zotero/storage/G3UY92LK/.zotero-ft-cache

THE UNIVERSE

ASTROPHYSICS AND SPACE SCIENCE LIBRARY
V O L U M E 244
EDITORIAL BOARD
Chairman W. B. BURTON, Sterrewacht, Leiden, P.O. Box 9513, 2300 RA Leiden, The Netherlands
Burton @ strw.leidenuniv.nl
Executive Committee J. M. E. KUIJPERS, Faculty of Science, Nijmegen, The Netherlands E. P. J. VAN DEN HEUVEL, Astronomical Institute, University ofAmsterdam,
The Netherlands H. VAN DER LAAN, Astronomical Institute, University of Utrecht,
The Netherlands
MEMBERS I. APPENZELLER, Landessternwarte Heidelberg-Königstuhl, Germany
J. N. B A H C A L L , The Institute for Advanced Study, Princeton, U.S.A. F. BERTOLA, Universitd di Padova, Italy
J. P. CASSINELLI, University of Wisconsin, Madison, U.S.A. C. J. CESARSKY, Centre d'Etudes de Saclay, Gif-sur-Yvette Cedex, France O. ENGVOLD, Institute of Theoretical Astrophysics, University of Oslo, Norway
R. McCRAY, University of Colorado, JILA, Boulder, U.S.A. P. G. MURDIN, Royal Greenwich Observatory, Cambridge, U.K.
F. PACINI, Istituto Astronomia Arcetri, Firenze, Italy V RADHAKRISHNAN, Raman Research Institute, Bangalore, India
K. SATO, School of Science, The University of Tokyo, Japan F. H. SHU, University of California, Berkeley, U.S.A.
B. V. SOMOV, Astronomical Institute, Moscow State University, Russia R. A. SUNYAEV, Space Research Institute, Moscow, Russia
Y. TANAKA, Institute of Space & Astronautical Science, Kanagawa, Japan S. TREMAINE, CITA, Princeton University, U.S.A. N. O. WEISS, University of Cambridge, U.K.

THE UNIVERSE
Visions and Perspectives
Edited by NARESH DADHICH Inter-University Centre for Astronomy and Astrophyics,
Pune, India and
AJIT KEMBHAVI Inter-University Centre for Astronomy and Astrophyics,
Pune, India
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.

A CLP. Catalogue record for this book is available from the Library of Congress.

ISBN 978-94-010-5784-4

ISBN 978-94-011-4050-8 (eBook)

DOI 10.1007/978-94-011-4050-8

Courtesy of Prof. W J . Couch, The University of New South Wales.
Printed on acid-free paper
All Rights Reserved © 2000 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 2000 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner.

Contents

Preface

ix

1

OBSERVATIONS AND THEORY

1

Halton Arp

2

EJECTION FROM ULTRALUMINOUS INFRARED GALAXIES

7

Halton Arp

3

QUANTUM MECHANICS OF GEOMETRY

13

A. Ashtekar

4

QUANTUM MECHANICS AND RETROCAUSALITY

35

D. Atkinson

5

INSTANTONS FOR BLACK HOLE PAIR PRODUCTION

51

Paul M. BranofJ and Dieter R. Brill

6

THE ORIGIN OF HELIUM AND THE OTHER LIGHT ELEMENTS

69

G. Burbidge F. Hoyle

7

SUPERLUMINAL MOTION AND GRAVITATIONAL LENSING

77

S. M. Chitre

8

DUAL SPACETIMES, MACH'S PRINCIPLE AND TOPOLOGI-

87

CAL DEFECT

Naresh Dadhich

v

VI THE UNIVERSE

9

NONCOSMOLOGICAL REDSHIFTS OF QUASARS

97

Prashanta Kumar Das

10

EXTRAGALACTIC FIRE-WORKS IN GAMMA-RAYS

105

Patrick Das Gupta

11

INSTABILITIES IN OPTICAL CAVITIES OF LASER INTER-

111

FEROMETRIC GRAVITATIONAL WAVE DETECTORS

S. V. Dhurandhar

12

THE EPISTEMOLOGY OF COSMOLOGY

123

George F. R. Ellis

13

MATHEMATICS AND SCIENCE

141

Fred Hoyle

14

RADIATION REACTION IN ELECTRODYNAMICS AND GEN-

145

ERAL RELATIVITY

Bala R. Iyer

15

GRAVITATIONAL COLLAPSE: THE STORY SO FAR

161

Pankaj S. Joshi

16

THOUGHTS ON GALACTIC MAGNETISM

169

K andaswamy Subramanian

17

THE BLACK HOLE IN MCG6-30-15

181

Ajit Kembhavi and Ranjeev Misra

18 INHOMOGENEOUS COSMOLOGICAL MODELS AND SYMMETRY 191 S. D. Maharaj

19

THE BLACK HOLE INFORMATION PARADOX: WHAT HAVE

201

WE LEARNT FROM STRING THEORY?

Samir D. Mathur

20

THE COUNTING OF RADIO SOURCES: A PERSONAL

213

PERSPECTIVE

Jayant V. Narlikar

Contents Vll

21

A VARIATIONAL PRINCIPLE FOR TIME OF ARRIVAL OF

227

NULL GEODESICS

Ezra T. Newman Simonetta Frittelli

22

CONCEPTUAL ISSUES IN COMBINING GENERAL RELATIV-

239

ITY AND QUANTUM THEORY

T. Padmanabhan

23

OPEN INFLATION IN HIGHER DERIVATIVE THEORY

253

B. C. Paul and S. Mukherjee

24

THE NON-HOMOGENEOUS AND HIERARCHICAL UNIVERSE

261

Jean-Claude Peeker

25

ELECTROMAGNETIC WAVE PROPAGATION IN GENERAL

277

SPACETIMES WITH CURVATURE AND/OR TORSION (U4)

A. R. Prasanna and S. Mohanty

26

A FRESH LOOK AT THE SINGULARITY PROBLEM

285

A. K. Rayehaudhuri

27

PROBING BEYOND THE COSMIC HORIZON

289

Tarun Souradeep

28

THE KERR-NUT METRIC REVISITED

301

P. C. Vaidya and L. K. Patel

29

BLACK HOLES IN COSMOLOGICAL BACKGROUNDS

309

C. V. Vishveshwara

30

ELEMENTARY PARTICLE INTERACTIONS

319

AND NONCOMMUTATIVE GEOMETRY

K ameshwar C. Wali

31

FROM INTERSTELLAR GRAINS TO PANSPERMIA

327

N. C. Wiekramasinghe

Preface
It is with great joy that we present a collection of essays written in honour of Jayant Vishnu Narlikar, who completed 60 years of age on July 19, 1998, by his friends and colleagues, including several of his former students. Jayant has had a long research career in astrophysics and cosmology, which he began at Cambridge in 1960, as a student of Sir Fred Hoyle. He started his work with a big bang, expounding on the steady state theory of the Universe and creating a new theory of gravity inspired by Mach's principle. He also worked on action-at-a-distance electrodynamics, inspired by the explorations of Wheeler, Feynman and Hogarth in that direction. This body of work established Jayant's reputation as a bold and imaginative physicist who was ever willing to take a fresh look at fundamental issues, undeterred by conventional wisdom. This trait, undoubtedly inherited from his teacher and mentor, has always remained with Jayant. It is now most evident in his untiring efforts to understand anomalies in quasar astronomy, and to develop the quasi-steady state cosmology, along with a group of highly distinguished astronomers including Halton Arp, Geoffrey Burbidge and Fred Hoyle. In spite of all this iconoclastic activity, Jayant remains a part of the mainstream; he appreciates as well as encourages good work along conventional lines by his students and colleagues. This is clear from the range of essays included in this volume, and the variety and distribution of the essayists.
After a long stay in Cambridge, Jayant moved to the Tata Institute of Fundamental Research in Mumbai (then Bombay) in 1972. There he inspired several research students to work in gravitational theory and its many classical and quantum applications to cosmology and astrophysics, and established collaborations with his peers, which led to a fine body of work over the next 15 years. But perhaps his most enduring contribution of this period was to forge a link between distinguished senior
ix

x THE UNIVERSE
relativists in India, and the younger generation of aspiring researchers. This has led to the formation of a warm and congenial community, spread throughout the country, working in relativity, cosmology and theoretical astrophysics. During this period Jayant also worked hard at the popularization of science, through the press, television and most importantly through talks to ever increasing audiences. This not only exposed people to good science, but it also helped to establish Jayant as one of the public figures of science in India. Jayant has used his formidable reputation and influence, developed during this period, for the advancement of science in India, always in a very quiet manner.
In 1988, inspired and aided by Professor Yashpal, then Chairman of the University Grants Commission, Jayant set up the Inter-University Centre for Astronomy and Astrophysics at Pune. Through this centre he has been able to open up for the university community avenues for excellent research in these areas. Jayant's broad vision, and his readiness to encourage every shade of opinion and to bring out the best in his colleagues, has enabled IUCAA to develop an international reputation. The centre is now seen as an example of how the energies of the research institutes and universities in India, usually considered disparate, could be harnessed together to excellent effect.
It is the general practice to list, in a volume of this kind, the scientific and other works of the person it seeks to honour. The list in the present case would have been rather unusually long, and we have therefore decided, in consultation with Jayant, that we will enumerate only his scientific books. These expose much of the work he has presented elsewhere in the form of research papers and review articles. They also present highly readable and often pedagogic accounts of modern astrophysics, and will surely continue to be read for a long time to come. Amongst the works that we will leave unlisted will be his contributions to the annals of science fiction, which have helped much to endear him to the general public. In this matter too Jayant has followed in the steps of Fred Hoyle.
Naresh Dadhich Ajit Kembhavi

Xl
Acknowledgments
We wish to thank Professor K. S. V. S. Narasimhan for a careful reading of the manuscript.

Chapter 1
OBSERVATIONS AND THEORyl
Halton Arp
Max-Planck Institut for Astrophysik Garching, Germany
1. INTRODUCTION
The most predictable observation concerning theories is that they will probably always turn out to be wrong. From Ptolemy to phlogisten these excercises have wasted untold model calculations and obsoleted endless sermons. Nevertheless, for the last 77 years, eschewing all humility, orthodox science has insisted on the theory that the entire universe was created instantaneously out of nothing. Observations for the last 33 years have shown this to be wrong - but these basic facts of science have been rejected on the grounds there was no theory to "explain" them.
Since 1977, however, there has not even been this feeble excuse for abandoning empiricism. That was the year in which Jayant Narlikar published a short paper in Annals of Physics (107, p325). The paper outlined how a more general solution of the equations of general relativity permitted matter to be "created" i. e. enter a black hole and remerge somewhere from a white hole without passing through a singularity where physics just broke down. This was not just another play with words because it turned out that the newly created matter would have to have a high intrinsic redshift. The latter is just what observations with optical and radio telescopes had been requiring since 1966!
As contradictory cases mounted over the years, the Big Bang theory had to be rescued by postulating an ever increasing number of adjustable parameters. As a consequence there is today a giant tsunami of evidence cresting above the Big Bang. It demonstrates continual creation of galaxies and evolution of intrinsic redshift in an indefinitely old and
1Editors' note: Dr. Halton Arp has requested that his contribution be presented as two separate articles, which we do in this chapter and the next.
N. Dadhich and A. Kembhavi (eds.), The Universe, 1-6.
© 2000 Kluwer Academic Publishers.

2 THE UNIVERSE
large universe. By now we can start anywhere with this evidence so let us start with new results on a class of objects called active galaxies.
2. ACTIVE GALAXIES
In the preceding paper, preliminary investigation of two Ultra Luminous Infrared Galaxies (ULIRG"s) are reported. It is clear that these very disturbed objects are being torn apart during the process of ejecting high redshift quasars. Empirical evolutionary sequences show that the ULIRG's themselves are very active galaxies recently evolved from quasars. Therefore they also possess an appreciable component of intrinsic redshift. Conventionally this redshift gives too large a distance and this is why these objects are considered to be so "overluminous". As we shall comment later, however, they do not look at all like the most luminous galaxies of which we have certain knowledge. Instead they resemble small, active companion galaxies to larger, older parent galaxies. For example, Markarian 273 is an obvious companion to the large, nearby spiral, Messier 101.
The defining characteristic of active galaxies is that they show enormous concentrations of energy inside very small nuclei. They also show optical, radio and X-ray jets and plumes of material emerging from their centers. The latter is not surprising since the concentrated energy must expand and escape somehow. It has been accepted for about 40 years that active galaxies eject radio material so it is difficult to understand why the ejections associated with quasars are not recognized. But the expulsion of material is clearly responsible for the disrupted appearance of the active galaxies. Why then does conventional astronomy make an enormous industry out of a completely different, ad hoc explanation for morphologically disturbed galaxies - namely mergers!
3. MERGERS?
What is the conventional view of disturbed galaxies and ULIRG's? It is that two independent galaxies are merging. One galaxy sees another and heads directly for it. In its excitement it forgets about angular momentum and unerringly scores a direct hit. To judge how reasonable this is one could ask how many comets are perturbed into the solar system and proceed to plunge directly into the sun?
In all honesty, however, I must admit that my long term scorn for the merger scenario has been tempered by recent evidence on ejection from active galaxies. For many years it was clear that there was a tendency for galaxies to eject along their minor axes. But recently there have been some cases where ejection has been aligned with striking accuracy

OBSERVATIONS AND THEORY 3
along the minor axis (6 quasars from NGC3516 , Chu et al. 1998, and five Quasars and four companion galaxies from NGC5985, Arp 1999). It is clear that proto galaxies ejected exactly out along the minor axis, and evolving into companion galaxies as they eventually fall back (Arp 1997;1998) will have little or no angular momentum and therefore move on plunging orbits. Their chances of colliding with the parent galaxy are therefore much greater than if they were field galaxies. So maybe there is some usefulness after all to those many detailed calculations which have been carried out on colliding galaxies.
But when the ejection of protogalactic material takes place in the plane or tries to exit through the substance of the parent galaxy then an entirely different scenario develops. Using the low mass creation theory, one can now begin to connect these events with previously uninterpretable observations.
4. SUPERFLUID
In 1957 the famous Armenian astronomer Ambartsumian concluded from looking at survey photographs that galaxies were formed by ejection from other galaxies. As an accomplished astrophysicist he realized that would require ejection in an initially non-solid form form but with properties different from a normal plasma. He called it "superfluid". In spite of general agreement that Ambartsumian was a great scientist his important conclusion about the formation of galaxies has been ignored.
But now the Hoyle-Narlikar variable mass theory is required to explain the high intrinsic redshifts of the quasars ejected from galaxies. The creation of mass in the centers of galaxies with this same variable mass theory then also solves the major problem which must have caused Ambartsumian to use the term "superfluid", namely that a normal, hot plasma expanding from the small dimensions of a galaxy nucleus would not have been able to condense into a new galaxy. In contrast, as the particles in the newly created plasma age they gain mass and, in order to conserve momentum, must slow their velocity. This means the plasma cools as it ages and also its self gravitation increases - both factors working in the direction of condensing the material into a proto galaxy.
The second major obstacle overcome by starting the particles off with near zero mass is the initial velocity of ejection. Observations have shown examples of ejected material in jets approaching closer and closer to the speed of light. Physicists believe that as a particle approaches the speed of light its mass must approach infinity. In other words one has to pump an enormous amount of energy into a huge number of particles to get

4 THE UNIVERSE

the velocities (gamma factors) which are implied by the observations. If the particles are initially near zero mass, however, they are almost all energy and are emerging naturally with near the signal velocity, c.
In M87, the very strong radio galaxy in the Virgo Cluster, knots in the jet have been measured by their proper motion to have apparent outgoing velocities of 5 to 6 c. But further out along this jet we find quasars and companion galaxies which the knots must evolve into. Now, however, all the calculations based on the assumption that the knots consist of normal plasma will have to be redone with a low mass plasma, e.g. the calculations of supposed shock fronts and containment envelopes. (See Arp 1998,1999)

5.

EXPLODING GALAXIES

There is a strong (and in some cases almost perfect) tendency for quasars to be initially ejected out along the minor axis and also ordered in descending redshift with angular separation. Nevertheless there are some cases where quasars are found close to their galaxy of origin but not ordered in redshift. The key to understanding this situation lies in the observation that the nearby galaxy of origin is usually spectacularly disrupted. What could cause this disruption? The obvious inference is that the process of ejection has, somehow, fragmented the galaxy when the ejection is not out along the minor axis.
At this particular point the usefulness of the variable mass theory becomes especially apparent. We are able to visualize a cloud of low particle mass material pushing out against the material of the galaxy, initially with velocity c. Low mass particle cross sections are large and eject and entrain the material of the galaxy into long, emerging jets. The initial pulse of energy concentrated at the center of the galaxy plus the sudden decentralization of mass explodes and tears asunder the parent galaxy. Moreover, the new material is retarded and fragmented so that it develops into many smaller new proto galaxies much closer to the, by now, thoroughly disrupted galaxy. This is the case where the new material does not exit along the minor axis. This is exactly what is observed as shown here in Figures 1 and 2.
Here the disrupted galaxy is 53W003 (a blue, radio, galaxy). As the picture shows it has been disrupted into at least three pieces. A pair of almost perfectly aligned quasars of z = 2.389 and z = 2.392 have apparently come out fairly unimpeded. (There are, as expected, brighter quasars of z = 1.09 and z = 1.13 about 7 arcmin further along in this direction). The rest of the quasars, about 18 similarly high redshift objects, have wound up in a cloud only about 1.5 arcmin from the disrupted

OBSERVATIONS AND THEORY 5
Figure 1.1 Part ofa4m PF-CCD field in the F410M filter (4150A, filter width 150A) . The WFPC2 search fields are outlined - plus signs show non-AGN Ly Q emitters. Quasars in the cluster are circled with z marked. From Keel et al. 1998.
Figure 1.2 Enlargement of z = .05 galaxy in Fig.1. Note how this blue radio galaxy,
53W003, has multiple components. Image courtesy W. Keel.

6 THE UNIVERSE

galaxy. Evidently they represent some low mass plasrna that was broken up into smaller clouds in its violent exit from the galaxy. In support of this scenario, high resolution, Hubble Telescope images of these high redshift objects show them to be blue and irregular. At their conven-
tional redshift distance they would have absolute magnitudes of M =
-24 mag. - well into the supposed quasar range of luminosity. Yet they have an extended morphology, whereas, in general, brighter quasars of the same redshift are point-like.
More broadly, this leads me to comment that the faint images in the famous Hubble Deep Field exposure which have such large redshifts are of predominantly blue, irregular morphology. At their conventional redshift distance they should be enormously luminous. But all our experience with genuinely luminous galaxies indicates such galaxies should be massive, relaxed, equilibrium forms - like E galaxies, or at least Sb's. These Hubble Deep Field objects have ragged, irregular looking dwarf morphology. Instead of a new kind of object suddenly discovered in the universe would it not be plausible that they are really relatively nearby dwarfs but simply have high redshifts because they are young?

6.

A USEFUL THEORY

Speaking for myself, the Narlikar general solution of the relativistic field equations has been a salvation. It has opened up possibilities of understanding the observational facts - facts which must be accounted for if we are to have a science. In the dogma of current astronomy, evidence no matter how many times confirmed, cannot be accepted if it does not fit Big Bang assumptions. With the the variable mass theory, however. essentially all the salient observational facts can be related to each other in a physically understandable, reasonable way. Even if it is only a stepping stone to a future, deeper theory - I must say, thank you Jayant.

References

[1] Arp, H., 1997, Joum. Astrophys. Astron., 393. [2] Arp, H., 1998, Seeing Red: Redshifts, Cosmology and Academic Sci-
ence, Apeiron, Montreal. [3] Arp, H. 1999, Astrophys. J. , submitted.
Keel W .. Windhorst R.. Cohen S.. Pascarelle S. & Holmes M. 1998. NOAO Newsletter 53, l.
Narlikar. J. V. 1977, Ann. Phys. 107, 325.

Chapter 2
EJECTION FROM ULTRALUMINOUS INFRARED GALAXIES

Halton Arp
Max-Planck Institut fur Astrophysik Garching, Germany

Abstract

Active galaxies, particularly Seyferts, have been shown to eject material in various forms including quasars with high intrinsic redshifts. A class of active galaxy which has so far not been analyzed from this standpoint is the so called Ultra Luminous Infrared Galaxies (ULRIG's). Here we report the very beginning of an analysis of the three most luminous examples of such galaxies. Aided by the availability of the new VLA all sky radio surveys it is clear that these ULRIG's show especially strong evidence for ejection in optical, radio and X-ray wavelengths . These ejections are strikingly connected with adjacent quasars, both with those of known redshifts and those which are candidate quasars waiting to be confirmed.

1.

MARKARIAN 273

This is a torn apart galaxy with a brilliant, long optical jet. At a conventional distance corresponding to its redshift (z = .038) it is one of the most luminous galaxies known in red wavelengths. Hence it is called an Ultra Luminous Infrared Galaxy (ULIRG). When observed in X-rays the galaxy has an active center. Only 1.3 arcmin NE, right at the end of a broad optical filament, lies another X-ray source (see Figure 2.1). When the spectrum of this companion (Mark273x) was taken it was reported as z = .038, the same as the central galaxy. Naturally this was interpreted as showing that Mark273x was a "dwarf" Seyfert interacting with Mark273. Fortunately the investigators checked the spectrum (Xia et al. [2], [3]). They found they had accidentally measured an HII region

7
N. Dadhich and A. Kembhavi (eds.), The Universe, 7-12.
© 2000 Kluwer Academic Publishers.

8 THE UNIVERSE

... ",

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...

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o

:0 .. l.'
_ .. ...15~ oa·rcsec .

