Table Of ContentESSA Y S IN
GENERAL RELATIVITY
A Festschrift for Abraham Taub
Edited by Frank J. Tipler
Center for Theoretical Physics
University of Texas at Austin
Austin, Texas
1980
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Library of Congress Cataloging in Publication Data
Main entry under title:
Essays in general relativity.
"Publication list of A. H. Taub" : p. 233
Includes bibliographical references.
1. General relativity (Physics)—Addresses, essays,
lectures. 2. Taub, Abraham Haskel, Date —Addresses,
essays, lectures. I. Taub, Abraham Haskel, Date
II. Tip 1er, Frank J.
QC173.6.E83 530.Π 80-517
ISBN 0-12-691380-3
PRINTED IN THE UNITED STATES OF AMERICA
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>■ «*
A
Abraham H. Taub
To Abe Taub, The Universe Man
He masters the biggest equation
of the biggest shock wave
of the biggest bang that ever was.
Hydrodynamics and entropy,
they phase him not;
they are his meat and drink.
He creates universes right and left.
To lesser men he leaves the job
of distinguishing between a left-handed
and a right-handed Taub universe.
May he keep waves and shocks,
stars and universes,
forever in happy pursuit of one another.
And may we continue to enjoy
the fruits of his work
for many more years.
Three cheers for Abe Taub,
The Universe Man!
John Archibald Wheeler
High Island
27 June 1978
H+m
ΗΛ?)
The Taub Universe. Taub's most important contribution to general relativity was his discovery
of the Taub universe [Annals of Mathematics 53, 472 (1951)]. This model has been a major
source of examples of strange behavior that can occur in strong gravitational fields. For example,
the above figure, which is a drawing of the causal structure of the Taub universe, pictures null
geodesies that enter but never leave a compact set. These geodesies wind around the Taub
universe an infinite number of times in a finite affine parameter length of the geodesies. Other
examples of peculiar behavior in a Taub universe are discussed in the essays of J. A. Wheeler and
L. C. Shepley. [Figure from "Structure of spacetime," by R. Penrose, in Batteile Rencontres,
edited by C. M. De Witt and J. A. Wheeler (Benjamin, New York, 1968); reprinted by
permission.]
List of Contributors
Numbers in parentheses indicate the pages on which the authors'contributions begin.
Dieter R. Brill (13), Department of Physics and Astronomy, University of
Maryland, College Park, Maryland 20742
Arthur E. Fischer (79), Department of Mathematics, University of Cali
fornia at Santa Cruz, Santa Cruz, California 95064
Robert T. Jantzen (97), Department of Physics and Astronomy, University
of North Carolina at Chapel Hill, Chapel Hill, North Carolina 27514
E. P. T. Liang (191), Institute of Theoretical Physics, Department of
Physics, Stanford University, Stanford, California, 94305
Andre Lichnerowicz (203), Collège de France, 75014 Paris, France
Lee Lindblom (13), Department of Physics, University of California at
Santa Barbara, Santa Barbara, California 93106
M. A. H. MacCallum (121), Department of Applied Mathematics, Queen
Mary College, London El 4NS, England
Jerrold E. Marsden (79), Department of Mathematics, University of Cali
fornia at Berkeley, Berkeley, California 94720
Charles W. Misner (221), Department of Physics and Astronomy, Uni
versity of Maryland, College Park, Maryland 20742
Vincent Moncrief (79), Department of Physics, Yale University, New
Haven, Connecticut 06520
A. Papapetrou (217), Laboratoire de Physique Théorique, Institut Henri
Poincaré, 75231 Paris, France
Roger Penrose (1), Mathematical Institute, Oxford University, Oxford,
England OX1 3LB and Department of Mathematics, University of
California, Berkeley, California
Tsvi Piran* (185), Center for Relativity, University of Texas at Austin,
Austin, Texas 78712
L. C Shepley (71), Department of Physics, University of Texas at Austin,
Austin, Texas 78712
♦Present Address: Institute for Advanced Study, Princeton University, Princeton, New Jersey
08540.
XV
XVI List of Contributors
Larry Smart* (157), Departments of Astronomy and Physics, Harvard
University, Cambridge, Massachusetts 02138
Clifford Taubes (157), Department of Physics, Harvard University, Cam
bridge, Massachusetts 02138
Kip S. Thorne (139), California Institute of Technology, Pasadena, Cali
fornia 91125
Frank J. Tipler (21), Center for Theoretical Physics, University of Texas at
Austin, Austin, Texas 78712
John Archibald Wheeler (59), Center for Theoretical Physics, Department
of Physics, University of Texas at Austin, Austin, Texas 78712
James R. Wilson (157), Lawrence Livermore Laboratory, University of
California, Livermore, California 94550
James W. York, Jr. (39), Department of Physics and Astronomy, University
of North Carolina at Chapel Hill, Chapel Hill, North Carolina 27514
Mark Zimmermann] (139), W. K. Kellogg Radiation Laboratory, Cali
fornia Institute of Technology, Pasadena, California 91125
♦Present Address: Departments of Astronomy and Physics, University of Illinois, Urbana,
Illinois 61801.
fPresent Address: IDA/SED, 400 Army-Navy Drive, Arlington, Virginia 22202.
