From Fluid Dynamics to Gravity and Back

There is an interesting connection between two of the best-studied nonlinear partial differential equations in physics: the equations of hydrodynamics and the field equations of gravity.

Let’s start with a brief review of hydrodynamics. At the microscopic level a tank of water is a collection of, say, 10²⁵ molecules that constantly collide with one another. The methods of physics may be used to model this collection of water molecules as follows: we set up equations that track the position and momentum of each of the water molecules and predict their time evolution. These conceptually complete equations have of order 10²⁵ variables and so are clearly too difficult to handle in practice.

Does it then follow that tanks of water cannot be usefully studied using the methods of physics? As every plumber knows, this conclusion is false: a useful description of water is obtained by keeping track of average properties of water molecules, rather than each individual molecule.

Think of a tank of water as a union of non-overlapping lumps of water. Each lump is big enough to contain a large number of molecules but small enough so that gross macroscopic properties of the water (energy density, number density, momentum density) are approximately uniform. The fundamental assumption of hydrodynamics is that under appropriate conditions, all the “average” properties of any lump are completely determined by its conserved charge densities (in the case of water, molecule number density, energy density, and momentum density). In particular, the conserved current for molecule number j^µ and the conserved current for energy and momentum T^µν are themselves dynamically determined functionals of local thermodynamical densities in a locally equilibrated system (fluctuations away from these dynamically determined values are suppressed by a factor proportional to the square root of the number of molecules in each lump). The equations that express conserved currents as functionals of conserved densities are difficult to compute theoretically but are easily measured experimentally and are known as constitutive relations.

When supplemented with constitutive relations, the conservation equations ∂_µ j^µ =0, and ∂_µ T^µν=0(2) turn into a well-posed initial value problem for the dynamic of conserved densities. They are the equations of hydrodynamics. Let me reemphasize that the effect of the ignored degrees on the evolution of conserved densities is inversely proportional to the square root of the number of molecules in a lump, and so is negligible in an appropriate thermodynamic limit, allowing the formulation of a closed dynamical system for conserved densities.

My research concerns how the equations of hydrodynamics pop up in an apparently completely unrelated setting: in the study of the long wavelength dynamics of black holes governed by Einstein’s equations with a negative cosmological constant.

Einstein’s gravitational equations describe the dynamics of the geometry of spacetime. The ripples of spacetime (gravitational waves) have interesting dynamics even in the absence of any matter. For most of this article, I will be referring to Einstein’s equations in the absence of matter.

The simplest solution of the most familiar Einstein equation G_µν=0 is simply flat Minkowskian spacetime. However, the usual Einstein equations can be deformed to admit the so-called cosmological constant term G_µν=λg_µν. This deformation, which was first suggested and later rejected by Einstein himself, appears to be needed to model the cosmological expansion of our universe. The observed accelerated expansion of our universe is plausibly explained by the existence of a positive cosmological constant (the equation above with a positive value of λ).

Recent theoretical investigations within string theory have focused attention on Einstein’s equations with a negative cosmological constant (negative value of λ). This equation does not have flat space as a solution. Its simplest solution is a highly symmetric spacetime called anti-de Sitter (AdS) space. In this article, I will explore asymptotically AdS solutions of Einstein’s equations with a negative cosmological constant in five spacetime dimensions.
Einstein’s equations, with or without a gravitational constant, admit a huge variety of black hole solutions. The equations with a negative cosmological constant also admit rather unusual related solutions called black brane. These solutions have finite energy and momentum density rather than a finite energy and momentum. Stationary black brane solutions are analytically well known, and appear in a four-parameter set, labeled by a uniform energy and momentum density.

It is often the case in physics that if a uniform distribution of “something” (a field, the orientation of a spin, etc.) leads to a time independent solution of the equations of motion, then slowly varying configurations of that thing result in slow dynamics. It turns out that this general expectation applies to the distribution of energy and momentum density on black branes. The exact four-parameter set of time-­independent black brane solutions may be generalized to an infinite number of approximate solutions of Einstein’s equations. These solutions are characterized by varying (rather than uniform) energy and momentum density fields. The fields are functions of spatial position as well as time, but are constrained to obey dynamical equations. It has been demonstrated that these equations take the form of conservation equations (conservation of the stress tensor in the case of the vacuum Einstein equations), with all components of the stress tensor determined to be a particular functional of the local energy density by an effective constitutive relation.¹ In other words, the long wavelength fluctuations of black branes are governed by the equations of hydrodynamics, with gravitationally determined constitutive relations. This fact is the so-called fluid-gravity correspondence.

The fluid-gravity correspondence was established constructively; there is an explicit construction of an approximate solution to Einstein’s equations dual to any fluid flow. The construction of these solutions proceeds in an expansion in derivatives. The procedure that determines the solutions to Einstein’s equations also simultaneously determines the constitutive relations to the given order in derivatives.

It turns out that the gravitational solutions that participate in the fluid-gravity correspondence all have regular event horizons; moreover, Hawking’s area theorem, which states that event horizons can only stay the same or increase but never decrease, may be used to determine a positive divergence entropy current for fluid flows. This establishes that the fluid flows generated by gravity obey a basic physical requirement; even locally, entropy in such flows never decreases.

Why does the equation that describes the average motion of water molecules in a water tank also govern the ripples of the event horizon of asymptotically AdS black brane? At least in some circumstances, we believe this is because Einstein’s equations in the presence of black branes actually describe the averaged dynamics of a large number of underlying microscopic variables. In particular, the AdS/conformal field theory correspondence of string theory proposes that the uniform black brane of negative cosmological constant gravity is dual, in a particular context, to a gas of gluons of a large U(N) gauge theory at large N. The hydrodynamical solutions of fluid gravity are presumably the duals to the fluid flows of this collection of gluons. Fluctuations about these hydrodynamical flows are suppressed by ¹⁄_N and may be thought of as quantum gravity fluctuations from the gravitational point of view.

At the conceptual level, the fluid-gravity correspondence suggests a novel view of the role of Einstein’s equations in the presence of event horizons. At a more practical level, this correspondence has had a completely unanticipated application in the reverse direction: to the theory of the equations of relativistic hydrodynamics, a subject that was thought to have been closed in the 1930s. Analyses by Lev Landau and Evgeny Lifshitz in the 1930s claimed to have determined the most general form of the constitutive relations of relativistic fluid at first order in the derivative expansion. Motivated by the fluid-gravity correspondence, it has been discovered that the Landau-Lifshitz constitutive relations must be generalized in the case of certain parity violating charged fluids.² In particular, if the fluid charge has a U(1)³ triangle anomaly, then there are new terms in the constitutive relations of this fluid––roughly in terms of the fluid current proportional to the vorticity––that are completely determined by the anomaly coefficient plus the thermodynamics of the fluid. This discovery may turn out to have experimental consequences (in the study of fluid flows in Brookhaven National Laboratory’s Relativistic Heavy Ion Collider experiment, for example, which aims to study the first few moments after the universe’s creation), surely a surprising application for the esoteric study of black hole physics in higher-dimensional gravity.³