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.Comment: v3: 50pp; minor corrections in Sec. 3.2; typos fixed; published in
JHE
