Alexander Friedmann and the origins of modern Ari Belenkiy

Citation: Phys. Today 65(10), 38 (2012); doi: 10.1063/PT.3.1750 View online: http://dx.doi.org/10.1063/PT.3.1750 View Table of Contents: http://www.physicstoday.org/resource/1/PHTOAD/v65/i10 Published by the American Institute of .

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Downloaded 17 Feb 2013 to 136.186.1.81. Redistribution subject to AIP license or copyright; see http://www.physicstoday.org/about_us/terms and the origins of modern cosmology Friedmann, who died young in Ari Belenkiy 1925, deserves to be called the father of cosmology. But his seminal contributions have been widely misrepresented and undervalued.

inety years ago, Russian Alexander Friedmann (1888–1925) demonstrated for the first time that ’s general theory of Nrelativity (GR) admits nonstatic solutions. It can, he found, describe a cosmos that expands, contracts, collapses, and might even have been born in a singularity. Friedmann’s fundamental equations de- scribing those possible scenarios of cosmic evo- lution provide the basis for our current view of the Big Bang and the accelerating . But his achievement initially met with strong re- sistance, and it has since then been widely mis- represented. In this article, I hope to clarify some persistent confusions regarding Fried- mann’s cosmological theory in the context of related work by contemporaries such as Einstein, , Arthur , and Georges Lemaître. Last year’s Nobel Prize in Physics was shared by three cosmological observers who discovered Figure 1. Alexander Friedmann in Petrograd, , in the early 1920s. that the cosmic expansion is currently accelerating (see PHYSICS TODAY, December 2011, page 14). Thus, papers.3 Regrettably, however, it significantly dis- one of the scenarios introduced by Friedmann in torts his contributions. 1,2 1922 and 1924 appears to correspond to reality. Already in 1922, Friedmann had set the appro- The Royal Swedish Academy of Sciences back- priate framework for a GR cosmology by introduc- ground essay for the 2011 prize cites Friedmann’s ing its most general metric and the “,” which describe the evolution of a per- Ari Belenkiy has taught at Bar Ilan University fect-fluid cosmos of uniform mass ρ. And he in Ramat-Gan, Israel and at the British Columbia Institute of elucidated all three major scenarios for a nonstatic Technology’s School of Business in Burnaby, British Columbia, universe consistent with GR. In fact, he introduced Canada. He lives in Richmond, British Columbia. the expression “expanding universe” and estimated the period of an alternative periodic universe that’s

