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Einstein’s Static : An Idea Whose Has Come Back? Aubert Daigneault and Arturo Sangalli

A Tribute to Irving Ezra Segal (1918–1998)

hat is the shape of ? Is the uni- spatial universe,” observed Theodore Frankel in verse finite or infinite? Did the world his 1979 introductory book to Einstein’s theory. have a beginning, or has it always In EU, space may be mathematically described existed? These fundamental ques- as a three-sphere S3 of fixed radius r, i.e., as the tions have intrigued and baffled hu- boundary of a four-dimensional ball, by the equa- Wmans since the most ancient . But it was only tion u2 + u2 + u2 + u2 = r 2 . In Einstein’s model, 1 2 3 4 in the twentieth century, with the development of time had no beginning: it is infinite in both direc- powerful tools for probing the immensity of the tions, and so the universe has always existed and skies, that it became possible to explore the cos- will always exist. Hence EU may be presented as mos beyond our galaxy’s neighborhood. the cartesian product R S3 where R is the whole Concurrently, advances in physics and mathemat- real timeline. ics provided the conceptual framework—the lan- In 1929 interpreted the galactic guage, so to speak—in which to formulate com- phenomenon detected by Vespo Slipher prehensive theories whose validity could be starting in 1912—the change in the observed fre- objectively put to the test. Scientific answers to the quency of light waves—as a Doppler effect, that is, above questions finally appeared to be at hand. as caused by the motion of a luminous source Space-time is defined as the totality of pointlike away from the observer. The Doppler interpreta- events—past, present, and future—in the history tion became known as the Expanding Universe of the world. A “pointlike event” means an “in- theory, whose most developed form is the cos- stantaneous event taking place at a point of space” mology of Friedmann and Lemaître and according as opposed to an “event” that is extended in either to which galaxies are moving away from each other. time or space, such as World War II. In 1917 Albert As told in 1997 by David Dewhirst and Michael Einstein proposed a model for space-time known Hoskin in The Cambridge Illustrated History of as the Einstein Universe (EU), in which the totality Astronomy, “There is no doubt that the nearly of physical space is finite and curved [3]. “Nothing simultaneous detection of the and the in has intrigued the lay public derivation of solutions of Einstein’s equations that more than Einstein’s possibility of a closed, finite suggested that the universe would be expected to expand greatly encouraged this interpretation.” Aubert Daigneault is professor of mathematics at the The Expanding Universe theory later begot the Université de Montréal, Canada. His e-mail address is theory, which maintains, in addition to [email protected]. universal expansion, that time and the universe Arturo Sangalli is professor of mathematics at Cham- had a beginning, the “Big Bang”. plain Regional College, Lennoxville, Quebec, Canada. His Hubble also stated his famous law: The e-mail address is [email protected]. galaxies recede from each other with a velocity

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whereas Minkowski’s time is only an approxima- tion of t. Which coordinates, those of EU or those of M, are actually measured in observations is empirically immaterial, as the two differ by unobservably small amounts. Indeed, assuming “as is commonly believed that r 108 lightyears,” the two “deviate by less than one part in 1015 out to distances of 1 lightyear, or of less than 1 part in 106 out to ©2000 Index Stock Photography, Inc. galactic distances. There is no apparent means to proportional to the distance separating them. But detect such differences in classical observations” he did not rule out that, by virtue of some unknown [13, p. 766]. mechanism, the redshift might result from space This two-time situation arises from the essen- being curved. His reservations led him to write tially uniquely defined conformal immersion of in 1936 that “the assumption that redshifts are not Minkowski space M = R R3, in which space is velocity shifts is more economical and less vul- ordinary three-dimensional Euclidean space, into nerable, except for the fact that, at the moment, EU = R S3 . This immersion is unique up to the no other satisfactory explanation [i.e., apart from group of causality-preserving motions of EU and the Doppler effect] is known.” Once Hubble’s is a relativistic variant of stereographic projec- expansion seemed to have been established as an tion. Minkowski space M can be thought of as empirical fact, Einstein was forced to abandon his being tangent to EU just as the complex plane is static universe model. tangent to the Riemann sphere. In spite of this, the The observation in 1964 of the so-called cosmic immersion of M into EU preserves neither the time background radiation (CBR) is often considered coordinate nor the space coordinate in the fac- to be the smoking gun supporting the assumption torizations of these space-times as “time space”. that there was a Big Bang. But in fact the CBR When units are chosen to make the might have other causes. Erwin Finlay-Freundlich c equal to 1, Einstein’s and Minkowski’s time and Max Born had already predicted the existence coordinates are related by the equation of such radiation in 1953, eleven years before its (1) x0 =2r tan(t/2r), detection, on the basis of a stationary universe. The interpretation of the CBR as a faint echo of the birth from which the relation