Figure 2.1 Copy of R film from POSSII. The X-ray contours around Mark273x (upper left) and Mark273 (center) are from Xia et al. [21. Redshifts of each object as measured by Xia et al. [31. Photographs to fainter surface brightnesses show luminous material extending in the direction of, and almost to, Mark273x.

in Mark273 and that Mark273x was actually a high redshift object of z = .458.
As in untold numbers of similar cases, as soon as the high redshift of the companion was discovered it was relegated to the background as an unassociated object. But, embarassingly, in this case it had already been claimed to be associated at the same distance. Tracking down the X-ray map of this system revealed at a glance that the z = .038 galaxy and the z = .458 companion were elongated toward each other! Moreover there was a significant excess of X-ray sources around the active central galaxy indicating further physically associated X-ray sources. Two of the brightest lay only 6.2 and 6.6 arcmin to the SE. The first was a catalogued quasar of z = .941 and the second an obvious quasar candidate whose redshift needs to be measured. As shown in Figure 2.2 there are both X-ray and radio jets emanating from Mark273 in the direction of these two additional quasars. Moreover the fainter radio emissions form two separate filaments leading directly to the two quasars. On a deep optical plate one can see the beginning of these two filaments starting SE from the strong optical jet which dominates

EJECTION FROM GALAXIES 9
o .,
Figure 2.2 Radio map from the NRAO VLA Sky Survey (NVSS). The four brightest X-ray sources in the region are marked with X's. The direction of the X-ray jet from Mark273 is indicated by an arrow. Faint radio filaments lead southeastward to the
quasar (z = 0.941) and the the quasar candidate (V = 1B.1 mag.) . This is generally
along the line of the main radio and X-ray extensions from Mark273. Note also the exact alignment of Mark273x and the strong radio source to the SW across Mark273.
Mark273. (See deep R photograph of Mark 273 on web page of John Hibbard, www.cv.nrao.eduj jhibbard)
This active galaxy appears to be ejecting optical, X-ray and radio material in two roughly orthogonal directions. (Note the exact alignment of 273x with the strong radio source to the SW of Mark273.) Associated with these ejections are high redshift quasars and quasar-like objects. Although all of these kinds of ejections have been observed many time before (see Arp [1] for a review), the ULIRG galaxies seem to be especially active. The authors of the original paper measuring Mark273x (Xia et al. [2]) report that in correlating ROSAT X-ray sources with ULIRG's: " .. .we find that some ULIRG's have soft X-ray companions within a few arcminutes of each source" and "This phenomenon was first mentioned by Turner, Urry and Mushotzky (1993) ...". Later (Xia et al. [3]) state: "It is interesting to note in passing that the X-ray companions of the three nearest ULIGs (Arp 220, Mrk 273 and Mrk 231) are all background sources..." .
Just a glance at two of the other most luminous ULIRG's (Mark231 and Arp220) shows similar evidence for ejection from these enormously

10 THE UNIVERSE

Figure 2.3 High resolution radio map centered on Mark231 (at 20 em from VLA FIRST). Note puff of radio material just below the ULIRG and double nature of radio sources paired across Mark231.

disturbed galaxies. I will show now some preliminary evidence for Mark231 but it is already clear that there appear to be strong X-ray sources, radio ejections and physically associated high redshift objects connected to all three of these ULIRG's.

2.

MARKARIAN 231

Figure 2.3shows a 30x30 arcmin radio map around Mark231. The images are high resolution 20cm from the VLA FIRST survey (www.nrao.edu) . The brightest object in the center is Mark231. There is a puff of radio material immediately below the galaxy. Forming a striking pair across Mark231 are radio sources both of which are close doubles. The multiplicity of these flanking sources is unusual and suggests secondary ejection. At the least these radio sources are strongly indicated to be associated with the central, active galaxy.
Figure 2.4 shows an approximately 19x19 arcmin continuum radio map at lower resolution but fainter surface brightness. Here we see a
continuous radio extension to the East of Mark231 including the multiple
source seen previously on the higher resolution map. In addition we see a radio extension to the West, in the direction of the strong, close double source. There is also a string of small sources extending northward

EJECTION FROM GALAXIES 11

@
57

MARK 231

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Figure 2.4 Contour maps of low surface brightness radio material around Mark231.
Continuous connection of radio material to the east of the galaxy contains blue quasar
candidates with the labeled, V apparent magnitudes. The remaining radio sources
have quasar candidates at the marked positions. The strong double source to the
west falls close to a quasar candidate of V = 16.4 mag. The only catalogued quasar in the field is faint and of z = 1.27.

from the central galaxy. We appear to be seeing another example of ejection in roughly orthogonal directions. (It is interesting to note that at FIRST resolution the strong radio source opposite Mark273x is also a close double.)
Two color APM finding charts have been centered at the positions of some of the radio sources indicated in Fig. 4. The charts reveal blue, candidate quasar images quite close to the radio positions. They are labeled in Fig. 4 with plus signs and the apparent visual magnitude of the candidate. They need to be analyzed spectroscopically but it can already be noted that the candidate at the position of the eastern radio lobe (V=19.3 mag.) is very blue and therefore highly probable. The strong western (double) source is close to a bright (V=16.4 mag.) candidate which has fainter candidates aligned across it - suggestive again of secondary ejection. The only catalogued quasar has z = 1.27 and is located in the direction of the western radio extension from Mark231. The X-ray maps are in the process of being analyzed and will undoubtedly add considerably to the understanding of the Mark231 region. Similarly, X-ray and radio maps of Arp220 are being analyzed and together with

12 THE UNIVERSE

Mark273 and Mark231 will form a representative sample of the most active infrared excess galaxies.

3.

CONCLUSION

In the case of the tendency for long lines of ordered quasars to come out along the minor axes of disk galaxies [1] it was suggested that ejections encountered the least resistance along this spin axis. It is suggested here that if the ejections try to penetrate any appreciable material in the parent galaxy that they will expel and entrain gas and dust and dynamically rupture the galaxy. The production of new material in the centers of such galaxies would then then be responsible for the energetic X-ray and radio jets, the explosive morphology and the numbers of high energy, intrinsically redshifted quasars found nearby.

References

[1] Arp, H., 1998, Seeing Red: Redshifts, Cosmology and Academic Science, Apeiron, Montreal.
[2] Xia, X.-Y., Boller, T., Wu, H., Deng, Z.-G., Gao, Y., Zou, Z.-L., Mao, S. and Boerner, G., 1998, Astmphys. J. 496, L99.
[3] Xia, X.-Y., Mao, S., Wu, H., Liu, X.-W., Deng Z.-G., Gao, Y., Zou, Z.-L., 1999, Astrophys. J. , in press.

Chapter 3 QUANTUM MECHANICS OF GEOMETRY
A. Ashtekar
Center for Gravitational Physics and Geometry Department of Physics, The Pennsylvania State University University Park, PA 16802, USA
It is a pleasure to dedicate this article to Professor Jayant Narlikar on the occasion of his 60th birthday.

Abstract

Over the past six years, a detailed framework has been constructed to unravel the quantum nature of the Riemannian geometry of physical space. A review of these developments is presented at a level which should be accessible to graduate students in physics. As an illustrative application, I indicate how some of the detailed features of the microstructure of geometry can be tested using black hole thermodynamics. Current and future directions of research in this area are discussed.

1. INTRODUCTION
During his G6ttingen inaugural address in 1854, Riemann [1] suggested that geometry of space may be more than just a fiducial, mathematical entity serving as a passive stage for physical phenomena, and may in fact have direct physical meaning in its own right. General relativity provided a brilliant confirmation of this vision: curvature of space
13 N. Dadhich andA. Kembhavi (eds.), The Universe, 13-34.
© 2000 Kluwer Academic Publishers.

14 THE UNIVERSE
now encodes the physical gravitational field. This shift is profound. To bring out the contrast, let me recall the situation in Newtonian physics. There, space forms an inert arena on which the dynamics of physical systems -such as the solar system- unfolds. It is like a stage, an unchanging backdrop for all of physics. In general relativity, by contrast, the situation is very different. Einstein's equations tell us that matter curves space. Geometry is no longer immune to change. It reacts to matter. It is dynamical. It has "physical degrees of freedom" in its own right. In general relativity, the stage disappears and joins the troupe of actors! Geometry is a physical entity, very much like matter.
Now, the physics of this century has shown us that matter has constituents and the 3-dimensional objects we perceive as solids are in fact made of atoms. The continuum description of matter is an approximation which succeeds brilliantly in the macroscopic regime but fails hopelessly at the atomic scale. It is therefore natural to ask: Is the same true of geometry? If so, what is the analog of the 'atomic scale?' We know that a quantum theory of geometry should contain three fundamental constants of Nature, c, G, Ii, the speed of light, Newton's gravitational constant and Planck's constant. Now, as Planck pointed out in his celebrated paper that marks the beginning of quantum mechanics, there is
a unique combination, £p = JIiG/ c3 , of these constants which has di-
mension oflength. (£p ~ 1O-33cm.) It is now called the Planck length. Experience has taught us that the presence of a distinguished scale in a physical theory often marks a potential transition; physics below the scale can be very different from that above the scale. Now, all of our welltested physics occurs at length scales much bigger than than £p. In this regime, the continuum picture works well. A key question then is: Will it break down at the Planck length? Does geometry have constituents at this scale? If so, what are its atoms? Its elementary excitations? Is the space-time continuum only a 'coarse-grained' approximation? Is geometry quantized? If so, what is the nature of its quanta?
To probe such issues, it is natural to look for hints in the procedures that have been successful in describing matter. Let us begin by asking what we mean by quantization of physical quantities. Take a simple example -the hydrogen atom. In this case, the answer is clear: while the basic observables -energy and angular momentum- take on a continuous range of values classically, in quantum mechanics their eigenvalues are discrete; they are quantized. So, we can ask if the same is true of geometry. Classical geometrical quantities such as lengths, areas and volumes can take on continuous values on the phase space of general relativity. Are the eigenvalues of corresponding quantum operators discrete? If so, we would say that geometry is quantized and the precise eigenvalues and

QUANTUM MECHANICS OF GEOMETRY 15
eigenvectors of geometric operators would reveal its detailed microscopic properties.
Thus, it is rather easy to pose the basic questions in a precise fashion. Indeed, they could have been formulated soon after the advent of quantum mechanics. Answering them, on the other hand, has proved to be surprisingly difficult. The main reason, I believe, is the inadequacy of standard techniques. More precisely, to examine the microscopic structure of geometry, we must treat Einstein gravity quantum mechanically, i.e., construct at least the basics of a quantum theory of the gravitational field. Now, in the traditional approaches to quantum field theory, one begins with a continuum, background geometry. To probe the nature of quantum geometry, on the other hand, we should not begin by assuming the validity of this picture. We must let quantum gravity decide whether this picture is adequate; the theory itself should lead us to the correct microscopic model of geometry.
With this general philosophy, in this article I will summarize the picture of quantum geometry that has emerged from a spp.cific approach to quantum gravity. This approach is non-perturbative. In perturbative approaches, one generally begins by assuming that space-time geometry is flat and incorporates gravity -and hence curvature- step by step by adding up small corrections. Discreteness is then hard to unravel1. In the non-perturbative approach, by contrast, there is no background metric at all. All we have is a bare manifold to start with. All fields -matter as well as gravity/ geometry- are treated as dynamical from the beginning. Consequently, the description can not refer to a background metric. Technically this means that the full diffeomorphism group of the manifold is respected; the theory is generally covariant.
As we will see, this fact leads one to Hilbert spaces of quantum states which are quite different from the familiar Fock spaces of particle physics. Now gravitons -the three dimensional wavy undulations on a flat metricdo not represent fundamental excitations. Rather, the fundamental excitations are one dimensional. Microscopically, geometry is rather like a polymer. Recall that, although polymers are intrinsically one dimensional, when densely packed in suitable configurations they can exhibit properties of a three dimensional system. Similarly, the familiar continuum picture of geometry arises as an approximation: one can regard the fundamental excitations as 'quantum threads' with which one can 'weave' continuum geometries. That is, the continuum picture arises upon coarse-graining of the semi-classical 'weave states'. Gravitons are no longer the fundamental mediators of the gravitational interaction. They now arise only as approximate notions. They represent perturbations of weave states and mediate the gravitational force only in the

16 THE UNIVERSE
semi-classical approximation. Because the non-perturbative states are polymer-like, geometrical observables turn out to have discrete spectra. They provide a rather detailed picture of quantum geometry from which physical predictions can be made.
The article is divided into two parts. In the first, I will indicate how one can reformulate general relativity so that it resembles gauge theories. This formulation provides the starting point for the quantum theory. In particular, the one-dimensional excitations of geometry arise as the analogs of 'Wilson loops' which are themselves analogs of the line
integrals exp if A.de of electro-magnetism. In the second part, I will
indicate how this description leads us to a quantum theory of geometry. I will focus on area operators and show how the detailed information about the eigenvalues of these operators has interesting physical consequences, e.g., to the process of Hawking evaporation of black holes.
I should emphasize that this is not a technical review. Rather, it is written in the same spirit that drives Jayant's educational initiatives. I thought this would be a fitting way to honor Jayant since these efforts have occupied so much of his time and energy in recent years. Thus my aim is present to beginning researchers an overall, semi-quantitative picture of the main ideas. Therefore, the article is written at the level of colloquia in physics departments in the United States. I will also make some historic detours of general interest. At the end, however, I will list references where the details of the central results can be found.
2. FROM METRICS TO CONNECTIONS
2.1 GRAVITY VERSUS OTHER
FUNDAMENTAL FORCES
General relativity is normally regarded as a dynamical theory of metrics -tensor fields that define distances and hence geometry. It is this fact that enabled Einstein to code the gravitational field in the Riemannian curvature of the metric. Let me amplify with an analogy. Just as position serves as the configuration variable in particle dynamics, the three dimensional metric of space can be taken to be the configuration variable of general relativity. Given the initial position and velocity of a particle, Newton's laws provide us with its trajectory in the position space. Similarly, given a three dimensional metric and its time derivative at an initial instant, Einstein's equations provide us with a four dimensional space-time which can be regarded as a trajectory in the space of 3-metrics 2.
However, this emphasis on the metric sets general relativity apart from all other fundamental forces of Nature. Indeed, in the theory of

QUANTUM MECHANICS OF GEOMETRY 17
electro-weak and strong interactions, the basic dynamical variable is a (matrix-valued) vector potential, or a connection. Like general relativity, these theories are also geometrical. The connection enables one to parallel-transport objects along curves. In electrodynamics, the object is a charged particle such as an electron; in chromodynamics, it is a particle with internal color, such as a quark. Generally, if we move the object around a closed loop, we find that its state does not return to the initial value; it is rotated by an unitary matrix. In this case, the connection is said to have curvature and the unitary matrix is a measure of the curvature in a region enclosed by the loop. In the case of electrodynamics, the connection is determined by the vector potential and the curvature by the electro-magnetic field strength.
Since the metric also gives rise to curvature, it is natural to ask if there is a relation between metrics and connections. The answer is in the affirmative. Every metric defines a connection -called the Levi-Civita connection of the metric. The object that the connection enables one to parallel transport is a vector. (It is this connection that determines the geodesics, i.e. the trajectories of particles in absence of non-gravitational forces.) It is therefore natural to ask if one can not use this connection as the basic variable in general relativity. If so, general relativity would be cast in a language that is rather similar to gauge theories and the description of the (general relativistic) gravitational interaction would be very similar to that of the other fundamental interactions of Nature. It turns out that the answer is in the affirmative. Furthermore, both Einstein and Schrodinger gave such a reformulation of general relativity. Why is this fact then not generally known? Indeed, I know of no textbook on general relativity which even mentions it. One reason is that in their reformulation the basic equations are somewhat complicated -but not much more complicated, I think, than the standard ones in terms of the metric. A more important reason is that we tend to think of distances, light cones and causality as fundamental. These are directly determined by the metric and in a connection formulation, the metric is a 'derived' rather than a fundamental concept. But in the last few years, I have come to the conclusion that the real reason why the connection formulation of Einstein and Schrodinger has remained so obscure may lie in an interesting historical episode. I will return to this point at the end of this section.
2.2 METRICS VERSUS CONNECTIONS
Modern day researchers re-discovered connection theories of gravity after the invention and successes of gauge theories for other interac-

18 THE UNIVERSE
tions. Generally, however, these formulations lead one to theories which are quite distinct from general relativity and the stringent experimental tests of general relativity often suffice to rule them out. There is, however, a reformulation of general relativity. itself in which the basic equations are simpler than the standard ones: while Einstein's equations are non-polynomial in terms of the metric and its conjugate momentum, they turn out to be low order polynomials in terms of the new connection and its conjugate momentum. Furthermore, just as the simplest particle trajectories in space-time are given by geodesics, the 'trajectory' determined by the time evolution of this connection according to Einstein's equation turns out to be a geodesic in the configuration space of connections.
In this formulation, the phase space of general relativity is identical to that of the Yang-Mills theory which governs weak interactions. Recall first that in electrodynamics, the (magnetic) vector potential constitutes the configuration variable and the electric field serves as the conjugate momentum. In weak interactions and general relativity, the configuration variable is a matrix-valued vector potential; it can be written as Am where Ai is a triplet of vector fields and Ti are the Pauli matrices. The conjugate momenta are represented by Em where Ei is a triplet of vector fields3. Given a pair (Ai, Ei) (satisfying appropriate conditions as noted in footnote 2), the field equations of the two theories determine the complete time-evolution, i.e., a dynamical trajectory.
The field equations -and the Hamiltonians governing them- of the two theories are of course very different. In the case of weak interactions, we have a background space-time and we can use its metric to construct the Hamiltonian. In general relativity, we do not have a background metric. On the one hand this makes life very difficult since we do not have a fixed notion of distances or causal structures; these notions are to arise from the solution of the equations we are trying to write down! On the other hand, there is also tremendous simplification: Because there is no background metric, there are very few mathematically meaningful, gauge invariant expressions of the Hamiltonian that one can write down. (As we will see, this theme repeats itself in the quantum theory.) It is a pleasant surprise that the simplest non-trivial expression one can construct from the connection and its conjugate momentum is in fact the correct one, i.e., is the Hamiltonian of general relativity! The expression is at most quadratic in Ai and at most quadratic in E i . The similarity with gauge theories opens up new avenues for quantizing general relativity and the simplicity of the field equations makes the task considerably easier.

QUANTUM MECHANICS OF GEOMETRY 19
What is the physical meaning of these new basic variables of general relativity? As mentioned before, connections tell us how to parallel transport various physical entities around curves. The Levi-Civita connection tells us how to parallel transport vectors. The new connection, Ai, on the other hand, determines the parallel transport of left handed spin- ~ particles (such as the fermions in the standard model of particle physics) -the so called chiral fermions. These fermions are mathematically represented by spinors which, as we know from elementary quantum mechanics, can be roughly thought of as 'square roots of vectors'. Not surprisingly, therefore, the new connection is not completely determined by the metric alone. It requires additional information which roughly is a square-root of the metric, or a tetrad. The conjugate momenta Ei represent restrictions of these tetrads to space. They can be interpreted as spatial triads, i.e., as 'square-roots' of the metric of the 3-dimensional space. Thus, information about the Riemannian geometry of space is coded directly in these momenta. The (space and) time-derivatives of the triads are coded in the connection.
To summarize, there is a formulation of general relativity which brings it closer to theories of other fundamental interactions. Furthermore, in this formulation, the field equations simplify greatly. Thus, it provides a natural point of departure for constructing a quantum theory of gravity and for probing the nature of quantum geometry non-perturbatively.
2.3 HISTORICAL DETOUR
To conclude this section, let me return to the piece of history involving Einstein and Schrodinger that I mentioned earlier. In the forties, both men were working on unified field theories. They were intellectually very close. Indeed, Einstein wrote to Schrodinger saying that he was perhaps the only one who was not 'wearing blinkers' in regard to fundamental questions in science and Schrodinger credited Einstein for inspiration behind his own work that led to the Schrodinger equation. Einstein was in Princeton and Schrodinger in Dublin. But During the years 1946-47, they frequently exchanged ideas on unified field theory and, in particular, on the issue of whether connections should be regarded as fundamental or metrics. In fact the dates on their letters often show that the correspondence was going back and forth with astonishing speed. It reveals how quickly they understood the technical material the other hand sent, how they hesitated, how they teased each other. Here are a few quotes:
The whole thing is going through my head like a millwheel: To take r
[the connection] alone as the primitive variable or the g's [metrics] and

20 THE UNIVERSE
r's? ...
-Schrodinger, May 1st, 1946.
How well I understand your hesitating attitude! I must confess to you that inwardly I am not so certain ... We have squandered a lot of time on this thing, and the results look like a gift from devil's grandmother.
-Einstein, May 20th, 1946
Einstein was expressing doubts about using the Levi-Civita connection alone as the starting point which he had advocated at one time. Schrodinger wrote back that he laughed very hard at the phrase 'devil's grandmother'. In another letter, Einstein called Schrodinger 'a clever rascal'. Schrodinger was delighted and took it to be a high honor. This continued all through 1946. Then, in the beginning of 1947, Schrodinger thought he had made a breakthrough. He wrote to Einstein:
Today, I can report on a real advance. May be you will grumble frightfully for you have explained recently why you don't approve of my method. But very soon, you will agree with me...
-Schrodinger, January 26th, 1947
Schrodinger sincerely believed that his breakthrough was revolutionary 4. Privately, he spoke of a second Nobel prize. The very next day after he wrote to Einstein, he gave a seminar in the Dublin Institute of Advanced Studies. Both the Taoiseach (the Irish prime minister) and newspaper reporters were invited. The day after, the following headlines appeared:
Twenty persons heard and saw history being made in the world of physics. ... The Taoiseach was in the group of professors! and students. .. [To a question from the reporter] Professor Schrodinger replied "This is the generalization. Now the Einstein theory becomes simply a special case ... "
-Irish Press, January 28th, 1947
Not surprisingly, the headlines were picked up by New York Times which obtained photocopies of Schrodinger's paper and sent them to prominent physicists -including of course Einstein- for comments. As Walter Moore, Schrodinger's biographer puts it, Einstein could hardly believe that such grandiose claims had been made based on a what was at best a small advance in an area of work that they both had been pursuing for some time along parallel lines. He prepared a carefully worded response to the request from New York Times:
It seems undesirable to me to present such preliminary attempts to the public. ... Such communiques given in sensational terms give the lay

QUANTUM MECHANICS OF GEOMETRY 21
public misleading ideas about the character ofresearch. The reader gets the impression that every five minutes there is a revolution in Science, somewhat like a coup d'etat in some of the smaller unstable republics.
Einstein's comments were also carried by the international press. On seeing them, Schrodinger wrote a letter of apology to Einstein citing his desire to improve the financial conditions of physicists in the Dublin Institute as a reason for the exaggerated account. It seems likely that this 'explanation' only worsened the situation. Einstein never replied. He also stopped scientific communication with Schrodinger for three years.
The episode must have been shocking to those few who were exploring general relativity and unified field theories at the time. Could it be that this episode effectively buried the desire to follow up on connection formulations of general relativity until an entirely new generation of physicists who were blissfully unaware of this episode came on the scene?
3. QUANTUM GEOMETRY
3.1 GENERAL SETTING
Now that we have a connection formulation of general relativity, let us consider the problem of quantization. Recall first that in the quantum description of a particle, states are represented by suitable wave functions 'lI(x) on the classical configuration space of the particle. Similarly, quantum states of the gravitational field are represented by appropriate wave functions 'lI(Ai) of connections. Just as the momentum operator
in particle mechanics is represented by p. 'lII = -iii (a'll / axI) (with I = 1,2,3), the triad operators are represented by Ei' 'lI = -iliG(8'l1/8Ai).
The task is to express geometric quantities, such as lengths of curves, areas of surfaces and volumes of regions, in terms of triads using ordinary differential geometry and then promote these expressions to well-defined operators on the Hilbert space of quantum states. In principle, the task is rather similar to that in quantum mechanics where we first express observables such as angular momentum or Hamiltonian in terms of configuration and momentum variables x and p and then promote them to quantum theory as well-defined operators on the quantum Hilbert space.
In quantum mechanics, the task is relatively straightforward; the only potential problem is the choice of factor ordering. In the present case, by contrast, we are dealing with a field theory, i.e., a system with an infinite number of degrees of freedom. Consequently, in addition to factor ordering, we face the much more difficult problem of regularization. Let me explain qualitatively how this arises. A field operator, such as

22 THE UNIVERSE

the triad mentioned above, excites infinitely many degrees of freedom. Technically, its expectation values are distributions rather than smooth fields. They don't take precise values at a given point in space. To obtain numbers, we have to integrate the distribution against a test function, which extracts from it a 'bit' of information. As we change our test or smearing field, we get more and more information. (Take the familiar Dirac o-distribution o(x); it does not have a well-defined value at x = O. Yet, we can extract the full information contained in
o(x) through the formula: Jo(x)f(x)dx = f(O) for all test functions
f(x).) Thus, in a precise sense, field operators are distribution-valued. Now, as is well known, product of distributions is not well-defined. If we attempt naively to give meaning to it, we obtain infinities, i.e., a senseless result. Unfortunately, all geometric operators involve rather complicated (in fact non-polynomial) functions of the triads. So, the naive expressions of the corresponding quantum operators are typically meaningless. The key problem is to regularize these expressions, i.e., to extract well-defined operators from the formal expressions in a coherent fashion.
This problem is not new; it arises in all physically interesting quantum field theories. However, as I mentioned in the Introduction, in other theories one has a background space-time metric and it is invariably used in a critical way in the process of regularization. For example, consider the electro-magnetic field. We know that the energy of the Hamilto-
nian of the theory is given by H = J(E . E + B . B) d3x. Now, in the
quantum theory, E and B are both operator-valued distributions and so
their square is ill-defined. But then, using the background flat metric, one Fourier decomposes these distributions, identifies creation and annihilation operators and extracts a well-defined Hamiltonian operator by normal ordering, i.e., by physically moving all annihilators to the right of creators. This procedure removes the unwanted and unphysical infinite zero point energy form the formal expression and the subtraction makes the operator well-defined. In the present case, on the other hand, we are trying to construct a quantum theory of geometry/gravity and do not have a flat metric -or indeed, any metric- in the background. Therefore, many of the standard regularization techniques are no longer available.

