The analysis of scattering by black holes first led us to consider many different formalisms for the wave perturbations in black hole spacetimes. When we had the perturbations in hand, we came against the problem of defining ‘plane waves’ in the long range Newtonian tail of the black-hole gravitational field. The analogy with Newtonian gravity gives the solution to this problem, and the solution to the integer-spin case for low frequencies; both embodied in the natural (Regge & Wheeler, 1957) radial variable r*.

The temptation exists to simply write down a partial sum expansion and allow high-speed computer technology to present you the answer. The results of such an approach are often unintelligible. Hence we were drawn to an extended study of the limiting forms of the cross sections, in the low frequency, in the high frequency, and in the high frequency glory limits. Finally in chapters 7 and 8 we arrive at the computational level, and agreement with the limiting and qualitative results of chapter 6 give us confidence in our result.

What have we profited? We have developed, and develop still, a variety of techniques of perturbation theory. We have learned much about scattering theory, and we have numerical predictions that one day may allow us to measure the inertial mass of a condensed object, perhaps proving the existence of a black hole.

We close with a brief conjecture on the qualitative features of off-axis black hole scattering. We will discuss each of the principal features seen in the scattering cross sections of axially incident gravitational waves: the forward divergence, the orbital dip, and the backward glory.

Introduction

There is no potential for the neutrino field, which means that one must work directly with the field quantities which have a more complicated asymptotic power-law behavior in (1/r) than either metric perturbations or the vector potential. The peeling theorem (cf section 2.4) predicts that we will have to deal with asymptotic solutions differing by one power of r at infinity.

This complicates the integrations necessary to perform the mode-expansions. In addition the neutrino fields transform under changes of coordinates and tetrads in a more complicated way than do vector or tensor quantities. Both of these features are due to the neutrino's intrinsic spin-½ character.

The interaction of neutrinos and gravitational fields was first studied by Brill & Wheeler (1957), who investigated several aspects of that problem including the bound states of neutrinos in a spherically symmetric gravitational field.

More recently neutrinos in the Kerr background have been studied by Unruh (1973), Teukolsky (1973), Chandrasekhar (1976) and Chandrasekhar & Detweiler (1977). The results of these investigations are summarized in Chandrasekhar (1979b; 1983). Briefly, the two-component neutrino and Dirac equations have been shown to be separable in the Kerr geometry, and it has been shown that unlike integer spin fields, neutrinos and electrons do not exhibit classical superradiance in the Kerr background.

In this chapter we expand neutrino plane waves in the normal modes appropriate to the Kerr geometry. We give an elementary account of electron and neutrino plane waves in the NP formalism in flat spacetime. We then transform the flat spacetime plane waves to a tetrad and coordinate system appropriate to the asymptotic Kerr geometry, expand in spin-½ spheroidal harmonics, and match to normal mode expansions.

Introduction

We now examine the details seen in the calculated quantities. Our aim is to supply a physically intuitive context in which to understand the scattering phenomenon as a whole. To that end we first discuss two simplified physically analogous problems, a square barrier and the null torpedo model. Most of the features of the calculated cross sections may be understood in terms of these simplified models.

We discuss in turn the calculated absorption of the incident wave as a function of l, the phase shifts as a function of l and the summed angular cross sections. The absorption as a function of l provides us with a measure of the apparent size of the hole as measured by the marginally trapped null trajectories for each incident mode. Further, by excluding the absorbed modes from the scattering cross section, we find we may anticipate certain features in the cross sections. The l modes which are not absorbed are summed in the angular cross section but contribute with their respective phase shifts. By examining the phase shifts we may understand the features of the angular cross sections in terms of interference of the l modes, governed by their phases. Finally, using physical intuition, we examine the detailed angular cross sections and find several interesting interference phenomena. These phenomena are analogous to similar phenomena seen in numerous classical and quantum scattering processes, in particular the glory phenomena described in section 6. Most of our results are for the scattering of gravitational radiation. However, we also include for comparison some very interesting results on the scattering of scalar waves in the Schwarzschild background, due to Sanchez (1978a, b).

Motivation, context and scope

When a physicist thinks of black holes, he may think of one of two substantially different concepts. There is the astrophysical black hole, and there is the black hole of the mathematical model.

Black holes as astronomical objects are the remnants of dead stars, or perhaps one of the remnants of the inhomogeneity spectrum of the early universe. Their detection as astronomical objects has so far only been by indirect means, by observations interpreted via the astrophysicists’ models. The plausible astronomical existence of black holes as X-ray sources, of black holes as the engine of quasi-stellar objects (QSO), of black holes contributing to the mass of the universe as hidden matter, makes them more interesting and more frustrating than one would expect from the mathematical description of a black hole in asymptotically flat space.

