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

215 Pages·1976·8.12 MB·English
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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 RECENT TITLES XIA DAO-XINGM. easure and Integration Theory of Infinite-Dimensional Spaces : Abstract Harmonic Analysis RONALDG . DOUGLASB.a nach Algebra Techniques in Operator Theory WILLARMDI LLERJ, R. Symmetry Groups and Their Applications ARTHURA . SAGLAEND RALPHE . WALDEIn troduction to Lie Groups and Lie Algebras T. BENNYR USHINGT. opological Embeddings JAMES W. VICK. Homology Theory: An Introduction to Algebraic Topology E. R. KOLCHIND. ifferential Algebra and Algebraic Groups GERALJD. JANUSZ. Algebraic Number Fields A. S. B. HOLLANIDnt.r oduction to the Theory of Entire Functions WAYNER OBERTAS ND DALEV ARBERGC.o nvex Functions A. M. OSTROWSKSIo. lution of Equations in Euclidean and Banach Spaces, Third Edition of Solution of Equations and Systems of Equations H. M. EDWARDRiSem. ann’s Zeta Function SAMUEELIL ENBEARuGto. mata, Languages, and Machines : Volume A. In preparation: Volume B MORRISH IRSCHA ND STEPHESNM ALED.i fferential Equations, Dynamical Systems, and Linear Algebra WILHELMM AGNUSN. oneuclidean Tesselations and Their Groups J. DIEUDONNET.r eatise on Analysis, Volume IV FUNGOTISR EVEBSa.si c Linear Partial Differential Equations WILLIAMM . BOOTHBYA. n Introduction to Differentiable Manifolds and Riemannian Geometry BRAYTOGNR AYH. omotopy Theory : An Introduction to Algebraic Topology ROBERAT . ADAMS.S obolev Spaces JOHN J. BENEDETTSOp.e ctral Synthesis D. V. WIDIBERT.h e Heat Equation IRVINEGZ RAS EGALM. athematical Cosmology and Extragalactic Astronomy In preparation WERNEGR REUBS, TEPHEHNA LPERINA. ND RAYV ANSTONEC.o nnections, Curvature, and Cohomology : Volume 111, Cohomology of Principal Bundles and Homogeneous Spaces J. DIEUDONTNr~ea tise on Analysis, Volume 11, enlarged and corrected printing I. MARTINI SAACCSh. aracter Theory of Finite Groups 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 TIUNS- IN ANY FORM OR BY ANY MEANS, ELECTBONK OR MECHANICAL, INCLUDINQ PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAGE A m R ETRIEVAL SYSTEM, WmOUT 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.

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