Mathematical Cosmology and Extragalactic Astronomy (Pure and Applied Mathematics 68)

215 Pages·1976·8.12 MB·English

by  Irving Ezra Segal

Mathematical Cosmology
and Extragalactic Astronomy
Pure and Applied Mathematicm
A Series of Monographs and Textbooks
Editors ~rmueEll lenberg and Hyman Errr
Columbia University, New York
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Mathematical Cosmology
-
and
Extragalactic Astronomy
IRVING EZRA SEGAL
Dejmrtment of Mathematics
Massachusetts Institute of Technology
Cambridge, Massachuseft s
A C A D E M I C P R E S S New York San Francisco London 1976
A Subsidiary of Harcourt Brace Jovanovich. Publishers
COPYRIQHT 0 1976, BY ACADEMPICRE SS, hC.
ALLRIQHTSRBBBRVED.
NO PART OF MIS PUBLICATION MAY BE RJ5PRODUCED OR
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OR MECHANICAL, INCLUDINQ PHOTOCOPY, RECORDING, OR ANY
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PERMISSION IN WRITINQ FROM THE PUBLISHER.
United Kingdom Edition published by
ACADEMIC PRESS, INC. (LONDON) LTD.
24/28 Oval Road. Londoll NWI
Library ol Congrea Cataloging in Publication Dats
Segal, Irving Ezra.
Mathematical cosmology and extragalactic astrono-
my.
(Pure and applied mathematics series ;
Bibliography: p.
Includes index.
1. Cosmology. 2. Astronomy-Mathematics.
I. Title. I€. Series: Pure and applied mathe-
matics : a series of monographs and textbooks ;
QA3.P8 vol. (QB9811 523.1'2 75-3574
ISBN 0-12-635250-X
AMS NOS) 1970 Subject Classification: 8SA40,83F05,22E70,
78A25
PRINTED IN THE UNITED SIATe8 OF AMBRICA
Contents
Prefme
I. Generalintroduction
1. Standpoint and purpose 1
2. Causality and geometry--historical 2
3. Conformality, groups, and particics-historical 5
4. Natural philosophy of chronogeOmetric cosmology 8
5. Theuniversalcosmos-sketch 12
6. The chronometric redshift theory 14
7. Theoretical rami6cations; the cosmic background radiation 19
II. Mathematical development
1. causalorientations 22
2. cadtyingroups 28
3. Causal morphisms of groups 34
4. Causality and confonnality 37
5. Relation to Minkowski space 41
6. Observers and clocks 44
7. Localobservefs 48
V
Contents
vi
1. TheCosmos 50
2. Postulational development 51
3. Physical observers 59
4. Conformal geometry and the unitary formafim 43
5. Causal symmetries and the energy 68
6. Theredshift 75
7. LocalLorentzframes 82
8. Cosmic background radiation 85
9. Special relativity as a limiting case of unispatial theory 88
IV. Astronomical applications
1. Introduction 93
2. The redshift-magnituder elation 94
3. Further cosmological tests 98
4. The aperture correction for galaxies 104
5. Statistical effect of the selection of the brightest objects 108
6. The Peterson galaxies 109
7. Markarian galaxies and N-galaxies 114
8. The redshift-magnitude relation for nearby galaxies 117
9. The redshift-magnitude relation for Sandage’s brightest cluster
Balaxies 134
10. Preliminary discussion of quasars 140
11. The N-z relation for quasars 147
12. The apparent magnitude distribution for quasars 152
13. The redshift-luminosity relation for quasars 154
14. The redshift-number relation for quasar subsamples 158
15. The Schmidt V/V, test for quasars in the chronometric theory 163
16. The angular diameter redshift relation for double radio sources 167
17. Observation versus theory for radio sources 168
18. The Setti-Woltjer quasar classes 170
19. Other observational considerations 174
V. Discussion
1. General conclusions 181
2. Theoreticalaspects 187
3. Further observational work 194
References 197
Index 201
Preface
The broad acceptance of the expansion of the universe as a physically real
phenomenon has been rooted in part in the apparent lack of an alternative
explanation of the redshift. Since its discovery more than a half-century
ago, many new observational phenomena have been uncovered, of which
quasars and microwave background radiation appear to be particularly
fundamental and striking. Nevertheless, there seem to have been few attempts
to rework the foundations of cosmology in a way that might tie these phe-
nomena together in a scientifically more economical way. This is probably
due more to the momentum of the theoretical studies based on the expansion
theory than to its agreement with observation, which has been quite limited
and increasingly equivocal.
In this book I present a new theory that is very different from the ex-
pansion theory, though equally rooted in the ideas of relativity going back to
Einstein, Minkowski, Robb, Veblen, and others. The specific germinal point of
the theory was the observation I made 25 years ago that a more natural oper-
ator to represent the physical energy than the conventional generator of
temporal translation in Minkowski space was a certain generator of the
conformal group that physically closely approximates the conventional
