Hydrostatic Newton-Cartan Membranes

TL;DR

This work develops a general hydrostatic, second-order theory for Newton-Cartan membranes embedded in flat Newton-Cartan spacetime, framed through a generating equilibrium partition function and the resulting constitutive relations. By encoding the geometry with TNC data $(\tau_\mu,h_{\mu\nu},m_\mu)$ and their submanifold counterparts, the authors derive equilibrium constraints and currents, including a bending moment and a generalized shape equation. Specializing to axisymmetric configurations with constant vorticity, they obtain an explicit axisymmetric Young-Laplace equation and show how the vorticity and chemical potential enter through a subset of transport coefficients, reproducing known discoid limits when $\omega=0$. The results suggest a link to poroelastic membrane physics via the finite temperature–chemical potential structure, offering a path to broader applicability in hydrostatic NC membrane systems. The framework provides a versatile tool for analyzing membrane shapes and their elastic–hydrodynamic responses in non-relativistic, geometrically rich backgrounds.

Abstract

We construct the most general equilibrium partition function for Newton-Cartan membranes of codimension-1 defined on a flat ambient Newton-Cartan spacetime at second order in the hydrodynamic derivative expansion. We find the equilibrium constraints and the corresponding set of equilibrium constitutive relations. We show the explicit form of the Young-Laplace equation for a subset of non-vanishing and non-constant transport coefficients in the presence of a rotating fluid background with a constant vorticity.