of the universe gave greater credence to the Big (2) z = tan2(t/2r) Bang theory, which became widely accepted as the standard astronomical gospel. of an observed redshift z to time of propagation t or, equivalently, geodesic distance on the A Two-Time 3-sphere, may be derived essentially by simple By the early 1970s the rapidly increasing mass of differentiation. statistical data on quasars and galaxies provided a Equation (1) implies that as t varies from r substantial basis for questioning the Big Bang to +r, ordinary time goes from minus infinity to cosmology. In 1972 Irving Ezra Segal, a professor plus infinity. Hence eternity (past and future) in the of mathematics at the Massachusetts Institute ordinary sense corresponds to a finite interval of of Technology, picked up where Einstein had left off cosmic time, which cosmic time t, nevertheless, and proposed a variant of special relativity [13]: varies over the whole real line. As for equation (2), chronometric cosmology (CC), so called it reveals that for small values of t (or, equiva- because it is based on the analysis of time. His 1976 lently, of the distance), the redshift varies as the book Mathematical Cosmology and Extragalactic square of t, in contradiction with Hubble’s law, Astronomy [14] contains a detailed presentation of which is linear. From (2) we also see that as r tends the theory. to infinity, z tends to 0. Hence, as envisaged as a According to CC, Einstein’s model is the correct possibility by Hubble himself, the curvature of one to understand the universe as a whole (i.e., space is the reason for the cosmic redshift in CC. global space-time), except that there are two kinds An application of l’Hospital’s rule to equation (1) of time: a cosmic or Einstein’s time t, and a local shows that, as expected, x0 tends to t as r tends or Minkowski’s time x0, which is (perhaps!) the time to infinity. The above-mentioned unobservably measured by existing techniques. In Segal’s words, small differences between the two times can eas- “The key point is that time and its conjugate ily be established from the series expansion of x0 variable, energy, are fundamentally different in in powers of t: the EU from the conventional time and energy in 3 2 5 4 the local flat Minkowski space M that approxi- (3) x0 t + t /(12r )+t /(120r )+.... mates the EU at the point of observation.” Simply Because the radius r of the universe is very put, Einstein’s cosmic time t is the “real” one, large, it follows from (3) that only extragalactic

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observations can be relevant for telling the two times apart. As Segal writes, “…in the absence of precise observations of masses and distances out- Irving Ezra side the solar system, only the redshift and related observations appear to fall in the category required Segal to distinguish, potentially at least, between the two clocks” [10, p. 857]. Equation (2) may also be derived [12, 17] in (and above) the observable frequency range by a rigor- ous analysis based on Maxwell’s equations gov- Photograph courtesy family of I. E. Segal. erning light and, more generally, all electromagnetic timeline and S, the space, is a three-dimensional radiation. These equations are defined primarily Riemannian manifold of constant curvature. in M, but extend uniquely, along with their solu- If d2 denotes the metric of S, the FRW space-time tions of finite Minkowskian energy, to the larger is endowed with the hyperbolic pseudo-Riemann- universe EU. It follows from this analysis that a free ian metric also known as the Lorentz metric will experience a redshift when propagated 2 2 2 2 2 over a very long period according to Einstein time. ds = c dt + r (t)d , In CC the Einstein universe EU is first thought of where c is the speed of light and r(t) is a function as empty of just like Minkowski space-time of time t in I known as the “radius of the universe M, which is the ordinary space-time of special at time t”. This is a mathematically unrestricted relativity, and appears as a natural alternative to scale factor; in particular, its time derivative r 0(t) the latter. In that sense EU belongs in the first place may very well exceed the speed of light without to special relativity rather than to general relativity special relativity being violated. Indeed, in an FRW (Einstein’s theory of gravitation) in whose frame- space the distance between two stationary points work it was presented by its initial proponent. EU is x and y of space increases at the rate r 0(t). The uniquely arrived at in CC by means unrelated to popular theory of Alan H. Guth [4], an gravitation. MIT physicist, claims that “…during the earliest Thus a distinction is introduced in CC between moments of time all space expanded far faster a space-time thought of as empty and a universe than the speed of light.” which is inhabited with matter and which enjoys In Big Bang models, r(t) tends to 0 as t tends only approximately the symmetries of the towards the lower limit of I, the instant of the Big underlying space-time. This distinction has no Bang (excluded from I), whereas in EU,r(t) is a counterpart in general relativity, where there is constant and I is the whole real line. Any such no a priori geometry and where it is precisely the FRW space-time is at each point (= event) locally introduction of matter that determines the approximated by its tangent space, which is geometry. essentially Minkowski space-time. A vector x =(x0,x1,x2,x3) in Minkowski space Causality 2 2 2 2 M is called a time vector if x0 x1 x2 x3 > 0; Space-time defined as the totality of all events— it is said to be oriented towards the future if x0 > 0. past, present and future—is, before anything else, The causality relation in M is defined by saying that a partially ordered set; the relation p

