Hydrodynamics and String Theory

where lmicro is a characteristic scale of microscopic processes in the system (e.g. a correlation length), and L is a typical size of the system. Hydrodynamic description becomes unreliable when the inequality (1) is not satisfied. Schwarzschild black holes do not correspond to any hydrodynamic regime in a (hypothetical) holographicaly dual description because in that case L ∼ 1/T ∼ lmicro. We also do not expect the linearized hydrodynamic description to hold near a point of a phase transition, where the correlation length is typically large. We shall see that in the case of LST, predictions of the linearized hydrodynamics are not in contradiction with the hydrodynamic limit of the correlation functions obtained in the microscopic theory at the tree level. To derive the dispersion relations for the shear and the sound modes, consider small deviations from equilibrium < Tμν > +T̃μν(t, x) in the stress-energy tensor of a theory in a D + 1 dimensional Minkowski space. The equations of linearized hydrodynamics follow from the conservation law ∂μT = 0, ∂0T̃ 00 + ∂iT 0i = 0 , ∂0T 0i + ∂j T̃ ij = 0 , (2)