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Downloaded 17 Feb 2013 to 136.186.1.81. Redistribution subject to AIP license or copyright; see http://www.physicstoday.org/about_us/terms made a risky record-breaking balloon flight to col- Figure 2. Friedmann’s fundamental cubic C x ( ), lect high-altitude data.4 given by equation 4 in the text, is plotted here x Einstein’s 1905 special was against cosmic radius for arbitrary values of the well known in . But awareness of Einstein’s λ A free cosmological parameters and . Their true 1915 paper introducing GR was delayed because of C x values determine whether ( ) has 0 or 2 positive the world war.5 But soon after the war, news of the roots, and they define three general scenarios of theory and of Eddington’s confirmatory 1919 solar- cosmic evolution. One gets additional limiting- eclipse observations caused tremendous excitement case scenarios if the two positive roots are degen- among scientists and the general public throughout erate. Changing the sign of the linear term in C(x) Russia. And in 1921, the resumed shipment of to describe a cosmos with negative European scientific publications provided scientists yields still more scenarios. in Petrograd with access to the literature. Further- more, physicist Vsevolod Frederiks brought insider Cx() information. Interned in Germany during the war, he worked at Göttingen as an assistant to , who independently proposed the GR equa- tions early in 1916, not long after Einstein. In collaboration with Frederiks, Friedmann wrote a mathematical introduction to GR. The first volume, devoted to tensor calculus, appeared in 1924. Another book, The World as Space and Time, written by Friedmann alone the year before, devel- x oped his philosophical interpretation of GR. But his fame rests on two Zeitschrift für Physik papers.1,2 In them he introduced the fundamental idea of mod- ern cosmology—that the geometry of the cosmos might be evolving, perhaps even from a singularity. before Friedmann The fundamental of GR are R − g R/2 − λg =−kT , (1) surprisingly close to what is believed now to be the μν μν μν μν time elapsed since the Big Bang. In 1924 he further where the indices μ,ν run from 1 to 4 and revolutionized the discourse by presenting the idea the constant k =8πG/c2. The spacetime distribution of of an infinite universe, static or nonstatic, with a the energy–momentum tensor Tμν determines the constant negative curvature. local geometry encoded in the metric tensor gμν, and the “Ricci tensor” Rμν is determined by gμν and its A short life spacetime derivatives. The Ricci scalar R, a contrac- Born and raised in , Friedmann tion of the Ricci tensor, is the actual local curvature of studied mathematics at the city’s university.4 There spacetime. The λ was intro- he attended the physics seminars of Vienna-born duced in 1917 by Einstein in the hope of finding a sta- Paul Ehrenfest, who had moved to St. Petersburg ble, static cosmological solution of the field equations. with his Russian wife in 1907. After Friedmann To seek a cosmological solution for a homoge- graduated in 1910, he worked primarily in mathe- neous universe approximated by a of matical physics applied to and aero- uniform density ρ and P, one takes 2 dynamics. T11 = T22 = T33 =−P and T44 = c ρ. All off-diagonal ele- Following the outbreak of in Au- ments are zero. For simplicity, Einstein considered gust 1914, Friedmann served with the Russian air the cosmological approximation P = 0. force on the Austrian front as a ballistics instructor. Finding cosmological solutions of the field He took part in several air-reconnaissance flights equations required great ingenuity. Before 1922 only and was awarded the military cross. After the Feb- two simple solutions were discovered, one by ruary 1917 revolution that deposed the czar, dozens Einstein and the other by de Sitter. The so-called of new universities were established across Russia “solution A,” found by Einstein in 1917, represented and Friedmann obtained his first professorship, in a finite three-dimensional spatial hyperspheric sur- Perm near the Ural Mountains. face of constant radius r embedded in 4D spacetime. At the end of the civil war that secured the Solution A couples the initially independent Bolshevik regime in 1920, Friedmann (pictured in parameters λ and ρ to the fixed cosmic radius r. It figure 1) returned to his hometown, renamed requires that λ = c2/r2 and ρ = 2/(kr2). In 1917, de Sit- Petrograd, and started working as a physicist at the ter invoked the second relation to arrive at r =8×108 Geophysical Observatory. He soon became the ob- light-years by estimating the mean cosmic mass servatory’s director. Most of his personal research density to be about 2 × 10−27 g/cm3. was oriented toward theories of turbulence and So it seemed for a moment that Einstein had aerodynamics. But in parallel, he also worked on GR achieved his goal of finding a finite, and quantum theory. A month before his untimely whose size is straightforwardly determined by its death from typhus in September 1925, Friedmann mass density. But de Sitter then found another www.physicstoday.org October 2012 Physics Today 39

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equations.1 Like Einstein’s solution A, Friedmann’s Figure 3. Three possible scenarios for cosmic evolution proposed solutions feature space as a 3D hypersphere. But its 1 by Alexander Friedmann are shown as temporal plots of the cosmic curvature changes in time with the hypersphere’s radius. In his first monotonic scenario, M1, the cosmos expands at a radius r(t). Now the field equations lead to a set of decelerating rate from a zero-radius singularity until an inflection two ordinary differential equations for r(t). t t point at f, after which the expansion accelerates. That curve, with 0 The first-order Friedmann differential equation

