This section adds details for several past orbiter missions to bring them up to date, and includes the discovery of the Beagle 2 lander apparently intact on the Martian surface.

This section examines planning for missions after Curiosity, including the process of landing site selection. It depicts the activities of NASA’s InSight lander and Perseverance rover, China’s Tianwen-1 lander and Zhurong rover, and orbiting spacecraft including MAVEN, Hope and the Trace Gas Orbiter. Plans for future human exploration of Mars are presented as they were imagined in this period.

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.