<div>Chapter I: INTRODUCTION</div><div>1. A brief review on essentials of General Relativity, from basic concepts, mathematical framework</div><div>and Einstein equations Einstein-Hilbert action and classical tests of GR;</div><div>2. Brief review of history and timeline of developments from Schwarzschild solution to black hole</div><div>mergers and to information paradox and rewall;</div><div>3. Gravitational collapse and Chandrasekhar mass bound;</div><div>4. Different schools of thought on black holes: high energy oriented, GR oriented and quantum</div><div>information theory oriented; open issue how to merge these schools</div><div><br></div><div>Chapter II: BASIC CONCEPTS and TOOLS</div><div>1. Schwarzschild metric and some basic facts and analysis;</div><div>2. Analysis of geodesics, notion of Killing horizon and near horizon Rindler geometry;</div><div>3. Kruskal coordinates, maximal extensions and Carter-Penrose diagram;</div><div>4. Einstein-Maxwell theory and Reisner-Nordström solution and its basic analysis;</div><div>5. Kerr solution and its basic analysis;</div><div>6. Black holes in (A)dS backgrounds.</div><div><br></div><div>Chapter III: CLASSICAL ASPECTS</div><div>1. Lensing and black hole shadows;</div><div>2. Super-radiance, Penrose process and black hole mining;</div><div>3. Gravitational wave emission in black hole mergers;</div><div>4. Accretion disk physics;</div><div>5. Extremal black holes, their near horizon and basic analysis.</div><div><br></div><div>Chapter IV: ADVANCED CONCEPTS</div><div>1. Mathematical defnition of black holes, notion of various different horizons, Killing, event,</div><div>cosmological, isolated; trapped surface.</div><div>2. Conjectures and theorems (Cosmic censorship; Penrose mass inequality, singularity, uniqueness</div><div>and topology theorems)</div><div>3. Raychaudhuri equation and area theorem (2nd law); energy conditions;</div><div>4. Linear and nonlinear stability of black hole solutions;</div><div>5. More detailed analysis of collapse, Choptuik exponents and critical exponents;</div><div>6. Canonical boundary charges (1st law), ADM, Brown-York, Regge-Teitelboim, Iyer-Wald-Zoupas,</div><div>Barnich-Brandt and Hajian-Sh-J charges.</div><div>7. Variation principle; Gibbons-Hawking-York boundary term; Brown-York stress tensor;</div><div>8. Quasi-normal modes and black hole perturbations;</div><div>9. Four laws of black hole thermodynamics and their new derivations a la Wald-Hajian-Sh-J;</div><div><br></div><div>Chapter V: SEMICLASSICAL ASPECTS</div><div>1. Quantization on black hole backgrounds;</div><div>2. Unruh effect;</div><div>3. Hawking effect;</div><div>4. Bekenstein entropy and the area law, the Bekenstein bound;</div><div>5. Parikh-Wilczek tunneling;</div>6. Black hole evaporation;<div>7. Membrane paradigm.</div><div><br></div><div>Chapter VI: EXPERT TOPICS</div><div>1. Gravity in lower dimensions (including various asymptotic symmetry algebras)</div><div>2. Gravity in higher dimensions (including a brief discussion on supergravity);</div><div>3. Higher dimensional black hole/ring/brane solutions.</div><div>4. Aspects of holography - holographic renormalization, correlation functions and asymptotic</div><div>symmetries</div><div>5. Extremal black holes and attractor mechanism</div><div>6. Kerr/CFT and related topics</div><div>7. Soft hair and black hole microstates.</div><div><br></div><div>Chapter VII: QUANTUM ASPECTS</div>1. Black holes in string theory;<div>2. Microstate counting;</div><div>3. Microstate identification/constructions, fuzzballs, fluffballs;</div><div>4. Information paradox and black hole complementarity and firewalls;</div><div>5. Black holes and quantum gravity;</div><div>6. Information paradox and the AdS/CFT;</div>7. Holography, Quantum information (entanglement entropy, Bousso bound, QNEC etc.) and<div>generalized laws of black hole thermodynamics.</div><div><br></div><div>Chapter VIII: OUTLOOK</div><div>1. Summary;</div><div>2. Outlook and open issues;</div><div> - Experimental/observational prospects</div><div> - Black holes as a window to Quantum Gravity</div><div> - gravity may be emergent | what does it emerge from?</div><div><br></div><div>Chapter IX: SOLUTIONS TO EXERCISES</div><div>We present numerous exercises throughout the book and in this chapter we give solutions to a</div><div>selected subset of them.</div><div><br></div><div>Appendices</div><div>We intend to have some appendices in which we present some details of crucial mathematical</div><div>frameworks and formulations not fitting into the main text, in particular</div><div> - Cartan formulation</div><div> - Basics of QFT in curved spacetime</div> - Covariant phase space method
