The Michelson experiment

At the end of the nineteenth century, it was a common belief that light needs and has a medium in which it propagates: light is a wave in a medium called ether, as sound is a wave in air. This belief was shattered when Michelson (1881) tried to measure the velocity of the Earth on its way around the Sun. He used a sensitive interferometer, with one arm in the direction of the Earth's motion, and the other perpendicular to it. When rotating the instrument through an angle of 90°, a shift of the fringes of interference should take place: light propagates in the ether, and the velocity of the Earth had to be added that of the light in the direction of the respective arms. The result was zero: there was no velocity of the Earth with respect to the ether.

This negative result can be phrased differently. Since the system of the ether is an inertial system, and that of the Earth is moving with a (approximately) constant velocity, the Earth's system is an inertial system too. So the Michelson experiment (together with other experiments) tells us that the velocity of light is the same for all inertial systems which are moving with constant velocity with respect to each other (principle of the invariance of the velocity of light). The speed of light in empty space is the same for all inertial systems, independent of the motion of the light source and of the observer.

The evolutionary phases of a spherically symmetric star

In our universe a star whose temperature lies above that of its surroundings continuously loses energy, and hence mass, mainly in the form of radiation, but also in explosive outbursts of matter. Here we want to sketch roughly the evolution of such a star which is essentially characterized and determined by the star's innate properties (initial mass and density, …) and its behaviour in the critical catastrophic phases of its life.

According to observation, stars exist for a very long time after they have formed from hydrogen and dust. Therefore they can almost always settle down to a relatively stable state in the interplay between attractive gravitational force, repulsive (temperature-dependent) pressure and outgoing radiation.

The first stable state is reached when the gravitational attraction has compressed and heated the stellar matter to such a degree that the conversion of hydrogen into helium is a long-term source of energy sufficient to prevent the star cooling and to maintain the pressure (a sufficiently large thermal velocity of the stellar matter) necessary to compensate thegravitational force. The average density of such a star is of the order of magnitude 1 g cm-3. A typical example of such a star is our Sun.

When the hydrogen of the star is used up, the star can switch over to other nuclear processes (possibly only after an unstable phase associated with explosions) and produce nuclei of higher atomic number. These processes will last a shorter time and follow one another more quickly.

What are far fields?

The linearized theory of gravitation is based on the presumption that over whole regions of space, at any rate in the vicinity of the sources of the field, the gravitational field is weak, and the metric deviates only slightly from that of a Minkowski space. In nature we often meet a situation in which a distribution of matter (a satellite near the Earth, the Earth, the planetary system, our Galaxy) is surrounded by vacuum, and the closest matter is so far away that the gravitational field is weak in an intermediate region. In the neighbourhood of the sources, however, the field can be strong.

If such an intermediate region exists, and far away sources are not present or their influence can be neglected, then we speak of the far field of the configuration in question (Fig. 28.1). Notice that here, by contrast, for example, to most problems in electrodynamics, we may not always assume an isolated matter distribution which is surrounded only by a vacuum. The assumption of a void (the ‘infinite empty space’) into which waves pass and disappear contradicts the basic conception of General Relativity; also the fact that we orient our local inertial system towards the fixed stars indicates that we must always in principle take into account the existence of the whole Universe whenever we examine the properties of a part of the Universe.

In the examples and applications considered up until now we have always correctly taken into account the non-linearity of the Einstein equations, but most of the properties and effects discussed do not differ qualitatively from those of other classical (linear) fields. Now, in the discussion of black holes and of cosmological models, we are going to encounter properties of the gravitational field which deviate clearly from those of a linear field. The structure of the space-time is essentially changed by comparison with that of Minkowski space, and essentially new types of questions arise.

How does one find the fundamental physical laws?

Before turning in the next chapter to the laws governing the gravitational field, that is, to the question of how the matter existing in the universe determines the structure of the Riemannian space, we shall enquire into the physical laws which hold in a given Riemannian space; that is to say, how a given gravitational field influences other physical processes. How can one transcribe a basic physical equation, formulated in Minkowski space without regard to the gravitational force, into the Riemannian space, and thereby take account of the gravitational force?

In this formulation the word ‘transcribe’ somewhat conceals the fact that it is really a matter of searching for entirely new physical laws, which are very similar to the old laws only because of the especially simple way in which the gravitational field acts. It is clear that we shall not be forced to the new form of the laws by logical or mathematical considerations, but that we can attain the answer only by observation and experiment. In searching for a transcription principle we therefore want our experience to be summarized in the simplest possible formulae.

In the history of relativity theory the principle of covariance plays a large rôle in this connection. There is no clear and unique formulation of this principle; the opinions of different authors diverge here.