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by a directed curve such that at each point the measurements of the wavelength or the frequency, forward-looking tangent is a time vector oriented in accordance with the Minkowski energy. However, towards the future. of extragalactic are necessarily of cosmic spatial support when observed. And “a Conservation of Energy direct measurement of the Einstein energy of an The Einstein universe and Minkowski space-time extragalactic photon may depend on more than the are the only two space-times satisfying general pointwise oscillation of the photon wave. Indeed conditions embodying three fundamental physical mathematical analysis shows that the Einstein principles: first, the cosmological principle, also energy of a photon depends not only on the known as the Copernican principle, which states oscillation frequency of the photon but also on the that there is no preferred direction or point in number of oscillations, which appears beyond space; second, the “principle of inertia”, the state- the scope of the present methods to observe” [17, ment that there is no preferred timelike direction, p. 316]. which implies the equivalence between observers As says, “The utility of the in relative motion at the same point; and third, the concept of energy, in general, arises from the fact statement that there is no preferred moment on that it is conserved.” In Einstein’s theory of the time axis, which is a precondition for the general relativity, as opposed to what is the case principle of conservation of energy to hold. Such in the special theory of relativity, there is only a principles should be abandoned only in the face differential (or infinitesimal) conservation law of compelling evidence to the contrary. Segal con- and not an integral conservation law.1 The former tended that not a shred of such evidence exists and implies the latter only in the presence of a group that, on the contrary, the verifiable consequences of temporal translations that are isometries of the of these principles are confirmed by all available space-time model. Energy is integrally conserved astronomical data. if the of energy through the boundary of any Segal’s explanation of the origin of the cosmo- compact 4-dimensional domain of space-time logical redshift is free from the adjustable para- vanishes. An example of such a compact domain meters that intervene in the Expanding Universe is a finite portion of space during a finite interval theory. Moreover, unlike the Big Bang cosmology, of time [5, pp. 61, 62]. CC does not need to resort to ad hoc scenarios— Aside from Minkowski space-time, the Einstein such as “” for quasars and radio sources, universe is the only one amongst the FRW spaces and “inflation” at the hypothetical beginning of the that has such a temporal group of isometries and universe—reminiscent of the epicycles introduced consequently the only one in which the principle by ancient astronomers in their futile efforts to of integral conservation of energy is valid. reconcile the observed planetary orbits with a The energy that is conserved is, in particular, geocentric universe. Guth’s still very fashionable that of electromagnetic radiation in vacuum “which “inflationary universe” [4] was invented in the early is basically all that is observable in large scale 1980s to resolve two outstanding problems of astronomy.” Both electromagnetic energies, the Big Bang theory, namely, the so-called “flatness” Minkowskian and the Einsteinian, are conserved in and “horizon” problems, which simply do not arise their respective cosmos M and EU. “They are both in CC. representable as the integral over space in the A major merit of CC is to reestablish the respective cosmos of the square of the electro- principle of global energy conservation, which magnetic field, in terms of spatially natural supporters of Big Bang cosmology are willing coordinates,…and the latter integral is always to relinquish. “The explicit time dependence of larger” [17, p. 317]. The free electromagnetic field the Friedmann-Lemaître [expansionary] models can be construed as a Hamiltonian system in break Lorentz invariance as well as conservation infinitely many dimensions whose total energy is of energy-momentum. The loss of such theoreti- as just said, and the dynamical group is induced cally fundamental as well as experimentally by the isometry group of temporal translations of well-established laws must be weighed in the space-time, both in M and in EU. balance against the simplicity of a Doppler expla- Challenging Hubble’s Law nation for the redshift” [11, p. 4805]. Just as there are two times in CC, there are correspondingly In 1929 Edwin Hubble made the sensational two concepts of energy. The photon frequency announcement that the galaxies were receding is a measure of the Minkowski energy, and the from each other. This phenomenon had already redshift reduces that energy, which is therefore been detected more than a decade earlier by Vespo not conserved. But in CC, thanks to its cosmic Slipher, an astronomer at the Lowell Observatory, time, this energy is not lost. who had reported that at least some of the faint Freshly emitted photons have small spatial 1 We purposely refrain from using the word “local” in this support, and the energy of light composed of such connection, as it sometimes means “differential” and photons is conventionally inferred from laboratory sometimes “integral”.

