We numerically calculate cross sections for gravitational waves axially incident on a Kerr black hole. The resulting detailed cross sections display a wealth of structure which can be linked conceptually to phenomena familiar from other scattering problems. The limiting cross sections in chapter 6 assist analysis by allowing truncation of numerical calculations where the limiting results become applicable and by pointing to parameter ranges of particular interest.

We examine the angular and radial equations in detail, finding in each case a form of the solution which allows efficient numerical integration. We consider two methods of solution for the angular equation; a perturbation calculation for insight and its continuation for the actual numerical work. For the radial equation, considered in the remainder of the chapter, we use a slow stable solution for small values of r and apply a JWKB approximation to integrate rapidly to large r.

Angular equation

We first consider the perturbation of the angular equation about aω = 0, where as before, a is the specific angular momentum and ω is the scattered wave frequency.

One reason for considering the perturbation calculation is that it lays the foundation for the actual method used. In particular we follow a technique of Press & Teukolsky (1973) to expand the spin-weighted spheroidal harmonics conveniently in the spin weighted spherical harmonics in both cases. The perturbation calculation only obtains spheroidal values to first or second order in aω. The continuation method utilizes the same expansion of the spheroidal functions, but finds expressions for their derivatives with respect to aω. The numerical solution then integrates these derivatives.

The scattering problem divides naturally into two major parts: the perturbation solution and the asymptology. For a given spacetime, the scattering problem is obtained by defining standard ingoing and outgoing states in the asymptotic regions of the spacetime, solving the perturbation equations, and matching (by adjusting the complex constant coefficients of the solution) the perturbation solutions to the asymptotic forms. Asymptology refers to the detailed description and normalization of the ingoing and outgoing states; it is treated in chapters 3 and 4. In this chapter we treat the derivation of the equations governing perturbations of the Kerr geometry.

The first approach taken to obtain perturbation equations was by perturbing the metric directly (in Schwarzschild) and solving for the resulting perturbed field solutions (Regge & Wheeler, 1957). This approach is the most intuitively physical approach, dealing throughout with metric quantities having direct, physical interpretations and therefore immediate connection to such quantities of interest as wave amplitudes and energy fluxes. As would be expected, the equations involved are manifestly real; in particular the scattering potentials are real. We will see that the reality of the scattering potential is intimately related to the parity of the solutions, interpretations of the wave scattering amplitudes, and numerical convenience in integrating the radial equations.

However, the above considerations are secondary in considering metric perturbations for the Kerr geometry. In Schwarzschild, direct solution for metric perturbations is a formidable task. In Kerr, the solution is very much more difficult; Chandrasekhar (1983) has given a review of this approach. Here we will concentrate on a Riemann tensor approach, based on the NP techniques.

Introduction

Decisions have to be made regarding the nature of the universe before observed properties of quasars can be transformed into intrinsic properties. We must adopt a cosmology. My objective is not to derive formally the equations of cosmology, as that has been done thoroughly many times. The ultimate purpose of this chapter is to justify these equations qualitatively while putting most of the quantitative effort into describing how to use them to study quasars. The cosmologies to be adopted were derived long before quasars were discovered, but their use became a much more serious affair once quasars had to be considered. This is because the redshifts of quasars are often sufficiently high that differences among different cosmologies become quite large. For most galaxies, by contrast, even the difference between newtonian and relativistic cosmologies can be ignored. We can certainly not yet guarantee the equations of favored cosmologies as applying to the real universe. Once a single cosmology is adopted for it, the universe is forced to become a simple place as regards the structure of spacetime. As long as such a simple universe fits what we see, it is appropriate to retain it. There is certainly little motivation for arbitrarily postulating increased complications; nevertheless, observers must forever be on the alert for those anomalies which would show the simple models to be valid no longer. There is a vested interest in having a simple universe, as that is the only kind we can understand.

Introduction

Astronomy in the early part of this century demonstrated that galaxies were systems made of tremendous numbers of stars. Spectroscopy of galaxies revealed the absorption lines that would be expected in the composite light from stars of different spectral classes. Galaxies showing dominant emission lines in their spectra were recognized as highly unusual. The first of these to be studied, NGC 1068, was commented upon even before the real size and nature of galaxies were understood (Slipher 1918). For several decades, because of their rarity, such galaxies were sufficiently outside mainstream research as to be given little attention. The subject of emission line spectroscopy for extragalactic objects suddenly became extremely important with the discovery of quasars, whose visible spectra are characterized by strong emission lines. Emission lines can provide diagnostics of velocities, temperatures and densities unavailable from any other technique. The lines which can be seen represent a wide range of ionization, so line fluxes also provide indirect measurements of unobserved portions of the continuum. Not least is the fact that emission lines are spectroscopically conspicuous, calling attention to locations where unusual events are occurring. The general similarities among the emission line spectra of quasars, and the scaling of these lines with the continuum source, means that the emission line spectrum is a characteristic quasar feature. To understand the origin of these lines, it is necessary to review the general physical concepts of spectroscopy.