3.2

GEOMETRIC OPERATORS

Fortunately, between 1992 and 1995, a new functional calculus was developed on the space of connections Ai -i.e., on the configuration space of the theory. This calculus is mathematically rigorous and makes no reference at all to a background space-time geometry; it is generally

QUANTUM MECHANICS OF GEOMETRY 23
covariant. It provides a variety of new techniques which make the task of regularization feasible. First of all, there is a well-defined integration theory on this space. To actually evaluate integrals and define the Hilbert space of quantum states, one needs a measure: given a measure on the space of connections, we can consider the space of squareintegrable functions which can serve as the Hilbert space of quantum states. It turns out that there is a preferred measure, singled out by the physical requirement that the (gauge covariant versions of the) configuration and momentum operators be self-adjoint. This measure is diffeomorphism invariant and thus respects the underlying symmetries coming from general covariance. Thus, there is a natural Hilbert space of states to work with5. Let us denote it by 1£. Differential calculus enables one to introduce physically interesting operators on this Hilbert space and regulate them in a generally covariant fashion. As in the classical theory, the absence of a background metric is both a curse and a blessing. On the one hand, because we have very little structure to work with, many of the standard techniques simply fail to carryover. On the other hand, at least for geometric operators, the choice of viable expressions is now severely limited which greatly simplifies the task of regularization.
The general strategy is the following. The Hilbert space 1£ is the space of square-integrable functions 'lI(Ai) of connections Ai. A key simplification arises because it can be obtained as the (projective) limit of Hilbert spaces associated with systems with only a finite number of degrees of freedom. More precisely, given any graph 'Y (which one can intuitively think of as a 'floating lattice') in the physical space, using techniques which are very similar to those employed in lattice gauge theory, one can construct a Hilbert space 1£7 for a quantum mechanical system with 3N degrees of freedom, where N is the number of edges of the graph6 . Roughly, these Hilbert spaces know only about how the connection parallel transports chiral fermions along the edges of the graph and not elsewhere. That is, the graph is a mathematical device to extract 3N 'bits of information' from the full, infinite dimensional information contained in the connection, and 1£7 is the sub-space of 1£ consisting of those functions of connections which depend only on these 3N bits. (Roughly, it is like focusing on only 3N components of a vector with an infinite number of components and considering functions which depend only on these 3N components, i.e., are constants along the orthogonal directions.) To get the full information, we need all possible graphs. Thus, a function of connections in 1£ can be specified by fixing a function in 1£7 for every graph 'Y in the physical space. Of course, since two distinct graphs can share edges, the collection of functions on 1£"1

24 THE UNIVERSE
must satisfy certain consistency conditions. These lie at the technical heart of various constructions and proofs.
The fact that tl is the (projective) limit of tl, breaks up any given problem in quantum geometry into a set of problems in quantum mechanics. Thus, for example, to define operators on tl, it suffices to define a consistent family of operators on tl, for each ,. This makes the task of defining geometric operators feasible. I want to emphasize, however, that the introduction of graphs is only for technical convenience. Unlike in lattice gauge theory, we are not defining the theory via a continuum limit (in which the lattice spacing goes to zero.) Rather, the full Hilbert space tl of the continuum theory is already well-defined. Graphs are introduced only for practical calculations. Nonetheless, they bring out the one-dimensional character of quantum states/excitations of geometry: It is because 'most' states in tl can be realized as elements of tl, for some, that quantum geometry has a 'polymer-like' character.
Let me now outline the result of applying this procedure for geometric operators. Suppose we are given a surface S, defined in local coordinates
J by X3 = const. The classical formula for the area of the surface is:
f As = d2X Ef Ef, where Ef are the third components of the vectors
Ei. As is obvious, this expression is non-polynomial in the basic variables Ei. Hence, off-hand, it would seem very difficult to write down the corresponding quantum operator. However, thanks to the background independent functional calculus, the operator can in fact be constructed rigorously.
To specify its action, let us consider a state which belongs to tl, for
some ,. Then, the action of the final, regularized operator As is as
follows. If the graph has no intersection with the surface, the operator simply annihilates the state. If there are intersections, it acts at each intersection via the familiar angular momentum operators associated with SU(2). This simple form is a direct consequence of the fact that we do not have a background geometry: given a graph and a surface, the diffeomorphism invariant information one can extract lies in their intersections. To specify the action of the operator in detail, let me suppose that the graph, has N edges. Then the state W has the form: w(Ai) = 1/J(gl, ...gN) for some function 1/J of the N variables gl, ···,gN, where gk (E SU(2)) denotes the spin-rotation that a chiral fermion undergoes if parallel transported along the k-th edge using the connection Ai. Since gk represent the possible rotations of spins, angular momentum operators have a natural action on them. In terms of these, we can introduce 'vertex operators' associated with each intersection point

QUANTUM MECHANICS OF GEOMETRY 25

v between Sand ,:

Ov . w{A) = 'L,k{J,L)JI ·JL . 'I/J{gl, ... ,gN)

(3.1)

I,L

where J, L run over the edges of , at the vertex v, k{J, J) = 0, ±1 depending on the orientation of edges J, L at v, and J I are the three angular momentum operators associated with the I-th edge. (Thus, lJ act only on the argument gI of'I/J and the action is via the three left invariant vector fields on SU(2).) Note that the the vertex operators
resemble the Hamiltonian of a spin system, k{I, L) playing the role of the coupling constant. The area operator is just a sum of the square-roots of the vertex operators:

(3.2)

Thus, the area 0perator is constructed from angular momentum-like operators. Note that the coefficient in front of the sum is just !£~, the square of the Planck length. This fact will be important later.
Because of the simplicity of these operators, their complete spectrum -i.e., full set of eigenvalues- is known explicitly: Possible eigenvalues as are given by

as =

£2 ;

'L, [2j~d)(j~d)+I)+2j~U)(j~U)+I)-j~d+U)(j~d+U)+1)r1

(3.3)

v

where v labels a finite set of points in Sand j(d), iu) and j(d+u) are nonnegative half-integers assigned to each v, subject to the usual inequality

(3.4)

from the theory of addition of angular momentum in elementary quantum mechanics. Thus the entire spectrum is discrete; areas are indeed quantized! This discreteness holds also for the length and the volume operators. Thus the expectation that the continuum picture may break down at the Planck scale is borne out fully. Quantum geometry is very different from the continuum picture. This may be the fundamental reason for the failure of perturbative approaches to quantum gravity.
Let us now examine a few properties of the spectrum. The lowest eigenvalue is of course zero. The next lowest eigenvalue may be called
1 the area gap. Interestingly, area-gap is sensitive to the topology of the
surface S. If S is open, it is £~. If S is a closed surface -such as
a 2-torus in a 3-torus- which fails to divide the spatial 3-manifold into

26 THE UNIVERSE
an 'inside' and an 'outside' region, the gap turns out to be larger, ~l~. If S is a closed surface -such as a 2-sphere in R3- which divides space into an 'inside' and an 'outside' region, the area gap turns out to be
even larger; it is 2fl~. Another interesting feature is that in the large
area limit, the eigenvalues crowd together. This follows directly from the form of eigenvalues given above. Indeed, one can show that for large
J eigenvalues as, the difference t::..as between consecutive eigenvalues goes
as t::..as ~ (exp - as/ l~ )l~. Thus, t::..as goes to zero very rapidly. (The
crowding is noticeable already for low values of as. For example, if S
is open, there is only one non-zero eigenvalue with as < O.5lp2, seven with as < l~ and 98 with as < 2l~.) Intuitively, this explains why the
continuum limit works so well.
3.3 PHYSICAL CONSEQUENCES: DETAILS MATTER!
However, one might wonder if such detailed properties of geometric operators can have any 'real' effect. After all, since the Planck length is so small, one would think that the classical and semi-classical limits should work irrespective of, e.g., whether or not the eigenvalues crowd. For example, let us consider not the most general eigenstates of the area
operator As but -as was first done in the development of the subject-
the simplest ones. These correspond to graphs which have simplest intersections with S. For example, n edges of the graph may just pierce S, each one separately, so that at each one of the n vertices there is just a straight line passing through. For these states, the eigenvalues are as = (v'3/2)nl~. Thus, here, the level spacing t::..as is uniform, like that of the Hamiltonian of a simple harmonic oscillator. If we restrict ourselves to these simplest eigenstates, even for large eigenvalues, the level spacing does not go to zero. Suppose for a moment that this is the full spectrum of the area operator. wouldn't the semi-classical approximation still work since, although uniform, the level-spacing is so small?
Surprisingly, the answer is in the negative! What is perhaps even more surprising is that the evidence comes from unexpected quarters: the Hawking evaporation of large black holes. More precisely, we will see that if t::..as had failed to vanish sufficiently fast, the semi-classical approximation to quantum gravity, used in the derivation of the Hawking process, must fail in an important way. The effects coming from area
quantization would have implied that even for large macroscopic black
holes of, say, a thousand solar masses, we can not trust semi-classical arguments.

QUANTUM MECHANICS OF GEOMETRY 27
Let me explain this point in some detail. The original derivation of Hawking's was carried out in the framework of quantum field theory in curved space-times which assumes that there is a specific underlying continuum space-time and explores the effects of curvature of this space-time on quantum matter fields. In this approximation, Hawking found that the classical black hole geometries are such that there is a spontaneous emission which has a Planckian spectrum at infinity. Thus, black-holes, seen from far away, resemble black bodies and the associated temperature turns out to be inversely related to the mass of the hole. Now, physically one expects that, as it evaporates, the black hole must lose mass. Since the radius of the horizon is proportional to the the mass, the area of the horizon must decrease. Thus, to describe the evaporation process adequately, we must go beyond the external field approximation and take in to account the fact that the underlying space-time geometry is in fact dynamical. Now, if one treated this geometry classically, one would conclude that the process is continuous. However, since we found that the area is in fact quantized, we would expect that the black hole evaporates in discrete steps by making a transition from one area eigenvalue to another, smaller one. The process would be very similar to the wayan excited atom descends to its ground state through a series of discrete transitions.
Let us look at this process in some detail. For simplicity let us use units with c = 1. Suppose, to begin with, that the level spacing of eigenvalues of the area operator is the naive one, i.e. with flas = (J3/2)e~. Then, the fundamental theory would have predicted that the smallest frequency, wo, of emitted particles would be given by ~o and the smallest possible change flM in the mass of the black hole would be given by flM = ~o. Now, since the area ofthe horizon goes as AH rv G2M2, we have flM "" flaH/2G 2M "" e~/G2M. Hence, ~o "" n/GM. Thus, the 'true' spectrum would have emission lines only at frequencies w = Nwo , for N = 1,2, ... corresponding to transitions of the black hole through N area levels. How does this compare with the Hawking prediction? As I mentioned above, according to Hawking's semi-classical analysis, the spectrum would be the same as that of a black-body at temperature T given by kT "" n/GM, where k is the Boltzmann constant. Hence, the peak of this spectrum would appear at wp given by ~p "" kT rv n/GM. But this is precisely the order of magnitude of the minimum frequency Wo that would be allowed if the area spectrum were the naive one. Thus, in this case, a more fundamental theory would have predicted that the spectrum would not resemble a black body spectrum. The most probable transition would be for N = 1 and so the spectrum would be peaked at Wp as in the case of a black body. However, there would be no emis-

28 THE UNIVERSE
sion lines at frequencies low compared with wp ; this part of the black body spectrum would be simply absent. The part of the spectrum for
w > wp would also not be faithfully reproduced since the discrete lines
with frequencies Nwo, with N = 1,2, ... would not be sufficiently near each other -i.e. crowded- to yield an approximation to the continuous black-body spectrum.
The situation is completely different for the correct, full spectrum of the area operator if the black hole is macroscopic, i.e., large. Then, as I noted earlier, the area eigenvalues crowd and the level spacing goes as
AaH ~ (exp -JaH/£~)£~. As a consequence, as the black hole makes
J transition from one area eigenvalue to another, it would emit particles
at frequencies equal to or larger than f"V wp exp - aH / £~. Since for a
macroscopic black-hole the exponent is very large (for a solar mass blackhole it is f"V 1038!) the spectrum would be well-approximated by a continuous spectrum and would extend well below the peak frequency. Thus, the precise form of the area spectrum ensures that, for large black-holes, the potential problem with Hawking's semi-classical picture disappears. Note however that as the black hole evaporates, its area decreases, it gets hotter and evaporates faster. Therefore, a stage comes when the area is of the order of £~. Then, there would be deviations from the black body spectrum. But this is to be expected since in this extreme regime one does not expect the semi-classical picture to continue to be meaningful.
This argument brings out an interesting fact. There are several iconoclastic approaches to quantum geometry in which one simply begins by postulating that geometric quantities should be quantized. Then, having no recourse to first principles from where to derive the eigenvalues of these operators, one simply postulates them to be multiples of appropriate powers of the Planck length. For area then, one would say that the eigenvalues are integral multiples of £~. The above argument shows how this innocent looking assumption can contradict semi-classical results even for large black holes. In the present approach, we did not begin by postulating the nature of quantum geometry. Rather, we derived the spectrum of the area operator from first principles. As we see, the form of these eigenvalues is rather complicated and could not have been guessed a priori. More importantly, the detailed form does carry rich information and in particular removes the conflict with semi-classical results in macroscopic situations.

QUANTUM MECHANICS OF GEOMETRY 29
3.4 CURRENT AND FUTURE DIRECTIONS
Exploration of quantum Riemannian geometry continues. Last year, it was found that geometric operators exhibit certain unexpected noncommutativity. This reminds one of the features explored by Alain Connes in his non-commutative geometry. Indeed, there are several points of contact between these two approaches. For instance, the Dirac operator that features prominently in Connes' theory is closely related to the connection Ai used here. However, at a fundamental level, the two approaches are rather different. In Connes' approach, one constructs a non-commutative analog of entire differential geometry. Here, by contrast, one focuses only on Riemannian geometry; the underlying manifold structure remains classical. In three space-time dimensions, it is possible to get rid of this feature in the final picture and express the theory in purely combinatorial fashion. Whether the same will be possible in four dimensions remains unclear. However, combinatorial methods continue to dominate the theory and it is quite possible that one would again be able to present the final picture without any reference to an underlying smooth manifold.
Perhaps the most striking application of quantum geometry has been to black hole thermodynamics. We saw in section 3.3 that the Hawking process provides a non-trivial check on the level spacing of the eigenvalues of area operators. Conversely, the discrete nature of these eigenvalues provides a statistical mechanical explanation of black hole entropy. To see this, first recall that for familiar physical systems -such as a gas, a magnet, or a black body- one can arrive at the expression of entropy by counting the number of micro-states. The counting in turn requires one to identify the building blocks that make up the system. For a gas, these are atoms; for a magnet, electron spins and for the radiation field in a black body, photons. What are the analogous building blocks for a large black hole? They can not be gravitons because the gravitational fields under consideration are static rather than radiative. Therefore, the elementary constituents must be non-perturbative in nature. In our approach they turn out to be precisely the quantum excitations of the geometry of the black hole horizon. The polymer-like one dimensional excitations of geometry in the bulk pierce the horizon and endow it with its area. It turns out that, for a given area, there are a specific number of permissible bulk states and for each such bulk state, there is a precise number of permissible surface states of the intrinsic quantum geometry of the horizon. Heuristically, the horizon resembles a pinned balloon -pinned by the polymer geometry in the bulk- and the surface states describe the permissible oscillations of the horizon subject to the given

30 THE UNIVERSE
pinning. A count of all these quantum states provides, in the usual way, the expression of the black hole entropy.
Another promising direction for further work is construction of better candidates for 'weave states', the non-linear analogs of coherent states approximating smooth, macroscopic geometries. Once one has an 'optimum' candidate to represent Minkowski space, one would develop quantum field theory on these weave quantum geometries. Because the underlying basic excitations are one-dimensional, the 'effective dimension of space' for these field theories would be less than three. Now, in the standard continuum approach, we know that quantum field theories in low dimensions tend to be better behaved because their ultra-violet problems are softer. Hence, there is hope that these theories will be free of infinities. If they are renormalizable in the continuum, their predictions at large scales can not depend on the details of the behavior at very small scales. Therefore, one might hope that quantum field theories on weaves would not only be finite but also agree with the renormalizable theories in their predictions at the laboratory scale.
A major effort is being devoted to the task of formulating and solving quantum Einstein's equations using the new functional calculus. Over the past two years, there have been some exciting developments in this area. The methods developed there seem to be applicable also to supergravity theories. In the coming years, therefore, there should be much further work in this area. More generally, since quantum geometry does not depend on a background metric, it may well have other applications. For example, it may provide a natural arena for other problem such as that of obtaining a background independent formulation of string theory.
So far, I have focussed on theoretical ideas and checks on them have come from considerations of consistency with other theoretical ideas, e.g., those in black hole thermodynamics. What about experimental tests of predictions of quantum geometry? An astonishing recent development suggests that direct experimental tests may become feasible in the near future. I will conclude with a summary of the underlying ideas. The approach one takes is rather analogous to the one used in proton decay experiments. Processes potentially responsible for the decay come from grand unified theories and the corresponding energy scales are very large, 1015 GeV -only four orders of magnitude below Planck energy. There is no hope of achieving these energies in particle accelerators to actually create in large numbers the particles responsible for the decay. Therefore the decays are very rare. The strategy adopted was to carefully watch a very large number of protons to see if one of them decays. These experiments were carried out and the (negative) results actually ruled out some of the leading candidate grand unified theories.

QUANTUM MECHANICS OF GEOMETRY 31
Let us return to quantum geometry. The naive strategy of accelerating particles to Planck energy to directly 'see' the Planck scale geometry is hopeless. However, as in proton decay experiments, one can let these minutest of effects accumulate till they become measurable. The laboratory is provided by the universe itself and the signals are generated by the so-called ,-ray bursts. These are believed to be of cosmological origin. Therefore, by the time they arrive on earth, they have traveled extremely large distances. Now, if the geometry is truly quantum mechanical, as I suggested, the propagation of these rays would be slightly different from that on a continuum geometry. The difference would be minute but could accumulate on cosmological distances. Following this strategy, astronomers have already put some interesting limits on the possible 'graininess' of geometry. Now the challenge for theorists is to construct realistic weave states corresponding to the geometry we observe on cosmological scales, study in detail propagation of photons on them and come up with specific predictions for astronomers. The next decade should indeed be very exciting!
Acknow ledgments
The work summarized here is based on contributions from many researchers especially John Baez, Alejandro Corichi, Roberto DePitri, Rodolfo Gambini, Chris Isham, Junichi Iwasaki, Jerzy Lewandowski, Renate Loll, Don Marolf, Jose Mourao, Jorge Pullin, Thomas Thiemann, Carlo Rovelli, Steven Sawin, Lee Smolin and Jose-Antonio Zapata. Special thanks are due to Jerzy Lewandowski for long range collaboration. This work was supported in part by the NSF Grant PHY95-14240 and by the Eberly fund of the Pennsylvania State University.
Notes
1. The situation can be illustrated by a harmonic oscillator: While the exact energy levels of the oscillator are discrete, it would be very difficult to "see" this discreteness if one began with a free particle whose energy levels are continuous and then tried to incorporate the effects of the oscillator potential step by step via perturbation theory.
2. Actually, only six of the ten Einstein's equations provide the evolution equations. The other four do not involve time-derivatives at all and are thus constraints on the initial values of the metric and its time derivative. However, if the constraint equations are satisfied initially, they continue to be satisfied at all times.
3. As usual, summation over the repeated index i is assumed. Also, technically each Ai is a I-form rather than a vector field. Similarly, each Ei is a vector density of weight one, i.e., natural dual of a 2-form.
4. The 'breakthrough' was to drop the requirement that the (Levi-Civita) connection be symmetric, i.e., to allow for torsion.
5. This is called the kinematical Hilbert space; it enables one to formulate the quantum Einstein's (or supergravity) equations. The final, physical Hilbert space will consist of states which are solutions to these equations.