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patches of light then called nebulae appeared to crucial test for this theory. In 1992 the statistician be receding from the Milky Way. When Hubble Bradley Efron, a creator of the bootstrap tech- plotted the observed redshift (or rather, in his nique, and Vahé Petrosian, a physicist, both at interpretation, the recession velocity v ) against the Stanford University, published a paper [2] which estimated distance d for a small number of galax- appears to disprove Segal’s claims. However, the ies, he claimed to have detected a “roughly linear” data on which they based their conclusions are pattern, corresponding to the relation v = Hd, not readily available and have been “criticized where the constant of proportionality H is known for being too inaccurate,” as Petrosian himself as Hubble’s constant. acknowledges. In addition, the criteria for select- As Hubble himself admitted, the estimated ing a subsample of 492 redshifts used to confirm distances from which his law was derived were in the Hubble law was not specified. part uncertain, but the best then available. In the In a 1997 article [6] Daniel Koranyi of the ensuing decades other astronomers followed up Harvard-Smithsonian Center for Astrophysics with their own verifications, and the linear redshift- and Michael Strauss of Princeton University claim distance relation is by now generally accepted. that since its original announcement by Hubble, But Segal denounced the Hubble school for hav- “observational evidence has mounted steadily” in ing based its confirmation of the law on observa- favor of the linear law. This opinion conflicts with tions of a relatively small class of objects, the bright Segal’s: “Careful investigation shows that the re- cluster galaxies, and without specifying a selection ported confirmation of Hubble’s Law has been criterion: “The central source list for such galaxies based on large components of wishful thinking….” has been the catalog of Abell, who stated: ‘In deter- At any rate, discussing an earlier paper [15] by Segal mining whether a cluster meets this criterion [for and his collaborators, the authors point to what inclusion in the catalog], it was assumed that the they perceive as a methodological pitfall affecting redshifts of clusters are proportional to their dis- Segal’s analysis against Hubble’s law: the sensitivity tances.’ […] For this [circular argument] and other of the estimation of the luminosity function (i.e., reasons, the observational case for the Hubble Law the probability distribution of the intrinsic lumi- on the basis of cluster galaxy samples is at best nosity of galaxies) to the number of bins into which inconclusive” [16, p. 295]. the sample used for that purpose is decomposed. Since the proposal of CC (chronometric cos- This, they contend, “could lead one to mistakenly mology), Segal and his collaborators, notably J. F. favor the quadratic law.” However, Koranyi and Nicoll of the Institute for Defense Analyses, have Strauss’s statistics suffer from other shortcom- carried out an extensive analysis of all published ings, some already pointed out by Segal in [15]. and reliable astronomical data on galaxies and Such challenges to Segal’s work are rare. Over quasars in accordance with modern statistical the years there has been very little response to his principles and in a reproducible manner. Their claims, and he has refuted all published criticism conclusions have invariably been consistent with the predictions of CC and at variance with those of CC. of the expansionary theory and Hubble’s law. As The Three Empirical Pillars of Big Bang Segal points out, “Older statistical methods were Theory dependent on parametrizations that developed from plausibility and convenience more than The tenants of the Big Bang theory interpret the direct observation. Rapid computer simulations redshift as a Doppler effect indicating the expan- now make possible efficient nonparametric sion of the universe. They interpret the cosmic methods that provide objective probabilities background radiation as an echo of the initial ex- of deviations of specified observations from plosion. The third empirical pillar of the theory is theoretical prediction, in circumstances far more called “”, which claims to complex and realistic than would be possible by predict the abundances of the four light elements: theoretical estimation of probabilities.” deuterium, helium-3, helium-4, and lithium-7. Using a sophisticated bootstrap statistical The much-trumpeted alleged concordance of technique that they call ROBUST, which takes into Big Bang predictions with observations goes es- account the so-called “observational cutoff bias” sentially as follows. What is called the abundance making faraway celestial objects less likely to be of a chemical element is the ratio of the amount observed than closer ones, Segal and Nicoll demon- of this element in the universe to that of hydro- strated that the quadratic redshift-distance law gen, which is the most prevalent. In his 1989 re- predicted by CC fits the available experimental view of Big Bang nucleosynthesis, Gary Steigman data on the more than 2,000 quasars of a reputed of Ohio State University [18] explains that the only comprehensive 1989 catalogue much better than parameter on which the predicted abundances de- does Hubble’s law [16]. pend is , the universal ratio of nucleons to pho- Because Hubble’s law is incompatible with tons. For a narrow interval of values of tightly chronometric cosmology, its status has become a constrained by the observed abundances, “the