indicating the present, looks much like what observations have 2 2 3 2 been revealing in recent decades. The second monotonic scenario, (r/c )(dr/dt) = A − r + λr /3c (2) M2, shows ever-accelerating expansion from a nonzero initial governs the dynamics of the universe. (Nowadays radius. The periodic scenario P shows evolution from and back to r(t) is regarded as an arbitrary scale length in a zero radius. presumably infinite universe.) Friedmann found that the integration constant A equals kρr3/3. Thus it’s proportional to the constant total mass of the cosmos. The rest of his 1922 paper is dedicated to ana-

rt M2 lyzing the evolutionary implications of equation 2, which after integration over the cosmic radius becomes M1 r 1 x tdxt=+. c λ 0 (3) r ∫ Ax−+ x3 i P 2 COSMIC RADIUS COSMIC ( ) √ 3c r 0 t tf 0 TIME t If one takes r0 to be the present value, then t0 desig- nates, in Friedmann’s words, “the time that has passed since Creation.” Three cosmic scenarios solution that hit Einstein like a cold shower.6 De Sit- The right side of equation 3 has physical meaning ter’s “solution B” presented a different sort of static only when the cubic denominator

universe with zero mass density and negative spa- 3 2 tial curvature. In Einstein’s solution A, all points in C(x)=A − x + λx /3c (4) space are equivalent. But de Sitter’s space has a under the square-root sign is positive (see figure 2). unique center. Only light rays passing through that That requirement defines three different scenarios center travel along geodesics. for cosmic evolution: Einstein found de Sitter’s solution unacceptable ‣ One gets the first scenario if C(x) has no positive because it violated philosopher Ernst Mach’s dictum roots and thus is positive for all positive x. That hap- that inertia cannot exist without . But the so- pens when λ >4c2/9A2, that is, when the cosmologi- lution had apparent virtues. It seemed to surround cal constant exceeds some critical value that de- every observer with a kind of horizon that might ex- pends on ρ. In that case, the cosmos starts at t =0 plain the in the spectra of distant galaxies from the singularity r = 0, and its expansion rate that had been reporting since 1912. changes from deceleration to acceleration at an in- Furthermore, de Sitter, Eddington, and his student flection point t , at which r =(3c2A/2λ)1/3. After that, f f ― Lemaître looked to solution B as a way of testing GR. r grows asymptotically like e t√λ/3. Friedmann called this scenario “the monotonic world of the first kind” Friedmann’s universe (see the curve labeled M1 in figure 3). Friedmann’s 1922 paper1 cited the original papers by ‣ The second situation occurs when 0 < λ <4c2/9A2.

Einstein and de Sitter, as well as Eddington’s 1920 In that case, C(x) has two positive roots, x1 < x2, and book, Space, Time and Gravitation, available to him in is negative between them. This condition admits a French edition. Instead of taking sides between two different scenarios, 2a and 2b. In 2a, expansion

Einstein and de Sitter, Friedmann approached the oscillates between r = 0 and r = x1. That gives the pe- problem of a cosmological solution from a wider riodic solution discussed below. In the 2b scenario,

viewpoint. expansion starts from a nonzero radius, ri= x2, and His interpretation of GR shows strong ground- expands forever with accelerating rate. Friedmann ing in . Friedmann continually called it the monotonic world of the second kind reminded his readers of the need to distinguish (curve M2 in figure 3). between intrinsic features of spacetime, such as the ‣ Friedmann called the third scenario “the peri- metric, and purely mathematical artifacts like the odic world” (curve P in figure 3). It results either choice of a particular coordinate representation. from 2a above or from λ ≤ 0. In either case, C(x) has

The physical requirement of spatial homogene- only one positive root, x1, and its interval of positiv- ity, he asserted, did not necessitate a static universe. ity is from 0 to x1. The cosmos starts from the singu- Focusing on the most general form of the GR metric larity r = 0, expands at a decelerating rate to maxi-

for a homogeneous and isotropic cosmos, Fried- mum radius x1, and then begins contracting back mann found, in addition to the static solutions A and down to zero. The life of the cosmos is finite, ending B, a new class of nonstatic solutions of the GR field in a . Assuming a total mass of 5 × 1021