32 THE UNIVERSE
6. The factor 3 comes from the dimension of the gauge group SU(2) which acts on Chiral spinors. The mathematical structure of the gauge-rotations induced by this SU (2) is exactly the same as that in the angular-momentum theory of spin- ~ particles in elementary quantum mechanics.
References
[1] Riemann, B., 1854, Uber die Hypothesen, welche der Geometrie zugrunde liegen. Monographs and Reviews on Non-perturbative Quantum Gravity:
[2] Ashtekar, A., 1991, Lectures on Non-perturbative Canonical Gravity, Notes prepared in collaboration with R.S. Tate. (World Scientific, Singapore).
[3] Gambini, R., Pullin, J., 1996 Loops, Knots, Gauge Theories and Quantum Gravity, Cambridge University Press, Cambridge.
[4] Ashtekar, A., 1995, Gravitation and Quantizations, ed B. Julia and J. Zinn-Justin Elsevier, Amsterdam.
[5] Rovelli, C., 1998, Gravitation and Relativity: At the Turn of the Millennium, ed N. Dadhich and J. Narlikar, IUCAA, Pune.
Background-independent Functional Calculus: [6] Ashtekar, A., Isham, C.J., 1992, Class. & Quan. Grav. 9, 1433. [7] Ashtekar, A., Lewandowski, J., 1994, Representation theory of an-
alytic holonomy C* algebras, in Knots and Quantum Gravity, ed Baez, J., Oxford University Press, Oxford. [8] Baez, J. 1994, Diffeomorphism invariant generalized measures on the space of connections modulo gauge transformations, Lett. Math. Phys., 31, 213, hep-th/9305045, in The Proceedings of the Conference on Quantum Topology, ed D. Yetter, World Scientific, Singapore. [9] Ashtekar, A., Lewandowski, J., 1995, J. Math. Phys. 36, 2170.
[10] Marolf, D., Mourao, J.M., 1995, Commun. Math. Phys. 170, 583. [11] Ashtekar, A. Lewandowski, J., 1995 J. Geo. & Phys. 17, 191.
[12] Baez, J., 1996, Adv. Math. 117, 253 (1996); "Spin networks in non-perturbative quantum gravity," gr-qc/9504036 in The Interface of Knots and Physics, ed L. Kauffman {American Mathematical Society, Providence.
[13] Rovelli, C., Smolin, L., 1995, Phys.Rev. D52, 5743. [14] Ashtekar, A., Lewandowski, J., Marolf, D., Mourao, J., Thiemann,
T., 1995, J. Math. Phys. 36, 6456.
[15] Baez, J., Sawin, S., 1997, J. Funct. Analysis 150, 1 .

QUANTUM MECHANICS OF GEOMETRY 33
[16] Zapata, J.A., 1998, Gen.Rel.Grav. 30, 1229.
Geometric Operators [17] Ashtekar, A., Rovelli, C., Smolin, L., 1992, Phys. Rev. Lett. 69,
237. [18] Iwasaki, J., Rovelli, C., 1993 Int. J. Modern. Phys. Dl, 533; Class.
Quant. Grav. 11, 2899 (1994). [19] Rovelli, C. Smolin, L., 1995, Nuel. Phys. B442, 593 . [20] Ashtekar, A., Lewandowski, J., Marolf, D., Mourao, J., Thiemann,
T., 1996, J. Funct. Analysis, 135, 519. [21] Loll, R., 1995, Phys. Rev. Lett. 75, 3084. [22] Ashtekar, A., Lewandowski, J., 1997, Class. & Quant. Grav. 14,
A55-A81. [23] Loll, R., 1997, Nucl.Phys. B500 405. [24] Thiemann, T., 1997, J.Math.Phys. 39, 3372. [25] Ashtekar, A., Lewandowski, J., 1997, Adv. Theo. Math. Phys. 1,
388. [26] Ashtekar, A., Corichi, A., Zapata, J.A., 1998, Class. & Quant. Grav.
15,2955.
Black Hole Thermodynamics: [27] Bekenstein, J.D., 1973, Phys. Rev. D7, 2333; Phys. Rev. D9, 3292
(1974). [28] Bardeen, J.W., B. Carter, Hawking, S.W., 1973, Commun. Math.
Phys. 31, 161. [29] Hawking, S.W., 1975, Comun. Math. Phys. 43, 199. [30] Bekenstein, J., Mukhanov, V.F., 1995, Phys. Lett. B360, 7. [31] Fairhurst, S., 1996, Properties of the Spectrum of the Area Operator
(unpublished Penn State Report). [32] Rovelli, C., 1996, Helv.Phys.Acta 69, 582. [33] Ashtekar, A., 1997, Geometric issues in quantum gravity, in: The
Geometric Universe, ed S. Hugget, 1. Mason, K.P. Tod, S.T. Tsou and N.M.J. Woodhouse, Oxford University Press, Oxford, . [34] Ashtekar, A., Baez, J., Coriehi, A., Krasnov, K., 1998, Phys. Rev. Lett. 80, 904. [35] Ashtekar, A., Krasnov, K., 1998, Quantum geometry and black holes in Black Holes, Gravitational Radiation and the Universe, ed B.
Bhawal and B. K. Iyer, Kluwer, Dodrecht.

34 THE UNIVERSE
[36] Ashtekar, A., Beetle, C., Fairhurst, S., A generalization of black hole mechanics, gr-qc/9812065, Class. & Quant. Grav.,in press.
[37] Ashtekar, A., Corichi, A., Krasnov, K., 1998, Isolated horizons: Classical Phase space CGPG pre-print.
[38] Ashtekar, A., Baez, J., Krasnov, K., 1998, Quantum geometry of isolated horizons and black hole entropy CGPG pre-print.
Experimental tests [39] Amelino-Camelina, G., Ellis, J., Marvomatos,N., Nanopoulos, D.,
Sarkar, S., 1998, Nature 393, 763. [40] Gambini, R, Pullin, J., Nonstandard optics from quantum space-
time, gr-qc/9809038. [41] Biller, S.D., Breslin, A.C., Buckley, J., Catanese, M., Carson, M.,
Carter-Lewis, D.A., Cawley, M.F. , Fegan,D.J., Finley,J., Gaidos, J.A. , Hillas, A.M. , Krennrich, F., Lamb, RC. , Lessard,R , Masterson, C. , McEnery, J.E., McKernan, B. , Moriarty, P., Quinn, J., Rose, H.J., Samuelson, F., Sembroski, G., Skelton, P., Weekes, T.C., "Limits to Quantum Gravity Effects from Observations of TeV Flares in Active Galaxies", gr-qc/9810044

Chapter 4
QUANTUM MECHANICS AND RETROCAUSALITY

D. Atkinson
Institute for Theoretical Physics University of Groningen NL-9747 AG Groningen The Netherlands

Abstract

The classical electrodynamics of point charges can be made finite by the introduction of effects that temporally precede their causes. The idea of retrocausality is also inherent in the Feynman propagators of quantum electrodynamics. The notion allows a new understanding of the violation of the Bell inequalities, and of the world view revealed by quantum mechanics.

1.

INTRODUCTION

Dirac was never happy with quantum electrodynamics, although it was in large part his own creation. In old age, during an after-dinner seminar in 1970 that I attended in Austin, Texas, he lambasted such upstarts as Feynman, Schwinger, Tomonaga, and their ilk, under the dismissive collective term 'people'. These "People neglect infinities in an arbitrary way. This is not sensible mathematics. Sensible mathematics involves neglecting a quantity when it is small - not neglecting it just because it is infinitely great and you do not want it." A timorous spirit among the chastened listeners asked: "But, Professor Dirac, what about 9 - 2?", referring of course to the g-factor in the expression for the magnetic moment of the electron. Dirac's own equation had predicted that this factor should be precisely 2, and the highly accurate quantum electrodynamical calculation of its deviation from 2 was, and is, one of the tour~ de force of modern physics. The agreement with painstaking experimental measurement of this quantity is phenomenal (the Particle Data Group gives on the World Wide Web ten digits of agreement after

35
N. Dadhich and A. Kembhavi (eds.), The Universe. 35-50.
© 2000 Kluwer Academic Publishers.

36 THE UNIVERSE

the decimal point[I]). But the old maestro had his own views about this remarkable result: "It might just be a coincidence," he remarked evenly.
Quantum mechanics, married to electromagnetism, has produced a very successful theory, as measured by its empirical adequacy. The matter is not so adequate, however, at a conceptual level. There are still many competing interpretations of what quantum mechanics is telling us about the nature of the world. Despite the early preoccupation with the breakdown of determinism, the serious difficulties have to do rather with causality, which is by no means the same thing. Classical electromagnetic theory is in fact not immune to such problems either: the only known way to remove disastrous infinities in the theory of point charges interacting through the electromagnetic field is by the introduction of retrocausal effects. Quantum electrodynamics inherits the diseases of causality and of divergence from both of its parents. Their nature is pervasive, the cure unknown.

2. ADVANCED POTENTIALS

An electrically neutral particle, of mass m, subject to a force F, sat-

isfies Newton's second law of motion, which may be expressed in the

form

ma=F,

(4.1)

where a = r is the acceleration, on condition that Ii- I << e, so that
relativistic corrections may be neglected. A similar charged particle cannot satisfy the same equation, because an accelerated charge emits electromagnetic waves, losing energy in the process. Newton's law may be repaired by adding an effective radiative damping force that accounts for this extra source of energy loss to space:

+ ma = F F rad ,

(4.2)

where one finds, for a point charge e,

(4.3)

We may rewrite Eq.(4.2)-(4.3) in the form

m(a-7a) =F,

(4.4)

where

2e2 7 = -m3e3'

QUANTUM MECHANICS AND RETROCAUSALITY 37

is called the Abraham-Lorentz relaxation time. For an electron it is about 6 x 10-24 sec., in which time light travels only about 1O-':l3 em., the size of a proton.
The general solution of Eq.(4.4) is
[e a(t) = _1_ dt' e{t-tl)/TF(t') , mT t

where c is an integration constant. Clearly a(t) blows up exponentially

as t --+ 00, the so-called runaway solution, unless c = 00. Accordingly,

we choose this latter value, and find we can rewrite the solution in the

form

10 ma(t) = 00 dse-SF(t+TS) ,

(4.5)

from which we derive the following Taylor series in T:

L00
ma(t) = TnF{n)(t).

(4.6)

n=O

The Newton law Eq.(4.1), as it applies to a neutral particle, corresponds
to the zeroth term only. From Eq.(4.5), the acceleration at time t is determined not only by the value of the applied force at time t, but also by the force at all times later than t.
For a simple force, one can evaluate Eq.(4.5) explicitly. For example,
if a force is turned on at time t = 0, after which it remains constant, i.e.
F(t) = 0 for t < 0 and F(t) = K for t ~ 0, then we find ma(t) = K for
t ~ 0, as we would for a neutral particle, but surprisingly ma(t) = K et/T
for t < O. This preacceleration violates a naive notion of causality,
according to which a cause precedes its effect, whereas here the force, which is not applied before time t = 0, produces (has already produced!) an acceleration before t = O.
Consider next a universe consisting of many particles, at positions
Xa , Xb,'" with masses ma, mb,'" and charges ea , eb,'" For particle a, the relativistic generalization of Eq.(4.2) for the four-momentum p~ is

dp~
dTa

=

ea

[PIi-
v

+

RIi-]
v

dx~
d Ta

.

(4.7)

Here Ta is the proper time of particle a, and pli-v is the retarded field tensor that gives rise to the usual Lorentz force. It may be written

FIi-v

=

"~ ' "

F re
b

'

Ii- v '

bi-a

38 THE UNIVERSE

where the sum is over all the contributions to the field from the particles other than a itself: there is no self-interaction. The term R/-!v is the radiation damping tensor: it corresponds to F rad in the nonrelativistic approximation (4.3). Dirac deduced the explicit form of this tensor and showed that it can be written

R/-! = 1 [Fre,/-! _ Fadv/-! ]

V 2 av

a v·

(4.8)

It is very interesting that this expression involves the advanced, as well
as the retarded fields arising from particle a. For the point particles that
we are considering, these fields are separately singular on the world-line
of a itself, but their difference (4.8) is finite.
To simplify the notation, we will henceforth suppress the Lorentz
indices. It is important to distinguish the sum l:b:;t:a' in which one sums
over all particles except a, in order to calculate the influence of the rest of
the universe on particle a, and the sum l:b' in which a is also included,
giving a quantity that refers to the universe in its entirety.

F+R

=

~
~

ge'
b

+

1
2

[Fre'
a

_

Fadv]
a

b-:j:.a

~
~

p,re'
b

_

Fre'
a

+

1
2

[Fre'
a

_

Fadv]
a

b

L Fi"' - ~ [F~e, + F~dV]

(4.9)

b

The essential assumption of Wheeler and Feynman[2] is that the universe is a perfect absorber: all radiation is absorbed somewhere and none escapes to infinity. Since a radiation field is of order 1/r for large distances r, to eliminate energy loss by radiation it is enough to require

L Fr/' = 0 (r-1)
b

LFtv = 0 (r-1) ,
b

for all times, i.e. the sum of all retarded (advanced) fields is assumed
always to vanish faster than 1/r at spatial infinity. However, l:b Fr' and l:b Fbdv each satisfies Maxwell equations with the same sources and
sinks (the charges). They are indeed two independent solutions of the
same second-order equations. Hence their difference,

L [Fb"' - F;dvl ,

(4.10)

b

satisfies a homogeneous system of equations, i.e. a system without sources or sinks. Such a system possesses nontrivial solutions, but they

QUANTUM MECHANICS AND RETROCAUSALITY 39

are radiation fields that decrease like r-1 at spatial infinity: there are
no 0 (r-1) nontrivial solutions. Thus the difference (4.10) is not merely
zero at spatial infinity, it must be identically zero everywhere. Hence

'~ "" Rrbe' --'~ "" p,abdv _-2 1 '"~" [Rb,e' + p,ab dv],

b

b

b

(4.11)

for all times. On combining this result with Eq.{4.9), we obtain
L F+R = ~ [Fbe' + Ftv]- ~ [F~e' + F;dV]
b
L~ [Fbet + F;dV]
bta

(4.12)

This is a stunning result: it says that to calculate the response of a charged particle to all the other charged particles in the universe, one has to sum over the fields emanating from all those other particles, on condition that one uses the time-symmetric solution of the Maxwell equation. In this approach there is no need, nor room, to add a further radiation damping term: it is all contained in the average of the retarded and advanced solutions of Maxwell's equations. Turning the argument around, one can say that the time-symmetric form is equivalent to, and so validates, the conventional calculation in which a retarded solution is supplemented, in a somewhat ad hoc manner, by a radiation damping field.
It must not be thought that we have hereby forged an arrow of time from a time-symmetric theory. This can be seen by complementing Eq.(4.9) by

F +R

L F;dv - ~ [F~et + F;dvl

b

L =

F;dV + ~ [F;dV - F~etl .

(4.13)

bta

This is an equally valid modus operandi, involving the full advanced potential, supplemented by a radiation damping term, but since it is precisely minus the corresponding term in the first line of Eq.{4.9), it might better be called a radiation boosting term.

3. BELL INEQUALITY
Let us turn now to the Einstein-Podolsky-Rosen scenario[3] in its
modern experimental avatar[4]. We will see that the violation of the

40 THE UNIVERSE

Bell inequality loses much of its impact once we entertain the notion of advanced fields.
Briefly, two photons are prepared with opposed spins by the sequential decay of a calcium atom from an excited S state, through an intermediate P state, to the ground state, which is also S. The state of linear polarization of one photon is measured by means of a birefringent calcite crystal and a photo-detector at location A, and that of the other photon by a similar arrangement at location B. The separation of A and B is several metres, and the measurement events are contained within small space-time hypervolumes that have a mutual spacelike separation. Thus the measurement events at A and B are independent of one another in the sense that no information about the result of the measurement at A can be transmitted to B in time to influence the result of the measurement there (and vice versa). This is true only if we limit ourselves to the usual retarded fields. The two photons are not independent, however, in the sense that their spins are correlated because of their common genesis in an atomic decay. The polarizations have, in the locution of Reichenbach, a common cause[5] .
If the optical axes of the calcite crystals at A and B are parallel, then whenever a photon at A is found to go in the direction ofthe ordinary ray, the same is found at B. Similarly, there is perfect correlation in the case that the photons are deflected along the extraordinary ray directions. The more general situation, in which the optical axis at A is at an angle a to the vertical, and the optical axis at B is at an angle (3 to the vertical, leads to the following joint probabilities:

Poo(a, (3) = ~ cos2(a - (3) = Pee(a, (3)
Poe (a, (3) = ~ sin2 (a - (3) = Peo(a, (3) .

(4.14)

Here PO~ is the probability that the photons at A and B both go into the ordinary rays, Pee that both photons go into the extraordinary rays, Poe is the probability that the photon at A goes into the ordinary ray but the photon at B goes into the extraordinary ray, and finally Peo is the probability that the photon at A goes into the extraordinary ray but the photon at B goes into the ordinary ray. The results Eq.(4.14) are predicted by quantum mechanics and confirmed by experiment.
The correlation coefficient is defined as follows:

C(o'., (3) = Poo(o'., (3) + Pee (a, (3) - Peo(o'., (3) - Poe (a, (3) = cos 2(0'. - (3) .
(4.15) If we suppose, with Bell[6], that the joint probabilities, and hence the
correlation coefficient, are separable, in the sense of classical probability

QUANTUM MECHANICS AND RETROCAUSALITY 41

theory, then we can write, for this correlation coefficient,
L C(a, (3) = p('x)C(al'x)C((3I'x),
oX

(4.16)

where ,X are hidden variables that account for the correlations between the two photon polarizations: they arise from the birth of the twin
photons in the de-exciting calcium atom. The weight p(,X) is supposed to be positive and normalized; and C(al'x) is the correlation coefficient

at location A, conditioned by the hidden variable 'x. Similarly, C((3I'x) is the conditional correlation coefficient at location B. Clearly each conditional correlation coefficient, being the difference between two probabilities, lies in the interval [-1,1].
The Bell coefficient is defined as the following combination of four correlation coefficients:

B = C(a,(3) + C(c/,(3) + C(a',(3') - C(a,(3').

(4.17)

It can be measured by combining the results of four separate runs of the experiment, with a choice of two possible orientations (a or a') of the
calcite optical axis at A, and two possible orientations ((3 or (3') at B.
One can show, under the assumption of separability, and

(4.18)

with p('x) ~ 0, that

IBI ~2.

(4.19)

However, by choosing the angles a, (3, a' and (3' suitably, one can arrange
that quantum mechanics yields B = 2V2 > 2. However,

C(a, (3) = cos 2a cos 2(3 + sin 2a sin 2(3,

so the normalization Eq.(4.18) is ruined! - on the right-hand side of
Eq.(4.18) we obtain 2 instead of I! We must conclude that something is
amiss; and we seem to have (at least) the following options:

1. No hidden variables can be found that screen off the common cause.

2. Classical probability theory is simply inapplicable in the quantum domain, in particular Kolmogorov's definition of stochastic independence is inappropriate[7] .

42 THE UNIVERSE
3. Advanced as well as retarded fields are present.
In this paper we will concentrate on the third possibility. If the absorption of the photon at A, after its passage through the calcite crystal at A, is accompanied by an advanced, as well as a retarded field, then information about the interaction of the photon at A, in particular details about the polarizer orientation at the moment of measurement, will ride the advanced wave back to the genesis of the photon pair, arriving at the calcium atom just at the moment that it de-excites. We can understand how, even if the orientation of the A polarizer is changed at the last moment before the polarization measurement, still the interaction can carry information back about the measurement configuration. This way of speaking about information being carried back and forth, as if there were a sort of internal biological time of the sort that science fictional time travellers seem to carry about with themselves, is imprecise and may be confusing. It is better to say that, in the advanced field approach, one has a self-consistent picture in which the state of the photon's polarization is correlated to its future, as well as to its past interactions. The notions of 'cause' and 'information' are replaced by that of 'correlation'.
In one variant of Aspect's experiment, the selection between the angles a and a' at A, and {3 and {3' at B, was changed randomly by two independent oscillators every few nanoseconds. Still the predictions of quantum mechanics were borne out and the Bell inequality violated. Most people interpret this as a demonstration of nonlocality (more soberly of nonseparability). With option 3 we can retain Lorentz covariance while achieving action at a distance. Is this action local or nonlocal? In a sense it is a semantic matter. It is not usual to call conventional retarded field theory nonlocal, the idea being that a particle is only influenced by a distant causal agent in the particle's past light cone. This influence is fleshed out by imputing a real existence to the field (in quantum theory to the field quanta). In this way the field serves as a messenger from afar, bringing influence and information at no more than light speed and delivering it in the vicinity of the particle. One might describe advanced action also as being local in an analogous manner: an influence is transmitted by the advanced field, also within the light cone, arriving in the vicinity of the particle to deliver its information, much on a par with the retarded case. However, this account, even after deanthropomorphization in terms of correlations rather than of causes and of influences, is incomplete. Since correlations can be established forwards and backwards in time, really the only logical requirement is one of consistency. The theory need only be such that it is impossible for an event in a space-time hypervolume both to occur and not to occur2.

