Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Advanced Lectures on General Relativity

Geoffrey Compère, Adrien Fiorucci

TL;DR

The notes present a structured tour of canonical surface charges in gravity via the covariant phase space formalism, highlighting generalized Noether theorems and Noether–Wald charges. They then explore 3D gravity, BTZ black holes, and AdS_3/CFT through a Chern–Simons lens, before turning to asymptotically flat spacetimes and the BMS_4 structure with memory effects. The Kerr sector closes the course by examining extremality limits, Kerr/CFT ideas, and black hole spectroscopy through Teukolsky’s perturbation theory and NP/Petrov formalisms. Collectively, the notes illuminate how asymptotic symmetries, holographic dualities, and perturbative analyses shape our understanding of gravitational charges, horizons, and black hole dynamics with broad implications for quantum gravity and gravitational wave phenomenology.

Abstract

These lecture notes are intended for starting PhD students in theoretical physics who have a working knowledge of General Relativity. The 4 topics covered are (1) Surface charges as conserved quantities in theories of gravity; (2) Classical and holographic features of three-dimensional Einstein gravity; (3) Asymptotically flat spacetimes in 4 dimensions: BMS group and memory effects; (4) The Kerr black hole: properties at extremality and quasi-normal mode ringing. Each topic starts with historical foundations and points to a few modern research directions.

Advanced Lectures on General Relativity

TL;DR

The notes present a structured tour of canonical surface charges in gravity via the covariant phase space formalism, highlighting generalized Noether theorems and Noether–Wald charges. They then explore 3D gravity, BTZ black holes, and AdS_3/CFT through a Chern–Simons lens, before turning to asymptotically flat spacetimes and the BMS_4 structure with memory effects. The Kerr sector closes the course by examining extremality limits, Kerr/CFT ideas, and black hole spectroscopy through Teukolsky’s perturbation theory and NP/Petrov formalisms. Collectively, the notes illuminate how asymptotic symmetries, holographic dualities, and perturbative analyses shape our understanding of gravitational charges, horizons, and black hole dynamics with broad implications for quantum gravity and gravitational wave phenomenology.

Abstract

These lecture notes are intended for starting PhD students in theoretical physics who have a working knowledge of General Relativity. The 4 topics covered are (1) Surface charges as conserved quantities in theories of gravity; (2) Classical and holographic features of three-dimensional Einstein gravity; (3) Asymptotically flat spacetimes in 4 dimensions: BMS group and memory effects; (4) The Kerr black hole: properties at extremality and quasi-normal mode ringing. Each topic starts with historical foundations and points to a few modern research directions.

Paper Structure

This paper contains 93 sections, 328 equations, 21 figures.

Table of Contents

  1. Surface charges in Gravitation
  2. Introduction : general covariance and conserved stress tensor
  3. Generalized Noether theorem
  4. Gauge transformations and trivial currents
  5. Lower degree conservation laws
  6. Surface charges in generally covariant theories
  7. Covariant phase space formalism
  8. Field fibration and symplectic structure
  9. Noether's second theorem : an important lemma
  10. Einstein's gravity
  11. Einstein-Maxwell electrodynamics
  12. Fundamental theorem of the covariant phase space formalism
  13. Cartan's magic formula
  14. Noether-Wald surface charge
  15. Proof
  16. ...and 78 more sections

Figures (21)

  • Figure 1: Elements from the variational bicomplex structure.
  • Figure 2: Penrose diagram of global $AdS_3$ spacetime.
  • Figure 3: Penrose diagram for a static BTZ black hole ($J=0$).
  • Figure 4: Penrose diagrams for non-extremal BTZ black holes
  • Figure 5: Penrose diagrams for extremal BTZ black holes.
  • ...and 16 more figures