The fluid gravity correspondence

Abstract:   In this set of four lectures, I aim to describe the interesting connection between solutions of a certain class of generalised Navier-Stokes equations (generalised by allowing higher derivatives of velocity) and a class of solutions of Einstein equations describing negatively curved spacetimes. Given my background, the talk would be from a physicist viewpoint but as an effort to address applied mathematicians.

 In the first lecture, I will begin by reviewing relativistic Navier-Stokes equations with higher derivative corrections and with a special property that the only scale in the equations is given by the fluid temperature. I will also review the Einstein equations describing negatively curved spacetimes (AdS spacetimes in physicists parlance) and their exact solutions which get mapped to rigidly rotating fluid configurations.

 In the second lecture, I will describe the simultaneous constctsruction order by order in derivative expansion of the following: 

a) A particular relativistic Navier-Stokes equation with higher derivative corrections

b) A map between its solutions and solutions of Einstein equations with negative curvature

I will comment on constructing an entropy principle for these relativistic fluid equations using the geometry of the corresponding Einstein solutions. I plan to end describing a check on this map by examining the known exact solutions.

In the third lecture, I will describe the limit of slow fluid motion and recover the non-relativistic fluid equations and how the fluid-gravity map behaves under this limit. I will end by describing various preliminary numerical studies which have used this map to argue that negatively curved Einstein equations should exhibit turbulence akin to the familiar fluid turbulence.

 In the fourth lecture, I will review the further developments and future questions that might be of interest to applied mathematicians.