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Newton-Cartan Submanifolds and Fluid Membranes

Jay Armas, Jelle Hartong, Emil Have, Bjarke Frost Nielsen, Niels A. Obers

TL;DR

The geometric description of submanifolds in Newton-Cartan spacetime is developed and it is argued that this is the natural geometrical framework to study fluid membranes in thermal equilibrium and their dynamics out of equilibrium.

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.

Newton-Cartan Submanifolds and Fluid Membranes

TL;DR

The geometric description of submanifolds in Newton-Cartan spacetime is developed and it is argued that this is the natural geometrical framework to study fluid membranes in thermal equilibrium and their dynamics out of equilibrium.

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.

Paper Structure

This paper contains 43 sections, 210 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. The geometry of Newton--Cartan submanifolds
  3. Newton--Cartan geometry
  4. Galilean boost-invariant structures
  5. Covariant differentiation and affine connection
  6. Absolute time and flat space
  7. Submanifolds in Newton-Cartan geometry
  8. Embedding map, tangent and normal vectors
  9. Timelike submanifolds and boost-invariance
  10. Induced Newton--Cartan structures
  11. The role of the transverse velocity $v^I$
  12. Covariant derivatives, extrinsic curvature and external rotation
  13. Integrability conditions
  14. Variations and dynamics of Newton--Cartan submanifolds
  15. Variations of Newton--Cartan objects on the submanifold
  16. ...and 28 more sections

Figures (2)

  • Figure 1: Graphical depiction of the embedding of timelike Newton--Cartan submanifolds. The vertical direction represents the time $t$, while the spatial directions are in the plane orthogonal to the $t$--axis. The spatial hypersurfaces of constant time are denoted by their corresponding value of $t$. Note in particular that the condition \ref{['eq:tauIzero']} implies that the submanifold does not “bend” away from the time direction in the ambient spacetime.
  • Figure 2: The three axisymmetric biophysical solutions to the Canham--Helfrich model and how they arise as surfaces of revolution. The coordinate $\rho$ measures the perpendicular distance to the $z$-axis (blue), while $\psi$ is the angle between the tangent of the contour and the $\rho$-axis.