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Downloaded 17 Feb 2013 to 136.186.1.81. Redistribution subject to AIP license or copyright; see http://www.physicstoday.org/about_us/terms solar masses, Friedmann found a lifetime (roughly ric properties alone do not resolve the problem. In- πA/c) of 1010 years for his periodic world. spired by Poincaré’s theory of Riemannian mani- In addition to the three principal scenarios, folds, he imagined the possibility of a spherical uni- Friedmann also considers two special limiting cases, verse of infinite size. The S³ geometry did not seem 2 2 when λ has precisely the critical value λc =4c /9A . to admit an infinite volume. Unabashed, however, Then C(x) is degenerate; it has a double positive root Friedmann suggested that the azimuthal coordi- at x =3A/2. In one limiting case, the periodic world’s nate ϕ of the spherical cosmos might run not from expansion period becomes infinitely long, asymp- 0 to 2π, but rather wind around over and over to totically approaching, from below, the static radius infinity. of Einstein’s solution A. In the other, Friedmann’s Friedmann found another, even more surpris-

M2 world at λc requires an infinitely long past to rise ing argument to undermine the notion of a neces- asymptotically from Einstein’s static radius, which sarily finite cosmos. On advice from his colleague is its ri. Whereas these limiting cases, like Einstein’s Yakov Tamarkin, he sought to discover whether GR solution, bind λ to ρ, in the general Friedmann sce- allowed solutions for a hyperboloid of infinite vol- narios they are independent free parameters. ume whose negative spatial curvature is given everywhere by −6/r2. In such a space, every point Einstein’s reaction would in effect be a saddle point. And indeed, When Friedmann’s 1922 paper first appeared, its Friedmann’s 1924 paper gives a positive answer, main ideas were mostly ignored or rejected. Ein- with both static and nonstatic solutions.2 stein’s immediate reaction illustrates how unwel- The static solution, like de Sitter’s solution B, ne- come the idea of a nonstatic universe was. In his cessitates zero density. The nonstatic scenario, with view, a proper theory had to uphold the evidently evolving negative curvature and r(t), has a nonvanish- static character of the cosmos. ing average density whose evolution is indistinguish- Therefore Einstein initially found Friedmann’s able from that of Friedmann’s positive-curvature solu- solution “suspicious.” In September 1922, he pub- tion. So one can’t determine the sign of the cosmic lished a short note in the Zeitschrift für Physik sug- curvature simply by measuring ρ. The only change in gesting that Friedmann’s derivation contained a the fundamental Friedmann equation for the negative- mathematical error. In fact, Einstein had mistakenly curvature case is that the sign of the linear term in concluded that Friedmann’s equations, in the ap- equations 2–4 becomes positive. Friedmann thus clar- proximation that neglects pressure, imply constancy ified the meaning of the linear term’s coefficient in C(x): of density and therefore a cosmos of fixed size. It gives the sign of the cosmic curvature. Learning of Einstein’s note, Friedmann wrote The 1924 paper was also ignored. Einstein paid him a long letter elaborating his derivations. But Einstein was on a world tour, returning to Berlin only in May 1923. Only then could he have read Friedmann’s letter.4 Later that month, Friedmann’s Russian colleague Yuri Krutkov met Einstein at Ehrenfest’s home in and clarified the confu- sion. So Einstein promptly published another short note in the Zeitschrift, acknowledging the mathemat- ical correctness of Friedmann’s results. He opined, however, that “the solution has no physical mean- ing.” But wisely, he crossed out that imprudent re- mark from the galley proofs at the last moment. Still, it would be another eight years before Einstein was ready to accept the idea of the expanding universe. Friedmann was the first to realize that GR alone cannot determine the geometry, topology, or kine- matics of the real cosmos. The choice of one cosmo- logical solution over another has to come from ob- servation. For example, the universe in the shape of a finite 3D hyperspheric surface (denoted S3 by topologists) admits “ghosts,” double images of the same object in opposite directions on the sky. In 1917, de Sitter had pioneered the idea that the space of directions must be considered as the basic space, with the opposite directions viewed as one. Fried- mann mentioned that view favorably,2 and Lemaître later applied it to compute the volume of his own Figure 4. A table of line-of-sight (“radial”) velocity components deduced model cosmos.7 by measurements for 41 spiral galaxies compiled by Vesto Slipher and included in ’s 1923 book on relativity theory.15 Each Infinite worlds galaxy is identified by NGC catalog number and celestial coordinates. In Friedmann’s major concern, however, lay with the 1927, Georges Lemaître plotted these velocities against ’s very notion of a finite cosmos, which was at the approximations of galaxy distances to produce the first estimate of the time firmly entrenched. He insisted that local met- Hubble constant.7 www.physicstoday.org October 2012 Physics Today 41

