General Relativity and Gravitation, Vol. 32, No. 9, 2000

0001-7701/ 00/ 0900-1961$18.00/ 0 2000 Plenum Publishing Corporation

1961

Book ReviewThe Geometric Universe: Science, Geometry andthe Work of Roger Penrose

S. A. Huggett, L. J. Mason, K. P. Tod, S. T. Tsou and N. M. J.Woodhouse, Eds.450p., Oxford Univ. Press 1998. GBP29.50, ISBN 0-19-850059-9

This book is a festschrift commemorating Roger Penrose’s 65th birthday. As onemight expect, given the scope of Penrose’s interests, and the breadth of the workhe has stimulated, the contributions cover a wide area. It would be impossibleto discuss all of them here, and I shall limit myself to a few, which are perhapsof greatest interest to readers of this journal. A list of the other articles appearsat the end of this review, and my omitting detailed comments on most of thecontributions should not be taken as criticism.

How is one to model an isolated general-relativistic system? This questionis difficult because the sense in which one “goes off to infinity” in order todefine “isolation” is determined by the dynamical field itself. The beginnings ofthe modern understanding of this go back to work of Bondi and Sachs, and thepresent mathematical formulation was first clearly given by Penrose. He sug-gested that physically interesting space-times modeling such isolated systemsshould admit conformal completions, the limits at null infinity representing radia-tion. For a long time, the existence theory of the Einstein equations was not goodenough to know whether any very large class of space-times did in fact satisfythese conditions. But now, largely due to work initiated by Friedrich, such pos-itive answers are available. There are still issues which remain to be cleared up(some suggested by the results of Christodoulou and Klainerman), but the overallpicture is that envisioned by Penrose. This is the subject of Helmut Friedrich’sadmirable status report, “Einstein’s Equation and Conformal Structure.”

There does remain a deep problem, which is only peripherally addressedby current approaches. This is the question of how to specify the gravitationaldegrees of freedom. The difficulty here is that in the standard 3 + 1 formalism

Book Review1962

there are constraints on what would most naturally be the initial data, the met-ric of the three-surface and its extrinsic curvature. While there are good mathe-matical theorems guaranteeing the existence of classes of solutions to these con-straints, the physical question of how to choose data corresponding to a givenphysical problem remains unsolved. In other words, solutions to the constraintsexist, but there is no known physically compelling way of parametrizing them.Which is the correct one for a given problem?

Consider, for example, the problem of modeling two colliding black holes.We expect to be able to specify the state of the holes at some fixed timein the distant past by just a few numbers: their masses, angular momenta,charges, separation and relative velocity. But we know there are infinitely manydegrees of freedom in the gravitational field. What physically convincing crite-rion determines these data? Part of the answer, which is naturally expressed in theBondi–Penrose–Sachs formalism, is that there should be no gravitational radia-tion coming in to the system. In principle, this condition is checked by evolvingthe data backwards to past null infinity, and requiring the radiation profile thereto vanish. But it is not known in practical terms how to implement this.

Abhay Ashtekar’s paper, “Quantum Gravity,” is an excellent sampler ofsome of what has been achieved with his “new variables” program. A clearsummary of the elements and some of the results is given, without overmuchformalism but with a sufficient explanation of the main issues which have to befaced. Even though some time has passed since the paper was written, and eventhough the paper was not meant to be a survey, it is still to be recommendedas a starting-point. The reader completing this paper will have a good sense ofthe general shape of the area, and will have a vantage from which to view laterdevelopments.

Some of the most intriguing contributions are those on quantum theory in itsmost vividly non-classical forms. Alan Ekert’s, “From Quantum Code-making toQuantum Code-breaking,” is not only stimulating but a model of accessibility.The ideas are brought out concisely and directly. Only an elementary knowl-edge of quantum theory and the simplest facts of number theory are assumed.Lev Vaidman’s “Interaction-Free Measurements,” is similarly accessible (withan account of a charming science-fair experiment, due to Marchie Van Voorthuy-sen). These papers could profitably be used in first-semester quantum theorycourses—perhaps they already are. Richard Jozsa’s “Entanglement and Quan-tum Computation” will give readers a sense of the difference between quantumand classical computation, although it is not intended as a survey and the inter-ested reader will have to go to other papers to get details.

Paul Steinhardt’s “Penrose Tilings and Quasicrystals Revisited” is a nicebrief on the area, from the point of view of one of its advocates. While the early(i.e., mid-1980s) experimental evidence for quasicrystals was somewhat equivo-cal, the quasiperiodic order of various materials is now more firmly established.

Book Review 1963

What is not so well understood is how they form. This is bound up with a deepand fascinating speculation of Penrose.

As originally conceived, the Penrose tilings were restricted by certain “match-ing rules,” and these were known to be non-local and non-computable. In otherwords, to be sure that adding on a few tiles at one point would not ultimately lead tocontradictions (the pattern not being extensible as a tiling, but bumping into itself)required non-local and non-computable information. So Penrose speculated thatthe existence of quasicrystals might be evidence that quantum reduction (of atomsto lattice sites) has an element of non-computability.

While this is not ruled out, there are now competing, more prosaic, expla-nations. New sorts of matching rules (which look to the vertices, and not just theedges) have been found which are local. And Jeong and Steinhardt have shownthat with certain assumptions Penrose tilings can be achieved by maximizingdensities of certain tile clusters.

