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Alexander Friedmann and the origins of modern cosmology

Ari Belenkiy

Citation: Phys. Today 6 (10), 38 (2012); doi: 10.1063/PT.3.1750

View online: http://dx.doi.org/10.1063/PT.3.1750

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Ninety years ago, Russian physicistAlexander Friedmann (1888–1925)demonstrated for the first time thatAlbert Einstein’s general theory ofrelativity (GR) admits nonstatic

solutions. It can, he found, describe a cosmos

that expands, contracts, collapses, and mighteven 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 ofthe Big Bang and the accelerating universe. Buthis achievement initially met with strong re-sistance, and it has since then been widely mis-represented. In this article, I hope to clarifysome persistent confusions regarding Fried-mann’s cosmological theory in the context ofrelated work by contemporaries such asEinstein, Willem de Sitter, Arthur Eddington, andGeorges Lemaître.

Last year’s Nobel Prize in Physics was shared by three cosmological observers who discoveredthat the cosmic expansion is currently accelerating(see PHYSICS TODAY , December 2011, page 14). Thus,one of the scenarios introduced by Friedmann in1922 and 1924 appears to correspond to reality.1,2

The Royal Swedish Academy of Sciences back-ground essay for the 2011 prize cites Friedmann’s

papers.3 Regrettably, however, it significantly dis-torts his contributions.

Already in 1922, Friedmann had set the appro-priate framework for a GR cosmology by introduc-ing its most general metric and the “Friedmannequations,” which describe the evolution of a per-fect-fluid cosmos of uniform mass density ρ. And heelucidated all three major scenarios for a nonstaticuniverse consistent with GR. In fact, he introducedthe expression “expanding universe” and estimatedthe period of an alternative periodic universe that’s

and the origins of moderncosmology

Ari BelenkiyFriedmann, who died young in

1925, deserves to be called the

father of Big Bang cosmology. But

his seminal contributions have

been widely misrepresented

and undervalued.

Ari Belenkiy has taught mathematics at Bar Ilan Universityin Ramat-Gan, Israel and at the British Columbia Institute of

Technology’s School of Business in Burnaby, British Columbia,Canada. He lives in Richmond, British Columbia.

Figure 1. Alexander Friedmann in Petrograd, Soviet Union, in the early 1920s.

Alexander Friedmann

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surprisingly close to what is believed now to be thetime elapsed since the Big Bang. In 1924 he furtherrevolutionized the discourse by presenting the ideaof an infinite universe, static or nonstatic, with aconstant negative curvature.

A short lifeBorn and raised in Saint Petersburg, Friedmannstudied mathematics at the city’s university.4 Therehe attended the physics seminars of Vienna-bornPaul Ehrenfest, who had moved to St. Petersburgwith his Russian wife in 1907. After Friedmanngraduated in 1910, he worked primarily in mathe-matical physics applied to meteorology and aero-dynamics.

Following the outbreak of World War I in Au-gust 1914, Friedmann served with the Russian airforce on the Austrian front as a ballistics instructor.He took part in several air-reconnaissance flightsand was awarded the military cross. After the Feb-ruary 1917 revolution that deposed the czar, dozensof new universities were established across Russiaand Friedmann obtained his first professorship, inPerm near the Ural Mountains.

At the end of the civil war that secured theBolshevik regime in 1920, Friedmann (pictured infigure 1) returned to his hometown, renamedPetrograd, and started working as a physicist at theGeophysical Observatory. He soon became the ob-servatory’s director. Most of his personal researchwas oriented toward theories of turbulence andaerodynamics. But in parallel, he also worked on GRand quantum theory. A month before his untimelydeath from typhus in September 1925, Friedmann

made a risky record-breaking balloon flight to col-lect high-altitude data.4

Einstein’s 1905 special theory of relativity waswell known in Russia. But awareness of Einstein’s1915 paper introducing GR was delayed because ofthe world war.5 But soon after the war, news of thetheory and of Eddington’s confirmatory 1919 solar-eclipse observations caused tremendous excitementamong scientists and the general public throughout

Russia. And in 1921, the resumed shipment ofEuropean scientific publications provided scientistsin Petrograd with access to the literature. Further-more, physicist Vsevolod Frederiks brought insiderinformation. Interned in Germany during the war,he worked at Göttingen as an assistant to DavidHilbert, who independently proposed the GR equa-tions early in 1916, not long after Einstein.

