This chapter contains detailed hints on how to solve the more difficult exercises and verify some other complicated calculations.

The plane- and hyperbolically symmetric counterparts of the L–T models (i.e. the Ellis solutions), and generalisations of all three classes to charged dust source are derived and discussed. It is shown that the most natural interpretation of the plane-symmetric Ellis metric is an expanding or contracting family of 2-dimensional flat tori. The proof of the Ori theorem that for a spherically symmetric weakly charged dust ball shell crossings will block the bounce through the minimal radius is copied in detail. A subcase left out by Ori is discussed, but it will also lead to a shell crossing, only at the other side of the minimal radius. In this special case, a peculiar direction-dependent singularity is present: at the centre the matter density becomes negative for a short period before and after the bounce. The Datt–Ruban solution, its generalisation to charged dust source and the matching of both these solutions to, respectively, the Schwarzschild and Reissner–Nordstr\“{o}m solutions are presented and discussed. In the matched configuration the DR region stays inside the Schwarzschild or RN event horizon.

It is shown how the assumption of symmetry implies the Killing equations (more generally, invariance equations of arbitrary tensors are derived and discussed). It is also shown how to find the symmetry transformations of a manifold given the Killing vectors. The Lie derivative is introduced, and it is shown that the algebra of a symmetry group always has a finite dimension, not larger than n(n+1)/2, where nis the dimension of the manifold. Conformal symmetries are defined and it is shown that the algebra of the conformal symmetry group has dimension not larger than (n+1)(n+2)/2. The metric of a spherically symmetric 4-dimensional manifold is derived from the Killing equations, and its general properties are discussed. Explicit formulae for the conformal symmetries of a flat space of arbitrary dimension are given.

The logical and observational problems of Newton’s theory of gravitation that led Einstein to think about general relativity are briefly presented. In particular, this was the anomalous orbital motion of Mercury and the failed attempts to explain it within Newton’s theory. The local equivalence of inertial and gravitational forces is demonstrated.

The need for differential geometry is explained by considering the construction of parallel straight lines running far from each other in Euclidean space. Generalisation of the notion of parallelism to curved surfaces is explained.

The description of motion of a continuous medium in curved spacetime is introduced and related to the corresponding Newtonian description. Expansion, acceleration, shear and rotation of the medium are defined and interpreted. The Raychaudhuri equation and other evolution equations of hydrodynamical quantities are derived. A simple example of a singularity theorem is presented. Relativistic thermodynamics is introduced and it is shown that a thermodynamical scheme is guaranteed to exist only in such spacetimes that have an at least 2-dimensional symmetry group.

The geometric optics approximation to Maxwell equations is derived. The redshift and the description of bundles of rays via expansion, shear and rotation are defined. Equations of propagation of these optical tensors are derived. The proofs of the Goldberg - Sachs theorem and of the reciprocity theorem are presented. The equations of the Fermi - Walker transport and of the position drift of light sources are derived.

The Lemaitre–Tolman class of cosmological models (spherically symmetric inhomogeneous metrics obeying the Einstein equations with a dust source) is derived and discussed in much detail, from the point of view of its geometry and its applications to cosmology. It is shown that these metrics can be used to describe the formation of cosmic voids and of galaxy clusters out of small perturbations of homogeneity at the time of emission of the cosmic microwave background radiation. Apparent horizons for central and noncentral observers, the formation of black holes, the existence and avoidance of shell crossings, the equations of redshift and the generation and meaning of blueshift are discussed. A simple example of a shell focussing singularity is derived. Among the cosmological applications are: solving the horizon problem without inflation, mimicking the accelerating expansion of the Universe by mass-density inhomogeneities in a decelerating model, drift of light rays, lagging cores of Big Bang, misleading conclusions drawn from observed mass distribution in redshift space.

The covariant derivative is introduced via its postulated properties (the same as of ordinary derivative, plus the requirement that it produces tensor densities when acting on tensor densities). It is shown that tensor densities of arbitrary rank can be represented by sets of scalars – the projections on vector bases in the tangent space to the manifold. The coefficients of affine connection are defined using these bases, and the explicit formula for a covariant derivative of an arbitrary tensor density is derived.

The family of the Szekeres–Szafron, shell solutions of Einstein’s equations, is derived and discussed in detail. The discussion contains, among other things, the interpretation of the Szekeres coordinates, the invariant definitions of the whole family, the class II ($\beta,_z = 0$) family as a limit of the class I family, matching the Szekeres metric to the Schwarzschild metric (the class II S metric can be matched to Schwarzschild only inside the event horizon), conditions for absence of shell crossings, the description of the mass dipole, the apparent horizons, the Goode–Wainwright representation and a brief listing of recommended further reading on geometric and astrophysical properties of these solutions.

This is an encyclopaedia of basic knowledge about the Kerr metric and related topics. It includes, among other things, the original Kerr derivation from Einstein’s equations via the Kerr–Schild metrics, the Carter derivation from the separability of the Klein–Gordon equation (a by-product thereof is the generalisation to nonzero cosmological constant), the derivation (with illustrations) of the formulae for the event horizons and stationary limit hypersurfaces, the derivation of Carter’s fourth first integral of geodesic equations, the discussion of properties of general geodesics and of geodesics in the equatorial plane, the maximal analytic extension by Boyer and Lindquist, the Penrose process of extracting angular momentum from a rotating black hole and the Bardeen proof of existence of locally nonrotating observers in a stationary-axisymmetric spacetime.

This is a brief description (following the work of N. Ashby) of the influence of relativistic effects on the working of the Global Positioning System. If these effects were neglected for 24 hours, the error in determination of the locator’s position would exceed 18 km.

Tensors and tensor densities of arbitrary rank on arbitrary differentiable manifolds are defined and described. The difference between covariant and contravariant vectors is explained and illustrated with examples. Mappings between manifolds and the associated mappings of tensors are discussed, and it is shown that coordinate transformations are examples of such mappings. The Levi-Civita symbols and multidimensional Kronecker deltas are defined, and their usefulness in calculations involving determinants is demonstrated.