energy. It has taken a long time to realize that the redness of the observed
shift follows from a law of conservation of the new, essentially curved,
energy, which necessarily involves a depletion in the old, essentially flat,
energy, which is all that can locally be measured and directly observed.
This book is however not merely, or even basically, the presentation of a
Viii Preface
new model. It is in large part an attempt to lay rational foundations for
cosmology on the basis of the most elementary types of causality and related
symmetry considerations. It is extraordinary how incisive these qualitative
desiderata turn out to be, when integrated with the modem theory of trans-
formation groups. On the purely physical side, the key concepts of time and of
its dual energy are given a new precision of definition and treatment that re-
moves much apparent mystery and, in particular, partially mechanizes the
murky but important matter of the correlation of mathematical with obser-
vational quantities. The title of my original abstract, Covuriunt chrono-
geometry and extreme distances, summarizes this natural philosophic stand-
point, but a corresponding treatment of very small distances (i.e., elementary
particles) will require much further exploration.
The new chronometric theory emerges in a unique way from this stand-
“ ”
point. It has been interesting to test it against virtually all available relevant
astronomical data and to find that, despite its lack of adjustable parameters
(other than the unit of distance), it is accepted, in the sense of the theory of
statistical hypotheses, by all large or objectively defined samples of galaxies
or quasars, indeed at notably high probability levels. In the cases of samples
less amenable to rigorous statistical treatment, it typically provides a dis-
dinctly better fit to the data than does the expansion theory, with its free pa-
rameter qo , with one equivocal exception. From an overall scientific point of
view, it has been reassuring to find that a fully rational approach to cosmology
can lead to physical predictions that conform to observation, and that modem
statistical theory is a vital aid in comparing theory with observation, rather
than, as appears to be the outlook of some astronomers, an annoying hind-
rance.
One reason for the delay in promulgating the new theory was that initially
one of its predictions appeared in flat contradiction with observation. It
implied a square-law redshift4stance dependence for sufficiently small
distances, whereas it was “well known” that the relation was observed to
be linear. But the mathematical unicity and simplicity of the model, together
with its immediate success in dealing with quasars, gave grounds for further
exploration of the theory. It has been quite reassuring to find confirmation
for the square law in a number of observational studies at moderate red-
shifts, and overwhelming evidence for a phenomenological square law in the
case of low-redshift galaxies. (Of course, actual distances are not directly
observable, but the implications of the respective laws for the relations be-
tween the redshift, number, apparent luminosity, and angular diameter of
luminous objects may be compared with actual observations.) Hubble’s
original (1929) derivation of the linear law was based on 22 of the more than
700 galaxies included in the low-redshift analysis, and it now appears that the
linear law was of a much more tentative character than has been generally
Preface ix
realized. The later observations by his associates and successors must be seen
in the context of a natural tendency to seek the validation and development
of a previously indicated hypothesis, rather than to explore possible alter-
natives.
I have had useful discussions with many astronomers and mathematicians,
and some physicists and statisticians. Particularly valuable assistance was
provided by J. F. Nicoll who contributed many corrections to the entire
manuscript, assisted in several computations, and kindly permitted the
inclusion here of jointly developed material, as noted below. I am also spe-
cially indebted to N. S. Poulsen for many corrections to Chapter I1 and to
L. Hormander, B. Kostant, J. W. Milnor, S. Sternberg, and J. L. Tits for
stimulating mathematical comment. Z am grateful for the astronomical
criticism and information conscientiously given by E. Holmberg, C. C. Lin,
D. Lynden-Bell, P. Morrison, G. Setti, B. Strarmgren, and others. More
formal thanks are due the Universities of Copenhagen and Lund,
where the present study was drafted in 1971-1972, the Scubla Normale of
Pisa, the I.H.E.S. of Bures-sur-Yvette, and the University of Warwick,
England, where it was continued, and the National Science Foundation
where my research was partially supported.