Adjoint L-Infinity Actions and Conserved Charges in GR

Abstract

In this work we compute the conserved currents and charges associated to the action of an infinitesimal isometry (Killing field) in Einstein--Cartan--Palatini gravity. We offer a new approach to these quantities through the formalism of $L_{\infty }$-algebras and the work of Ćirić, Giotopoulos, Radovanović, and Szabo, and Costello and Gwilliam. We demonstrate our approach by computing the entropy of the Schwarzchild and Kerr black holes. Along the way, we prove a purely algebraic result about the existence and utility of a higher (a full $\infty$) version of the adjoint action of an $L_{\infty }$-algebra.

v2: clarified conventions, e.g., shifts, throughout; significant revision of Sections 6 and 7; addition of application to Kerr spacetime