The Penrose Inequality and the Fluid/Gravity Correspondence

TL;DR

This paper explores a fluid dynamics analogue of the Penrose inequality from general relativity, analyzing its validity in relativistic and nonrelativistic flows and implications for singularity formation.

Contribution

It derives a new Penrose-like inequality for fluid flows and investigates its validity across different fluid regimes and viscous orders.

Findings

Inequality always holds at ideal fluid order.

Relativistic viscous flows can violate the inequality.

Nonrelativistic incompressible flows always satisfy the inequality.

Abstract

Motivated by the fluid/gravity correspondence, we consider the Penrose inequality in the framework of fluid dynamics. In general relativity, the Penrose inequality relates the mass and the entropy associated with a gravitational background. If the inequality is violated by some Cauchy data, it suggests a creation of a naked singularity, thus providing means to consider the cosmic censorship hypothesis. The analogous inequality in the context of fluid dynamics can provide a valuable tool in the study of finite-time blowups in hydrodynamics. We derive the inequality for relativistic and nonrelativistic fluid flows in general dimension. We show that the inequality is always satisfied at the ideal fluid order. At the leading viscous order, the inequality may be violated by relativistic fluid flows, while it is always satisfied by nonrelativistic incompressible flows. The inequality may be violated at the next to leading viscous order by both relativistic and nonrelativistic flows.