# Compere and Fiorucci Chapter 1: Surface charges ## 1.1 Introduction - Killing equation - Conservation of stress tensor <- general covariance ## 1.2 Generalised Noether theorem - Noether first theorem - bijection: equivalent class of *global* continuous symmetries of L <-> equivalent class of conserved vectors fields $J^\mu$ - equivalent: $J^\mu_2=J^\mu_1+\partial_\nu k^{[\mu\nu]}+t^\mu$ where $t^\mu \approx 0$ on shell - problem: for pure gauge theories there is no non-trivial current - e.g. Einstein-Hilbert-Matter $T^{\mu\nu}=\frac{2}{\sqrt{-g}}\frac{\delta L}{\delta g_{\mu\nu}}=-\frac{1}{8\pi G}(G^{\mu\nu}-8\pi G T^{\mu\nu}_M)\approx 0$ - n.b. $T^{\mu\nu}$ is the conserved stress tensor, not the matter stress tensor, which is $T^{\mu\nu}_M$ - lower degree conservation laws ## Refs - defines [[0060 Asymptotic symmetry]]