In collaboration with Frederiks, Friedmannwrote a mathematical introduction to GR. The firstvolume, devoted to tensor calculus, appeared in1924. Another book, The World as Space and Time ,written by Friedmann alone the year before, devel-oped his philosophical interpretation of GR. But his

fame rests on two Zeitschrift für Physik papers.1,2 Inthem he introduced the fundamental idea of mod-ern cosmology—that the geometry of the cosmosmight be evolving, perhaps even from a singularity.

General relativity before FriedmannThe fundamental Einstein field equations of GR are

Rµν − gµνR/2 − λgµν = − kT µν , (1)

where the spacetime indices µ,ν run from 1 to 4 andthe constant k = 8πG/c2. The spacetime distribution ofthe energy–momentum tensor T µν determines thelocal geometry encoded in the metric tensor gµν , andthe “Ricci tensor” R

µν

is determined by gµν

and itsspacetime derivatives. The Ricci scalar R , a contrac-tion of the Ricci tensor, is the actual local curvature ofspacetime. The cosmological constant λ was intro-duced in 1917 by Einstein in the hope of finding a sta-

ble, static cosmological solution of the field equations.To seek a cosmological solution for a homoge-

neous universe approximated by a perfect fluid ofuniform density ρ and pressure P , one takesT 11 = T 22 = T 33 = −P and T 44 = c2ρ. All off-diagonal ele-ments are zero. For simplicity, Einstein consideredthe cosmological approximation P = 0.

Finding cosmological solutions of the fieldequations required great ingenuity. Before 1922 onlytwo simple solutions were discovered, one byEinstein and the other by de Sitter. The so-called“solution A,” found by Einstein in 1917, representeda finite three-dimensional spatial hyperspheric sur-face of constant radius r embedded in 4D spacetime.

Solution A couples the initially independentparameters λ and ρ to the fixed cosmic radius r. Itrequires that λ = c2/r2 and ρ = 2/(kr2). In 1917, de Sit-ter invoked the second relation to arrive at r = 8 × 1 08

light-years by estimating the mean cosmic massdensity to be about 2 × 10−27 g/cm3.

So it seemed for a moment that Einstein hadachieved his goal of finding a finite, static universewhose size is straightforwardly determined by itsmass density. But de Sitter then found another

C x( )

x

Figure 2. Friedmann’s fundamental cubic C ( x ),given by equation 4 in the text, is plotted hereagainst cosmic radius x for arbitrary values of thefree cosmological parameters λ and A. Their truevalues determine whether C ( x ) has 0 or 2 positiveroots, and they define three general scenarios of cosmic evolution. One gets additional limiting-case scenarios if the two positive roots are degen-

erate. Changing the sign of the linear term in C ( x )to describe a cosmos with negative curvatureyields still more scenarios.

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solution that hit Einstein like a cold shower.6 De Sit-ter’s “solution B” presented a different sort of staticuniverse with zero mass density and negative spa-tial curvature. In Einstein’s solution A, all points inspace are equivalent. But de Sitter’s space has aunique center. Only light rays passing through thatcenter travel along geodesics.

Einstein found de Sitter’s solution unacceptable because it violated philosopher Ernst Mach’s dictum

that inertia cannot exist without matter. But the so-lution had apparent virtues. It seemed to surroundevery observer with a kind of horizon that might ex-plain the redshifts in the spectra of distant galaxiesthat astronomers had been reporting since 1912.Furthermore, de Sitter, Eddington, and his studentLemaître looked to solution B as a way of testing GR.

Friedmann’s universeFriedmann’s 1922 paper1 cited the original papers byEinstein and de Sitter, as well as Eddington’s 1920

book, Space, Time and Gravitation , available to him ina French edition. Instead of taking sides betweenEinstein and de Sitter, Friedmann approached theproblem of a cosmological solution from a widerviewpoint.

His interpretation of GR shows strong ground-ing in Riemannian geometry. Friedmann continuallyreminded his readers of the need to distinguish

between intrinsic features of spacetime, such as themetric, and purely mathematical artifacts like thechoice of a particular coordinate representation.

The physical requirement of spatial homogene-ity, he asserted, did not necessitate a static universe.Focusing on the most general form of the GR metricfor a homogeneous and isotropic cosmos, Fried-mann found, in addition to the static solutions A andB, a new class of nonstatic solutions of the GR field

equations.1 Like Einstein’s solution A, Friedmann’ssolutions feature space as a 3D hypersphere. But itscurvature changes in time with the hypersphere’sradius r(t). Now the field equations lead to a set oftwo ordinary differential equations for r(t).

The first-order Friedmann differential equation

(r/c2)(dr/dt)2 = A − r + λr3/3c2 (2)

governs the dynamics of the universe. (Nowadays

r(t) is regarded as an arbitrary scale length in apresumably infinite universe.) Friedmann foundthat the integration constant A equals kρr3/3. Thusit’s proportional to the constant total mass of thecosmos.

