Hydrostatic Newton-Cartan Membranes

Abstract

An application of the Newton-Cartan framework to the study of membranes is presented. Specifically, for membranes of co-dimension one in hydrostatic equilibrium embedded in a flat ambient Newton-Cartan spacetime. For such membranes, the corresponding equilibrium partition function at second order in the hydrodynamic derivative expansion is shown. Equilibrium constraints and the corresponding set of equilibrium constitutive relations are found. For the generically non-constant elastic subset of thermodynamic coefficients, the Young-Laplace equation is presented for the case of two-dimensional axisymmetric closed membranes embedded in a flat three-dimensional spacetime with constant ambient vorticity. Some numerical solutions to this Young-Laplace equation are examined, and some analytic solutions for particular choices of the thermodynamic coefficients are also discussed.