After a general description of event horizons, and the definition of the surface gravity of a horizon, we find some formulae valid at horizons. Finally, we calculate the Raychaudhuri equation for the convergence of a black hole, defining the fact that gravity is always convergent. We apply it to horizons, defined by null geodesics, where the equation is slightly modified.

We define the laws of black hole thermodynamics by first reviewing the laws of regular thermodynamics, and seeing what the analog of the zeroth, first, second, and third laws are. After stating them, we show some partial proofs. As part of this, we show a simple proof and a general argument for the Hawking radiation and the Hawking temperature of a black hole, and the corresponding Bekenstein–Hawking entropy of the black hole. We finish by defining the gravitational thermodynamic potential.

We define gravitoelectric and gravitomagnetic fields, that is, splitting the gravitational field into “electric” and “magnetic” components. We first use an electromagnetic analogy for perturbative fields, which however only works in the static case. Next, we use a covariant formulation, first defining it by using Weyl tensor components. But the best definition, that is found to be in complete analogy to the electromagnetic case, is by using the Riemann tensor components, to define tidal tensor, for the tidal effect on neighboring geodesics. To define the analogy, we first define tidal tensors in electromagnetism and write the Maxwell’s equations in terms of them, then define the tidal tensors in gravity, and find that the Einstein’s equations are also written in terms of the gravitational tidal tensors. As a first application, we find the Lense–Thirring effect, for the precession of satellites in orbit due to “frame-dragging,” and as a second application, the clock effect, for the effect of the period of the clocks in orbit.

We define the canonical formalism for gravity. After a quick review of the Dirac formalism for constrained systems, we use it for gravity, and find the Hamiltonian constraint and the momentum constraint. We use them for defining the Wheeler–de Witt equation, the quantum version of the Einstein equations, and their solution, the wave function of the Universe, in the Hawking “no-boundary boundary condition” and the Villenkin “tunneling from nothing” versions, with their corresponding interpretations. We also define the Brown–York stress tensor in AdS background. Finally, we define Ashtekar variables, and the corresponding quantization in Dirac formalism for canonical gravity.

We describe the Newman–Penrose formalism for gravity in four dimensions. We first define some relations for covariant derivatives, then define some basis vectors and the spin coefficients, for the spin connection in this basis. Then commutation relations and the transport relations for basis vectors, and the Newman–Penrose field equations, for the action of covariant derivatives on spin coefficients. We then show how we can change null frames, and the important case of the spinorial notation for the Newman–Penrose formalism. Finally, we describe some applications of the formalism.

We describe the fluid-gravity correspondence. After defining the equations for viscous relativistic fluids and for conformal fluids in particular, we consider the case of most interest, of conformal fluids described by black holes in asymptotically AdS space, that is, the fluid-gravity correspondence. We also describe it via the membrane paradigm, which was initially defined in asymptotically flat space, but makes sense in asymptotically AdS space. Finally, we take the nonrelativistic Navier–Stokes scaling limit of the equations, obtaining the Navier–Stokes equations.

The (Riemannian) curvature is based on the notion of a Riemann tensor. Actions in general relativity are found as a generalization of special relativity actions. The action for gravity, the Einstein–Hilbert action, is the simplest nontrivial action compatible with general relativity. Matter is described by the energy-momentum tensor, generalized from special relativity. The equations of motion obtained from the action are the Einstein’s equations.

We first define energy conditions, which are gravitational analogs of the positivity of the energy in nongravitational theories. After defining the notion of singularity more precisely, we state (without proof) the singularity theorems of Hawking and Penrose, and a “counterexample,” which evades all of their assumptions. Then we define wormholes, traversable wormholes, and give as example the Morris–Thorne wormhole, with its embedding diagram.

We first describe the Newtonian limit for gravity, in particular in the case of spherical symmetry. We use it to help with the ansatz in the case of the vacuum solution of the Einstein’s equations with spherical symmetry. After calculating the equations of motion on the ansatz, we find the most general such solution, the Schwarzschild solution. We also define what a Schwarzschild black hole is.

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.