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, exhibit a structure very similar to the nonrelativistic Navier-Stokes equation. I show that this result arises quite naturally when gravitational dynamics is viewed as an emergent phenomenon. Extremizing 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 equation. This is in contrast to the usual description of the Damour-Navier-Stokes equation in a general coordinate system, in which there appears a Lie derivative rather than a 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 extremization principle that leads to the Damour-Navier-Stokes equation, which makes the hydrodynamical analogy with gravity completely natural and obvious. Several implications of these results are discussed.