The fluid–gravity correspondence establishes a detailed connection between solutions of relativistic dissipative hydrodynamics and black hole spacetimes that solve Einstein’s equations in a spacetime with negative cosmological constant. The correspondence can be seen as a natural corollary of the holographic anti–de Sitter (AdS)/conformal field theory (CFT) correspondence, which arises from string theory. The latter posits a quantum duality between gravitational dynamics in AdS spacetimes and that of a CFT in one dimension less. The fluid–gravity correspondence applies in the statistical thermodynamic limit of the CFT but can be viewed as an independent statement of a relation between two classic equations of physics: the relativistic Navier–Stokes equations and Einstein’s equations. The general structure of relativistic fluid dynamics is formulated in terms of conservation equations of energy–momentum and charges, supplemented with constitutive relations for the corresponding current densities. One can view this construction as an effective field theory for these conserved currents. This intuition applied to the gravitational equations of motion allows the solutions of relativistic hydrodynamics to be embedded as inhomogeneous, dynamical black holes in AdS spacetime.