QUANTUM MECHANICS AND RETROCAUSALITY 43
4. RETROCAUSALITY
According to David Hume, causality is based on nothing more than the observed constant conjunction of two or more kinds of events, say A and B. It is a mere habit we have to call the earlier of the occurrences, say A, the cause, and the later, B, the effect; no relation of necessity, nor even of likelihood, of a B's succeeding an A in the future can be deduced. If we replace the word 'habit' by 'theory', then we may reconstrue Hume's admonition as the trite Scottish verity that we have no proof that a theory, based on the results of observations in the past, will yield reliable predictions in the future, no matter how numerous the observations in question are. Indeed, we neither have, nor expect to be able to provide, such a proof concerning empirical matters. Moreover, if it is a mere habit, a mere linguistic convention, to call the temporal antecedent a cause, and the successor an effect, why should we not expand our horizons, generalize our theories, and envisage causes that can occur later than their hypothesized effects?
In his intriguing article "Bringing About the Past", Michael Dummett has indeed claimed that the temporal asymmetry of the causal relation is contingent rather than necessary[8] . He describes two situations in which one might speak of a voluntary action performed with the intention of bringing about a past event. Nevertheless, stringent conditions must be satisfied to ensure the coherence of such a standpoint. In particular, Dummett claims that it is incoherent to hold all of the following claims:
1. There is a positive correlation between an agent's performing an action of type A at time t A and the occurrence of an event of type
B at time tB, where tA > tB.
2. It is entirely within the power of the agent to perform A at time tA, if he so chooses.
3. It is possible for the agent to find out, at time tA, whether B has or has not already occurred, independently of his performing A.
One of the two examples that Dummett describes concerns a tribe that has the following custom: "Every second year the young men of the tribe are sent, as part of their initiation ritual, on a lion hunt: they have to prove their manhood. They travel for two days, hunt lions for two days, and spend two days on the return journey; ... While the young men are away from the village the chief performs ceremonies-dances, let us say-intended to cause the young men to act bravely. We notice that he continues to perform these dances for the whole six days that the party is away, that is to say, for two days during which the events that the

44 THE UNIVERSE
dancing is supposed to influence have already taken place. Now there is generally thought to be a special absurdity in the idea of affecting the past, much greater than the absurdity of believing that the performance of a dance can influence the behavior of a man two days' journey away; ... " Ref.[8], pages 348-9. In physicists' terms, retrocausality seems even more absurd than action at a distance.
The chief is a wise and rational man: he believes the first of the abovementioned three claims, at any rate as a statement of the significant statistical efficacity of his magic dancing. Let us further suppose that he does not believe that he is somehow hindered from dancing, or perhaps caused to dance inadequately, during the last two days, in the case that his young men have been cowardly. Then he must deny the third claim: he must assume that there is no way that he can find out, during the crucial days 5 and 6, what in fact has happened during days 3 and 4. For if it were possible to find it out, he could bilk the correlation. That is to say, he could choose to dance properly if, and only if, he knew that his men had not been brave. Then there would not be a positive correlation of the sort envisaged in claim 1.
It seems that we, as anthropologists, would at any rate accept claim 3, and thus conclude incoherence. With the aid of radio communication and a field worker, we could always arrange a bilking scenario, so that A could not count, even stochastically, as a cause of the earlier event B. But is there a situation in which claim 3 could defensibly be denied? There seem indeed to be such cases in subatomic physics. For example, the state of polarization of a photon, which has passed through one polarizer, and will pass through a second polarizer, is a property that we can only test by passing it through the next polarizer that it will encounter. If we choose to insert a calcite crystal in the path of the photon in such a way that it effects a polarization measurement, then this crystal is the next polarizer. Ifit be claimed that the state of polarization of a photon is correlated, not only with the orientation of the polarizer in its past, but also with that of the polarizer in its future trajectory, no bilking of the claim is possible. Here is indeed a clear candidate for retrocausal effects.
5. THE VIEW FROM NOWHEN
Is there a way to fit the notion of retrocausality into a general theoretical framework, rather than merely to permit its fugitive occurrence when all bilking scenarios are impossible? The Australian philosopher Huw Price elaborates a Weltanschauung that he calls the view from nowhen[9]. His point of departure is the time reversal (T) invariance of

QUANTUM MECHANICS AND RETROCAUSALITY 45
microscopic processes3. When two inert gases of different colours, initially segregated and at different temperatures, are allowed to mix, the approach to an equilibrium mixture, of an intermediate colour and at an intermediate temperature, is irreversible, although the dynamics of the molecular collisions is T-invariant. A reversed video recording of the process would not look queer at the level of individual collisions, seen one by one, but it would appear odd at the macro-level, where it would show an apparently spontaneous segregation of the two gaseous components. It is generally agreed that the Stofizahlansatz of Boltzmann, an example of what Price calls P13 , or the principle of the independence of incoming influences, is not acceptable as an explanation of the irreversibility in question. For if P13 holds, why should not PI0 I hold, the principle of the independence of outgoing influences? If one suggests that P101 breaks down because correlations are generated by a collision, then one must ask whether after all P13 is justified. That is, if correlations are generated in a collision process, may they not be present before as well as after the scattering? There seems in fact to be no good reason for adopting a double standard in this matter. Indeed, to do so in the search for a thermodynamic arrow of time is a flagrant example of petitio principii.
A convincing case can be made that the the master arrow of time is cosmological, and the major task lies in explaining why the cosmos had such a low entropy in what for us is the distant past. The thermodynamic arrow follows readily: there is no need for an ad hoc P 13 without a PIOI. The Wheeler-Feynman time symmetric treatment of electromagnetic radiation implicitly appeals ultimately to cosmology, for the effective retardation arises from the assumption of perfect future absorption. This absorption is treated as a matter of irreversible thermodynamics, in terms in fact of a phenomenological absorptive (complex) refraction index. The thermodynamic arrow is tied to the cosmological one, and Wheeler and Feynman reason that radiation appears to us to be retarded because of thermodynamic processes in the future universe. The reason for the direction of the thermodynamic arrow itself seems to lie in the statistical properties of the early universe, i.e. in the fact that it was in such a low entropy condition.
If the arrow of radiation ultimately derives from cosmological considerations, it would be desirable to show this directly, in terms of the properties of a cosmological model, rather than indirectly, via thermodynamics. This is precisely what Hoyle and Narlikar have done[lO]. Suppose that the future is not a perfect absorber, but only works at
efficiency f, in the sense that the reaction of the universe, on particle
a, is not the full Dirac radiation damping of ~ [F~et - F;dv], but only

46 THE UNIVERSE
f times this quantity. Analogously, suppose that the past is also not
perfect as an absorber, but has efficiency p. That is, the boosting is not minus the Dirac term, but rather -p times that quantity. Let us write the symmetric sum over all the fields acting on particle a as a general linear superposition of retarded and advanced contributions, each with its damping or boosting terms:

with A + B = 1. This leads to

(1- 2A) LF[/t + (1- 2B) LFrv =

b

b

(1- 2A + Af - + Bp)F~et (1 - 2B - + Af Bp)F~dv(A.21)

The system is consistent if the coefficients of F~·t and F~dV vanish:

I-p
A 2-f-p

B

=

I-f
2-f-p'

(4.22)

and this is indeed consistent with A + B = 1.
The Hoyle-Narlikar relation Eq.(4.22) is interesting. Unless the past and the future are both fully absorbing, the values of A and Bare
uniquely defined. For p < 1 and f < 1, since neither A nor B is zero, the
radiation from an accelerated charge is effectively neither retarded nor
advanced, but a superposition of the two, and the radiation damping
is a definite fraction of the Dirac value. The special case in which the
future is a perfect, but the past an imperfect absorber, f = 1 but P < 1,
leads to A = 1 and B = 0, which is the empirically satisfactory situation
of effectively retarded radiation, together with the full strength Dirac
radiation damping. With p = 1 but f < 1, on the other hand, we obtain
B = 1 and A = O. That is, in the situation in which the big bang acts
as a perfect absorber but the future is not fully absorbing-in an open
Friedmann model, for instance-one finds the unacceptable effectively
advanced solution, with a radiation boosting term, i.e. minus the Dirac
radiation damping. The main point to be made here is that, while the
basic emission is time symmetric, the effective radiation is not symmet-
ric if and only if p =I f. That is, the radiative temporal symmetry is
broken by an asymmetry in the absorptive properties of the past and
future universe, in short by a cosmological asymmetry.

QUANTUM MECHANICS AND RETROCAUSALITY 47

It seems that Feynman himself, after he had elaborated quantum electrodynamics (QED) in the form that we still use today, rejected only part ofthe credo of symmetric action at a distance[ll]: "It was based on two assumptions:
1. Electrons act only on other electrons
2. They do so with the mean of retarded and advanced potentials
The second proposition may be correct but I wish to deny the correctness of the first." The reason given for accepting that a charged particle can interact with its own field was precisely the success of the calculation of the anomalous magnetic moment of the electron-the famous 9 - 2 to which we alluded at the beginning.
The close similarity between the Wheeler-Feynman account of radiation and that given in QED-and also the crucial difference-can be appreciated by looking at the Green's functions of the theories. The electromagnetic field tensor may be expressed in terms of the four-potential, AJl(x), by
PJlV = 8JlAv - 8vAJl ,
and the Maxwell equations can be written

82AJl = jJl'

(4.23)

in the Lorentz gauges, for which 8JlAJl = O. Here jJl is the four-current density. A solution of Eq.(4.23) is expressible as an integral,

where DJlv is a Green's function that satisfies
o 82DJlv(x) = gJlV 4 (x)

The relations between the different theories can be appreciated by comparing the various choices of Green's function. 'The standard classical choice is the retarded one:

DJrlevt ( X )

=

-

(2gJ)4lV! 7r

d 4 P

(p 0

+

ue.:-)2ipx-

p.p

=

-g/-2lv O(xo)8(x2 ). 1f

The if. prescription means that the Green's function is to be interpreted as a distribution on a space of analytic functions: the implicit limit f. --+ 0 through positive values is equivalent to a small deformation of the ko-integration contour in the appropriate direction. The advanced
Green's function is obtained from the above by changing the sign of E,

48 THE UNIVERSE

which implies that O{xo) is replaced by O{-xo). The Green's function of
the Wheeler-Feynman theory is

D;:{x) = ~ [D~~{x) + D~~{x)]

= _ gJ.lv !d4pe-iPX p _ gJ.lv 8{x2)

(27r)4

p2 - 47r

'

(4.24)

where P means the principal value in the sense of Cauchy.

The QED Feynman propagator, defined through the vacuum expec-

tation value of the time ordered product of two fields, is in QED

! + DF x) _ J.lv{ -

-(-2g7J.rl- v)4

d4

p

e -
p2

i

p

x
if.

_ -

1 -4giJ7.-lrv2 x 2 -

if. .

Now we can write

[P - DF (x) = _ gJ.lv !d4pe-iPX

i7r8(p2)]

J.lV

(27r)4

p2

On comparing this with Eq.(4.24), we see that the Wheeler-Feynman Green's function is the real part of the Feynman Green's function. The extra piece, the imaginary part of the Feynman propagator, corresponds to the mass-shell contribution in momentum space, and has to do with the self-interaction of a charged particle that is coupled to the electromagnetic field. It guarantees the meromorphy of scattering amplitudes on the principal sheet of a suitably cut p2-plane.
Microcausality, as it is now understood in quantum field theory, is expressed by the vanishing of (anti-)commutators of fields outside the light-cone; and this leads to analyticity of scattering amplitudes with respect to momenta. However, this new style causality is perfectly consistent with, indeed requires, retrocausality on the same footing as ordinary (Humean) causality. However, the heavy price that we must pay is the introduction of self interaction. This gives rise to divergences that are only provisionally hidden in the renormalization programme. Feynman was not satisfied with what he had achieved[ll]: "I invented a better way to figure, but 1 hadn't fixed what 1 wanted to fix ... The problem was how to make the theory finite ... 1 wasn't satisfied at all."
Hoyle and Narlikar also add a self-action term to their quantized action at a distance theory, almost as an afterthought, and clearly against their better inclination[lO]. As Dirac had done before them, they simply introduce an ultraviolet cut-off that breaks Lorentz covariance. Dirac writes, at the end of the fourth edition of his classic book, Quantum Mechanics[12]: "It would seem that we have followed as far as possible the path of logical development of the ideas of quantum mechanics as

QUANTUM MECHANICS AND RETROCAUSALITY 49
they are at present understood. The difficulties, being of a profound character, can be removed only by some drastic change in the foundations of the theory, probably a change as drastic as the passage from Bohr's orbit theory to the present quantum mechanics."
Could it be that the change to the view from nowhen, following in the footsteps of Wheeler, Feynman, Hoyle, Narlikar and Price, is sufficiently drastic to cure the malaise of electromagnetism and of quantum mechanics? As we have shown, retrocausality was built into the very foundations of QED. Yet the T-symmetry of quantum mechanics is routinely squandered in the projection postulate, with its attendant mystique of the measurement process. Might a rigorously atemporal viewpoint lead to a physical picture closer to Einstein's than to Bohr's, and might it be that the infinite self interaction is somehow a, mistake induced by our time-bound viewpoint?
Notes
= = = = = 1. peA) 1, A {I, 2}. CC'Y11) cos 2-y, C(-Y12) sin 2-y, -y {a, ,B}.
2. We leave out of consideration the science fiction scenario of many worlds. This option is logically flabby and it carries moreover an unwieldly metaphysical baggage.
3. This must be generalized to PCT invariance for some electroweak interactions, for example those responsible for KO-decay.
References
[1] http://pdg.lbl.gov/1998/contents_tables.html Kinoshita, T. 1981, Phys. Rev. Lett. 47 1573.
[2] Wheeler, J.A., Feynman, R.P. 1945, 1949, Rev. Mod. Phys. 17 157; 21 425.
[3] Einstein, A., B. Podolsky, B., Rosen, N. 1935,Phys. Rev. 47777. [4] Aspect, A., Dalibard, J., Roger, G. 1982, Phys. Rev. Lett. 49 1804. [5] Reichenbach, H. 1956, The Direction of Time, Univ. of Calif. . [6] Bell, J.S. 1987, Speakable and Unspeakable in Quantum Mechanics,
Cambridge Univ. Press. [7] Atkinson, D. 1998, Dialectica 52 103. [8] Dummett, M.A.E. 1964, Philos. Rev. 73 338. [9] Huw Price, 1996, Time's Arrow and Archimedes' Point, Oxford. [10] Hoyle, F., Narlikar, J.V. 1974, 1995, Action at a Distance in Physics
and Cosmology, Freeman and Co. ; Rev. Mod. Phys. 67 113. [11] Schweber, S.S., 1986, Rev. Mod. Phys. 58449; see page 501. [12] Dirac, P.A.M. 1958, Quantum Mechanics, Oxford (4th edition); see
page 310.

50 THE UNIVERSE [13] Dirac, P.A.M. 1978, Directions in Physics, Wiley, New York.

Chapter 5
INSTANTONS FOR BLACK HOLE PAIR PRODUCTION

Paul M. Branoff and Dieter R. Brill
Department of Physics, University of Maryland College Park, MD 20742, USA

Abstract

Various ways are explored to describe black hole pair creation in a universe with a cosmological constant that do not rely on an intermediate state of "nothing".

1. INTRODUCTION
Of Jayant Narlikar's many important contributions to astrophysics and cosmology, none is more creative and imaginative than the idea, developed with Fred Hoyle, that particles may be created as the universe expands. Stated long before quantum effects of gravity could be treated, this proposal has new meaning today. Methods are now available to analyze quantum particle production in dynamic spacetimes, and even black hole creation can be understood semiclassically as a tunneling process. The latter process is the main subject of this paper.
Although a complete theory of quantum gravity does not yet exist, examples of gravitational tunneling have been studied for a number of years, including such processes as pair creation of black holes and vacuum decay of domain walls. In each case the treatment is based on an instanton (solution of the Euclidean field equations) that connects the states between which tunneling is taking place. However, there are some nucleation processes of interest where the standard instanton method is not effective, for example because no solutions exist to the Euclidean Einstein equations that smoothly connect the spacelike sections representing the initial and final states of the tunneling process. It is therefore an interesting challenge to adapt the "bounce" method, most suitable for vacuum decay calculations, to deal with non-static initial states and
51
N. Dadhich andA. Kembhavi (eds.), The Universe, 51-68.
© 2000 Kluwer Academic Publishers.

52 THE UNIVERSE
background fields such as a positive cosmological constant or domain walls typically present when particle-like states are created.
A positive cosmological constant (and other strong gravitational sources, such as a positive energy density domain wall) acts to increase the separation of timelike geodesics. It is therefore expected to "pull particles out of the vacuum" by favoring creation of pairs over their annihilation. The analogous creation of black hole pairs in de Sitter space can be treated in WKB approximation by the "no boundary" realization of quantum cosmology [1]. The first (and usually only) step in such a treatment consists of finding a solution of the Euclidean field equations containing the initial state (pure de Sitter universe) and the final state (Schwarzschild-de Sitter space) as totally geodesic boundaries. Such a solution exists only if we accept it in two disconnected pieces. If the cosmological constant is large enough one then obtains an appreciable probability of creating in each Hubble volume a pair of black holes comparable to the volume's size; if these break up into smaller ones (see, for example, Gregory and Laflamme [2]) one has, within pure gravity, a model of continuous creation not too far removed in spirit from that of Hoyle and Narlikar.
This model is, however, not fully satisfactory in several respects. For example, it is not clear how to calculate the "prefactor" of the exponential in the transition probability, which would define the dimensionful rate of the process. When it can be calculated from the fluctuations about the instanton [3], a "negative mode" is necessary for a non-vanishing rate. But this negative mode would have to connect the two parts of the instanton, and therefore cannot be treated as a small perturbation. A discontinuous instanton is of course also conceptually unsatisfactory, because the usual composition rules assume that histories are continuous.
Each of the two parts of the disconnected instanton has the universe's volume reaching zero. By forbidding arbitrarily small volumes one can connect the two parts. The exploration of modifications of Einstein gravity in which this is possible is still in its infancy. For example, Bousso and Chamblin [4] have used virtual domain walls to construct interpolating instantons. A similar technique using 'pseudomanifolds' has also been used to construct such solutions [5].
Modifications of Einstein's theory that have been proposed in other contexts may also give continuous instantons, if the change from Einstein's theory becomes important at small volumes. For this reason it is natural to consider higher curvature gravity theories.
Another promising modification of Einstein gravity is Narlikar's Cfield [6]. This field can describe reasonable energetics of particle produc-

INSTANTONS FOR BLACK HOLE PAIR PRODUCTION 53

tion in a context where quantum mechanics plays no essential role, and it is therefore interesting to explore, as we will below, whether it can also solve the disconnectedness problem in the instanton treatment.
So we ask whether these modifications of the Einstein-Hilbert action allow continuous paths from an initial to final cosmological state when calculating amplitudes for cosmological black hole production in the context of closed universes. We will outline a modified version of the calculation of Bousso and Chamblin concerning the use of virtual domain walls in constructing interpolating instantons. We next discuss the existence of continuous instantons in higher curvature gravity theories whose Lagrangians are nonlinear in the Ricci scalar. Finally, we consider the case of general relativity with a cosmological constant and a Narlikar C-field.

2.

GRAVITATIONAL TUNNELING

Processes such as black hole pair creation can be analyzed semiclassically through the use of instanton methods. One can think of such a process as a tunneling phenomenon. The initial state consists of a universe with some background metric and no black holes, and the final state consists of a universe with two black holes supplementing the background metric. Classical dynamics is prevented from connecting the two states by a generalized potential barrier. The quantum process can "penetrate" the barrier with some probability, and the same barrier makes it improbable for the final state, once created, to "annihilate" back to the initial state. In problems that can be treated by instantons, the non-classical transition from initial to final state can be described approximately as an excursion in imaginary time. A solution that goes from the initial state to the final state and back again is called a bounce solution; an instanton is a solution which goes from the initial state to the final state, i.e., half a bounce. In the WKB interpretation the excursion into imaginary time simply signifies an exponentially decreasing wavefunction that is large only near configurations contained in the instanton. In the sum over histories interpretation the instanton is a saddle point by means of which the propagator is to be be evaluated.
The exponential of the instanton's classical Euclidean action is the dominant factor in the transition probability, provided it is normalized so that the action vanishes when there is no transition. That is, we are really comparing two instantons, one corresponding to the background alone in which initial and final states are the same, and the instanton of the bounce, in which they are different. If the initial state is static, it is typically approached asymptotically by the bounce, and therefore

54 THE UNIVERSE
the normalization of the action can be achieved by a suitable surface term. If the initial state is only momentarily static, as in the case of the de Sitter universe, we must find the two instantons explicitly and evaluate their actions. In the context of the disconnected instanton the background instanton corresponds to two disconnected halves of a 4sphere: a de Sitter space fluctuating into nothing and back again. A first test whether a modification of Einstein's theory can have connected instantons is therefore to see whether the background instanton can be connected (Fig. 1).
The rate of processes like black hole pair creation is calculated by subtracting from the action of the bounce, f bc , the action corresponding to the background state, fbg. The pair creation rate is then given as
(5.1)
where A is a prefactor, which is typically neglected in most calculations because it involves fluctuations about the classical instantons that are difficult to calculate. Without this dimensionful prefactor one can find the relative transitions to different final states, but the actual the number of transitions per spacetime volume to a given final state can only be estimated, for example as 1/(instanton four-volume) for finite volume instantons.
The connected background instanton as described above is closely related to a Euclidean wormhole, or birth of a baby universe [7]: if the two parts are connected across a totally geodesic 3-surface, we can, according to the usual rules, join a Lorentzian space-time at that surface, passing back to real time. An instanton with this surface as the final state would then describe the fluctuation of a large universe into a small one, with probability comparable to that of the creation of a black hole pair. Thus whatever process provides a connected instanton is likely to lead not only to the pair creation but also to formation, of a baby universe. (In section 5 we will see how the latter can be avoided)
An instanton calculation has been used by a number of authors to find the pair creation of black holes on various backgrounds (see, for example, Garfinkle et al [8]). The instantons involved a continuous interpolation between an initial state without black holes and a final state with a pair of black holes. By contrast, in cosmological scenarios where the universe closes but Lorentzian geodesics diverge, as in the presence of a positive cosmological constant or a domain wall, there are Lorentzian solutions to Einstein's equations with and without black holes (such as de Sitter and Schwarzschild-de Sitter spacetimes, respectively), but there are no Euclidean solutions that connect the spacelike sections of these geometries [4]. (For the related case of baby universe creation the

INSTANTONS FOR BLACK HOLE PAIR PRODUCTION 55

absence of such solutions is understood, for it is necessary that the Ricci tensor have at least one negative eigenvalue [9].)
The No-Boundary Proposal of Hartle and Hawking [1] can be modified
to provide answers in these cases. The original proposal was designed to eliminate the initial and final singularities of cosmological models by obtaining the universe as a sum of regular histories, which may include intervals of imaginary time. One can think of the Euclidean sector of the dominant history as an instanton that mediates the creation of a (typically totally geodesic) Lorentzian section from nothing. By calculating the action corresponding to these instantons, one can calculate the wave function for this type of universe, i.e.,
(5.2)
where IinstW) = ~Ibc is the action corresponding to a saddlepoint so-
lution of the Euclidean Einstein equations whose only boundary is the 3-dimensional geometry q. The probability measure associated with this universe is then given by
(5.3)
To relate the probability measure to the pair creation rate of black holes given in Eq. (5.1) one writes

r - R - - Pbh _ exp[- ( 2Iibnhst - 2Iibngst )] bg

(5.4)

so the ratio of the probability of a universe with black holes to the

probability of a background universe without black holes is taken to be

also the rate at which an initial cosmological state can decay into a final cosmological state, that is, the pair creation rate. In the latter sense the two disconnected instantons together describe the tunneling process.
Although this formalism allows one to calculate, in principle, the rates of nucleation processes, there is no well-justified reason why Eq. (5.4) should be identified with this quantity. The straightforward interpretation of the instanton concerns the probability for one universe to annihilate to nothing and for a second universe to be nucleated from nothing. This second universe can either contain a pair of black holes, or it can be identical to the initial universe, but it retains no "memory" of the initial state. It would clearly be preferable to have a continuous interpolation between the initial and final states. (This would allow degrees of

freedom that interact only weakly with the dynamics of gravity to act as a memory that survives the pair creation.) In the following sections we

will consider several ways in which this continuity of spacetime can be

achieved, the first of which involves matter fields that can form virtual domain walls.

56 THE UNIVERSE

3. CONTINUOUS INSTANTONS VIA VIRTUAL DOMAIN WALLS
In this section, we will consider the method by which the authors of [4] use virtual domain walls to construct continuous paths between two otherwise disconnected instantons. They illustrated the method for the nucleation of magnetically charged Reissner-Nordstrom black holes in the presence of a domain wall. We will confine attention to nucleation of uncharged black holes in a universe with a cosmological constant. The initial state is the de Sitter universe and the final state is the extremal form of a Schwarzschild-de Sitter universe known as the Nariai universe [10], which is dictated by the requirement that the Euclidean solution be non-singular. To understand virtual domain walls we will need some elementary properties of real domain walls. These have been discussed extensively in [4, 11, 12, 13, 14, 15].

