Newton-Cartan submanifolds and fluid membranes

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 generalization of the Canham-Helfrich bending energy for lipid vesicles that takes into account the requirements of thermal equilibrium.

Oddity in nonrelativistic, strong gravity

“…The work on the large c expansion, which by construction provides a nonrelativistic gravity theory descendant from GR, is also of value to the recent explorations of more general theories of nonrelativistic gravity. In that context we can mention for example recent work on 3d nonrelativistic gravity [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27], Lifshitz Holography [28][29][30][31], nonrelativistic string theory [32][33][34][35][36][37][38][39][40][41][42][43][44][45][46][47] and condensed matter and fluid mechanics applications [48][49][50][51][52][53][54][55][56][57]. In parallel the symmetries underlying such theories have been further investigated …”

Non-boost invariant fluid dynamics

“…In another direction, it would be worthwhile to examine submanifolds and fluids living on them in the spirit of [24]. In particular, it was shown in the context of Newton-Cartan geometry that the normal projection of v µ can be interpreted as the transverse velocity of the submanifold.…”

Non-relativistic gravity and its coupling to matter

“…Finally, it is relevant to mention that for fixed backgrounds, non-relativistic geometry has proven to be useful for understanding aspects such as energy-momentum tensors, Ward identities, hydrodynamics and anomalies in the context of non-relativistic field theories, which are ubiquitous in condensed matter and biological systems (see e.g. [54][55][56][57][58][59]).…”