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 with 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 stress-tensor 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.