We consider the other classical tests of general relativity. The first is the gravitational redshift (the change in the frequency of light). Next, we consider the geodesic radial motion, and we use the equations to find the time delay of a radar signal (or any light signal) moving in a gravitational field (the classic test is for bouncing a signal from a satellite in orbit, or on the Moon). We use the same geodesic equations to find the precession of the perihelion (closest distance to the Sun) of the ecliptic (motion of the planets around the Sun), specifically for the perihelion of Mercury’s ecliptic. Finally, we analyze the possible motions in the Schwarzschild metric, by using a nonrelativistic analogy (effective potential), both in the massive and in the null cases.

We describe gravitational waves. We start with radiation in the TT gauge, then the gravitational field of a mass distribution: after using the electromagnetic multipole expansion as an analogy, we describe the gravitational multipole expansion. We calculate gravitational radiation emitted from a source, then describe the pseudotensor of the gravitational field for the quadratic approximation, and use it to calculate the power radiated from through gravitational waves (the Einstein formula). Finally, we describe the exact, non-perturbative solution for gravitational waves with cylindrical symmetry found by Einstein and Rosen.

This text on general relativity and its modern applications is suitable for an intensive one-semester course on general relativity, at the level of a PhD student in physics. Assuming knowledge of classical mechanics and electromagnetism at an advanced undergraduate level, basic concepts are introduced quickly, with greater emphasis on their applications. Standard topics are covered, such as the Schwarzschild solution, classical tests of general relativity, gravitational waves, Arnowitt, Deser, Misner parametrization, relativistic stars, and cosmology, as well as more advanced standard topics such as vielbein–spin connection formulation, trapped surfaces, the Raychaudhuri equation, energy conditions, the Petrov and Bianchi classifications, and gravitational instantons. More modern topics, including black hole thermodynamics, gravitational entropy, effective field theory for gravity, the parametrized post- Newtonian expansion, the double copy, and fluid-gravity correspondence are also introduced using the language understood by physicists, without mathematics that is too abstract mathematics, proven theorems, or the language of pure mathematics.

We consider relativistic stars and find the equations of gravitational collapse. In particular, we write the Tolman–Oppenheimer–Volkov (TOV) equation. We define general stellar models. We find the Chandrasekhar limit for white dwarfs to break electron degeneracy pressure and collapse to a neutron start, and the TOV limit for neutron stars to break neutron degeneracy pressure and collapse to a black hole. Finally, we describe a simple model (Oppenheimer–Snyder) for collapse to a black hole, and the resulting Penrose diagram.

We consider some general properties of black holes and event horizons, of causality and topology. We define trapped surfaces, congruence, convergence, and show an example of a marginally trapped surface different than the event horizons. We prove the existence of an horizon for de Sitter spacetime, via its Penrose diagram. We then define Rindler spacetime, as the accelerated Minkowski spacetime, that gains an event horizon and mimics what happens for a black hole.

We consider parallel plane (pp) waves, solutions of Einstein’s equations for which the linearized equation is exact. We describe the Penrose theorem, for the Penrose limit, saying that in the neighborhood of a null geodesic, any space looks like a pp wave. We exemplify it for AdS3 × S3. We then consider gravitational shockwaves, an example of pp waves, in flat space, and in other backgrounds. Finally, we describe the Khan–Penrose interacting solution, for the head-on collision of two gravitational shockwaves.

We consider inflationary cosmology, but only general relativistic aspects of it. We first show some of the important problems with standard (Hot Big Bang) cosmology before inflation. Then we describe the general paradigm of inflation, and how it solves the cosmological problems. We specialize to inflation with a single scalar field, and moreover to slow-roll inflation. Finally, we calculate the fluctuations spectrum during inflation.

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.