmetric solutions of General Relativity (GR) coupled to a dynamical unit timelike vector are studied. We find there is a three-parameter family of solutions with this symmetry. Imposing asymptotic flatness restricts to two parameters, and requiring that the aether be aligned with the timelike Killing field further restricts to one parameter, the total mass. These "static aether" solutions are given analytically up to solution of a transcendental equation. The positive mass solutions have spatial geometry with a minimal area 2-sphere, inside of which the area diverges at a curvature singularity occurring at an extremal Killing horizon that lies at a finite affine parameter along a radial null geodesic. Regular perfect fluid star solutions are shown to exist with static aether exteriors, and the range of stability for constant density stars is identified.

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.

A generally covariant extension of general relativity (GR) in which a dynamical unit timelike vector field is coupled to the metric is studied in the asymptotic weak field limit of spherically symmetric static solutions. The two post-Newtonian parameters known as the Eddington-Robertson-Schiff parameters are found to be identical to those in the case of pure GR, except for some non-generic values of the coefficients in the Lagrangian.

We study gravitational collapse of a spherically symmetric scalar field in Einstein-aether theory (general relativity coupled to a dynamical unit timelike vector field). The initial value formulation is developed, and numerical simulations are performed. The collapse produces regular, stationary black holes, as long as the aether coupling constants are not too large. For larger couplings a finite area singularity occurs. These results are shown to be consistent with the stationary solutions found previously.

We consider the dynamics of a d + 1 space-time dimensional membrane defined by the event horizon of a black brane in (d + 2)-dimensional asymptotically Anti-de-Sitter space-time and show that it is described by the d−dimensional incompressible Navier-Stokes equations of non-relativistic fluids. The fluid velocity corresponds to the normal to the horizon while the rate of change in the fluid energy is equal to minus the rate of change in the horizon cross-sectional area. The analysis is performed in the Membrane Paradigm approach to black holes and it holds for a general non-singular null hypersurface, provided a large scale hydrodynamic limit exists. Thus we find, for instance, that the dynamics of the Rindler acceleration horizon is also described by the incompressible Navier-Stokes equations. The result resembles the relation between the Burgers and KPZ equations and we discuss its implications.The dynamics of fluids has remained an unsolved problem for centuries. Unlocking its main features, chaos and turbulence, is likely to provide an understanding of the principles and non-linear dynamics of a large class of systems far from equilibrium. The fundamental formulation of the problem of the nonlinear dynamics of fluids is given by the incompressible Navier-Stokes (NS) equations [1, 2]whereis the fluid pressure divided by the density, ν is the (kinematic) viscosity and f i (x, t) are the components of an externally applied force. The equations can be studied mathematically in any space dimensionality d, with two and three space dimensions having an experimental realization. Among the central questions posed by the NS equations is the existence of singularities in the solutions and the statistics of the solutions in the limit of small ν, both with and without forcing.The scarce progress achieved in understanding the NS equations prompts one to look for other frameworks of viewing the dynamics that might offer new insights. In this letter we explore a new viewpoint that can lead to a development of geometrical methods to study the NS equations. We represent the whole spatio-temporal picture of the velocity field in terms of a structure of a null hypersurface (whose normal vector is also a tangent vector), embedded in a bulk space-time and evolving according to the General Relativity field equations. Thus, the dynamics of the NS equations is related to the dynamics of the geometry, as described by the relativistic Einstein equations.A particular case of a geometric representation of the NS equations has been emerging recently based on the AdS/CFT correspondence relating conformal field theories (CFTs) in a flat (d + 1)-dimensional space-time to gravity (string) theory on asymptotically (d + 2)dimensional Anti de Sitter (AdS) space-time [3] (for a review see [4]). Here the spatio-temporal pattern of fluid motion is mapped onto a black brane solution (a black hole with planar topology) in an asymptotically AdS space-time. The map is based on the realization that at large spatial and temporal scales the CFT dynamics r...