Dennis Sciama’s contribution, “Decaying Neutrinos and the Geometry ofthe Universe,” is the only one to attempt to directly link experiment with space-time geometry. Sciama shows that a remarkably consistent account can be madeof such diverse phenomena as the dark-matter problems associated with cos-mology and galactic rotation, as well as the ionization of hydrogen in opaqueregions, by assuming a t neutrino with mass ∼27 eV and life-time ∼2 × 1023

sec.Dorje Brody and Lane Hughston’s “Geometric Models for Quantum Statis-

tical Inference” shows how a systematic refinement of the uncertainty relationis possible. Their results take the form

DPDQ ≥ (h/ 2)(1 + f )1/ 2,

where DP, DQ are the standard deviations associated to canonically conjugatevariables, and f is a function of moments of P (say). The expressions for f arecomplex, but it should be of general interest that there is a systematic way ofderiving such results.

There are two omissions which are noteworthy. There is no article ondifferential-topological techniques in general relativity, although important workcontinues to be done in this area more than thirty years since its power wasdemonstrated by Penrose’s singularity theorem. And there is nothing on quasi-local kinematics. It is understandable that the editors did not ask for a contribu-tion on the latter, as little progress has been made in recent years, but the issueis of such importance that it is worth taking up here.

The fact that we do not have a good understanding of how energy-momen-tum and angular momentum are exchanged between general-relativistic systemsis not just an embarassing limitation on the depth of our understanding; it is

Book Review1964

probably a major restriction on our ability to do gravitational physics. It is rarethat one can by brute force analyze complicated systems. One almost alwaysrelies on basic principles to do so, and in non-gravitational physics the mostimportant of these are conservation laws. Almost nothing like this is known ingeneral relativity. How much energy is there in a box in space-time? We don’tknow.

The fundamental obstacle to progress is of course that the kinematic invari-ants (energy-momentum and angular momentum) are usually defined as quan-tities conjugate to symmetries, and a general space-time will have no symme-tries. While occasional attempts have been made to use diffeomorphisms (or, atinfinity, Bondi–Metzner–Sachs motions) in the formal role of symmetries, theseattempts have not gotten us any closer to satisfactory definitions of the kinematicinvariants. And the reason is that in such very large classes of motions one hasnot said which, if any, are the key kinematic degrees of freedom. The main goal,remember, is to isolate finitely many kinematic invariants.

Penrose proposed a “quasilocal twistor program” for attacking this issue.While far from complete, it is worthwhile pointing out that it has had a significantsuccess and seems to contain a depth of physical insight. To my knowledge, itis the only program so far which provides a satisfactory definition of angularmomentum at null infinity. The results there are so good they alone would justifyfurther investigation. It should be emphasized that these very good properties ofthe twistor definition arise precisely because it is not closely related to BMSmotions: the twistor definition almost magically factors out the spurious BMSdegrees of freedom. Besides good results, there is another point in the twistordefinition’s favor: it seems to have considerable physical depth. This is boundup with its non-locality. We know from other arguments that energy in generalrelativity is a non-local quantity; a non-locality of the same nature is at the rootof the twistorial definition. This work should not be abandoned.

The book closes with an Afterword by Penrose, partly surveying recentwork aimed at extending twistor theory to curved space-time, and partly reflect-ing on the directions in which his speculations on geometry and quantum theoryhave gone. There is perhaps a note of wistfulness at some points, where not asmuch has been accomplished as he would have liked. This is all to his credit:science is at its best when its workers are not only bold and enthusiastic, buthard-headed and honest in their assessments of their own and others’ work. Muchhas been accomplished by Penrose and by others he has stimulated, and he hashad a large part in bringing difficult questions about the natures of quantum the-ory and consciousness into the mainstream of scientific discourse. It is as alwaysenjoyable to read him, lucid, precise and tinged with enthusiasm, both for whathas been done and for those questions whose answers still elude us.

Articles not discussed above: “Laudatio” by John A. Wheeler, “RogerPenrose—A Personal Appreciation,” by Michael Atiyah; “Hypercomplex Man-

Book Review 1965

ifolds and the Space of Framings,” by Nigel Hitchin; “Gauge Theory in HigherDimensions,” by S. K. Donaldson and R. P. Thomas; “Noncommutative Differ-ential Geometry and the Structure of Space–Time,” by Alain Connes; “Twistors,Geometry, and Integrable Systems,” by R. S. Ward; “On Four-Dimensional Ein-stein Manifolds,” by Claude LeBrun; “Loss of Information in Black Holes,” byStephen Hawking; “Funda-mental Geometry: The Penrose–Hameroff ‘Orch OR’Model of Consciousness,” by Stuart Hameroff; “Implications of Transience forSpacetime Structure,” by Abner Shimony; “Quantum Geometric Origin of AllForces in String Theory,” by Gabriele Veneziano; “Space from the Point of Viewof Loop Groups,” by Graeme Segal; “The Twistor Diagram Programme,” byAndrew P. Hodges; “Spin Networks and Topology,” by Louis H. Kauffman; “ThePhysics of Spin Networks,” by Lee Smolin; “The Sen Conjecture for DistinctFundamental Monopoles,” by Gary Gibbons; “An Unorthodox View of GR viaCharacteristic Surfaces,” by Simonetta Frittelli, E. T. Newman, and Carlos Koza-meh; “Amalgamated Codazzi-Raychaudhuri Identity for Foliation,” by BrandonCarter; “Abstract/ Virtual/ Reality/ Complexity,” by George Sparling; “QuantumMeasurement Problem for the Gravitational Field,” by Jeeva Anandan; “Pen-rose Transform for Flag Domains,” by Simon Gindikin; “Twistor Solution ofthe Holonomy Problem,” by S. A. Merkulov and L. J. Schwachhofer; “The Pen-rose Transform and Real Integral Geometry,” by Toby N. Bailey; “PythagoreanSpinors and Penrose Twistors,” by Andrzej Trautman.

Adam HelferDepartment of MathematicsUniversity of Missouri at ColumbiaColumbia, Missouri 65211, USA