The rest of his 1922 paper is dedicated to ana-lyzing the evolutionary implications of equation 2,which after integration over the cosmic radius

becomes

(3)

If one takes r0 to be the present value, then t0 desig-nates, in Friedmann’s words, “the time that haspassed since Creation.”

Three cosmic scenariosThe right side of equation 3 has physical meaningonly when the cubic denominator

C(x) = A − x + λx3/3c2 (4)

under the square-root sign is positive (see figure 2).That requirement defines three different scenariosfor cosmic evolution:‣ One gets the first scenario if C(x) has no positiveroots and thus is positive for all positive x. That hap-pens when λ > 4c2/9 A2 , that is, when the cosmologi-cal constant exceeds some critical value that de-pends on ρ. In that case, the cosmos starts at t = 0from the singularity r = 0, and its expansion ratechanges from deceleration to acceleration at an in-flection point tf , at which rf = (3c2 A/2λ)1/3. After that,r grows asymptotically like e t √― λ/3. Friedmann calledthis scenario “the monotonic world of the first kind”(see the curve labeled M1 in figure 3).‣ The second situation occurs when 0 < λ < 4c2/9 A2.In that case, C(x) has two positive roots, x1 < x2 , andis negative between them. This condition admitstwo different scenarios, 2a and 2b. In 2a, expansionoscillates between r = 0 and r = x1. That gives the pe-riodic solution discussed below. In the 2b scenario,expansion starts from a nonzero radius, ri= x2 , andexpands forever with accelerating rate. Friedmanncalled it the monotonic world of the second kind(curve M2 in figure 3).‣ Friedmann called the third scenario “the peri-odic world” (curve P in figure 3). It results eitherfrom 2a above or from λ ≤ 0. In either case, C(x) hasonly one positive root, x1 , and its interval of positiv-ity is from 0 to x1. The cosmos starts from the singu-larity r = 0, expands at a decelerating rate to maxi-mum radius x1 , and then begins contracting backdown to zero. The life of the cosmos is finite, endingin a Big Crunch. Assuming a total mass of 5 × 1021

t dx t= +0

.1

λ

x

c

3c 2 A x x− + 3 ∫ √

r0

r

40 October 2012 Physics Today www.physicstoday.org

Alexander Friedmann

C O S M I C R A D I U S

( ) r

t

TIME t

tf

ri

t0

P

M1

M2

Figure 3. Three possible scenarios for cosmic evolution proposedby Alexander Friedmann1 are shown as temporal plots of the cosmicradius. In his first monotonic scenario, M1, the cosmos expands at adecelerating rate from a zero-radius singularity until an inflectionpoint at t f , after which the expansion accelerates. That curve, with t 0

indicating the present, looks much like what observations havebeen revealing in recent decades. The second monotonic scenario,M2, shows ever-accelerating expansion from a nonzero initial

radius. The periodic scenario P shows evolution from and back tozero radius.

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solar masses, Friedmann found a lifetime (roughlyπA/c) of 1010 years for his periodic world.

In addition to the three principal scenarios,Friedmann also considers two special limiting cases,when λ has precisely the critical value λc = 4c2/9 A2.Then C(x) is degenerate; it has a double positive rootat x = 3 A/2. In one limiting case, the periodic world’sexpansion period becomes infinitely long, asymp-totically approaching, from below, the static radius

of Einstein’s solution A. In the other, Friedmann’sM2 world at λc requires an infinitely long past to riseasymptotically from Einstein’s static radius, whichis its ri. Whereas these limiting cases, like Einstein’ssolution, bind λ to ρ , in the general Friedmann sce-narios they are independent free parameters.

Einstein’s reactionWhen Friedmann’s 1922 paper first appeared, itsmain ideas were mostly ignored or rejected. Ein-stein’s immediate reaction illustrates how unwel-come the idea of a nonstatic universe was. In hisview, a proper theory had to uphold the evidentlystatic character of the cosmos.

Therefore Einstein initially found Friedmann’ssolution “suspicious.” In September 1922, he pub-lished a short note in the Zeitschrift für Physik sug-gesting that Friedmann’s derivation contained amathematical error. In fact, Einstein had mistakenlyconcluded that Friedmann’s equations, in the ap-proximation that neglects pressure, imply constancyof density and therefore a cosmos of fixed size.

