A survey of spacecraft results and mission planning for the Martian satellites, Phobos and Deimos, since 2014. Images and other observations by many spacecraft are included, as well as plans for future missions.

We consider solutions that can be obtained via dimensional reduction. We first consider the domain wall, both the perturbative nonrelativistic solution and the exact relativistic solution, first directly in four dimensions, and then show how it can be described via dimensional reduction. Then we consider the cosmic string solution, first directly in four dimensions, and then via dimensional reduction, and finally deriving it at weak field. Finally, we consider the BTZ black hole solution in 2+1 dimensions, deriving it directly, and then show how the BTZ solution and AdS space are continuously related.

We describe nontrivial topologies. First, we describe the Taub–NUT solutions. Then the Taub–NUT of Hawking and the Taub solution, as gravitational instantons. Then the Eguchi–Hanson metric, obtained from a Yang–Mills like instanton ansatz. Then the Gibbons–Hawking multi-instanton. The KK monopole is shown to be an example of application of the Taub–NUT instanton. Finally, we describe the Gödel Universe, a rotating solution with closed timelike curves (CTCs), even though the source is standard, just dust matter and cosmological constant.

We describe cosmological solutions. First, we consider the Friedman–Lemaitre–Robertson–Walker (FLRW) ansatz and find the resulting Friedmann equations. Then we find the cosmological solution and cosmological models corresponding to types of matter. Finally, we describe in details the cosmologies of de Sitter and Anti-de Sitter space.

We consider the deflection of light by the Sun (or a massive object) in general relativity. We first find it by analogy of the geodesic equation with the motion of light in a medium with small, position-dependent index of refraction, and then by the formal method of the Hamilton–Jacobi equation, which is first reviewed, before being used. Finally, we compare with the special relativity result, and find the famous ½ factor distinguishing between the two.

We consider the rotating black hole, the Kerr solution, and the rotating black hole with charge, the Kerr–Newman solution. We describe their symmetries and causal structure, including the new features of the ring singularity and the ergosphere, with frame-dragging (observers are forced to rotate with the black hole) and calculate the Penrose diagram. Finally, we describe the Penrose process of extracting energy and angular momentum from the rotating black hole.

We find out how to write general relativity solutions as double copies of gauge theory solutions. As a motivation, we first consider the KLT relations and the BCJ relations, for graviton quantum amplitudes as double copies of gluon quantum amplitudes. Then we consider the double copy for solutions in Kerr–Schild coordinates. As examples, we consider the Schwarzschild black hole, the Kerr black hole, pp waves, and the Taub–NUT solution. We define the Weyl double copy and write it for the general Petrov type D solution.

The kinematics of general relativity is described. General relativity is given by intrinsically curved spacetimes, for non-Euclidean geometry, based on two assumptions, leading to two physical principles for the kinematics (plus one equation for the dynamics). The kinematics is based on the same parallel transport as for gauge theories. The motion of free particles is on geodesics in the curved spacetime.

We consider the black hole with charge, the Reissner–Nordstrom black hole. We describe the solution, and the BPS bound and its saturation, for extremal black holes, of mass = charge. We describe properties of the event horizon. Finally, we calculate the Penrose diagram of the Reissner–Nordstrom black hole, in the non-extremal and extremal cases.

We describe the Fierz–Pauli action, the quadratic action coming from the Einstein–Hilbert action. Then we introduce gauge conditions. After a discussion of general gauges, we described the de Donder gauge condition, then the transverse-traceless (TT) gauge, and the synchronous gauge.

We describe the Petrov classification of spacetimes, by the number of independent principal null directions (PNDs), eigenvectors of the Weyl tensor. The Petrov types are defined, and they are described in the Newman–Penrose formalism. Finally, examples of the various Petrov type metrics are given.

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.