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physical meaning,11 that of a “primeval atom” Figure 5. At Caltech in 1933 are Robert Millikan (chair of Caltech’s blowing up—what Fred Hoyle later dismissively executive council), Georges Lemaître, and Albert Einstein. Lemaître, called “the Big Bang.” an ordained priest, was a physics professor at the Catholic University Unable to foresee the 1998 discovery of cos- of Louvain, in Belgium. mic acceleration, Einstein no longer saw any need for a cosmological constant. He thought Friedmann’s expanding solution with λ = 0 might be the answer. Starting with the 1946 edition of his popular exposition The Meaning of Relativity, Einstein interjected that “the mathematician Friedmann found a way out of this [cosmological constant] dilemma. His result then found a sur- prising confirmation in Hubble’s discovery of the expansion of the stellar system.... The following [15-page exposition] is essentially nothing but an exposition of Friedmann’s idea.”12 Unfortunately, Einstein attributed to Hubble alone what properly belongs to several people, among them de Sitter and Vesto Slipher. Later generations have bestowed the title “father of modern cosmology” primarily on Lemaître or Hubble.13 The debate among historians has focused on the portion of Lemaître’s 1927 paper7 that con- tains his introduction of the Hubble constant. Strangely, that portion was omitted in the 1931 CALTECH English translation.14 Still, the consensus is that the “Hubble” constant was solely Lemaître’s idea. it no attention. On meeting Lemaître in 1927, he Although Lemaître was unaware of Fried- called the idea of an expanding universe “abom- mann’s 1922 and 1924 papers, he appeared on the inable.” But his mind was gradually changed by scene just when the shortcomings of the static solu- growing evidence, most notably Edwin Hubble’s ob- tions A and B were becoming clear in the light of the servations of distant galaxies in 1929 and Eddington’s data coming from the new 100-inch telescope on 1930 proof that Einstein’s static solution A is unstable, Mount Wilson near Los Angeles. Unlike Friedmann, even with a cosmological constant. Lemaître was in possession of Hubble’s 1926 galaxy- distance data and Eddington’s 1923 book on GR and In 1931 Einstein recognized Friedmann’s 15 achievement and suggested that his old nemesis, the cosmology. That book brought Lemaître’s atten- tion to the spectral redshifts of 41 spiral galaxies cosmological constant, be expunged from GR. measured by Slipher (see figure 4). Einstein and de Sitter soon wrote a paper8 promot- Plotting the line-of-sight velocity component de- ing a flat cosmos that is just a limiting case of the duced from each galaxy’s redshift against Hubble’s Friedmann scenarios. Modern observations have as estimate of its distance d, Lemaître postulated that yet found no evidence of departure from Euclidean they were proportional to each other, and he found flatness on cosmological scales. And indeed such a best fit for the proportionality constant H. His flatness is preferred by today’s widely accepted in- major contribution was to connect H to the evolving flationary Big Bang scenario. But finer observations nonstatic cosmic radius via rH = dr/dt. Thus, meas- might eventually reveal either the positive or nega- uring the Hubble constant yields an estimate of the tive curvature Friedmann put forward. .7 Certainly a great achievement, Friedmann’s 1922 and 1924 papers, sadly often but not, I would argue, meriting the paternity of Big misquoted, have become part of the established his- Bang cosmology. torical narrative of the Big Bang and the accelerating In fact, Lemaître missed the Big Bang solution in universe. In his 1923 book, he suggests measuring his 1927 paper. Having rediscovered the Friedmann cosmic curvature by triangulation of distant objects equations, he failed to consider all classes of solu- like Andromeda. But the book remains largely un- tions. Instead, he considered only the limiting case known, despite a recent German translation.9 discussed above, in which C(x) has a double positive root. He identified that root with r , a finite initial Lemaître and Hubble i cosmic radius like that of Friedmann’s M2 scenario. The years between Friedmann’s seminal papers But unlike the M2 scenario, Lemaître’s solution re- and the crowning revelation of accelerating cos- quired a critical value of λ specified by the total mic expansion in 1998 witnessed two other mass of the cosmos. groundbreaking achievements essential to the Lemaître stuck with the finite-initial-radius story: the discoveries of the Hubble constant and limiting case for years after Einstein (shown with . Hubble’s 1926 estimates of distances Lemaître in figure 5) introduced him to Fried- to distant galaxies10 led Lemaître to formulate the mann’s papers in 1927. Only in 1931 did he begin Hubble constant H the following year.7 In 1931, to consider the Big Bang scenario.11 So it’s puzzling Lemaître first gave Friedmann’s singularity a that historians Harry Nussbaumer and Lydia Bieri