3.1 BRIEF OVERVIEW OF DOMAIN WALLS

A vacuum domain wall is a (D - 2)-dimensional topological defect in

a D-dimensional spacetime that forms as a result of a field </> undergoing

the spontaneous breaking of a discrete symmetry. If we let M denote the

manifold of vacuum expectation values of the field </>, then a necessary

condition for a domain wall to form is that the vacuum manifold is not
connected (11"0 (M) =J 0). An example of a potential energy function U(</»

of the field </> giving rise to domain walls is the double-well potential.

Throughout this section, we will be dealing with a domain wall in

the "thin-wall" approximation, which means that the thickness of the

domain wall is negligible compared to its other dimensions, and it is

homogeneous and isotropic in its two spacelike dimensions, so that the

spatial section of the wall can be treated as planar, and the spacetime

geometry as reflection symmetric with respect to the wall.

The action of a real scalar field </>, interacting with gravity, that may

form a domain wall is given by

! Idw = d4xH [Lmat + Rl~:A]

(5.5)

with matter Lagrangian

Lmat = -~ gp.v 8p.</>8v</> - U{</»

(5.6)

and stress-energy tensor

Tp.v = 8p.</>8v</> - gp.v [~ga{38a</>8{3</> + U{</»]

(5.7)

INSTANTONS FOR BLACK HOLE PAIR PRODUCTION 57
Here U (¢) is a potential function with two degenerate minima ¢_ and ¢+, at which U = 0; 9 is the determinant of the 4-metric gJjIl; and R is the Ricci scalar. (We have neglected boundary terms in the action since the instantons we will be considering are compact and have no boundary.)
The trace of the Einstein equations (resulting from the variation of Idw with respect to gJjIl) gives
(5.8)
which can be used to simplify the action (5.5) when evaluated on a solution:
(5.9)

The ¢-field is essentially constant away from a domain wall, with
¢ = ¢_ on one side and ¢ = ¢+ on the other. In Gaussian normal coordinates {(i, z) with the wall at z = 0, ¢ depends only on z, and the
field equation for ¢ implies that Tzz of Eq. (5.7) is negligible. The rest of the components of the stress-energy tensor differ from zero only near the wall, where ¢ changes rapidly from ¢_ to ¢+:

Tt' = ac5(z)diag(l, 1, 1,0)

(5.10)

where u can be related via the ¢-field equation to the ¢-potential alone,

!u = 2U{¢(z))dz.

(5.11)

Thus u is the surface energy density of the wall. For such surface distri-

butions the Israel matching condition imply that the intrinsic geometry

hij of the domain wall is continuous, and that the extrinsic curvature

jumps according to

Ktj - Kij = 41fuhij .

(5.12)

Here the normal with respect to which Kij is defined points from the +

side of the surface to the - side. Outside the wall we have the sourceless

Einstein equations.

3.2 JOINING INSTANTONS BY DOMAIN WALLS
The jump (5.12) in extrinsic curvature across a domain wall can be used to join the two parts of a disconnected instanton (Fig. 1) by "surgery": We remove a small 4-ball of radius", from each instanton.

58 THE UNIVERSE

(a)

(b)

(c)

Figure 5.1 Two-dimensional analog of de Sitter instanton. Imaginary time runs horizontally. Because no significant change can be shown in two dimensions, this is a "background" instanton with identical initial and final states. (a) The disconnected instanton. (b) "Yoyo" instanton connected by domain wall (heavy curve labeled DW). (c) Instanton connected by a "virtual baby universe" (BU).

Their two 3-surface boundaries have the same intrinsic geometry, and their extrinsic curvatures are proportional to the surface metric. They can therefore be joined together in such a way as to satisfy the Israel matching conditions, Eq. (5.12), thereby inserting a domain wall.
However, the surface energy density aof the domain wall used to join
the instanton must be negative: As we approach the domain wall from the initial state, heading towards annihilation, successive 3-spheres are
shrinking, Kij < o. After we pass through the domain wall, successive 3-
spheres are expanding, Kij > O. Because of the negative energy density
the authors of [4] call this a virtual domain wall, but it is not virtual in the sense that it corresponds to a Euclidean solution of the equations of section 3.1, for the (7 of Eq. (5.11) remains positive when passing to imaginary time. Within this scheme the only way to achieve a "yoyo" instanton as a saddle point of the Euclidean action is to have a scalar field with a negative energy also in the real domain, that is, a Lagrangian with the opposite sign as that of Eq. (5.6). As we will see in section 5, in that case a plain scalar field, without the domain-waIl-forming potential U(¢), will do as well and is preferable.
By how much does the Euclidean action change when we introduce a domain wall whose radius 'T/ is small compared to the radius y'3/A of the instanton itself? The extrinsic curvature of the connecting 3-sphere is then nearly the same as what it would be in flat space, Kij = hij/'T/, and the jump in curvature is twice that; hence the size of the domain wall is determined from Eq. (5.12),

1 'T/=--_.
211"(7

(5.13)

INSTANTONS FOR BLACK HOLE PAIR PRODUCTION 59
The Euclidean version of Eq. (5.9) is
(5.14)
We have taken a 4-ball with scalar curvature R = 4A away from each part of the original instanton, for a total change in action (including a boundary term) by 37rrJ2/2 - A7rrJ4/8; this is comparable to that due to the added domain wall with action given by Eqs. (5.14) and (5.11), Idw = -7r2(jrJ3 = ~7rrJ2, which is small compared to the total action -37r/ A. Thus the Euclidean action increases when we add the domain wall, and the connected instanton therefore has a relatively smaller probability measure (although the difference is small compared to the total action), and the disconnected instanton will dominate. If the path integral is extended over continuous histories only, the domain wall provides the only saddle point, with action very close to what the discontinuous history would have given, thus justifying the calculation using the discontinuous history alone. But in that case a path integral without a domain-wall-forming scalar field gives a very similar value of the action, as shown in [4].
Introducing this scalar field may therefore be considered a high price to pay for gaining a saddle point, particularly because it entails other, less desirable processes. For example, the "center" z = 0 of the domain wall is totally geodesic with 8¢/8z = 0, that is, a possible place to revert from imaginary time back to real time. This corresponds to the formation of a baby universe of size comparable to rJ and smaller Euclidean action than that for the black hole formation.
If a field exists that can form small domain walls, any two instanton parts can be connected by such surgery across one or several small 3-spheres, with a change in action as estimated above for each; the dominant history will have the fewest connections.
Finally, recall that the periodicity in imaginary time of each part of the disconnected instanton is well defined by the requirement that conical singularities should be absent from each part. If the parts are connected where there would otherwise be a conical singularity, one such requirement is eliminated. Thus there are connected instantons for which the final state is not Nariai but Schwarzschild-de Sitter geometry with black hole and cosmological horizons of unequal size.

60 THE UNIVERSE
4. CONTINUOUS INSTANTONS IN HIGHER CURVATURE THEORIES
Higher curvature theories have a long history and have been proposed in several different contexts. For example, they arise naturally in theories describing gravity by an effective action [16, 17].
In this section we will explore whether higher order theories can pass the "first test" of Section 2, namely whether there is a continuous instanton describing the annihilation and rebirth of de Sitter space (generalized to these theories). Adding higher order terms to the action does not, however, immediately eliminate disconnected instantons; for example, de Sitter space (that is, a spacetime of constant curvature) is a solution of many higher-order theories. In fact, if the universe without and with black holes can originate by tunneling from nothing, a disconnected instanton will also exist. Therefore connected instantons may again co-exist with the de Sitter-like, disconnected instantons.
The Euclidean action we will be considering has the form

(5.15)

where

(5.16)

R is the Ricci scalar, A is the cosmological constant, and Cl', ,,(, etc., are coupling constants whose value we leave unspecified for the moment. The metric has the Euclidean Robertson-Walker form appropriate to three-dimensional space slices of constant positive curvature:!

(5.17)

Here T is imaginary time determined from the analytic continuation t ~ iT, N is the lapse function, a is the universe radius and dn~ is the metric on the unit three-sphere. Having the metric depend on Nand a allows us to obtain all the independent Einstein equations by varying only these functions in the action (5.15): variation with respect to a gives us the one independent spacelike time development equation, and variation with respect to N yields the timelike constraint equation, as in ordinary Einstein theory. A further variation that is easily performed is a conformal change of the metric, giving the trace of the field equations, which is not independent of the other equations but involves only the function f{R):

o (5.18)

INSTANTONS FOR BLACK HOLE PAIR PRODUCTION 61

a3 f+-aNf a 3 - -d [-a.UNa3) ]

aN

dT aN

2R!' + 6\72!, - 4f

o (5.19)
o (5.20)

where

\72 _ d2 + 3iL ~
- dT2 a dT'

(5.21)

a dot denotes d/dT, and a prime denotes d/dR. Equation (5.18) is a fourth order ordinary differential equation, and
Eq. (5.19) is a third order first integral of this equation. The trace equation (5.20) shows that we can regard R as an independent variable, satisfying a second order equation. In this view Eq. (5.20) replaces

Eq. (5.18) (to which it is equivalent), and a also satisfies a second-order differential equation, namely its definition in terms of R,

(5.22)

In addition we still have the constraint, Eq. (5.19), a first order relation between a and R.
A general Hamiltonian analysis (c.f. [18J and references therein), not confined to the symmetry of Eq. (5.17), bears out the idea that, as a second-order field theory, this is Einstein theory coupled to a nonstandard scalar field [19J. For example, for a Lagrangian quadratic in the Ricci scalar with no cosmological constant, the relationship between R and the non-standard scalar field ¢ is given by [20J

¢= {faR

(5.23)

where the ¢>-field has the standard stress energy tensor multiplied by
(1 + 4(7r/3)1/2¢)-2.
Can this effective scalar field form a domain wall in four dimensions? If the coefficients up to , in Eq. (5.16) are non-zero, then the "force term" R!, - 2f occurring in the trace equation (5.20) vanishes at three equilibrium states for R, where \72 f'(R) = 0, approximately at R =
±1/vIT and at R = 4A, for small,. But in order to have a macroscopic
universe on either side of the wall we need R = 4A on either side, so the usual wall formation where the scalar field changes from one equilibrium to another is unsuitable in this case. A solution of the bounce type may appear possible, since the equilibrium at R = 4A is unstable. At the turning point the time-dependent R would then have to "overshoot" the
stable equilibrium near R = -1/v1T' A negative R is required there so

62 THE UNIVERSE

that the universe radius can turn around at the same moment. This
synchronization, if possible at all (numerical calculations have failed to reveal it to us; see however ref. [21]), would require fine tuning that does
not appear natural in this context. Furthermore, if we had a bounce for both a and R, half of it would be an instanton describing the formation of a baby universe of size '" 'Y~1/4, which would then continue to collapse
classically, and this process would be exponentially more probably than
the black hole formation. For these reasons the effective scalar field that
derives from higher curvature Lagrangians of the form (5.15) does not
appear promising for connected instantons.
We therefore consider solutions to Eqs (5.18) - (5.20) when R is con-
stant, R = Ro. To allow a continuous transition to imaginary time at
7 = 0 we make the usual ansatz that all odd time derivatives of a{7)
vanish at 7 = O. With the choice N = 1, the above equations at 7 = 0
take the form

Rof' - 2f = 0
aof + 6aof' 0
a5f - 2f'{aoao - 2) = O.
Eliminating f from these equations we get the condition

(5.24) (5.25) (5.26)

(a6Ro - 12)f' = O.

(5.27)

Thus, we have two classes of solutions. The first class is described by the condition Ro = 12/a5. The second class is described by the condition
f = f' = O. If R = Ro = 12/a5 and ao = 0 then the unique regular solution of
Eq. (5.22) is a de Sitter-like solution, a(7) = aOcos{7/ao), leading only to the disconnected instanton.
The second condition indeed allows periodic, non-collapsing solutions with any amplitude A of the form

V 2
a =

6
Ro

+ A cos

fRo
37

where

f{Ro) = 0 = !,(Ro). (5.28)

If we want this Ro to be close to that of the de Sitter universe, Ro = 4A, then at least one of the higher-order coefficients (a, 'Y ... ) in f{R) of Eq. (5.16) has to be large and rather fine tuned. Furthermore, because the action for all of these solutions vanishes, we should integrate over all values of A, which includes some disconnected instantons, so this problem is not really avoided by these solutions. {They appear pathological also in other ways, for example they would allow production of baby universes of any radius. They would also tend to be unstable in the

INSTANTONS FOR BLACK HOLE PAIR PRODUCTION 63
Lorentzian sector, although this can be confined to the largest scale by judiciously choosing f(R) = R - 2A except near Ro '" 2A.)
5. CONTINUOUS INSTANTONS IN C-FIELD THEORY
Except for boundary terms, which describe classical matter creation and which we neglect in the present context, the C-field Lagrangian is similar to the usual scalar field Lagrangian without self-interaction (Eq. (5.6) with U = 0), but with the important difference that the
coupling constant - f of the C-field has the opposite sign from the usual
one [6]. Thus the total action of gravity with cosmological constant and C-field has the form, for Lorentzian geometries
(5.29)
(We have not included ordinary matter fields here because we are confining attention to pair production of black holes as purely geometrical objects.)
5.1 SOURCELESS C-FIELD IN LORENTZIAN COSMOLOGY
The field equations that follow from this action by varying C and 9J.tv are, for the C-field:
(5.30)
and for the geometry,
The stress-energy tensor Tffv gives a negative energy density (for f > 0).
Narlikar [6] has given reasons why this violation of the energy condition is not an objection when the C-field is coupled to Einstein gravity of an expanding universe.
For Lorentzian cosmology we make a Robertson-Walker ansatz analogous to (5.17),
(5.32)
In agreement with the homogeneous nature of this geometry we assume that C is homogeneous in space and hence depends only on t. The field

64 THE UNIVERSE

O~~=-~~-------------a
-1 ' c

Figure 5.2 The effective potential for de Sitter-like universes: a universe with only cosmological constant (curve a) , one with a real C-field (curve b) , and one with a virtual C-field (curve c).

equations, derived by varying a, N, and C, and then setting N = 1, are

ii 0,2 + 1

2 a

-

+

-a2-

-

A

a. 2 + 1 - _Aa2 3

d ( a 3( : )

a3 dt

47rJ(:2
_ 47rJ (:2a2 3
O.

(5.33) (5.34) (5.35)

The second equation, as usual, is a first integral of the first (time development) equation, and it implies the latter except for extraneous solutions a = const. The third equation has the integral

. K C=-
a3

(5.36)

where K is a constant. By eliminating (: we obtain an equation of the "conservation of energy" type for a:

. 2

. 2

a +v'ff=a

e

A 2 - -3a

+

47r JK2
3a4

=-1.

(5.37)

This is the usual de Sitter equation supplemented by a term in l /a4,
which is unimportant at late times when a is large and does not change the qualitative Lorentzian time development at any time (Fig. 2).

INSTANTONS FOR BLACK HOLE PAIR PRODUCTION 65
5.2 SOURCELESS C-FIELD IN EUCLIDEAN COSMOLOGY
The effective potential in Eq. (5.37) increases monotonically as a decreases below the minimum classically allowed value. The corresponding Euclidean motion in such a potential therefore does not bounce; instead, a would continue to decrease and reach a = 0 in a finite Euclidean time. This is a geometrical singularity if K -=f:. 0 because, for example, it follows
from Eqs. (5.33) and (5.36) that R = 4A + 81fj(K2/a6 ).
However, a different potential is obtained if the motion of both a and C is continued to imaginary time,2 thereby describing a virtual process that involves both of these variables, so that we take into account fluctuations in C as well as in a. Then the K of Eq. (5.36) becomes imaginary, K = ik, and the Euclidean "conservation of energy" equation becomes

-a. 2 - -Aa2 - - 41f- jk2 = -1.

3

3a4

(5.38)

It is easily seen that, for sufficiently small k, this equation does have bounce solutions, with a turning point at a f'V k1/ 2 j1/4 (Fig. 2). Thus the C-field theory passes the "first test": it has a continuous instanton describing a fluctuation with identical initial and final state. It is reasonable to suppose that the theory will also have continuous instantons describing the creation of a black hole pair, because for small k the turning point occurs at small a, so that two disconnected instantons can be joined by surgery similar to that of section 3.
It is essential that the fluctuation of the C-field be virtual, that is, that the coupling constant j have the opposite sign from the usual, positive energy density scalar field. If the C-field were real, time could revert to real values at the minimum radius of the bounce and continue in a small, Lorentzian universe [23] that we have above described as a baby universe. This transition would be the most probable if allowed. By contrast, in the case of the virtual C-field this transition is not allowed, The reason is that at the moment of the bounce, the C-field's effective potential dominates. A return to real time (K changing from imaginary
to real) would make a large change in Veff of Eq. (5.37), violating this
Lorentzian Hamiltonian constraint. A much smaller violation is involved at the first change to imaginary time, at large a. This can occur if the background is not exactly de Sitter-like, but contains some gravitational wave excitation that can supply the necessary small energy difference in the local region where the black hole will form. Thus the C-field makes a continuous instanton possible, but avoids forming a baby universe.3

66 THE UNIVERSE
5.3 BLACK HOLES IN C-FIELD COSMOLOGY
As a final step we exhibit as an endstate of the particle creation instanton an expanding universe in C-field theory of spatial topology 8 1 x 8 2. This describes a universe with an extremal black hole pair in the same sense that the Nariai solution [10, 25] describes such a universe in Einstein's theory. The metric has the homogeneous form

where X has periodicity appropriate to 8 1, () and <p are coordinates on 8 2, and a and b are functions only of t. The C-field likewise is a function only of t and therefore obeys the conservation law analogous to (5.36),

. K C= ab2 '

(5.40)

The field equations then take the form

. '2
Gt + A = _ 2iLb _ b + 1 + A

t

ab

b2

Gx

x

+

A

=2bb-

-

-b2-+-
b2

+1

A

() Go

+ A =

-

ii -;;:

-

iLb ab -

bb + A

41l'J K2
a2b4 41l'JK2
a 2 b4 41l'J K2
a2 b4 .

(5.41) (5.42)
(5.43)

If the universe volume expands similar to the Nariai solution, the effects
the C-field will become negligible at late times. It is therefore reason-
able to solve the field equations with the condition that the solution
be asymptotic to the Nariai universe, a(t) = (l/..JA) cosh ..JAt, b(t) = l/..JA. We also require a moment of time-symmetry (to enable the
transition from imaginary time). The solution to first order in £ =
41l'JK2 A3/2 is

a(t)

b(t)

These functions do not differ much from those for the Nariai solution
for any time t. However, the differences would become large in the
continuation to imaginary time, as the volume decreases. In order to reach a minimum volume we again need an imaginary K (virtual C-
field). This minimum volume, like all t = const. surfaces, has topology

INSTANTONS FOR BLACK HOLE PAIR PRODUCTION 67

Sl X S2 and would therefore not fit directly on the minimum-a surface of a de Sitter-like metric, Eq. (5.17); a solution with less symmetry in both spaces would be needed to make the match.

6.

CONCLUSIONS

In Einstein's theory of gravity with a cosmological constant, typical Euclidean solutions describe a universe originating from "nothing," or decaying into nothing, but there are no equally simple solutions corresponding to quantum processes, such as creation of a pair of black holes, which change a universe that is already present. According to the simple interpretation of Euclidean solutions in Einstein's theory, the most probable path to black hole creation is discontinuous via nothing as an intermediate state. In the present paper we have considered several modifications of Einstein's theory that allow continuous histories as saddle points of the Euclidean action between two finite universes. Considered as a matter source, these modifications involve extreme forms of the stress-energy tensor because the Ricci tensor will typically have at least one negative eigenvalue. Therefore the formation of baby universes is a possible competing process.
A matter field that can form sufficiently small domain walls is a universal connector, replacing the intermediate state of nothing with at least a small three-sphere. Higher-order Lagrangians in the scalar curvature have to be fine tuned to allow the desired continuous histories. In many ways the most successful solution involves a scalar C-field of negative (but small) coupling constant.

Notes
1. The cases of zero or negative curvature present additional normalization problems because the naive Euclidean action would be infinite. Therefore we confine attention to the positive curvature case.
2. We assume that this transition is the most probable; this would not be so if a transition were possible in the potential of Eq. (5.37). For example in penetrating radially a spherically symmetric potential barrier the most likely transition maintains the real angular momentum [22].
3. We also note that, as remarked in [24], a real change in C (if K were real) during the instanton could be interpreted as a change in the gravitational constant after the pair creation, which would be undesirable.
References

[1] J. B. Hartle and S. W. Hawking, Phys. Rev. D 28, 2960 (1983).
[2] R. Gregory and R. Laflamme, Phys. Rev. Lett. 70, 2837 (1993); NucJ. Phys. B428, 399 (1994).

68 THE UNIVERSE
[3] C. G. Callan, Jr., and S. Coleman, Phys. Rev. D 16, 1762 (1977). [4] R. Bousso and A. Chamblin, "Patching up the No-Boundary Pro-
posal with Virtual Euclidean Wormholes" gr-qc/9803047. [5] E. Farhi, A. H. Guth, and J. Guven, Nuc1. Phys. B 339, 417 (1990). [6] J. V. Narlikar, J. Astrophys. Astr. 5, 67 (1984); J. V. Narlikar and
T. Padmanabhan, Phys. Rev. D 32, 1928 (1985). [7] S. B. Giddings and A. Strominger, Nuc1. Phys. B 306, 890 (1988). [8] D. Garfinkle, S. B. Giddings, and A. Strominger, Phys. Rev. D 49,
958 (1994). [9] J. Cheeger and D. Grommol, Ann. Math. 96, 413 (1972). [10] H. Nariai, Sci. Rep. Tohoku Univ. Series 1 34, 160 (1950); ibid. 35,
62 (1951). [11] A. Vilenkin and E. P. S. Shellard, Cosmic Strings and Other Topo-
logical Defects, Cambridge University Press, Cambridge (1994). [12] R. R. Caldwell, A. Chamblin, and G. W. Gibbons, Phys. Rev. D
53, 7103 (1996). [13] S. J. Kolitch and D. M. Eardley, Phys. Rev. D 56, 4651, (1997). [14] A. Vilenkin, Phys. Lett. B 133, 177 (1983). [15] J. Ipser and P. Sikivie, Phys. Rev. D 30, 712 (1984). [16] R. C. Myers, Nuc1. Phys. B 289, 701 (1987). [17] S. Deser and N. Redlich, Phys. Lett. 176B, 350 (1986). [18] Y. Ezawa, M. Kajihara, M. Kiminami, J. Soda, and T. Yano,
"On the Canonical Formalism for a Higher-Curvature Gravity" grqc/9801084. [19] J. D. Barrow and S. Cotsakis, Phys. Lett. B 214, 515 (1988); K. Maeda, Phys. Rev. D 39, 3159 (1989). [20] S. W. Hawking and J. C. Luttrell, Nuc1. Phys. B 247, 250 (1984). [21] H. Fukutata, K. Ghoroku, and K. Tanaka, Phys. Lett. B 222, 191 (1990); O. Bertolami, ibid. 234, 258 (1990). [22] K. Lee, Phys. Rev. Lett. 61, 263 (1988) and Phys. Rev. D 48, 2493 (1993). [23] J. D. Brown, Phys. Rev. D 41, 1125 (1990). [24] S. Cotsakis, P. Leach, and G. Flessas, Phys. Rev. D 49, 6489 (1994). [25] R. Bousso and S. W. Hawking, Phys. Rev. D 54, 6312 (1996).

