| Summary: | Condensed matter theory is the study of systems at finite density. In this thesis we will attempt to argue that gauge-gravity dualities can give deep and meaningful insights into the behaviour of strongly coupled condensed matter systems. The first three chapters will be a review of material already available in literature. Chapter 1 will introduce holography and the AdS-CFT correspondence. Particularly, in this chapter, the technique for the extraction of diffusion constants for charge and shear stress-energy-momentum fluctuations in a field theory with a holographic dual will be demonstrated. Chapter 2 will summarise relevant literature on the relativistic fluid-gravity correspondence. In the first half of the chapter it will be shown how to calculate the transport coefficients and Navier-Stokes equations for a suitable thermal field theory. The second half of chapter 2 will then be dedicated to extracting the transport coefficients for a strongly coupled field theory dual to a Reissner-Nordstrøm AdS spacetime. In chapter 3 a scaling of the metric and gauge field found in chapter 2 will be taken such that the boundary field theory admits Galilean, as opposed to relativistic, symmetry. Consequently, the governing hydrodynamic equations will be the non-relativistic, incompressible Navier-Stokes. Chapters 4 and 5 represent novel work. In chapter 4 the transport coefficients for a particular strongly coupled thermal field theory with underlying Schrodinger symmetry will be extracted from a charged, asymptotically Schrodinger spacetime. The governing hydrodynamic equations will be compressible with non-relativistic symmetry as opposed to those found via the scaling limit of chapter 3. In chapter 5 we show how knowledge of the transport coefficients of a thermal field theory can be used as a test-bed for numerical methods to explore beyond the hydrodynamic (long wavelength and low frequency) regime. With this in mind we consider Reissner-Nordstrøm AdS4 and determine the two point correlators at arbitrary frequency and momentum. Finally in chapter 6 we summarise the work discussed in this thesis and speculate about further applications of hydrodynamic techniques to strongly coupled condensed matter theories.
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