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Downloaded 17 Feb 2013 to 136.186.1.81. Redistribution subject to AIP license or copyright; see http://www.physicstoday.org/about_us/terms recently concluded that “Lemaître owes nothing and misrepresentation of his contributions to to Friedmann.”5 Indeed, “nothing” except for the modern cosmology. idea that the cosmological constant is a fully inde- It’s clear that in adumbrating Big Bang cosmol- pendent parameter and that the universe was born ogy, Friedmann went much further than his prede- in a singularity. cessors or early successors like Lemaître. He liked Ironically, the idea of an initial singularity was to quote Dante’s line: “L’acqua ch’io prendo già mai subverted for decades by early attempts to measure non si corse” (The sea I am entering has never yet H. Greatly underestimating the distances to remote been crossed). His approach, the first correct appli- galaxies, Hubble understated the age of the uni- cation of GR to cosmology, introduced the idea of an verse by an order of magnitude. Einstein, in his last expanding universe, possibly born from a singular- years, despaired of finding a way out of the paradox ity. Moreover, realizing that GR admits a variety of of a cosmological age of less than 2 billion years and cosmological metrics, Friedmann first alerted physi- a geological age that exceeded 4 billion! Only after cists to the possibility that the cosmos might be neg- Einstein’s death in 1955 did the magnitude of atively curved and infinite in size. Hubble’s error become clear. Still, after the 1930s, Lemaître received almost all the credit for the Big Bang theory. But the voices Confirmation and legacy of Russian speaking out on behalf of The early favorite among Friedmann’s three princi- Friedmann’s achievements were ultimately heard. pal scenarios was the periodic world (P in figure 3). One of them, , wrote that It allowed multiple cosmic births and deaths— Friedmann published his works in reminiscent of Greek and Asian philosophies of 1922–1924, a time of great hardships. reincarnation. But by the early 1990s, cosmologists In the issue of the 1922 journal that generally assumed that the cosmos was flat, and carried Friedmann’s paper, there was an that its expansion rate was asymptotically slowing appeal to German scientists to donate to zero, with no cosmological constant to resist the scientific literature to their Soviet pull of . So the 1998 results that revealed an colleagues, who were separated from it accelerating expansion came as a great surprise, during the revolution and the war. deemed worthy of the 2011 Nobel Prize. Friedmann’s discovery under those Teams led by the three laureates—Saul conditions was not only a scientific but Perlmutter, Adam Riess, and — also a human feat!16 discovered the acceleration by exploiting distant type 1a supernovae as standard candles. But the 1998 I thank Larry Horwitz, Alexei Kojevnikov, Zinovy results, by themselves, could not discriminate be- Reichstein, Robert Schmidt, Reinhold Bien, Dierck Lieb- tween Friedmann’s M1 and M2 monotonic worlds. scher, Leos Ondra, Evgeny Shapiro, and Eduardo Vila The litmus test invokes Friedmann’s formula Echagüe for helpful discussions. for the cosmic radius r at the inflection point in M1. f References In modern notation, it reads 1. A. Friedmann, Z. Phys. 10, 377 (1922). 