Abstract
We construct the theory of dissipative hydrodynamics of uncharged fluids living on embedded space-time surfaces to first order in a derivative expansion in the case of codimension-1 surfaces (including fluid membranes) and the theory of non-dissipative hydrodynamics to second order in a derivative expansion in the case of codimension higher than one under the assumption of no angular momenta in transverse directions to the surface. This construction includes the elastic degrees of freedom, and hence the corresponding transport coefficients, that take into account transverse fluctuations of the geometry where the fluid lives. Requiring the second law of thermodynamics to be satisfied leads us to conclude that in the case of codimension-1 surfaces the stress-energy tensor is characterized by 2 hydrodynamic and 1 elastic independent transport coefficient to first order in the expansion while for codimension higher than one, and for non-dissipative flows, the stress-energy tensor is characterized by 7 hydrodynamic and 3 elastic independent transport coefficients to second order in the expansion. Furthermore, the constraints imposed between the stress-energy tensor, the bending moment and the entropy current of the fluid by these extra non-dissipative contributions are fully captured by equilibrium partition functions. This analysis constrains the Young modulus which can be measured from gravity by elastically perturbing black branes.
Highlights
Directions to the submanifold.1 It was shown that such fluids were characterized by a family of 3 hydrodynamic, 4 elastic and 1 mixed fluid-elastic transport coefficient2 in the case of codimension-1 surfaces and by a family 3 hydrodynamics, 3 elastic and 1 spin transport coefficient in the case of codimension higher than one, ignoring certain dimension specific contributions [1]
We construct the theory of dissipative hydrodynamics of uncharged fluids living on embedded space-time surfaces to first order in a derivative expansion in the case of codimension-1 surfaces and the theory of non-dissipative hydrodynamics to second order in a derivative expansion in the case of codimension higher than one under the assumption of no angular momenta in transverse directions to the surface
It was shown recently that in the case of space-filling uncharged fluids, the equilibrium partition function, which only applies to fluids in stationary motion, captures 3 hydrodynamic transport coefficients [3, 4]
Summary
We review the necessary tools for dealing with the geometry of embedded surfaces and the tensor structures that characterize it. We present the equations of motion that any material living on a surface must satisfy in the probe approximation when the surface is taken to have a finite thickness. These equations of motion are determined in terms of a set of tensors structures which, in order to construct a generic theory of dissipative hydrodynamics, need to be classified in terms of independent components. These components consist of all possible contributions which are allowed by symmetry and are on-shell independent. This classification is given at the end of this section and it will be the starting point for imposing the second law of thermodynamics and constraining the allowed contributions
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