How the Movement of Water Molecules Corresponds to Ripples in Spacetime
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 jµ and the conserved current for energy and momentum Tµν 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 jµ =0, and ∂µ Tµν=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.