# Comparing the complexity of Einstein's field equations with the Navier-Stokes equations

I know litte about the Navier-Stokes equations, but it seems to me that questions related to Navier-Stokes are more fundamental (e.g. do solutions exist for a set of initial conditions) than the ones that are usually aked for the equations of gravity.

Also, a lot of non-linear solutions for general relativity are known and classified according to their symmetries (Stephani, et al.: Exact Solutions of Einstein's Field Equations). Do similar classifications exist for known solutions of the Navier-Stokes equations?

Is it possible to give arguments, why one set of equations is more difficult than the other, if this is actually the case?

• To get any reasonable answers for this question, you need to provide a definition of what "more difficult" is actually supposed to mean. What, specifically, makes one equation more difficult than another? – probably_someone Oct 16 at 13:30
• Interesting question. A couple of differences: (1) the Einstein field equations generically produce singularities from typical initial conditions (i.e., in some sense this has probability 1); (2) in the EFE, you don't have an underlying spacetime manifold on which this happens -- the manifold only comes about once you've already solved the equations. – Ben Crowell Oct 16 at 13:58

An interesting perspective on this question could be offered by works on fluid/gravity correspondence.

For example, see this paper:

Abstract

We show by explicit construction that for every solution of the incompressible Navier-Stokes equation in $$p+1$$ dimensions, there is a uniquely associated “dual” solution of the vacuum Einstein equations in $$p+2$$ dimensions. The dual geometry has an intrinsically flat timelike boundary segment $$\Sigma_c$$ whose extrinsic curvature is given by the stress tensor of the Navier-Stokes fluid. We consider a “near-horizon” limit in which $$\Sigma_c$$ becomes highly accelerated. The near-horizon expansion in gravity is shown to be mathematically equivalent to the hydrodynamic expansion in fluid dynamics, and the Einstein equation reduces to the incompressible Navier-Stokes equation. For $$p=2$$, we show that the full dual geometry is algebraically special Petrov type II. The construction is a mathematically precise realization of suggestions of a holographic duality relating fluids and horizons which began with the membrane paradigm in the 70's and resurfaced recently in studies of the AdS/CFT correspondence.

This allows one to establish a “dictionary” relating problems concerning Navier–Stokes equations with problems posed for Einstein equations in a higher dimensional spacetime. As the paper suggests:

… cosmic censorship could be related to global existence for Navier-Stokes or the scale separation characterizing turbulent flows related to radial separation in a spacetime geometry …

On the other hand, not every question that could be asked about Einstein equations is about the near-horizon expansion, so for some rather loose interpretation, the “complexity” of Einstein's field equations (in higher dimensions) is strictly greater than that of Navier–Stokes equations.

• Thank you for the nice reference. This will take me a step further. I'm tempted to say that Einstein's field equations are "more complex", if geometric perturbations on a particular background exhibit dynamics described by Navier-Stokes equations. I somehow expected something like this, knowing that the Einstein equations of higher dimensional compactifications can incorporate even classical Yang-Mills field equations. – p6majo Oct 17 at 13:06