General relativity is the currently accepted theory of gravitation. Under this heading one could include a huge amount of material. For the needs of this theory an elaborate mathematical apparatus was created. It has partly become a self-standing sub-discipline of mathematics and physics, and it keeps developing, providing input or inspiration to physical theories that are being newly created (such as gauge field theories, supergravitation, and, more recently, the brane-world theories). From the gravitation theory, descriptions of astronomical phenomena taking place in strong gravitational fields and in large-scale sub-volumes of the Universe are derived. This part of gravitation theory develops in connection with results of astronomical observations. For the needs of this area, another sophisticated formalism was created (the Parametrised Post-Newtonian formalism). Finally, some tests of the gravitational theory can be carried out in laboratories, either terrestrial or orbital. These tests, their improvements and projects of further tests have led to developments in mathematical methods and in technology that are by now an almost separate branch of science – as an example, one can mention here the (monumentally expensive) search for gravitational waves and the calculations of properties of the wave signals to be expected.

As stated in the introduction, it would not be possible to include the whole of relativity in a book of manageable size. We chose to go into several selected topics in depth, but omitted some other topics completely or nearly so. This short chapter is a list of the topics we omitted, with some suggestions to the reader for further reading.

The following topics were covered inadequately or not at all:

1. Gravitational waves. Speaking most generally, a gravitational wave is any gravitational field that propagates through space independently of matter. It may, but need not, be periodic. There exists a large collection of exact solutions of Einstein's equations describing waves, for these see Stephani et al. (2003). There exists also an elaborate theory of nearly-linear waves, a relatively good source for it is the book by Ohanian and Ruffini (1994), and also the classic MTW course (Misner, Thorne and Wheeler, 1973). The theory of generation and detection of gravitational waves is worked out rather well, but progress in it is still going on, so current knowledge can be gained only from papers. A sophisticated and elaborate experimental technology is already in place, but to keep up with this one has to attend conferences in addition to reading the literature. The pioneer of the search for gravitational waves was Weber (1961); his small book can be recommended to readers interested in the history of the subject.

2. The Cauchy problem. In each coordinate system, the set of Einstein's equations can be separated into those equations that contain at most the first-order time derivatives of the metric components and those that are of second order in time.

What are tensors good for?

In Newtonian physics, a preferred class of reference systems is used. They are the inertial systems – those in which the three Newtonian principles of dynamics hold true. However, it may be difficult in practice to identify the inertial systems. As we have seen in Chapter 1, the inertial force imitates the gravitational force, so it may not be easy to make sure whether a given object moves with acceleration or remains at rest in a gravitational field. Hence, the laws of physics should be formulated in such a way that no reference system is privileged. The choice of a reference system, even when it is evidently convenient (e.g. the centre of mass system), is an act of human will, while the laws of physics should not depend on our decisions.

Tensors are objects defined so that no reference system is privileged. For the beginning, we will settle for a vague definition that we will make precise later. Suppose we change the coordinate system in an n-dimensional space from {xα}, α = 1, 2, …, n to, α′ = 1, 2, …, n. A tensor is a collection of functions on that space that changes in a specific way under such a coordinate transformation. The appropriate class of spaces and the ‘specific way’ in which the functions change will be defined in subsequent sections.

Differentiable manifolds

As already stated, in relativity we will be using non-Euclidean spaces. The most general class of spaces that we will consider are differentiable manifolds. This is a generalisation of the notion of a curved surface for which a tangent plane exists at every point of it.

A continuous medium as a model of the Universe

When describing the Universe as a whole, one assumes that it is filled with a continuous medium (fluid or gas), whose state can be described by physical fields (scalar fields such as mass density and pressure, vector fields such as the velocity of flow, or tensor fields, e.g. an electromagnetic field). This is a rather crude approximation, since our real Universe has a ‘granular’ structure. Its basic units are stars, and the relevant information from the point of view of observational cosmology is, for example, the number of stars in a given volume rather than the average mass density in that volume. The less-than-perfect adequacy of the fluid approximation is also demonstrated by the fact that the view on which objects should be considered the ‘elementary cells’ of the cosmic fluid has been changing with time. In the times of Hubble (1920s and 1930s), these were the galaxies. In later times, when galaxy clusters and proper motions of galaxies in clusters were observed, the galaxy clusters took over. In still later years, it was found that galaxies and galaxy clusters tend to occupy edges of large volumes of space that are almost empty inside (called voids). According to current beliefs, the elementary units of the Universe should be groups of voids. These changes in the definition of the elementary unit of the Universe were, characteristically, adopted in order to save the assumption of homogeneity and isotropy of the Universe ‘in the large’.

Special versus general relativity

The name ‘relativity’ covers two physical theories. The older one, called special relativity, published in 1905, is a theory of electromagnetic and mechanical phenomena taking place in reference systems that move with large velocities relative to an observer, but are not influenced by gravitation. It is considered to be a closed theory. Its parts had entered the basic courses of classical mechanics, quantum mechanics and electrodynamics. Students of physics study these subjects before they begin to learn general relativity. Therefore, we shall not deal with special relativity here. Familiarity with it is, however, necessary for understanding the general theory. The latter was published in 1915. It describes the properties of time and space, and mechanical and electromagnetic phenomena in the presence of a gravitational field.

Space and inertia in Newtonian physics

In the Newtonian mechanics and gravitation theory the space was just a background – a room to be filled with matter. It was considered obvious that the space is Euclidean. The masses of matter particles were considered their internal properties independent of any interactions with the remaining matter. However, from time to time it was suggested that not all of the phenomena in the Universe can be explained using such an approach. The best known among those concepts was the so-called Mach's principle. This approach was made known by Ernst Mach in the second half of the nineteenth century, but had been originated by the English philosopher Bishop George Berkeley, in 1710, while Newton was still alive.