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standard hot Big Bang model predicts the indi- classic principle of the conservation of energy, vidual abundances to lie precisely in this range.” and is not at all uniquely indicative of a big bang.” But, as Steigman says, “Abundances are not ob- Back in 1926 A. E. Eddington had already cal- served. Abundances are derived from the obser- culated that the “temperature of space” produced vational data, often following a long and tortuous by the radiation of starlight would be found to be path involving theory.…Errors (or uncertainties), 3 degrees Kelvin, as is the case of the CBR. often systematic, may be introduced at many steps In a private communication to one of the au- in the overall process of deriving abundances thors, Bertram Kostant, a mathematician at MIT, from observational data. Furthermore we are here summed up some recent joint work of his and concerned with primordial abundances. Even if Nolan Wallach’s of the University of California at present day universal abundances were known to San Diego as follows: “One of the major props for arbitrary accuracy (which they are not!), we still the Big Bang theory is the black body radiation pro- would have to employ theory and observation to file of the background radiation. Cosmologists say extrapolate back to obtain primordial (or at least that only an enormous heat source could account pregalactic) abundances. Additional errors (un- for that profile. One of the things that we have certainties) are surely introduced here too.” shown is that there exist solutions of Maxwell’s N. Yu Gnedin and Jeremiah Ostriker of Prince- equations in the Segal universe which have that ton University, who are also Big Bang supporters, profile and have been around for all times. In declared in 1992: “Light element nucleosynthesis particular they did not require a heat source to has been a central pillar supporting the standard exist. In fact we have classified all such solutions.” FRW hot Big Bang cosmological model.…But there According to Segal and Zhengfang Zhou, a are several confusing and apparently inconsistent mathematician at Michigan State University, the elements in the canonical picture which have led treatment of nucleosynthesis is similar in CC and to ‘patches’ which are quite ad hoc and are accepted in Big Bang theory to first order. “The difference only because of our familiarity with them and our is only that a stochastic sequence of mini Bangs, basic belief that the underlying standard model is associated with, e.g., the formation of galaxy accurate.” clusters, replaces the unique Big Bang. Cluster Steigman still expresses some reservations in formation would be expected to be accompanied a 1996 paper: “Indeed, the success of standard by extremely high temperatures, which, as in Big Bang nucleosynthesis in predicting the abun- Big Bang theory, would be productive of light dances of the light nuclides with only one elements” [17, p. 323]. adjustable parameter restricted to a narrow The discovery in 1999 of apparently abundant range while the abundances range over some 9 molecular in the universe, a possibility orders of magnitude is impressive indeed. However, already envisaged by physicist Paul Marmet in as with the nineteenth century standard model 1989, may complicate even further for [Newtonian ], some clouds have now Big Bang nucleosynthesis while at the same time emerged on the horizon. Recent analyses…point rendering unnecessary the concept of (to this day to a crisis unless the data are in error, or the ex- undetected) nonbaryonic except for trapolation of the data are in error, or there is new maintaining Big Bang theory. physics.” In more recent articles valiant efforts are made to alleviate the crisis, which nevertheless Which Model for the Universe? perdures. Can Segal be right against almost the entire as- These current difficulties of Big Bang cosmology tronomical community? Isn’t there an experiment may ultimately lead to an “agonizing reappraisal” that would settle the question? The problem is of its basic tenet, the Doppler interpretation of that a direct, empirical confirmation of CC—or of the redshift. But there is no sign of this yet, as