Surface Charges in Gravitation

Abstract

The main purpose of this first lecture is to introduce the concept of canonical surface charges in a generally covariant theory of gravity, whose General Relativity is the most famous representative. We will first show that a conserved stress tensor can be generated for any classical field theory. We will then motivate why we cannot define conserved currents and charges in Noether’s fashion for generally covariant theories, and more globally, for theories that include gauge transformations. This will lead us to extend Noether’s first theorem to formulate lower degree conservation laws, which will be exploitable for theories such as Einstein’s gravity. On the way, we will discuss about the symplectic structure of abstract spaces of fields, and use the covariant phase space formalism to derive a powerful result linking this structure and the lower degree conserved forms. We will then be able to compute surface charges associated to these quantities and study their properties and their algebra. Along the text, some pedagogical examples will be provided, namely for pure Einstein’s gravity, and Maxwells electrodynamics. Finally, we will present another possible definition of these surface charges and use the latter definition as an efficient tool to derive the conserved charges of Chern-Simons theory.