# The Einstein and the Navier-Stokes Equations: Connecting the Two

 Title: The Einstein and the Navier-Stokes Equations: Connecting the Two Author: Bredberg, Irene Citation: Bredberg, Irene. 2012. The Einstein and the Navier-Stokes Equations: Connecting the Two. Doctoral dissertation, Harvard University. Access Status: Full text of the requested work is not available in DASH at this time (“dark deposit”). For more information on dark deposits, see our FAQ. Full Text & Related Files: Bredberg_gsas.harvard_0084L_10214.pdf (4.652Mb; PDF) Abstract: This thesis establishes a precise mathematical connection between the Einstein equations of general relativity and the incompressible Navier-Stokes equation of fluid dynamics. We carry out a holographic analysis which relates solutions to the Einstein equations to the behaviour of a dual fluid living in one fewer dimensions. Gravitational systems are found to exhibit Navier-Stokes behaviour when we study the dynamics of the region near an event horizon. Thus, we find non-linear deformations of Einstein solutions which, after taking a suitable near horizon limit and imposing our particular choice of boundary conditions, turn out to be precisely characterised by solutions to the incompressible Navier-Stokes equation. In other words, for any solution to the Navier-Stokes equation, the set-up we present provides a solution to the Einstein equations near a horizon. We consider the cases of fluids flowing on the plane and on the sphere. Fluid dynamics on the plane is analysed foremost in the context of a flat background geometry whilst the spherical analysis is undertaken for Schwarzschild black holes and the static patch of four-dimensional de Sitter space. Citable link to this page: http://nrs.harvard.edu/urn-3:HUL.InstRepos:10330319 Downloads of this work:

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