From Petrov-Einstein to Navier-Stokes in Spatially Curved Spacetime

TL;DR

This paper extends the fluid/gravity correspondence to spatially curved spacetimes by deriving the Navier-Stokes equations from Petrov type I conditions on hypersurfaces with intrinsic curvature, using the Brown-York stress tensor.

Contribution

It generalizes previous work by incorporating intrinsic curvature of the hypersurface and derives curved-space Navier-Stokes equations from gravitational conditions.

Findings

Navier-Stokes equations derived in curved spacetime

Petrov type I condition reduces to fluid dynamics

Framework applicable to non-flat hypersurfaces

Abstract

We generalize the framework in arXiv:1104.5502 to the case that an embedding may have a nonvanishing intrinsic curvature. Directly employing the Brown-York stress tensor as the fundamental variables, we study the effect of finite perturbations of the extrinsic curvature while keeping the intrinsic metric fixed. We show that imposing a Petrov type I condition on the hypersurface geometry may reduce to the incompressible Navier-Stokes equation for a fluid moving in spatially curved spacetime in the near-horizon limit.