1/3 21 rf = r0 (ΩM/2ΩΛ) , (5) 2. A. Friedmann, Z. Phys. , 326 (1924). 3. R. Swedish Acad. Sci., “The Accelerating Universe” where ΩM and ΩΛ are the present mean cosmic en- (2011), available at http://www.nobelprize.org/ ergy due, respectively, to matter and λ, nobel_prizes/physics/laureates/2011/advanced both normalized to the critical total energy density -physicsprize2011.pdf. required by inflationary cosmology. With the ap- 4. Biographical details can be found in E. A. Tropp, proximate values—ΩM = 0.3, ΩΛ = 0.7, t0 = 14 billion V. Y. Frenkel, A. D. Chernin, Alexander A. Friedmann: years—determined by a reassuring convergence of The Man Who Made the Universe Expand, U. cosmological data, equation 5 says that the inflec- Press, London (1993). tion from deceleration to acceleration should have 5. References to general-relativity papers not individu- ally cited can be found in H. Nussbaumer, L. Bieri, happened about 5.6 billion years ago. Only in 2004 Discovering the Expanding Universe, Cambridge U. was the issue resolved, when Riess and coworkers Press, New York (2009). confirmed the M1 prediction by measuring 6. W. de Sitter, Mon. Not. R. Astron. Soc. 78, 3 (1917). ultrahigh-redshift supernovae from the epoch of 7. G. Lemaître, Ann. Soc. Sci. Brux. 47A, 49 (1927). 18 cosmic deceleration (see PHYSICS TODAY, June 8. A. Einstein, W. de Sitter, Proc. Natl. Acad. Sci. USA , 2004, page 19). 213 (1932). There has been a tendency to present Fried- 9. A. Friedmann, Mir kak prostranstvo i vremia, Academia, Petrograd (1923); German trans., Die Welt als Raum mann as merely a mathematician, unconcerned und Zeit, G. Singer, trans., Harri Deutsch, Frankfurt 13 with the physical implications of his discovery. (2006). But such a view is belied even by Friedmann’s con- 10. E. Hubble, Astrophys. J. 64, 321 (1926). siderable achievements in meteorology and aero- 11. G. Lemaître, Nature 127, 706 (1931). dynamics. The wide spectrum of the problems he 12. A. Einstein, The Meaning of Relativity, 5th ed., 16 Princeton U. Press, Princeton, NJ (1951). solved, as seen in his collected works, leaves no 41 doubt that he cared about verification of his theo- 13. H. Kragh, R. W. Smith, Hist. Sci. , 141 (2003). 14. M. Livio, Nature 479, 171 (2011). ries. His death at age 37 prevented him from see- 15. A. S. Eddington, The Mathematical Theory of Relativity, ing any of the observational triumphs of his pio- Cambridge U. Press, London (1923). neering cosmological ideas. I contend that his 16. A. A. Friedmann, Collected Works, Nauka, Moscow early death has contributed to the undervaluation (1966). ■ www.physicstoday.org October 2012 Physics Today 43

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