the Hubble’s law for that matter—is not feasible notion of an expanding universe is by now such an given the current technological means, the entrenched belief in the scientific community and greatest difficulty being to determine the distances the population at large. to astronomical objects. These distances are not In any case, chronometric cosmology has re- observed but are estimated in a complex manner sponses to these contentions often presented as that depends on the model chosen. What can definite “proofs” of the Big Bang theory. In CC the be observed, apart from the redshift, are the cosmic background radiation is interpreted as the apparent brightness and the angular diameter. equilibrium state of free photons that have been Consequently, any comparison of the merits of scattered, reemitted, or absorbed many times in conflicting cosmological theories must hinge the course of possibly thousands of circuits of the not on direct observation but on a statistical 3-sphere. In Segal’s words: “The observed black- analysis of experimental data which would hardly body form of the cosmic microwave background be tractable without high-speed computers. is simply the most likely disposition of remnants Irving Ezra Segal died suddenly on August 30, of light on a purely random basis, assuming the 1998, a few days before his eightieth birthday. He

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was still actively engaged in research, pursuing, ignores, in particular, any possible electromag- amongst other mathematical endeavors, his lifetime netic part in the tensor. passion: to understand the universe. A 1940 Yale Segal expressed considerable skepticism [14, p. graduate, he taught at Harvard, Princeton, and 189]: “…Probably still less justified physically is the Chicago before becoming an MIT professor in 1960. application of general relativistic hydrodynamics He is the author of several monographs and count- to extragalactic questions such as the mass den- less scientific articles in prestigious journals and sity and the stability of the entire Cosmos. The ap- was a member of the American and the Royal Dan- proximation of the distribution of galaxies by ish Academies of Science. The width and depth of a fluid is quite uncontrolled and open-ended; his knowledge of physics and mathematics, at best, conclusions drawn in this way are merely and of the interaction between the two, was suggestive.” truly astonishing. His work may be summarized Nobel laureate Hannes Alfvén, the founder of as the search for the right mathematical models to modern plasma physics, i.e., the physics of elec- describe certain physical theories, such as cos- trically conducting gases, advocates what he calls mology and quantum field theory. a “plasma universe”. Eric Lerner has championed “Why has this work not received an adequate and popularized this idea in The Big Bang Never evaluation?” asks Edward Nelson, a Princeton Happened: “The plasma universe…is formed and University mathematician and former Ph.D. stu- controlled by electricity and magnetism, not just dent of Segal’s, in a memorial article [1]. And, gravitation—it is, in fact, incomprehensible with- answering his own question: “Part of the reason out electrical currents and magnetic fields.” lies in Segal’s style of scientific exchange—at times “Hannes Alfvén and his colleagues have shown it resembles that of Giordano Bruno (later burned that a nonexpanding universe can be stabilized at the stake), who very shortly after his arrival by electric and magnetic forces,” writes Gerald S. in Geneva issued a pamphlet on Twenty Errors Hawkins of the Harvard-Smithsonian Observatories. Already in 1920 Hermann Weyl had speculated Committed by Professor De la Faye in a Single on the consequences of a spatially closed universe, Lesson. But part of the fault lies with cosmologists observing that if such were the case, then “it would and particle physicists intent on defending turf. The become possible for an observer to see several time for polemics is past. Segal’s work on the pictures of one and the same star. These depict the Einstein universe as the arena for cosmology star at epochs separated by enormous intervals of and particle physics is a vast unfinished edifice, time (during which light travels once entirely round constructed with a handful of collaborators. It is the world).” rare for a mathematician to produce a life work In the December 1998 issue of this journal, that at the time can be fully and confidently Neil J. Cornish, a research fellow at the University evaluated by no one, but the full impact of the of Cambridge, UK, and Jeffrey R. Weeks, a geome- work of Irving Ezra Segal will become known only ter working as an independent scholar, took up to future generations.” Weyl’s idea in the context of the Big Bang theory: Segal never let up in his crusade against the “When we look out into the night sky, we may be