Basic test for evolution

Determining whether the properties of the universe have changed as a function of its age is a major concern of observational cosmology. Not without logic, a universe maintaining the same characteristics through all of time has a satisfying nature. If we could understand it now, we would by definition understand it always. Even those who do not adhere to such a steady state universe have been loath to invoke changing characteristics to the observable galaxies in the universe. Another one of the ironies of the history of astronomy is that the cosmological tests utilized to prove that we inhabit an evolving Friedmann universe, tests applied using the bright elliptical galaxies as distance probes for cosmological purposes, could not allow evolution of those same galaxies (Sandage 1961). Constancy of the galaxy properties was a necessary prerequisite to using them for cosmological purposes. It is presumed now that such galaxies do change, even over observable time scales, and our ignorance about the proper evolutionary corrections to apply has removed much of the stimulus for drawing cosmological conclusions (Tinsley 1977).

Yet, astronomers who two decades ago accepted little evolution for galaxies were never hesitant to accept a lot of evolution for quasars. Even now, it is necessary to invoke far more evolution in quasars than in galaxies to explain the data seen. Few are troubled by this inconsistency, but it is not too surprising that some are.

Optical surveys for quasars

Quasars cannot be studied until they are found. The purpose of any quasar survey is simply to provide an efficient method of discovering quasars. This efficiency is greatly enhanced if many quasars can be found with a single observation by the detecting instrument, so it is preferable if the observation has a wide enough field of view to include many detectable quasars. Furthermore, it is desirable but usually not feasible to identify a quasar with the survey observation alone, without the necessity of a subsequent observation with another instrument. Because of their characteristic signatures in many different parts of the spectrum, quasars can be surveyed for using various techniques. Much of the subsequent research effort goes into comparison of results from various techniques, to determine whether the same quasars are being found in different ways, or whether there are categories of quasars conspicuous to one form of observation but invisible to another.

Color based surveys

Quasars are easy to find with optical telescopes; a summary compilation by Smith (1984) lists over 40 surveys. The reason is because their spectral characteristics are so different from most stars and galaxies that broad band optical surveys using only three effective wavelengths can differentiate most quasars from other objects. Despite many other techniques for quasar surveys, including X-ray and radio surveys, the great majority of quasars have been discovered optically, and it is very probable that this will continue to be so.

Galactic envelopes of quasars

The discovery of quasars heralded the present era of astrophysics, characterized by wide ranging investigations of every part of the spectrum, whether easily accessible or not. Observers were stimulated to open new spectral windows, primarily in the hope of finding something as extraordinary and unexpected as the quasars. None succeeded. Even when observations were pushed to X-ray wavelengths, quasars stood out. When discovered, quasars came as a stunning surprise to the small community of theoreticians who dabbled in extragalactic astrophysics. The quasars seemed so unlike galaxies that it was not clear whether their redshifts should be interpreted with the same cosmological relations that applied to galaxies. Doing so gave unbelievable answers; the quasars were just too luminous to explain. Furthermore, surprise piled upon surprise, these luminosities arose in volumes so small that luminosity variations could be seen in times of less than a year. It was fair, even necessary, to question any assumptions made for quasars, including assumptions about cosmological redshifts. I have argued at some length (Weedman 1976), so will not repeat much of it here, that this bewilderment arose as a consequence of the sequence of discovery for quasars. Had quasars been discovered initially as events in the nuclei of galaxies, the nature of their redshifts would have never been questioned. As it happened, it was only realized after the fact that identical phenomena can be observed in galactic nuclei.

Fundamentals of a luminosity function

Understanding the true distribution of objects in space has always been a basic objective of astronomy. Highly sophisticated statistical techniques were developed for determining the distribution of stars in our Galaxy (Trumpler & Weaver 1953). Many of these techniques have recently resurfaced for application to quasars. In many respects, there are great similarities between quasar counting, as done today, and the star counting in the early part of this century that led to an understanding of the structure of our Galaxy. Let us hope that similarly significant results may eventually arise from current quasar surveys. It would be convenient to apply the older techniques directly to quasars, just plugging in a few new numbers. This cannot be done, however. Determining the distributions of interest requires dealing with three dimensions, and the cosmological equations that relate distance for quasars to the observable redshift are much more complex than the euclidean geometry usable by galactic astronomers. Furthermore, quasars are not distributed uniformly in the universe, so statistical techniques based upon homogeneous distributions will not work. Finally, all of the equations of statistical stellar astronomy use magnitude units. This is still the case for most optical astronomy of quasars, but not so for quasar counts based on radio or X-ray observations. So we must deal with the additional complication of discussing both magnitude and flux units. It is necessary, therefore, to build a discussion of ‘statistical astronomy’ for quasars from first principles.

