Non-Lorentzian geometry of fluids and strings

Abstract

Non-Lorentzian geometry is a branch of geometry where, roughly speaking, the notion of a metric is replaced by something else. We begin by providing an overview of non-Lorentzian geometries, and we describe how they as Cartan geometries which emphasises their connection to kinematical symmetries. We also review their construction as G-structures, and the formalism of 1/c² expansions.

Examples of non-Lorentzian geometries include Carrollian, Newton–Cartan and Aristotelian geometries. After reviewing relativistic fluid dynamics, we develop a theory of boost-agnostic fluids using Lagrangian methods. These fluids couple to Aristotelian backgrounds, and their non-dissipative transport is described by an action coupled to Aristotelian geometry. In the final part of the thesis, we describe nonrelativistic approximations of string theory using 1/c² expansions, and we show that this generalises the Gomis–Ooguri nonrelativistic string. The target space geometry of these string theories is a string Newton–Cartan-like geometry that arises from a stringy 1/c² expansion of Lorentzian geometry.