Scattering from Black Holes

203 Pages·2009·3.05 MB·English

by  J. A. H. Futterman, F. A. Handler, Richard Alfred Matzner

CAMBRIDGE MONOGRAPHS ON
MATHEMATICAL PHYSICS
General editors: P. V. Landshoff, D. W. Sciama, S. Weinberg
Scattering from black holes
SCATTERING FROM BLACK
HOLES
J.A.H. FUTTERMAN, & F.A. HANDLER
Lawrence Livermore National Laboratories, Livermore, California
and
R.A. MATZNER
Department of Physics, The University of Texas at Austin
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© Cambridge University Press 1988
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permission of Cambridge University Press.
First published 1988
This digitally printed version 2009
A catalogue record for this publication is available from the British Library
Library of Congress Cataloguing in Publication data
Futterman, J. A. H.
Scattering from black holes.
(Cambridge monographs on mathematical physics)
Bibliography:
Includes index.
1. Black holes (Astronomy) 2. Scattering (Physics)
3. Astrophysics. I. Handler, F. A. II. Matzner,
Richard A. (Richard Alfred), 1942- . III. Title.
IV. Series.
QB843.B55F88 1987 523 86-28322
ISBN 978-0-521-32986-6 hardback
ISBN 978-0-521-11210-9 paperback
Contents
Foreword
1 Introduction 1
1.1 Motivation, context and scope 1
1.2 Definition of black-hole scatering 5
1.3 Historical outline 7
1.4 Partial wave analysis: the scalar wave equation in the
Schwarzschild field 9
2 Perturbations of black hole spacetimes 13
2.1 NP formalism: perturbation of the vacuum Kerr geometry 14
2.2 Relation of NP quantities to potential perturbations 18
2.3 Discussion: analytical properties of the separation functions 22
2.4 Asymptotic behavior, normalization and conservation
relations 24
3 Integral spin plane waves in black hole spacetimes 29
3.1 Scalar waves, flat spacetime 29
3.2 Scalar waves, spherically symmetric (Schwarzschild)
spacetime 31
3.3 Scalar waves, Ker background (on-axis) 32
3.4 Scalar waves, Kerr background, arbitrary incidence 33
3.5 Electromagnetic plane waves 34
3.6 Gravitational plane waves 40
3.7 Aside: incident plane waves in the Reissner-Nordstr^m
spacetime 48
4 Neutrino plane waves 52
4.1 Introduction 52
4.2 Neutrino waves in flat background spacetimes 53
4.3 Neutrino waves in curved spacetimes 5
4.4 Neutrino waves in Kerr background; separation of variables 56
4.5 Properties of neutrino plane waves 58
4.6 Transformation of plane waves; normal mode expansion 62
5 Scatering 70
5.1 Scatering of scalar waves 70
vi Contents
5.2 Scatering of electromagnetic waves 71
5.3 Scatering of gravitational radiation 72
5.4 Scatering of neutrinos 74
5.5 Absorption cros section 75
5.6 Scatering and phase shifts 76
6 Limiting cros sections 7
6.1 Low frequency cros sections 7
6.1.1. Low frequency scalar cros sections 7
6.1.2. Low frequency neutrino wave cross sections 83
6.1.3 Low frequency limits of electromagnetic and gravita-
tional scatering 85
6.1.3.1. Low frequency gravitational scatering 87
6.1.3.2. The electromagnetic amplitude a(0) 92
6.2 High frequency cros sections 93
6.2.1. Eikonal approximation for integer spin fields 93
6.2.2. Eikonal approximation for neutrino fields 95
6.2.3. On-axis observers: glories in high frequency scattering 97
6.2.3.1. The attractive Coulomb problem 107
6.2.3.2. Attractive Coulomb plus short-range attractive
potential 109
6.2.3.3. Glory scattering for long-range forces 111
6.2.3.4. The scattering of massless scalar waves by a
Schwarzschild 'Singularity' 112
6.2.3.5. Polarized wave glory scattering by black holes 115
7 Computation of cros sections 120
7.1 Angular equation 120
7.2 Radial equation 123
7.2.1. The method of Press & Teukolsky and the JWKB
approximation 124
7.2.1.1. JWKB approximation for radial equation 126
7.2.2. The Chandrasekhar-Detweiler method and the JWKB
approximation 128
7.3 The Newtonian analogy revisited 133
8. Absorption, phase shifts and cross sections for gravitational
waves 136
8.1 Introduction 136
8.2 The square barrier and nul torpedoes 136
8.3 Absorption cros sections 140
8.3.1. Absorption cross sections: results for scalar waves 140
8.3.2. Absorption cross sections: results for gravity waves 141
Contents vi
8.4 Phase shifts 145
8.5 Scatering cros sections 155
8.5.1. The Schwarzschild scattering of massless scalar waves 155
8.5.2. The Schwarzschild scattering of gravitational waves 156
8.6 Scattering in the Schwarzschild and Kerr geometries -
interpretation: glories and spirals 158
9 Conclusion 167
Appendix Al: Integrals used to express plane waves in terms of
spin-weighted spheroidal functions 170
Appendix A2: Addition formulae for spin-weighted spherical
angular functions 174
Apendix B 176
References 17
Index 182
Acknowledgements
This work has benefited immeasurably from the advice of our colleagues,
Cecile DeWitt-Morette, John A. Wheeler, and Bruce L. Nelson. Much of
the research reported here has been supported by National Science
Foundation grant #PHY81-07381 to the University of Texas. Most of the
computations were done on the University of Texas Computation Center
facilities, whose generous support is gratefully acknowledged. Much post-
processing of the results was done on home/personal computers.
Foreword
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 formul-
ation 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. Numer-
ical computations are not always of less fundamental importance than
mathematical investigations; they also suggest new analytic approaches.
The presence of oscillatory numerical cross sections as functions of the
scattering angle, for instance, suggested an analytic calculation that
identified the oscillations near the backward direction as due to glory
scattering.
The scattering of massless polarized waves by Schwarzschild and Kerr
black holes raises many issues which need to be confronted to be