We first define the notion of Wald gravitational entropy, defined in a more general setting than the Bekenstein–Hawking one. Then we define Sen’s entropy function formalism, in a general gravity theory, that defines the entropy function, whose minimization at the event horizon gives the entropy. Finally, we define the effective potential of the event horizon, defined in a theory with scalars, and show that the horizon is an attractor for the equations of motion involving scalars.

We first describe the classification of three-dimensional Lie algebras. Then we show how that implies a classification of three-dimensional Lie invariant Riemann spaces, as Bianchi spaces, associated to cosmologies. Then, we describe examples of homogenous Bianchi cosmologies, the Kasner spacetime and the Mixmaster Universe.

We first review general concepts of quantum field theory, like Feynman diagrams and path integrals. Then we define the worldline formalism for quantum field theory, and apply it to gravity, in the case of classical gravity with sources. We describe first an electromagnetic analogy of how to write down the worldline action and obtain physics from it, and then use it in the gravity case, for a black hole or neutron star. We then consider tidal forces in the nonrelativistic regime and find how to describe Love numbers and radiated power, for instance. As a simple example, we consider scalar gravity interacting with a source current and how to obtain information from Feynman diagrams. Then, define Non-Relativistic General Relativity (NRGR), a formalism for the gravitational inspiral of two bodies (e.g., two black holes) and show how to calculate the two-body (or many-body) action from Feynman diagrams and how to obtain the radiated power via gravitational waves. We find the Einstein–Infeld–Hofman Lagrangian for many bodies in the NRGR case.

We define the ADM parametrization of gravitational actions. Then, we define extrinsic curvature for both a spacelike hypersurface in the ADM parametrization and for a general surface embedded in a larger space. The Gauss–Codazzi equations for the embedding of a surface are defined and explained, as well as the Israel junction conditions. After defining the notion of Killing vectors, we consider asymptotically flat spacetimes, with the possible definitions of mass (Kumar, ADM, linearized) and the BMS group of asymptotic symmetries. Finally, we calculate and define the boundary term of the gravitational action, the Gibbons–Hawking–York term.

We describe the geometric formulation for gravity, without using an inverse metric, only the vielbein and spin connection as gauge fields. We define the general Lanczos–Lovelock Lagrangian, in particular Chern–Simons (in odd dimensions) and Born–Infeld (in even dimensions) gravity, and the topological gravity terms and their dimensional extensions. We end by an application to anomalies, specifically gravitational anomalies.

We define Penrose diagrams, which keep the causal and topological properties of gravitational spacetimes, while moving infinity to a finite distance on the diagram. We use the examples of Minkowski space, in two dimensions and dimensions greater than two, then describe Anti-de Sitter spacetime in Poincaré coordinates (the Poincaré patch), and finally consider the Schwarzschild black hole.

We define the vielbein–spin connection formulation of general relativity and describe what happens in the presence of fermions (which can only be described in this formulation). Then, we see how close is general relativity to being described by a gauge theory, in three dimensions, which is special, and in four or higher dimensions.

We describe the parametrized post-Newtonian (PPN) formalism for expansion around the Newtonian limit. First we describe the formalism for generic gravity theories, for the equations of motion and the energy-momentum tensor. Then we consider it for many-body systems (like binary inspirals) and the coefficients for PPN in general relativity in this case. We describe the effective field theory approach in this case, show how to calculate the post-Newtonian (PN) order, and how it fits with the EIH Lagrangian. Finally, we consider metric frames, the usual Einstein frame versus the generic Jordan frame, and the original Brans–Dicke theory.