Newton-Cartan Submanifolds and Fluid Membranes

TL;DR

This paper develops a geometric framework for studying fluid membranes in Newton-Cartan spacetime, enabling covariant analysis of their equilibrium and out-of-equilibrium dynamics, including elastic wave dispersion and bending energies.

Contribution

It introduces a geometric description of submanifolds in Newton-Cartan spacetime for Galilean-invariant hydrodynamics on curved surfaces, extending the understanding of fluid membrane physics.

Findings

Perturbations from equilibrium reproduce elastic wave dispersion.

A generalized Canham–Helfrich energy accounts for thermal equilibrium.

A simple membrane model depending on surface tension is presented.

Abstract

We develop the geometric description of submanifolds in Newton--Cartan spacetime. This provides the necessary starting point for a covariant spacetime formulation of Galilean-invariant hydrodynamics on curved surfaces. We argue that this is the natural geometrical framework to study fluid membranes in thermal equilibrium and their dynamics out of equilibrium. A simple model of fluid membranes that only depends on the surface tension is presented and, extracting the resulting stresses, we show that perturbations away from equilibrium yield the standard result for the dispersion of elastic waves. We also find a generalisation of the Canham--Helfrich bending energy for lipid vesicles that takes into account the requirements of thermal equilibrium.