Abstract
Hydrodynamics can be consistently formulated on surfaces of arbitrary co-dimension in a background space-time, providing the effective theory describing long-wavelength perturbations of black branes. When the co-dimension is non-zero, the system acquires fluid-elastic properties and constitutes what is called a fluid brane. Applying an effective action approach, the most general form of the free energy quadratic in the extrinsic curvature and extrinsic twist potential of stationary fluid brane configurations is constructed to second order in a derivative expansion. This construction generalizes the Helfrich-Canham bending energy for fluid membranes studied in theoretical biology to the case in which the fluid is rotating. It is found that stationary fluid brane configurations are characterized by a set of 3 elastic response coefficients, 3 hydrodynamic response coefficients and 1 spin response coefficient for co-dimension greater than one. Moreover, the elastic degrees of freedom present in the system are coupled to the hydrodynamic degrees of freedom. For co-dimension-1 surfaces we find a 8 independent parameter family of stationary fluid branes. It is further shown that elastic and spin corrections to (non)-extremal brane effective actions can be accounted for by a multipole expansion of the stress-energy tensor, therefore establishing a relation between the different formalisms of Carter, Capovilla-Guven and Vasilic-Vojinovic and between gravity and the effective description of stationary fluid branes. Finally, it is shown that the Young modulus found in the literature for black branes falls into the class predicted by this approach - a relation which is then used to make a proposal for the second order effective action of stationary blackfolds and to find the corrected horizon angular velocity of thin black rings.
Highlights
The relation between the toy model of a fluid brane and gravitational physics can be thought of in the following way
It is shown that the Young modulus found in the literature for black branes falls into the class predicted by this approach - a relation which is used to make a proposal for the second order effective action of stationary blackfolds and to find the corrected horizon angular velocity of thin black rings
Each of these response coefficients is associated with a particular term which is added to the action and from our analysis we conclude that the 3 elastic terms are coupled to 4 of the hydrodynamic terms due to geometric constraints
Summary
We will show how some of the ideas of the previous section can be pushed to second order in a derivative expansion. To second order in the derivative expansion, focusing on the elastic corrections, we can add a total of 5 different terms to the action (2.26) which are quadratic in the extrinsic curvature. Spin corrections to O ε2 in the derivative expansion are the terms that can be added to the action which are quadratic in the extrinsic twist potential These do not contribute to the dipole moment Dabi. We examine the possible 2nd order hydrodynamic corrections to the action (2.26) and the stress-energy tensor T ab as well as the coupling between these modes and the ones found in the previous sections For all these corrections the dipole moment Dabi vanishes. The presence of elastic degrees freedom introduces further non-trivial response coefficients than those analyzed in [46, 47, 51] for both stationary and non-dissipative space-filling fluid flows
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