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

Abstract

Hydrodynamics is nowadays understood as an effective field theory that describes the dynamics of the long-wavelength and slow-time fluctuations of an underlying microscopic theory. In this work we extend the relativistic hydrodynamics to third order in the gradient expansion for neutral fluids in a general curved spacetime of d dimensions. We find 58 new transport coefficients, 19 due to third-order scalar corrections and 39 due to tensorial corrections. In the particular case of a conformal fluid, the number of new transport coefficients is reduced to 19, all of them due to third-order tensorial corrections. The dispersion relations of linear fluctuations in the third-order relativistic hydrodynamics is obtained, both in the rest frame of the fluid and in a general moving frame. As an application we obtain some of the transport coefficients of a relativistic conformal fluid in three dimensions by using the AdS/CFT correspondence. These transport coefficients are extracted from the dispersion relations of the linear fluctuations. The gravity dual of the fluctuations in this conformal fluid is described by the gravitational perturbations of fourdimensional anti-de Sitter black branes, which are solutions of the Einstein equations with a negative cosmological constant. To find the hydrodynamic quasinormal modes (QNMs) of the scalar sector we use the SUSY quantum mechanics of the gravitational perturbations of four-dimensional black branes. Such a symmetry allows us to find the wavefunction of the scalar (or sound) sector in the hydrodynamic limit directly from the wavefunction of the vector (or shear) sector, which is usually easier to be found because the perturbation wave equations for the vector sector are much simpler than the ones for the scalar sector.

Effective field theory of stochastic diffusion from gravity

“…33 For now we simply note the fact that there are corrections with simple coefficients predicted above. We note in passing that the dispersion relation has been used to obtain certain third order transport coefficients in [62,76], but we find this puzzling in light of the aforementioned mixing with fourth order transport data. 34 Stress tensor correlators.…”

“…The proof of these statements will be given in sections 8 and 9 and appendices B and C. The reader familiar with the study of linearized gravity in terms of the diffeomorphism invariant combinations starting from the early work of Regge-Wheeler-Vishweshwara-Zerelli [1,[56][57][58] and the more recent analysis of Kodama-Ishibashi [29,30] will find our parameterization natural. These have been adopted in the AdS/CFT literature in the course of the study of quasinormal modes and linearized hydrodynamics [12,[59][60][61][62]. We will outline the connection with these works when we explain the derivation of the designer system.…”

“…It should be possible to resum this expression in terms of polylogs (see example, [62] for results in d = 3), but we will settle for quoting the numeric values in special cases.…”

“…The quartic term was also computed in [5] for N = 4 SYM, while [62] obtained the analogous expression for d = 3 (ABJM plasma). The dispersions accurate to quartic order…”

“…33 We thank Temple He and Julio Virrueta for useful discussions on this issue. 34 Explicit expressions for tensor structures that could appear at third order in the stress tensor are given in the recent work of [62] which updates the original computation of [76]. However, these structures need to be further constrained by the second law, eliminating terms that comprise the forbidden class HF in the terminology of [31].…”