Monday, May 02, 2011

Constraints from Einstein's equations as Navier-Stokes equations

In January, Andy Strominger and friends posted a very interesting paper
Geometry for every solution of Navier-Stokes (TRF)
Today, Andy Strominger and Vyacheslav Lysov offer their extraordinary concise related preprint
From Petrov-Einstein to Navier-Stokes (hep-th)
In the paper, they start with a flat p+1-dimensional hypersurface that has 1 temporal direction and p spatial directions. However, it may be embedded into a p+2-dimensional curved spacetime.

Strominger and Lysov impose the Petrov type I condition for this p+2-dimensional spacetime - something that has been studied in GR for purely relativistic reasons, and something that uses Newman-Penrose vector fields, among other things - and they find out that it reduces the degrees of freedom near the hypersurface to the hydrodynamical degrees of freedom.

Now, they look at Einstein's equations near this p+1-dimensional hypersurface. Normally, we evolve the geometry in time. Whenever we're doing so with some partial differential equations with local symmetries, we find out that some of the equations are "constraints" which means that they restrict possible initial states - but they don't tell us anything else about the evolution because they don't contain time derivatives.

Maxwell's equations contain the equation "div D = rho" which is a typical constraint equation. It may be linked to the electromagnetic U(1) gauge invariance of the states in the quantized version of the theory of electromagnetism; after all, you get it from the variation of the action with respect to A_0. A similar constraint part exists for Einstein's equations - variations with respect to g_{0,mu}, roughly speaking - and it restricts the possible exterior curvature on the "t=const" slice.

However, Lysov and Strominger look at the constraint equations in a different way. Instead of studying a purely spacelike hypersurface - a "t=const" slice - they study a (flat) hypersurface of the type "x=const" that has a time-like direction in it. It's still possible to separate Einstein's equations to the "dynamical" ones (in space, in this case) and the "constraint" part.

And the two physicists find out that in the limit in which the extrinsic curvature goes to infinity, the constraint part of Einstein's equations reduces to the Navier-Stokes equations! So this is a pretty natural holographic embedding of hydrodynamics in general relativity, indeed.


Of course, general relativity expects us to extrapolate the geometry to long distances away from the p+1-dimensional hypersurface. This is a procedure that people studying hydrodynamics would probably not do but it is a pretty natural one, and assuming it can be done, it should tell us something about the Navier-Stokes equations - and perhaps even lots of things about turbulence.

This picture is still "unusual" in the realm of stringy dualities because both sides of the "duality" are completely classical. I still don't know how to think about it. Is it really analogous to the usual string dualities where one side always assumes the quantum effects of the other side to be huge? And if it is not a duality in this sense, shouldn't it mean that it's vacuous in some sense?


  1. Lubos,

    In regards to QGP, is there relevance then to Stromingers papers?


  2. Dear Plato, I don't know of any way to connect these insights to QGP - it really seems as a different approach to me but maybe I am wrong. Thanks, no one. LM

  3. I think calling the resulting equations of the paper Navier-Stokes equations is confusing. The equations they get are mathematically similar to the non-relativistic, incompressible Navier-Stokes equations in the limit Re -> 1. However their definitions of velocity and pressure in terms of the extrinsic curvature seem to be quite artifical. How they define viscosity in this context is also unclear to me, since they are starting from GR, which is a non-dissipative field theory.
    It seem the similarity of NS- equations to the equations they found in their paper is just a mathematical curiosity. The underlying physics has no connection to real fluids at all. I think it would be better to call their result simply a nonlinear, partial differential equation of second order.