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Hydrodynamics is nowadays understood as an effective field theory thatdescribes the dynamics of the long-wavelength and slow-time fluctuations of anunderlying microscopic theory. In this work we extend the relativistichydrodynamics to third order in the gradient expansion for neutral fluids in ageneral curved spacetime of $d$ dimensions. We find 58 new transportcoefficients, 19 due to third-order scalar corrections and 39 due to tensorialcorrections. In the particular case of a conformal fluid, the number of newtransport coefficients is reduced to 19, all of them due to third-ordertensorial corrections. The dispersion relations of linear fluctuations in thethird-order relativistic hydrodynamics is obtained, both in the rest frame ofthe fluid and in a general moving frame. As an application we obtain some ofthe transport coefficients of a relativistic conformal fluid in threedimensions by using the AdS/CFT correspondence. These transport coefficientsare extracted from the dispersion relations of the linear fluctuations. Thegravity dual of the fluctuations in this conformal fluid is described by thegravitational perturbations of four-dimensional anti-de Sitter black branes,which are solutions of the Einstein equations with a negative cosmologicalconstant. To find the hydrodynamic quasinormal modes (QNMs) of the scalarsector we use the SUSY quantum mechanics of the gravitational perturbations offour-dimensional black branes. Such a symmetry allows us to find thewavefunction of the scalar (or sound) sector in the hydrodynamic limit directlyfrom the wavefunction of the vector (or shear) sector, which is usually easierto be found because the perturbation wave equations for the vector sector aremuch simpler than the ones for the scalar sector.

Third-order relativistic hydrodynamics: dispersion relations and transport coefficients of a dual plasma