The mathematically defined black hole is the picture of simplicity. It depends only on three parameters: mass, angular momentum and charge (Schwarzschild, 1916; Reissner, 1916; Nordstrøm, 1918; Kerr, 1963, Newman et al., 1965. In this work we will largely ignore charged black holes.) It is the ultimate abstraction of a physically gravitating body. One is spared the complexity of describing matter degrees of freedom, and can concentrate on the behavior of the gravitational modes.

This work treats mathematical black holes. We consider scattering of massless waves by black holes embedded in asymptotically flat spacetime. Because of the simplicity of the problem, it is to a large extent explicitly soluble; and where explicit analytic solutions are not possible, a variety of qualitative methods can be applied.

The title says it all. Scattering, a powerful tool conceptually as well as experimentally, is applied to the simplest gravitational system, a black hole.

This study benefits gravitation, our best known and least understood phenomenon. Gravitational studies are often isolated from the mainstream of physics. This is how it should be occasionally; one needs to develop a consistent formalism per se; but one needs also to confront it with particle physics, cosmology, astrophysics. A knowledge of cross sections for the scattering of waves of arbitrary polarization by Schwarzschild and Kerr black holes contributes to the physical understanding of gravitation theory.

This study also benefits scattering theory. Starting with the simplest case, the scattering of massless scalar waves by a Schwarzschild black hole, the authors identify the scattering problems which have to be solved: How does one formulate scattering theory in curved spacetime? Can one define an incident plane wave in the long-range Newtonian field of a black hole? How does radiation propagate near a black hole? How does one handle the black hole horizon? How does one compute cross sections for polarized waves propagating in curved space-time etc…? The authors introduce several methods for solving these problems: wave mechanical scattering, partial wave decomposition, semiclassical methods, Newman–Penrose formulation of wave propagation (made powerful by Teukolsky's and Press’ separation into radial polar and axial harmonics of the equation describing the evolution of wave perturbations in black hole background), and Chandrasekhar's and Detweiler's metric perturbation formalism. Numerical computations are not always of less fundamental importance than mathematical investigations; they also suggest new analytic approaches.

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.

Radiation Pressure.

21. Radiant energy or radiation consists of electromagnetic waves in the aether. Maxwell's electromagnetic theory showed that these waves possess momentum. If E is the energy of the waves, c the velocity of light, the momentum is E/c in the direction in which the waves are travelling.

According to the modern view energy and mass are inseparable, c2 ergs corresponding to 1 gm. This leads immediately to the same result. For the energy E ergs indicates a mass E/c2 gm., and since the velocity is c the momentum is (E/c2) × c = E/c.

A material screen which absorbs the waves absorbs also their momentum. Thus the momentum of the screen changes, which is another way of saying that it is acted on by a force. Suppose that waves containing E ergs per cu. cm. impinge normally on a perfectly absorbing surface. A column of radiation of height c passes into and is absorbed by each sq. cm. of the surface per sec.; this column contains Ec ergs and the momentum is thus Ec/c or E units. The force on the screen is thus E dynes per sq. cm.

For imperfect absorbers we must deduct the proportion of the momentum which is not passed on to the material screen, viz. that of the transmitted, scattered or reflected waves. For example, a perfect reflector would experience a pressure 2E; half of this is due to its stoppage of the incident waves and half is the recoil due to the projection of the train of reflected waves.

Interaction of Radiation and Matter.

34. The theory of the equilibrium of matter and radiation at constant temperature depends on a principle which is a generalisation of the theory of exchanges (§ 29). After equilibrium is reached no visible change occurs; the density and constitution of the radiation, the proportion of atoms in various states of combination and ionisation, the number of free electrons, the proportion of molecular velocities between given limits, all remain steady; but beneath this statistical changelessness there is continual change happening to the individual atoms, electrons, and elements of radiation.

Consider the atoms of a particular element which are uncombined and in their normal neutral state. The number n of these atoms in the system will remain constant (apart from chance fluctuations) when equilibrium is reached. But the individuals composing this number continually change. New atoms appear in this state owing to the dissolution of chemical molecules containing them, neutralisation of ionised atoms by the capture of free electrons, relapse of excited atoms to the normal state. Atoms in the given state disappear owing to the converse processes—combination to form chemical molecules, ionisation by the expulsion of an electron, excitation by absorption of radiation or collision with electrons or atoms. The steadiness of n is due to an average balancing of gains and losses.

But the principle above mentioned is not content with formulating this general balance of gain and loss—a mere translation of the word “equilibrium.”