expansionary theory, alternating sound scientific seeing multiple images of the same finite set of arguments with emotional tirades. In a 1992 galaxies.” They even went one big step further letter to The New York Times, he reacted to the and, referring to an artist’s impression drawn on interpretation of detected “ripples” in the CBR as the cover of that issue of the Notices, they wrote: support for the Big Bang: “In this day of budding “It is possible that one of the galaxies seen in mass movements that appear dangerously irra- this image is our own Milky Way, and the light we tional, it is deplorable that some scientists, as receive from it has made a complete trip around well as the media, indirectly support detachment the universe.” from rationality by effectively (if with the best of This possibility can be contemplated in CC as intentions) catering to the popular fondness for the well: “The transparency of cosmic space implies that Big Bang.” photons in EU will typically make many circuits of A major objection often put forward against EU S3 before being absorbed or undergoing interac- as a realistic model of the universe, aside from its tion. A free photon will be infinitely redshifted at alleged inability to explain the observed redshift, the antipode of S3 to its point P of emission, but is its purported and much decried instability: if EU on returning to P it will be in its original state, as were somewhat perturbed, it would either expand a consequence of the periodicity of free photon or collapse. However, the proof of such instability wave function in EU” [17, p. 324]. is based on the widespread and rather naive We believe that Segal’s chronometric model for notion that the energy-momentum tensor of cosmology merits serious consideration. It is true Einstein’s gravitational equation can realistically that Albert Einstein, the most famous advocate of be represented by that of a perfect fluid, a notion a static, unevolving universe, later changed his that is only a natural starting point and that mind when he disavowed the “cosmological

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constant” (lambda), a kind of antigravity factor to fairly big “bangs”, and the same would be in that he had introduced in his equations, just as stock for the future. C. Neumann and H. Seelinger had done in a New- tonian context at the end of the nineteenth cen- References tury, to prevent the universe from collapsing under [1] J. C. BAEZ et al., Irving Ezra Segal (1918–1998), gravity [9]. Einstein is said to have referred to the Notices of the AMS 46 (June/July 1999), 659–668. introduction of lambda as his “greatest blunder,” [2] B. EFRON and V. PETROSIAN, A simple test of indepen- dence for truncated data with applications to and yet... redshift surveys, Astrophys. J. 399 (November 1992), General relativity reduces gravitation to curva- 345–52. ture. Einstein’s original equation implies that space [3] A. EINSTEIN, Kosmologische Betrachtungen zur allge- is flat, in the absence of matter, whereas meinen Relativitätstheorie, Sitzungsberichte der the modified equation that includes the -term Preussischen Akademie der Wissenschaften (1917), allows empty space to have a curvature of its own 142–152. Also with the title “Cosmological consid- in addition to that resulting from the presence of erations on the general relativity theory”, pp. 177–188 matter. The modified equation is the most general in The Principle of Relativity, a collection of original memoirs on the special and general theory of rela- obeying some minimal conditions and which allows tivity, by H. A. Lorentz, A. Einstein, H. Minkowski, and empty space to be curved or flat [8]. H. Weyl, with notes by A. Sommerfeld; translated by Saying that is necessarily zero in the absence W. Perrett and G. B. Jeffery. Published by Methuen of matter can be compared to saying that if it were & Co. Ltd., London, 1923; reprinted in 1990 (MR 94a: not for mountains, rivers, and valleys which define 01043); also published unaltered and unabridged local curvature on a piece of land, the earth would by Dover Publications, Inc., in 1952. be flat. The curvature defined by local topography [4] ALAN H. GUTH, The Inflationary Universe, Addison- masks the global curvature of our planet, but, of Wesley, Reading, MA, 1997. [5] S. W. HAWKING and G. F. R. ELLIS, The Large-Scale course, it is the other way around if one looks at Structure of Space-Time, Cambridge Univ. Press, it from a sufficiently large distance. 1973. “Theorists are already scrambling to understand [6] D. M. KORANYI and M. A. STRAUSS, Testing the Hubble law what 20 years ago would have been unthinkable: with the IRAS 1.2 Jy Redshift Survey, Astrophys. J. 477 a greater than zero yet much (1997), 36–46. smaller than current quantum theories predict,” ob- [7] LAWRENCE M. 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16 NOTICES OF THE AMS VOLUME 48, NUMBER 1