The Kerr solution

Most known stars are rotating relative to their local inertial system (relative to the stars) and are therefore not spherically symmetric; their gravitational field is not described by the Schwarzschild solution. In Newtonian gravitational theory, although the field certainly changes because of the rotational flattening of the star, it still remains static, while in the Einstein theory, on the other hand, the flow of matter acts to produce fields. The metric will still be time-independent (for a timeindependent rotation of the star), but not invariant under time reversal. We therefore expect that the gravitational field of a rotating star will be described by an axisymmetric stationary vacuum solution which goes over to a flat space at great distance from the source. Depending on the distribution of matter within the star there will be different types of vacuum fields which, in the language of the Newtonian gravitational theory, differ, for example, in the multipole moments of the matter distribution. One of these solutions is the Kerr (1963) solution, found almost fifty years after the discovery of the Schwarzschild metric. It proves to be especially important for understanding the gravitational collapse of a rotating star. To avoid misunderstanding we emphasize that the Kerr solution is not the gravitational field of an arbitrary axisymmetric rotating star, but rather only the exterior field of a very special source.

We shall now discuss the Kerr solution and its properties. Since its mathematical structure is rather complicated, we shall not construct a derivation from the Einstein field equations.

Gravitational forces are the only forces presently known which are long range (in contrast to the nuclear forces, for example) and which cannot be compensated (there are no negative masses). It is therefore to be expected that, for large quantities of matter distributed over wide regions of space, they will be the decisive forces, and hence the gravitational forces will determine the evolution and dynamics of the universe.

Physical laws get their importance from the fact that a single law describes many very different situations. Technically this comes out by writing the laws as differential equations (usually of second order), which admit a multitude of initial or boundary conditions. The law itself has often been found by extracting some common rules from the observed variety of effects. All these features are also present in the theory of gravitation.

In cosmology, however, we encounter a very different situation. There is only one realization of a cosmos, that which we are living in. And if there was an extra physical law for this cosmos, we could not find and prove it the usual way. That is to say, if we find a surprising new phenomenon, we cannot easily decide which of its properties are a new law, and which are due to initial conditions. Sometimes it is claimed that, in a proper theory, initial conditions should be excluded, the cosmos must not depend on them (this was one of the assumptions of the ‘inflation’ theory). Or one claims that the new law can be obtained from other principles, such us the anthropic principle.

Special Relativity originally dealt with the symmetries of the electromagnetic field and their consequences for experiments and for the interpretation of space and time measurements. It arose at the end of the nineteenth century from the difficulties in understanding the properties of light when this light was tested by observers at rest or in relative motion. Its name originated from the surprise that many of the concepts of classical non-relativistic physics refer to a frame of reference (‘observer’) and are true only relative to that frame.

The symmetries mentioned above show up as transformation properties with respect to Lorentz transformations. It was soon realized that these transformation properties have to be the same for all interacting fields, they have to be the same for electromagnetic, mechanic, thermodynamic, etc. systems. To achieve that, some of the ‘older’ parts of the respective theories had to be changed to incorporate the proper transformation properties. Because of this we can also say that Special Relativity shows how to incorporate the proper behaviour under Lorentz transformation into all branches of physics. The theory is ‘special’ in that only observers moving with constant velocities with respect to each other are on equal footing (and were considered in its derivation).

Although the words ‘General Relativity’ indicate a similar interpretation, this is not quite correct. It is true that historically the word ‘general’ refers to the idea that observers in a general state of motion (arbitrary acceleration) should be admitted, and therefore arbitrary transformation of coordinates should be discussed.

Are there gravitational waves?

The existence of gravitational waves was disputed for a long time, but in recent years their existence has been generally accepted. As often in the history of a science, the cause of the variance of opinions is to be sought in a mixture of ignorance and inexact definitions. Probably in the theory of gravitation, too, the dispute will only be completely settled when a solution, for example, of the two-body problem, has been found, from which one can see in what sense such a double-star system in a Friedmann universe emits waves and in what sense it does not, and when the existence of such waves has been experimentally demonstrated.

Waves in the most general sense are time-dependent solutions of the Einstein equations; of course such solutions exist. But this definition of waves is, as we can see from experience with the Maxwell theory, rather too broad, for a field which changes only as a result of the relative motion of the source and the observer (motion past a static field) would not be called a wave. Most additional demands which a gravitational wave should satisfy lead, however, to the characterization ‘radiation or transport of energy’, and this is where the difficulties begin, as explained in the previous chapter, starting with the definition of energy.

In order to make the situation relatively simple, in spite of the non-linearity of the field equations, one can restrict attention to those solutions which possess a far-field zone in the sense of Section 28.1.

This volume is the first part of notes that evolved during my teaching of a small class for junior and senior physics students at MIT. The course focused on a physical, analytical approach to astronomy and astrophysics. The material in this volume presents methods, tools and phenomena of astronomy that the science undergraduate should incorporate into his or her knowledge prior to or during the practice and study of quantitative and analytical astronomy and astrophysics.

The content is a diverse set of topics ranging across all branches of astronomy, with an approach that is introductory and based upon physical considerations. It is addressed primarily to advanced undergraduate science students, especially those who are new to astronomy. It should also be a useful introduction for graduate students or postdoctoral researchers who are encountering the practice of astronomy for the first time. Algebra and trigonometry are freely used, and calculus appears frequently. Substantial portions should be accessible to those who remember well their advanced high school mathematics.

Here one learns quantitative aspects of the electromagnetic spectrum, atmospheric absorption, celestial coordinate systems, the motions of celestial objects, eclipses, calendar and time systems, telescopes in all wavebands, speckle interferometry and adaptive optics to overcome atmospheric jitter, astronomical detectors including CCDs, two space gamma-ray experiments, basic statistics, interferometry to improve angular resolution, radiation from point and extended sources, the determination of masses, temperatures, and distances of celestial objects, the processes that absorb and scatter photons in the interstellar medium together with the concept of cross section, broadband and line spectra, the transport of radiation through matter to form spectral lines, and finally the techniques used in neutrino. cosmic-ray and gravitational-wave astronomy.

What we learn in this chapter

Introduction

The systems that extract information from faint signals about distant celestial bodies are the source of essentially all our astronomical knowledge. Telescopes collect and concentrate the radiation, and the instruments at their foci analyze one or more properties of the radiation. The systems used for the various frequency bands (e.g., radio, optical, and x-ray) differ dramatically from one another.

The faint signals must compete with background noise from the cosmos, the atmosphere, the earth's surface, and the detectors themselves. These noise sources differ with the frequency of the radiation. Advances in astronomy often follow from improved rejection of noise so that fainter signals can be detected.

What we learn in this chapter