Introduction

In the chapters that follow, nearly a quarter century of intensive research and substantial progress in understanding the quasars will be summarized. From the perspective of an astronomer, it would be satisfying to report that this progress was primarily attributable to the cleverness and diligence of astronomers in attacking the problem. To be honest, however, most of the progress should be credited to the engineers and physicists who have developed the tools that allow our wide ranging probes into the mysteries of quasars. Trying to understand quasars forced astronomers into realizing that observations must extend over the broadest possible spectral coverage, that we must learn how to use X-ray, ultraviolet, optical, infrared and radio astronomy. It is now possible for an individual astronomer to have access to telescopes that access all of these spectral regions, and I am convinced that the definition of a ‘good’ observer in the next few decades will weigh heavily on the ability to be comfortable with all of these techniques. To realize how much this has changed the science of astronomy, one need only recall that the analogously important talents in 1960 were the ability to work efficiently at night, to withstand cold temperatures, and to develop photographic emulsions without accidently turning on the lights. At that time, the technically sophisticated astronomer was one who could use a photomultiplier tube.

The discovery of quasars nearly a quarter of a century ago made a new science out of astronomy. There were two factors in this invigorating revolution. One was the conceptual shock of learning that some very important sources of energy exist in the universe that are not related to the nuclear fusion processes in stars. The other was the fact that the discovery was made with a new technology, in this case radio astronomy. For the theorist, there was suddenly an open season for wide ranging and creative speculations on cosmological processes, energy generation, and radiation physics. For the technologist, there was proof that opening new observational spectral windows could reveal extraordinary and totally unanticipated things. Radio astronomy was quickly followed by ultraviolet, infrared and X-ray astronomy.

That first phase of the theoretical and technical regeneration of astronomy is now complete. Astronomers are more open to heretical theoretical suggestions and unconventional observational techniques. Exceptional telescope facilities are at our disposal worldwide and in space. Two decades of effort have not answered all of the fundamental questions about quasars, nor have they led to the discovery of objects any more puzzling. We still have the problems, but we now have the tools, and so can get on with the work of learning what, where, when, and why are the quasars.

My initiation into the subject began during a few spare hours left over from another project while observing with the 36-inch telescope at McDonald Observatory, in the fall of 1967.

Introduction

It should be clear from the discussions in the preceding chapters that an overwhelming amount of information is now available for describing quasar properties. Observationally, the study of quasars has been a great success. Also, it should be no surprise that, as the data have accumulated, it becomes more difficult to produce models that can explain everything. As might be expected for the most energetic objects in the universe, quasars are complex. This should not be a source of discouragement. It is not necessary to understand all details of the solar surface to know why the Sun shines. It is not necessary to understand all sedimentary rocks to know why continents drift. It is not necessary to memorize the taxonomy of all living creatures to realize why evolution occurs. When we are after the fundamental understanding of why something happens, all of the details are not required. In the study of quasars, we are still struggling to the point of knowing which details can be safely ignored, and it is for guidance in this regard that existing models are most useful.

The single most significant observational datum about quasars is that their spectra are so extraordinarily similar, even over ranges of 107 in luminosity, for objects separated by more than ten billion light years in the universe.

Introduction

Quasars are unique among objects of the universe in the observable span of their continuous spectra. In some cases, the same quasar can be seen with existing instruments at wavelengths from X-rays to radio, including everything in between. The quasar continuous spectrum is deceptive. Order-of-magnitude agreement over all wavelengths, from tens of centimeters to fractions of an angstrom, covering a range of >1011 in frequency, can be obtained by fitting a single power law spectrum, of form fν∞να, where a is ˜ – 1. It is tempting in the face of such a result to attribute all parts of the spectrum to related mechanisms. As has become very clear from more careful examination of spectra, that is not valid. Different components of the continuous spectra are produced by drastically different mechanisms, and there are sometimes no physical relations among these mechanisms. It is nevertheless assumed that all of these mechanisms are basically set in motion by a single underlying engine, such as gravitational accretion, but the radiation which comes out represents many ways of transforming gravitational to radiative energy. The greatest success of the intensive observational effort has been to show the exceptional similarities among spectroscopic properties for quasars covering a factor approaching 107 in luminosity. This is the single key fact to be explained by theoretical models of quasars. Whatever processes control the radiation must be capable of scaling over this range of energy release without fundamentally changing character.

This book introduces the reader to research work on a particular aspect of rotating fields in general relativity. It should be accessible to someone with an elementary knowledge of general relativity, such as that obtained in an undergraduate course on general relativity at a British university. A person with some maturity in mathematical physics may be able to follow it without knowing general relativity, as I have given a brief introduction to the relevant aspects of general relativity in Chapter 1.

My intention has been to write a short book which can provide a relatively quick entry into some research topics. I have therefore made only a brief mention of some topics such as the important group theoretic generation of solutions by Kinnersley and others. A significant part of this book deals with interior solutions, for which these techniques are not yet applicable. I have also not touched upon Petrov classification of solutions as this is marginal to the problems considered in this book. The connecting link of the topics considered here is the Weyl–Lewis–Papapetrou form of the stationary axially symmetric metric, which is derived in detail in Chapters 1 and 2.

A significant part of the book is based on my own work and for this reason the book may be considered as too specialized. However, all research is specialized and I believe it is instructive for the beginning research worker to be shown a piece of work carried out to a certain stage of completion.