Chapter 6
THE ORIGIN OF HELIUM AND THE OTHER LIGHT ELEMENTS

G. Burbidge
Department of Physics and Center for Astrophysics and Science University of California, La Jolla, CA 92093, USA.

F. Hoyle
102 Admiral's Walk Bournemouth BH2 5HF, Dorset, UK

Abstract

The energy released in the synthesis of cosmic 4He from hydrogen is almost exactly equal to the energy contained in the cosmic microwave background radiation. This result strongly suggests that the 4He was produced by hydrogen burning in stars and not in the early stages of a big bang. In addition, we show that there are good arguments for believing that the other light isotopes, D, 3He, 6Li, 7Li, gBe, lOB and 11 B were also synthesized in processes involving stars. By combining these results with the earlier, much more detailed work of Burbidge et al. and of Cameron, we can finally conclude that all of the chemical elements were synthesized from hydrogen in stars over a time of about lOll yr.

1.

INTRODUCTION

There are more than 320 stable isotopes in the periodic table. In our original work ([1], hereafter B2FH; see also [11]), we showed that nearly all of them, with the possible exception of the helium isotopes and D, Li, Be, and B, were synthesized by nuclear processes in stellar interiors. In the 1950s, there appeared to be several problems associated with explaining the observed abundances of these remaining nuclides, which we discuss in turn. We shall show here that another approach leads

69
N. Dadhich and A. Kembhavi (eds.). The Universe. 69-76.
© 2000 Kluwer Academic Publishers.

70 THE UNIVERSE

to the conclusion that very likely all of them have been synthesized in processes involving stars.

2.

4HE

In the 1950s, it appeared to us that there were two problems associated with explaining the origin of helium in its measured abundance through hydrogen burning. Assuming that the time-scale of the universe is "'"
HOI, there was not enough time for a 4He/H ratio of about 0.24 to be
built up, if the luminosities of the galaxies remained at normal levels for 1010 yr. Second, there appeared to be no evidence that the energy released by this amount of hydrogen burning was present. The energy density of starlight of about 10-14 ergcm-3 is well below the energy released in hydrogen burning, which, for a 4He/H ratio of 0.243 [33, 25] that we assume to be universal, is 4.37x 10-13 erg cm-3 . In deriving this quantity, we have taken the mean density of baryonic matter associated with galaxies to be 4.31 x 10-31 gm cm-3. This number has been obtained from the counts of galaxies, and we assume that baryonic dark matter in the form of massive halos, etc. (with 10 times the visible mass), is present. Here we have put Ho = 60kmsec-1 Mpc-1.
In the 1950s, Bondi, Gold & Hoyle [5] argued that the large amount of undetected energy, which must be present if the helium has been synthesized in stars, must reside in the far-infrared spectrum, while Burbidge [8] speculated that perhaps there was an earlier short-lived phase in the evolution of galaxies in which they were much more luminous, or else possibly the true helium abundance was lower than 0.24, because most
of the mass is tied up in low-mass stars in which HelH < 0.24.
Of course, the solution to the He problem that became popular was that which Gamow, Alpher, & Herman proposed earlier [1]), that the helium was made in a hot big bang some 1010 yr ago. Several calculations following this work and starting with Hoyle & Tayler [24], Peebles [34], and Wagoner, Fowler, & Hoyle [46] demonstrated this. We have now reached the stage where it is argued that the existence of He and the other light isotopes is taken, together with the microwave background radiation, as primary evidence in favor of the standard, hot, big bang cosmological model. However, this argument is only powerful if there is no other way to explain the helium abundance and the microwave background radiation.
In 1941, McKellar [30] showed that there must be a radiation field present in the Galaxy with a temperature between 1.8 and 3.4 K. Penzias & Wilson's [35] measurements, followed by others and culminating in the COBE observations by J. Mather and his colleagues [16] have shown

THE ORIGIN OF ELEMENTS 71
that the cosmic microwave background (CMB) has a blackbody form at least out to radio wavelengths with T = 2.728 K. The hot big bang cosmological model is not able to predict the temperature [43]. But what is remarkable about the result that we have described here is that the energy density of the observed blackbody radiation is extremely close to the energy density expected from the production of helium from hydrogen burning. We showed earlier that this energy is 4.37x 10-13 erg s-l cm-3 and when this energy is thermalized, the temperature turns out to be T = 2.76K.
While the value of the baryonic density in galaxies and their environs is not known with anything like the precision with which the blackbody temperature is measured, it is clearly not very different from
p = 3xlO-31 gmcm-3 (Ho = 60kmsec- 1 Mpc- 1, and dark-to-luminous
baryon ratio'" 10) and of course, the calculated temperature is only proportional to p1/4. Indeed, it might be argued that the CMB temperature gives a more precise measure of the true mass density of baryonic matter in the universe than can be obtained by estimating the mass in galaxies.
We conclude that this result, based on two simple observational arguments, strongly suggests that the helium and the CMB were produced by hydrogen burning in stars. This requires a time much greater than 1010 yr, and there must be a physical mechanism operating that is able to thermalize the radiation that is initially released through hydrogen burning as ultraviolet photons from hot stars in starburst situations in galaxies. We have shown elsewhere that both of these conditions are fulfilled within the framework of the quasi-steady state cosmology (QSSC) (Hoyle, Burbidge, & Narlikar [20, 21, 22, 23]). In the QSSC, the universe is in a sequence of oscillations of period Q superposed on a general universal expansion of period P. In our model Q ~ 1011 yr and P ~ 1012 yr. These timescales correspond to the lifetimes of mainsequence dwarf stars with masses less than 0.7M8 and 0.4M8 , respectively, thereby greatly enhancing the importance of dwarf stars in cosmogony. We conclude that 4He in the cosmos is most likely a result of stellar nucleosynthesis. Given that this most abundant nucleus among the light elements is a result of stellar activity, it is then natural to ask whether the other light isotopes can also be due to processes involving stars.
3.
Much work has been done on these nuclides in recent years. It is generally accepted that 6Li, 9Be, lOB and 11B were produced in spallation

72 THE UNIVERSE
reactions of high-energy protons on 12C and 160 with energy ultimately coming from galactic processes as we originally proposed B2FH. Reeves,
Fowler & Hoyle [37] showed that galactic cosmic rays are an important
ingredient. The most modern work shows that it is the C and 0 that bombard the protons and a-particles. The Be and B abundances are proportional to the Fe/H ratio is subdwarfs, and Vangioni-Flam et al.
[45] have shown that spallation by high-energy C and 0 can account
for this. The high-energy C and 0 nuclei are ejected in the winds from massive stars and supernovae.
What about 7Li? The early suggestion [37] that spallation is respon-
sible gives a 6Li/sevenli ratio f'V 1, but in the solar system, 6Lij1Li ~ 10. This is one of the reasons why it has been argued that 7Li at least is due to big bang nucleosynthesis. This argument has been supported by the claim that there is a "plateau" at 7Li/H = 1.7x 10-10 in a sample of
Population II stars that are > 1010 yr old [41]. However, it is now known that this plateau is breached and that several stars have 7Li/H < 10-10 [6]. Ryan et al. [38] conclude that there is an intrinsic spread in the
7Li abundance due to influences other than uniform nucleosynthesis in a big bang. We must also not forget that while it is generally believed that susceptibility to destruction prevents 7Li from being synthesized in stars, the observation that there is a class of lithium-rich supergiants (cf. WZ Cas; [30]) shows that stellar processes may be responsible, as was
suggested in a complicated scenario by Cameron & Fowler [12]. Boesgaard & Tripicco [4] looked at the Li abundance as a function of
[Fe/H] for both Population I and old disk stars. They found that the Li abundance could be very different in stars where the [Fe/H] abundance has the solar value but that there is an absence of stars that are Li rich but have low values of [Fe/H] (see also [36, 3]). The abundances and isotope ratios of Li in the interstellar gas have been determined most recently by Lemoine, Ferlet, & Vidal-Madjar [27]. They have concluded that there must be an extra source of 7Li in the Galaxy. It is now clear from the observations that there may be at least three possible effects that have contributed to the observe Li abundance. They are (a) stellar processing, which tends to deplete Li, (b) galactic production which tends to build Li and (c) big bang nucleosynthesis. From the observations, the relative importance of (a), (b),and (c) is not yet clear. However, in view of our earlier arguments concerning the origin of 4He, we consider it likely that (c) is not operating. Thus, we believe that (a) and (b) alone can explain the Li abundance and that further observational investigations will show this.

THE ORIGIN OF ELEMENTS 73
4. D AND 3HE
The light isotope 3He is produced in large quantities in dwarf stars where the masses are not large enough for it to be destroyed by 3HeeHe, 2p)4He. It is also the case that there is a class of stars in which it has been shown from measurements of the isotope shift that most of the helium in their atmospheres is He. These stars include 21 Aquilae, three Centaurus A and several others [1, 39, 18, 42]. The stars are peculiar A, F, and B stars having He/H abundance that are 1/10 of the normal helium abundance. The 3He/4He ratio can range from 2.7 to 0.5. These stars occupy a narrow strip in the llogg, Tefr) plane between the B stars with strong helium lines that shows no evidence for the presence of 3He. However, the detection of 3He from the isotope shift will fail if the 3HerHe ratio is ::; 0.1. Thus, many of the weak helium-line stars may well have 3He/4 He abundance ratios far higher than abundance ratio that is normally assumed to be present, namely, 3He/4He:::::; 2x10-4 . The high abundance of He in these stars has been attributed by G. Michaud and his colleagues to diffusion ([32] 1979 and earlier references). Whether or not this is the correct explanation, what these results do tell us is that stellar winds from such stars will enrich the interstellar gas with 3He in large amounts. This 3He is in addition to the 3He that will be injected from dwarf stars. The final abundance required is 3He/H :::::; 2xlO-5. It has been argued by those who believe that 3He is a product of big bang nulceosynthesis that there has not been time to build up the required abundance by astrophysical processes. However, not only do we not know what the rate of injection from stars is, but in the QSSC, the time scale for all of this stellar processing is '" 1011 rather than HOI:::::; lOlD yr. Thus, we believe that 3He may very well have been produced by stellar processes.
We turn finally to the production of deuterium. It has been arguead that D cannot be synthesized by spallation or photo-disintegration in supernova outbursts [15, 40]. Recently, however, Fuller & Shi [17] have argued that antineutrinos ve can give rise to deuterons through
ve + P -t n + e+ followed by n(p, 'Y)D -reactions in the collapse of su-
permassive stars (M 2: 5 x 104M8) in the early history of galaxies. This
mechanishm may be important but in view of the fact that the 3He/H and D/H ratios are very similar, and because we believe that the 3He is likely to be produced by low-mass stars, we believe that the most likely source of the cosmic deuterium is the dwarf stars.
It is known that the dwarf M stars are a major constituent of normal galaxies. They have extensive convective envelopes, and thus they are likely to have outer layers in which extensive flare activity takes place. A very good example is the large UV flare in the red dwarf AU Microscopii,

74 THE UNIVERSE
which has just been reported [26]. In our view, it is the cumulative effect of stellar winds and flares from these low-mass stars that has led to the build up of the deuterium.
It is easily shown that the smount of energy required to generate a D/H ratio '" 10-5 through flaring and ejection from dwarf stars is not very large. The energy required to produce D in stellar flares through the generation of neutrons and the subsequent capture by protons turns out to be close to 6x1018 erggm-1 D, which is much the same as the energy release involved in hydrogen burning to 4He. For a universal mass density of 3x 10-31 gm cm-3, the energy requirement is then 1.8 x 10-17 erg cm-3. This is very small compared with the energy of starlight, which at present, is '" 10-14 erg cm-3 and which, in the QSSC will build up to '" 10-13 erg cm-3 in the full cycle. Thus, the energy requirement in the production of D is for a small fraction of the available energy that is to into the generation of neutrons.
Deuterium is known to be produced in solar flares [13, 2] and early work by Coleman & Worden [14] has shown how much mass can be ejected from the dwarf stellar component. They estimated that for a typical galaxy containing 1010 - 1011 dwarf M stars, the mass-loss rate will amount to about 0.lM0 yr-1 from the dwarfs. If we add to this the fact that the programs now underway to detect faint stars through microlensing are now showing that the number of dwarf stars is very large, and the fact that in the QSSC cosmology, the timescale for the buildup of D in the interstellar gas is much greater than 1010 yr, a large amount of interstellar gas that is enriched in deuterium will be produced in a timescale corresponding to a cycle of oscillation Q in the QSSC i.e. in 1011 yr
Of course, in the same period, the deuterium contained in gas that is recondensed into stars will be destroyed, so that the final abundance will depend on how much uncondensed gas remains. More measurements are required of D/H both in the gas in our Galaxy [28, 29] and elsewhere. Much has been made recently of the D/H ratio determined in the absorption-line spectra of QSOs with large redshifts. The value obtained by D. TyUer and his colleagues [44, 10], D/H ;S2xlO-5, is the best estimate that has been made so far for extragalactic material, and this has been discussed only in the context of big bang cosmology. In the QSSC, the absorbing clouds that give rise to the absorbtion spectrum may also lie at an earlier epoch in the cycle. However, as we have discussed elesewhere [19]' there is independent evidence that many QSOs may not lie at the distances indicated by their redshifts, so the epoch to which these values of D/H correspond is not clear. Our prediction is that with the deuterium made largely in stellar flares, there will be a

THE ORIGIN OF ELEMENTS 75
range of values of the D/H ratio. With values of D/H '" 10-5 at the high end. We do not expect that the D/H ratio will have a constant value throughout an individu al galaxy or throughout a cycle of the QSSC. Thus, a possible test is to look for difference in the D/H ratio both inside and outside our Galaxy.
5. CONCLUSION
We have shown that there are good reasons to argue that 4He has been produced by astrophysical processes following stellar activity. Thus, provided that a timescale much greater than HOI is available, as is the case in the QSSC, all of the chemical elements may well have been synthesized in stellar processes. The fact that the great majority of the 320 stable isotopes have been generated astrophysically has always made the idea that all of the isotopes were made this way very attractive.
References
[1] Alpher, R. A. & Herman R. 1950, Rev. Mod. Phys. 22. 153. [2] Anglin J. D. Dietrich, W. & Simpson J. 1973, Astrophys. J. 186,
L41. [3] Balachandran S. 1990, Astrophys. J. 354, 310. [4] Boesgaard, A. & Tripicco, M. J. 1986, Astrophys. J. 303, L42. [5] Bondi, H., Gold, T. & Hoyle F. 1955, The Observatory75, 80. [6] Bonifacio, P. & Molero P. 1997, Mon. Not. Roy. astr. Soc. 285, 547 [7] Burbidge E. M., Burbidge G., Fowler, W. A. & Hoyle F. 1957, Rev.
Mod. Phys. 29,547 (B2FH). [8] Burbidge G.1958, Proc. Astr. Soc. Pacific 70, 83. [9] Burbidge M. & Burbidge G. 1956, Astrophys. J. 124, 655. [10] Burles, S. & Tytler, D. 1996, Astrophys. J. 460, 584. [11] Cameron, A. G. W. 1957, Chalk River Rep. CRL - 41. [12] Cameron, A. G. W. & Fowler, W.A. 1971. Astrophys. J. 164, 111. [13] Chupp, E. et al. 1973, Nature 241, 333. [14] Coleman, G. & Worden P. 1976, Astrophys. J. 205, 475. [15] Epstein R. Lattimer J. & Schramm,D. 1976, Nature 263, 198 [16] Fixsen D. et a11996, Astrophys. J. 473, 576. [17] Fuller F. & Shi, X 1997, Astrophys. J. 487, L25. [18] Hartoog M. & Cowley A. 1979, Astrophys. J. 228, 229. [19] Hoyle F., Burbidge G. & 1996, Astro. Astrophys. 309, 335.

76 THE UNIVERSE
[20] Hoyle F., Burbidge, G. & Narlikar, J. V. 1993, Astrophys. J. 410, 437.
[21] Hoyle F., Burbidge, G. & Narlikar, J. V. 1994a, Mon. Not. Roy. astr. Soc. 267, 1007.
[22] Hoyle F., Burbidge, G. & Narlikar, J. V. 1994b, Astro. Astrophys. 289,729.
[23] Hoyle F., Burbidge, G. & Narlikar, J. V. 1995, Proc. Roy. Soc. Lond.
A 448,191. [24] Hoyle F. & Tayler R 1964, Nature 203, 1108. [25] Isotov, Y., Thuan, T. & Lipovetsky, V. 1997, Astrophys. J. Suppl.
108, 1. [26] Katsova M., Drake, J. & Livshits, M. 1998 CFA preprint 4704.
[27] Lemoine M., Ferlet. R & Vidal-Madjar, A. 1995, Astro. Astrophys.
298,879. [28] Linsky J., et a11993, Astrophys. J. 402, 695. [29] Linsky J., et al 995 Astrophys. J. 451, 335. [30] McKellar A. 1940 Proc. Astr. Soc. Pacific 52, 407. [31] McKellar A. 1941, Publ Dom, Astrophysics Obs. Victoria 7,15. [32] Michaud G. et al. 1979, Astrophys. J. 234, 206 [33] Pagel, B.E.J.1997, in The Universe at Large, ed. G. Munch, A.
Mampaso, & F. Sanchez, Cambridge Univ. Press, 343. [34] Peebles, P.J.E. 1966, ApJ, 146, 542. [35] Penzias, A., & Wilson, R 1965, Astrophys. J. 142, 419. [36] Rebolo, R, et al. 1988, Astro. Astrophys. 193, 193. [37] Reeves, H., Fowler, W.A., & Hoyle, F. 1970, Nature 226, 727. [38] Ryan, S. et al. 1996, Astrophys. J. 458, 543. [39] Sargent, W.L.W. & Jugaku, J. 1961, Astrophys. J. 134, 777. [40] Sigl,G., Jedamsik, K., Schramm, D., & Berezinsky, V. 1995, Phys.
Rev. D 52, 6682. [41] Spite, M., & Spite, F. 1985, Ann. Rev. Astron. Astrophys. 23, 225. [42] Stateva, I., Ryabchikov, T, T., & Iliev, I. 1998, Preprint. [43] Thrner, M., 1993, Science 262, 861. [44] Tytler, D., Fan, X.M., & Burles, S. 1996, Nature 381, 207. [45] Vangioni-Flam, et al. , 1996, Astrophys. J. 468, 199. [46] Wagoner, R, Fowler, W.A., & Hoyle, F. 1967. Astrophys. J. 148,
3.

Chapter 7
SUPERLUMINAL MOTION AND GRAVITATIONAL LENSING

S. M. Chitre
Tata Institute of Fundamental Research Homi Bhabha Road, Bombay 400 005

Abstract

The role of gravitational bending of light in generating ohserved apparent superluminal motions of VLBI components in the compact cores of some of the AGNs and quasars is highlighted.

1. INTRODUCTION
In the early part of 1970s, the very long base line interferometry (VLBI) enabled radio astronomers to probe the internal structure of radio sources at milliarcsecond scales. There was an understandable feeling of disbelief, therefore, when several radio sources monitored with VLBI over a number of years, revealed components in their nuclei separating at speeds exceeding that of light. The first hint of a superluminal motion in quasars was contained in observations of the sizes of variable components in quasars 3C273 and 3C279 ([1], [2]). More observational evidence for such motions accumulated through the 1970s when two distinct components apparently separating with a linear speed, f3app = vapp/c ~ 5 -10, over a period of a few months, were detected ([3], [4], [5]). It has now been convincingly demonstrated for several dozens of sources, from the high-resolution VLBI observations, that the compact radio sources in the active galactic nuclei exhibit striking superluminal motion associated with several components ([6], [7], [8]. Since their discovery, the superluminal sources have remained one of the most intriguing themes in radio astronomy.
Even prior to the detection of apparent superluminal motion, the observations of some quasars had indicated the presence of fast bulk motions through their rapid intensity variations ([9]). The feasibility of
77
N. Dadhich andA. Kembhavi (eds.), The Universe, 77-86.
© 2000 Kluwer Academic Publishers.

78 THE UNIVERSE
superluminal motion was in fact, suggested by Rees ([10]) in a prescient paper where he had argued that the relativistic expansion of a source at
speed v can result in its size increasing at an apparent speed of ')'vb == (1_v2/c2)-1/2).. The arrival time differences ofthe signals from different
parts of the source can then lead to the apparent size expansion at a
» transverse speed '" ')'c(')' 1). The VLBI measurements of the compact
core regions of quasars suggest a typical Lorentz factor, ')' ~ 10 for the relativistically moving components, corresponding to a typical proper motion of ~ 1 milliarcsec yr-1. Over the past quarter of a century, the superluminal sources have been observed with VLBI and VLBA to establish a number of striking features ([8]):
(i) The superluminalmotion appears to be common amongst the brighter radio sources and generally exhibits properties such as rapid variability of intensity and polarization, although, not all well-surveyed sources display superlight motion (e.g. 3C84).
(ii) The expansion speeds are on the average larger for the core-dominated sources compared to the lobe-dominated sources.
(iii) The compact sources exhibit superluminally expanding relative motion of the components, with the emergence of new components from the core.
(iv) The superluminal motion is largely uniform, but there are cases of acceleration and deceleration, and in some cases there are instances of bent trajectories as well.
(v) The VLBI jets associated with the superluminal sources are invariably curved and misaligned with the large-scale symmetry axis of the extended lobes.
2. THEORETICAL SCENARIOS
There are two ways in which it is possible to account for the observed superluminal motion of VLBI components in the nuclei radio sources. The obvious way out to explain the velocities of components apparently exceeding the speed of light, c is to argue that these radio sources are located at distances considerably smaller than what is implied by Hubble's law. Since all the observations depend on measurements of angular separation between components, their conversion into linear transverse motion would necessarily require a knowledge of the distance to the objects. Should the sources be situated closer than what is indicated by the Hubble interpretation of their redshifts, the observed motion would turn out to be subluminal after all ([11], [12]). Indeed, the AGNs and

SUPERLUMINAL MOTION AND GRAVITATIONAL LENSING 79

quasars are now widely accepted to be located at cosmological distances. It, therefore, becomes necessary to imagine that the observed superluminal speeds are not physical, but rather, the result of cosmic illusions.
A number of ingenious proposals, mostly based on the kinematics of the source have been advanced for explaining such cosmic illusions. These include:

(a) Christmas-tree model proposed by Dent ([13]) invokes independent flares erupting all over at random locations in the source. Such random flaring could mimic a regular superlight motion, though it was realized that the observed motions were highly systematic and indicated only expanding motions of the components ([14]).