Preface
This volume is a collection of essays to honor Professor Abraham H. Taub
on the occasion of his retirement from the mathematics faculty of the
University of California at Berkeley. I am happy to report that this does not
mark his retirement from research; he is still "covering page after page with
ink," as he puts it. This means that his ideas will continue to play a seminal
role in the development of general relativity, as they have in the past.
Abraham Taub was born in Chicago in 1911 and obtained his under
graduate degree from the University of Chicago at the age of twenty. He then
went to Princeton for graduate study in mathematical physics. His thesis
advisor was the famous cosmologist H. P. Robertson. Taub's first out
standing achievements are contained in a series of papers on spinors and the
Dirac equation written between 1934, when he had just received his Ph.D.,
and 1950. Many of the ideas underlying twistor theory are implicit in these
early papers. In particular, what is now known as local twistor theory was
developed in the 1930s by Veblen, von Neumann, and Taub. He also did
fundamental work on the action of the conformai group on spinors; this work
is basic to modern developments in twistor theory.
Relativistic hydrodynamics has always been a subject dear to Taub's heart.
In fact, many basic results on special relativistic fluid flows are due to him,
and he has been a major contributor to the study of fluid flows near shocks.
The importance of Taub's work in relativistic hydrodynamics is pointed out
in the paper by Smarr, Taubes, and Wilson in this book. His interest in
hydrodynamics led him to develop variational methods in general relativity
and fluid dynamics. His research on variational methods began with his
famous 1954 paper on perfect fluids and still continues with several key review
articles written in recent years. The so-called Taub conserved quantities arose
from Taub's work on variational methods, and these quantities are playing an
essential role in the development of linearization stability theory, as discussed
in the contribution by Fischer, Marsden, and Moncrief to this volume.
Taub's single most famous paper is his 1951 Annals of Mathematics article
entitled "Empty spacetimes admitting a three parameter group of motions."
This paper gives several spatially homogeneous vacuum solutions to
Einstein's equations, one of which, when extended to "Taub-NUT" space, is
well known to every relativist. This solution has played a key role in the
development of general relativity, for in the words of C. W. Misner, it "is a
counter-example to almost anything." By providing examples of pathologies
XVII
XVIII Preface
that can occur in Einstein universes, Taub-NUT space has been crucial to
the understanding of singularities. Taub space is the simplest known exact
vacuum solution with three-sphere spatial topology and thus provides an
easily understood model of possible behavior in closed universes, as the
papers by Wheeler and Shepley demonstrate. More generally, Taub's 1951
paper gave the essential methods used in discusssing homogeneous aniso-
trophic cosmological models. See for example the papers by Jantzen and
MacCallum.
In addition to his work in classical general relativity, Taub has been active
in computer science and analysis; he was instrumental in setting up the
computer systems at the University of Illinois and at Berkeley. He was one of
the founders of the now burgeoning field of numerical general relativity.
Throughout his career, Taub has grappled with the problem of singularities
in general relativity. He has persuasively argued that the time has come to
attack the problem of singularities from the point of view of distribution
theory and that action integrals can be used to define space-time structures at
singularities. Some of my own current work is concerned with this action
integral idea of Taub's, and I feel that Taub's approach to singularities will be
an important complement to the more popular approach of Hawking and
Penrose.
Taub has also had an important influence on general relativity through his
penetrating discussions with his colleagues. These discussions are often
heated, for Taub is very definite about expressing his point of view. This
definiteness has occasionally disconcerted scientists who do not know him
well, and who thus are not aware of his great inner gentleness. In the words of
his last graduate student, Robert Jantzen, "Professor Taub has a crusty
exterior, but underneath he is just one big marshmallow!"
Some of the papers in this Festschrift were delivered at the Berkeley
Symposium on Relativity and Cosmology, held in August 1978 to honor
Professor Taub. I am grateful to my colleagues on the organizing committee
ofthat Symposium, D. Eardley, E. Liang, J. Marsden, and V. Moncrief for
their assistance in preparing this volume.
FRANK J. TIPLER
June 15, 1979
ESSAYS IN GENERAL RELATIVITY
1
On Schwarzschild Causality—A
Problem for "Lorentz Covariante
General Relativity
Roger Penrose
Mathematical Institute
University of Oxford
Oxford, England
and
Department of Mathematics
University of California
Berkeley, California
I. Introduction
There appears to be a viewpoint, prevalent among some physicists
[1] (cf. [2]), that while a geometrical approach to general relativity may
have merits on aesthetic grounds and may have appeal for those whose
interests are, perhaps, essentially pure mathematical, a strong emphasis on
curved-space geometry is nevertheless to be rejected if real physical under
standing and important future progress are to be achieved in gravitation
theory. Abe Taub, however, is clearly not of this way of thinking. His many
important contributions to both the physical and geometrical aspects of
general relativity bear strong witness to the fact that far from being an
obstacle to progress, differential geometry is an efficient and essentially
indispensable tool in this highly significant aspect of physical insight.
As a way of honoring Abe's retirement, I shall present here a result that
lends strong additional support to this geometrical viewpoint. It is directed,
particularly, against the idea that general relativity might be adequately
described as though it were a Lorentz-covariant (or, more correctly, Poincaré-
covariant) field theory according to which the physical metric tensor is to be
1
Copyright © 1980 by Academic Press, Inc.
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ISBN 0-12-691380-3
