Entropy density of spacetime and the Navier-Stokes fluid dynamics of null surfaces

Abstract

It has been known for several decades that Einstein's field equations, when projected onto a null surface, exhibits a structure very similar to non-relativistic Navier-Stokes equation. I show that this result arises quite naturally when gravitational dynamics is viewed as an emergent phenomenon. Extremising the spacetime entropy density associated with the null surfaces leads to a set of equations which, when viewed in the local inertial frame, becomes identical to the Navier-Stokes (NS) equation. This is in contrast to the usual description of Damour-Navier-Stokes (DNS) equation in a general coordinate system, in which there appears a Lie derivative rather than convective derivative. I discuss this difference, its importance and why it is more appropriate to view the equation in a local inertial frame. The viscous force on fluid, arising from the gradient of the viscous stress-tensor, involves the second derivatives of the metric and does not vanish in the local inertial frame while the viscous stresstensor itself vanishes so that inertial observers detect no dissipation. We thus provide an entropy extremisation principle that leads to the DNS equation, which makes the hydrodynamical analogy with gravity completely natural and obvious. Several implications of these results are discussed.

The holographic fluid dual to vacuum Einstein gravity

“…In a recent paper [6] (see [7][8][9][10][11][12][13][14] for earlier relevant work), a remarkable relation between incompressible non-relativistic fluids in (d + 1) dimensions satisfying the Navier-Stokes equation and (d + 2)-dimensional Ricci flat metrics was found. The construction of [6] starts from considering the portion of Minkowski spacetime between a flat hypersurface Σ c , given by the equation X 2 − T 2 = 4r c , and its future horizon H + , the null surface X = T .…”

“…Here, we are restricting to flat space, 11 and we have defined the relativistic shear and vorticity according to…”

Emergent gravity paradigm: Recent progress

“…This allows us interpret h g as the (fictitious [23] but useful) viscous dissipation rate of the null surface which is being minimized in the extremum principle. Another equivalent expression [3] for the thermodynamic extremum principle, obtained by ignoring another total divergence which does not contribute to the variation, can be based on the following integral over the null surface:…”

Phase transition and scaling behavior of topological charged black holes in Hořava–Lifshitz gravity

“…One specific implementation of this idea considers the field equations of the theory to be 'emergent' in a well-defined sense, rather than use that term in a more speculative vein like e.g., considering the space and time themselves to be emergent etc. The evidence for such a specific interpretation comes from different facts like the possibility of interpreting the field equation in a wide class of theories as thermodynamic relations [5], the nature of action functional in gravitational theories and their thermodynamic interpretation [6], the possibility of obtaining the field equations from a thermodynamic extremum principle [7], application of equipartition ideas to obtain the density of microscopic degrees of freedom [8], the equivalence of Einstein's field equations to the NavierStokes equations near a null surface [9] etc. The important fact in the emergent paradigm is that one does not need to know about the details of the microscopic description of the theory.…”