(b) Light-echo model of Lynden-Bell ([15]) attributed the superluminal motion to an outward propagating relativistic blast curve that can cause a progressive brightening of the region of the source with increasingly large size. If such an oppositly directed signal along an axis making a small angle with the sight-line can lead to a superluminal expansion. The model does not seem to be compatible with the observed core-jet structures in these sources.

(c) Gravitational screen model was proposed by Chitre and Narlikar ([16], [17]) as a plausible explanation of superluminal motion in AGNs, prior to the discovery of the first gravitational lens system,
the twin quasar 0957 + 561 A,B, by Walsh, Carswell & Weymann
([18]). This model envisages the presence of a gravitational screen in the form of an intervening galaxy or a cluster of galaxies, between the source and the observer. Owing to the gravitational bending produced by the deflecting mass en route, the observer would see the components in the nucleus of the background source, not in their real positions, but at virtual transversely separated locations, thus creating an illusion of superlight motion.

The effect is due to the differential gravitational deflection caused

by the intervening mass with the increasing impact parameter distance,

from the centre, of the light rays emanating from the background source.

For a spherically symmetric matter distribution in a galaxy, G, with

mass, M and radius, R, the external gravitational bending of a typical

ray is given by

~ = 4GM for r 2:: R.
c2r '

It turns out the interesting effects are produced from light paths that

go through the inner regions of G. It can be demonstrated ([17]) that

value of the relativistic bending angle is exactly twice the Newtonian

80 THE UNIVERSE
value in the case of weak gravity. The important feature of the bending
angle is that ll.'(a) > 0 in the inner regions of most physical mass-
distributions associated with galaxies or galaxy-clusters. For external
bending, on the other hand, ll.'(a) < O. The most striking aspect is the
effect of gravitational bending on the apparent velocity of separation. For this purpose let us denote by Dd, Dds and Ds the distances between the observer and the deflector, the deflector and the source, and the observer and the source respectively. Let V..L be the transverse speed of separation between two components in the nuclear region of a stationary background source. Then, the apparent separation velocity as seen by the observer is
Vapp = 1- D1fsds ll.'(a) .
It is clear that we can get a large maBnification of the real transverse
velocity provided, ll.'(a) > 0 and D~8 ds ll.'(a) :::: 1, a condition that
is satisfied when the source and the observer are situated at conjugate points with respect to the deflector.
It is, thus, essential for the manifestation of apparent superluminal motions to have a suitably placed gravitating intervenor between the source and the observer. The presence of an intervening deflector for producing the superluminally separating images is a requirement for this scenario. A test of the gravitational screen model would, therefore, be the detection of an actual deflecting object, which has, unfortunately, not been borne out in all the known superluminal sources.
There are certain features associated with the gravitational lensing effect which may even stand the scrutiny of future observations in the case of the superluminal sources. A notable feature of the gravitational bending of light is that the amount of deflection is independent of wavelength and we therefore expect the superluminal separation of components to be the same at all observing wavelengths. A definitive characteristic associated with the lensing phenomenon is the non-uniform amplification in directions perpendicular to the line of sight. Thus, the image of a linear trajectory would appear curved or bent, and it is only to be expected that the VLBI jets should be misaligned in relation to the extended structures. Such a misalignment property has, indeed, been noted in some of the quasars exhibiting superluminal motions. The superluminal acceleration or deceleration of the separating components is yet another consequence of the gravitational screen model, this could result from changes in the amplification of the light beams when the amount of relativistic bending varies with the density of the intervening matter. Furthermore, the local inhomogeneities in the deflecting object is also liable to produce short-term (8 '" 1 yr) changes in the velocity;

SUPERLUMINAL MOTION AND GRAVITATIONAL LENSING 81

the angular separation as a function of epoch is, therefore, unlikely to be a smooth curve, but rather should have a scatter around the linear trend. The apparent separation velocity observed in the source 3C345, for ex-
ample, shows an increase from 7.5c to 12.2c (for Ho = 50 km 8- 1 Mpc 1)
which is a genuine case of superluminal acceleration. A clear-cut prediction of the gravitational screen model would be
the detection of superluminal separation of the VLBI components in the cores of AGNs and quasars which have been unambigously estab-
lished gravitational lens systems. Thus, the twin quasar 0957 + 5cI
should show a magnificationi of velocity by a factor of 2-3 and consequently, should there be relativistically separating components in the source-quasars, we should see apparent superluminal motion, vapp ::::; 3c.
Likewise, the triple radio source 2016 + 112 should reveal an apparent
superluminal speed, vapp exceeding 10c. Indeed, there are reported cases of highly magnified gravitational lens systems whose cores exhibit structures at submilliarcsecond scales. The VLBA features are detected in one of the images first, followed by their appearance in the second image of the lens system with a time-delay of several weeks to months. There is some observational evidence for the existence of such a superluminal motion in the lens system 1830-211 (Patnaik, Private Communication).
d) Relativistic beaming model was proposed by Rees ([19]) and later elaborated by Blandford and Konigal ([20]). In this kinematic picture the superluminal motion is simulated by one or more blobs or plasmacomponents moving at a relativistic speed, v away from the core that is stationary in the rest frame of the observer. The transverse velocity of separation of the plasma blobs from the core is then given by

(3 sin ()
= vapp 1 - (3 cos () c,

((3 = vic).

For the manifestation of apparent superluminal motion, the angle () of the beaming plasmoid with the sightline has to be very small. The
expression for the apparent velocity attains a maximum value, v~~x =
H-=1 c ~ 'YC. This model makes an ingenious use of the kinematic
effect, and was in fact, advanced even before the apparent faster-thanlight phenomenon was discovered on the VLBI scale in the cores of AGNs and quasars.
The observed superluminal motions may be best interpreted in the framework of bulk relativistic motion beamed towards the observer. This is by far the most attractive model to explain the observed phenomena associated with the superluminal sources. However, the simple relativistic beaming model is not without its difficulties in accommodating various observational features.

82 THE UNIVERSE
Thus a successful model must be able to explain the emission characteristics of the superluminal sources such as the spectrum, polarization, flux variability and features like the curved trajectories of superluminal components, and their variable angular speeds, bent jets on the parsec scales and misalignment property of the extended structures. Should the relativistic beaming model be the correct description of superluminal sources, we would expect at least some quasars to show two-sided largescale jets, unless the one-sided jet is an intrinsic property of quasars on both small and large scales. In any case the bright hot spots are expected to be on the jetside. Furthermore, because of the Doppler boosting the flux density of the approaching components is expected to exceed that of the receding (or stationary) component by several orders of magnitude, in conflict with the comparable flux densities of components.
The X-ray emission from superluminal radio sources is supposed to have provided a strong indication for the occurrence of relativistic beaming in their compact cores. For this purpose it is argued that the synchrotron radiation in a compact volume would produce X-ray flux, by an inverse Compton scattering of radio and infrared photons. The basic question to be addressed is whether the inverse Compton process is the underlying cause for the X-ray flux from superluminal sources, for which it is possible to constrain the physical parameters associated with the radio sources. It turns out that the observed X-ray emission is much smaller than what is expected from the parameters of the radio compo-
nents. Essentially, the VLBI measurements determine vic and the X-ray
fluxes set limits on the Doppler factor, 6 = 1/,(1 - (3 cos fJ), thus providing valuable constraints on the geometry and motion of the emitting components. It is usually argued (cf. [6]) that the observed superluminal motions, weak X-ray emission and variability of the sources are taken to provide strong evidence in support of the relativistic beaming model.
Marscher ([21]) has, however, pointed out certain difficulties encountered by the synchrotron-Compton emission process when applied to realistic radio sources. It turns out because of the complex nature of the compact sources, they are composed of a number of discrete components, and these could conspire to become self-absorbed at different frequencies to produce a remarkably flat composite spectrum. For the sake of simplicity, each component is assumed to have a spectrum of a uniform synchrotron source, but, then the resultant inverse-Compton X-ray flux density and the total energy requirement have a very strong dependence on the turnover frequency, and the angle of the bulk velocity vector with the line of sight. Based on the simplifying assumptions, the evidence for the inverse-Compton process generating the observed X-ray flux is favorable, though not overwhelming. But the discrepant

SUPERLUMINAL MOTION AND GRAVITATIONAL LENSING 83
time scales of variability in the wavebands ranging from millimeter (~ weeks) to X-rays ('" day) for the bright quasar 3C273 certainly casts a shadow on the tenability of the inverse-Compton hypothesis.
3. SPECULATIONS ON MICROLENSING AND SUPERLUMINAL MOTIONS
The phenomenon of gravitational lensing has been effectively used to gain valuable information about the masses and sizes of intervening deflectors. In most studies the lensing objects are generally assumed to be stationary, except in those cases where the effects of motion on the light curve have been important while crossing the caustics like in the microlensing events (cf. [22], [23], [24]). The usefulness of astronomic diagnostic properties of moving lenses was discussed by Chitre and Saslaw ([25]). It was demonstrated that with a suitable placement of the background source within the cone of inversion, the source velocities could conceivably be magnified by an order of magnitude or more and part of the image may even exhibit an apparent superluminal motion (cf. [26]; [27]).
A striking feature associated with moving lenses is the conversion of linear proper motion into rotational motion, since the lensing effect magnifies the velocities by different amounts in different directions. Consequently, we expect the conversion of uniform linear source motion to be accompanied by an apparent acceleration of the individual components in the source. Equally, the radial component of the source motion is also influenced by the moving lenses by converting it into a transverse component of the image motion.
One of the fascinating challenges in galactic astronomy is to surmise the presence of a putative massive black hole residing at the centre of our Galaxy. One obvious way to infer its existence and physical properties would be to search for its gravitational influence on the background sources such as maser complexes, relativistic jets of 'microquasars', lying on the far side of the galactic nucleus from us. Thus, it is tempting to imagine an individual relativistically moving source in a maser complex, or a relativistic beam of a microquasar located in the background to be lensed by the black hole in the galactic centre. This should almost certainly generate the resulting velocities which could apparently mimic superluminal motions. Such a suitable positioning of the background microquasar along the line of sight passing through the nuclear region should create an image morphology that could provide a valuable handle to infer the mass of the lensing black hole (cf. [25], [28]).

84 THE UNIVERSE
A possible velocity manifestation would consist of nearly circularly moving superluminal components, resulting from the lensing of a background relativistic jet by the massive black hole; the typical state of the velocity pattern would be of the order of several arcseconds. A definitive observation of superluminal motion in the direction of the galactic centre would provide further support to the existence of a massive black hole in the nucleus of our Galaxy.
A remarkable aspect of superluminality has been stressed by GopalKrishna and Subramaniam ([29]). This involves a superluminal microlensing scheme which combines beaming with the phenomenon of gravitational lensing. The microlensing of compact sources such as quasars by brown dwarfs has been invoked to account for their intensity variability on timescales of the order of several months to a few years. But some of the active quasars, in particular, blazars are known to show variability on a time-scale as short as hours in the optical waveband and days in the radio. The blazars have relativistically beaming jets composed of bright components which are known to make a small angle with the line of sight ([20]) and these knots are expected to exhibit apparent superluminal motions. Gopal Krishna and Subramanian ([30]) have invoked the superluminal microlensing of such ultra-rapidly moving components which causes an amplification of both the flux and velocity, over and above that resulting from the relativistic beaming or lensing phenomenon alone. Such a composite beaming-lensing scheme would also lead to the requisite short time-scale intensity variations. Furthermore, for the case of knots crossing a caustic this would lead to extraordinarily large apparent superluminal velocities exceeding 20-30c.
Thus, if the microlensing by a million solar mass black hole of a quasar or a relativistic jet were to happen, this will almost certainly lead to significant morphological distortions, variations in the flux ratios and velocities of the images over a very short time-scale (~lhr). Clearly, the VLBA monitoring of the galactic nuclear region and of the cores of compact radio sources (e.g. AGNs, quasars, blazars) should reveal the existence of massive and supermassive black holes in the nuclei of galaxies. A definitive observation of superluminal motion in the direction of the Galactic centre would provide further support to the existence of a massive black hole in the nucleus of our Galaxy.
Acknowledgments
It was a pleasure to collaborate with Jayant Narlikar, some two decades ago, on problems relating to superluminal motions and physics of radio sources. Thanks are due to D. Narasimha for useful comments on the manuscript and valuable discussions.

SUPERLUMINAL MOTION AND GRAVITATIONAL LENSING 85
References
[1] Gubbay, J., et a11969, Nature, 224, 1094 [2] Moffet, A.T. et al 1972 in IAU Symp. 44: External Galaxies and
Quasi-Stellar Objects ed. D.S. Evans (Dordrecht: Reidel) [3] Cohen, M.H. et al. 1971, Ap. J. 179. 207 [4] Whitney, A.R et al 1971, Science, 173, 225 [5] Kellermann, KI. 1978, Physica Scripta 17, 257 [6] Cohen, M.H. & Unwin, S.C. 1984 in IAU Symp. 110: VLBI and
Compect Radio Sources, ed. R Fanti, K Kellermann and G. Setti (Dordrecht: Reidel) [7] Fanti, R, Kellermann, K & Setti, G. 1984, IAU Symp. 110:VLBI and Compact Radio Sources (D. Reidel, Dordrecht) [8] Vermeulen, RC. & Cohen, M.H. 1994, Ap. J. 430, 467 [9] Burbidge, G. & Burbidge, E.M. 1967, Quasistellar Objects (Freeman: San Francisco) [10] Rees, M. J. 1966, Nature 211, 468 [11] Burbidge, G. 1978, Physica Scripts, 17, 281 [12] Arp H. 1983, in Liege ColI. 24: Quasars and Gravitational Lenses ed. J.-P. Swings (Univ. Liege) [13] Dent, W.A. 1972, Science, 175, 1105 [14] Cohen, M.H. et al 1977, Nature, 268, 405 [15] Lynden-Bell, D. 1977, Nature, 270, 396 [16] Chitre, S.M. & Narlikar, J. V. 1979, MNRAS, 189,655 [17] Chitre, S.M. & Narlikar, J.V. 1980, Ap. J. 235, 335 [18] Walsh, D., Carswell, RF. & Weymann, RJ. 1979 Nature, 279, 381 [19] Rees, M. J. 1967, MNRAS, 135,345 [20] Blandford, R & Konigal, A. 1979, Ap. J. 232, 34 [21] Marscher, A.P. 1987 in Superluminal Radio Sources, Ed. J.A. Zensun & T.J. Pearson (Cambridge University Press) [22] Birkinshaw, M. & Gull, S.F. 1983, Nature 302, 315 [23] Mitrofanov, J.G. 1981. Sovt.Astr. Lett. 7, 39 [24] Gott, J.R 1981, Ap. J. 243, 140 [25] Chitre, S.M. & Saslaw, W.C. 1989, Nature, 341, 38 [26] Liebes, S. 1964, Phys.Rev.B 133, 835 [27] Saslaw, W.C., Narasimha, D. & Chitre, S.M. 1985, Ap. J. 292, 348

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[28] Narasimha, D. & Chitre, S.M. 1991 in Gravitational Lens, Hamburg, Ed., R. Kayser, T. Schramm, L. Nieser, p378
[29] Gopal-Krishna & Subramanian, K. 1991, Nature, 349, 766 [30] Gopal-Krishna & Subramanian, K. 1996, Astron. Ap. 315, 343

Chapter 8
DUAL SPACETIMES, MACH'S PRINCIPLE AND TOPOLOGICAL DEFECTS
Naresh Dadhich
Inter- University Centre for Astronomy and Astrophysics Ganeshkhind, Pune 411 007, India
It is a matter of great pleasure and privilege to have known Jayant Narlikar and worked with him closely in setting up IUCAA. With deep affection and feeling I dedicate this work to him on his completing 60 years.

Abstract

By resolving the Riemann curvature relative to a unit timelike vector into electric and magnetic parts, we define a duality transformation which interchanges active and passive electric parts. It implies interchange of roles of Ricci and Einstein curvatures. Further by modifying the vacuum/fiat equation we construct spacetimes dual to the Schwarzschild solution and fiat spacetime. The dual spacetimes describe the original spacetimes with global monopole charge and global texture. The duality so defined is thus intimately related to the topological defects and also renders the Schwarzschild field asymptotically non-fiat which augurs well with Mach's Principle.

1. INTRODUCTION
In analogy with the electromagnetic field, it is possible to resolve the gravitational field; i.e. Riemann curvature tensor into electric and mag-
87 N. Dadhich and A. Kembhavi (eds.), The Universe, 87-96.
© 2000 Kluwer Academic Publishers.

88 THE UNIVERSE
netic parts relative to a unit timelike vector [1-2]. In general, a field is produced by charge (source) and its menifestation when charge is stationary is termed as electric and magnetic when it is moving. Electromagnetic field is the primary example of this general feature, which is true for any classical field. In gravitation, unlike other fields, charge is also of two kinds. In addition to the usual charge in terms of nongravitational energy, gravitational field energy also has charge. Thus electric part would also be of two kinds corresponding to the two kinds of charge, which we term as active and passive.
The Einstein vacuum equation, written in terms of electric and magnetic parts is symmetric in active and passive electric parts. We define the duality relation as interchange of active and passive electric parts... Then it turns out that the Ricci and the Einstein tensors are dual of each-other. That is, the non-vacuum equation will in general distinguish between active and passive parts and we could have solutions that are dual of each-other [3]. In particular it follows that perfect fluid space-
times with the equation of states, p - 3p = 0 and p + p = 0 are self dual
(/\ -+ -1\) while the stiff fluid is dual to dust. The question is, can we obtain a dual to a vacuum soltuion? Since
the equation is symmetric in active and passive parts, it would remain invariant under the duality transformation. However it turns out that in obtaining the well-known black hole solutions not all of the vacuum equations are used. In particular, for the Schwarzschild solution the equation Roo = 0 in the standard curvature coordinates is implied by the rest of the equations. If we tamper this equation, the Schwarzschild solution would remain undisturbed for the rest of the set will determine it completely. However this modification, which does not affect the vacuum solution, breaks the symmetry between active and passive electric parts leading to non-invariance of the modified equation under the duality transformation. Now we can have solution dual to vacuum which is different. This is precisely what happens for the Schwarzschild solution.
The Schwarzschild is the unique spherically symmetric vacuum solution, which means it characterizes vacuum for spherical symmetry. It is true that not all the equations are used in getting to the solution. In fact it turns out that ultimately the equations reduce to the Laplace equation and its first integral [4-5]. That means the Laplace equation becomes free as it would be implied by its first integral equation. Without disturbing the Schwarzschild solution we could introduce some energy density on the right which would be wiped out by the other equations. The modified equation would turn out to be not invariant under the duality transformation, yet however it admits the Schwarzschild solution as the unique solution. Now the dual set of equations also admits the unique

DUAL SPACETIMES 89
solution, which could be interpreted as representing the Schwarzschild particle with global monopole charge [6]. The static black hole with and without global monople charge are hence dual of each-other.
Similarly it turns out that flat spacetime could as well be characterized by a duality non-invariant form of the equation. The static solution of the dual equation will represent massless global monopole (putting the Schwarzschild mass zero in the above solution) and the non-static homogeneous solution will give the FRW metric with the equation of
state p + 3p = 0, which is the characterizing property of global texture
[7-8]. The former could as well be looked upon as spacetime of uniform relativistic gravitational potential [4-5]. Global monopoles and textures are stable topological defects which are produced in phase transitions in the early universe when global symmetry is spontaneously broken [7-10]. In particular a global monopole is produced when the global 0(3) symmetry is broken into U(l). A solution for a Schwarzschild particle with global monopole charge has been obtained by Barriola and Vilenkin [6]. It therefore follows that the Schwarzschild and the Barriola-Vilenkin solutions are related through the duality transformation. They are dual of each- other. Like the Schwarzschild solution, the global monopole solution is also unique. Applications to cosmology and properties of global monopoles [10-14] and of global textures [7-8,11,15-19] have been studied by several authors. What dual solution signifies is restoration of gauge freedom of choosing zero of relativistic potential which was not permitted by the vacuum equation that implied asymptotic flatness. This means that the dual solution breaks asymptotic flatness of the Schwazschild filed without altering its basic physical character. The relativistic potential is now given by ¢ = k - M / r instead of ¢ = - M / r. This is precisely what is required to make the Schwarzschild field consistent with Mach's principle. The constant k brings in the information of the rest of the Universe, say for solar system moving towards the great attractor [20]. The important difference between the Newtonian and relativistic understanding of the problem is that constant k produces non-zero curvature and hence has non-trivial physical meaning. This is the most harmless way of making the field of an isolated body consistent with Mach's principle. In sec. 2, we shall give the electromagnetic decomposition of the Riemann curvature, followed by the duality transformation and dual spacetimes in Section 3 and concluded with discussion in Section 4.

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2. ELCTROMAGNETIC DECOMPOSITION
We resolve the Riemann curvature tensor relative to a unit timelike vector [1-2] as follows:
(8.1)

(8.2) where

H(ae) = *CabedUbud

(8.3)

H[ae] = 21rJabeeRedubud.

(8.4)

Here Cabed is the Weyl conformal curvature, rJabed is the 4-dimensional
volume element. Eab = Eba, Eab = Eba, (Eab, Eab, Hab)Ub = 0, H = Hg = 0 and uaua = 1. The Ricci tensor could then be written as

Rab

=

Eab

+

E

-

ab+

(E

+

E)UaUb

-

Egab

+

1 2

Hmnue(rJaemn

Ub +

rJbemnUa)

(8.5)
where E = Eg and E = Eg. It may be noted that E = (E + ~T)/2

defines the gravitational charge density while E = -TabUaub defines the energy density relative to the unit timelike vector u a .

3. DUALITY TRANSFORMATION AND DUAL SPACETIMES
The vacuum equation, Rab = 0 is in general equivalent to

E or E = 0, H[ab] = 0 = Eab + Eab

(8.6)

which is symmetric in Eab and Eab.

We define the duality transformation as

(8.7)
Thus the vacuum eqaution (6) is invariant under the duality transformation (7). From eqn. (1) it is clear that the duality transformation would map the Ricci tensor into the Einstein tensor and vice-versa. This is because contraction of Riemann is Ricci while of its double dual is Einstein.