Learning of Einstein’s note, Friedmann wrotehim a long letter elaborating his derivations. ButEinstein was on a world tour, returning to Berlinonly in May 1923. Only then could he have readFriedmann’s letter.4 Later that month, Friedmann’sRussian colleague Yuri Krutkov met Einstein atEhrenfest’s home in Leiden and clarified the confu-sion. So Einstein promptly published another shortnote 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 wasready to accept the idea of the expanding universe.

Friedmann was the first to realize that GR alonecannot 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 ofa finite 3D hyperspheric surface (denoted S3 bytopologists) admits “ghosts,” double images of thesame object in opposite directions on the sky. In1917, de Sitter had pioneered the idea that the spaceof directions must be considered as the basic space,with the opposite directions viewed as one. Fried-mann mentioned that view favorably,2 and Lemaîtrelater applied it to compute the volume of his ownmodel cosmos.7

Infinite worldsFriedmann’s major concern, however, lay with thevery notion of a finite cosmos, which was at thetime firmly entrenched. He insisted that local met-

ric properties alone do not resolve the problem. In-spired by Poincaré’s theory of Riemannian mani-folds, he imagined the possibility of a spherical uni-verse of infinite size. The S³ geometry did not seemto admit an infinite volume. Unabashed, however,Friedmann suggested that the azimuthal coordi-nate ϕ of the spherical cosmos might run not from0 to 2π , but rather wind around over and over toinfinity.

Friedmann found another, even more surpris-ing argument to undermine the notion of a neces-sarily finite cosmos. On advice from his colleagueYakov Tamarkin, he sought to discover whether GRallowed solutions for a hyperboloid of infinite vol-ume whose negative spatial curvature is giveneverywhere by −6/r2. In such a space, every pointwould in effect be a saddle point. And indeed,Friedmann’s 1924 paper gives a positive answer,with both static and nonstatic solutions.2

The static solution, like de Sitter’s solution B, ne-cessitates zero density. The nonstatic scenario, withevolving negative curvature and r(t), has a nonvanish-ing average density whose evolution is indistinguish-

able from that of Friedmann’s positive-curvature solu-tion. So one can’t determine the sign of the cosmiccurvature simply by measuring ρ. The only change inthe fundamental Friedmann equation for the negative-curvature case is that the sign of the linear term inequations 2–4 becomes positive. Friedmann thus clar-ified the meaning of the linear term’s coefficient in C(x):It gives the sign of the cosmic curvature.

The 1924 paper was also ignored. Einstein paid

www.physicstoday.org October 2012 Physics Today 41

Figure 4. A table of line-of-sight (“radial”) velocity components deducedby redshift measurements for 41 spiral galaxies compiled by Vesto Slipherand included in Arthur Eddington’s 1923 book on relativity theory.15 Eachgalaxy is identified by NGC catalog number and celestial coordinates. In1927, Georges Lemaître plotted these velocities against Edwin Hubble’sapproximations of galaxy distances to produce the first estimate of theHubble constant.7

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it no attention. On meeting Lemaître in 1927, hecalled the idea of an expanding universe “abom-inable.” But his mind was gradually changed bygrowing evidence, most notably Edwin Hubble’s ob-servations of distant galaxies in 1929 and Eddington’s1930 proof that Einstein’s static solution A is unstable,even with a cosmological constant.

In 1931 Einstein recognized Friedmann’sachievement and suggested that his old nemesis, the

cosmological constant, be expunged from GR.Einstein and de Sitter soon wrote a paper8 promot-ing a flat cosmos that is just a limiting case of theFriedmann scenarios. Modern observations have asyet found no evidence of departure from Euclideanflatness on cosmological scales. And indeed suchflatness is preferred by today’s widely accepted in-flationary Big Bang scenario. But finer observationsmight eventually reveal either the positive or nega-tive curvature Friedmann put forward.

Friedmann’s 1922 and 1924 papers, sadly oftenmisquoted, have become part of the established his-torical narrative of the Big Bang and the acceleratinguniverse. In his 1923 book, he suggests measuring

cosmic curvature by triangulation of distant objectslike Andromeda. But the book remains largely un-known, despite a recent German translation.9

Lemaître and HubbleThe years between Friedmann’s seminal papersand the crowning revelation of accelerating cos-mic expansion in 1998 witnessed two othergroundbreaking achievements essential to thestory: the discoveries of the Hubble constant anddark matter. Hubble’s 1926 estimates of distancesto distant galaxies10 led Lemaître to formulate theHubble constant H the following year.7 In 1931,Lemaître first gave Friedmann’s singularity a

physical meaning,11 that of a “primeval atom” blowing up—what Fred Hoyle later dismissively

called “the Big Bang.”Unable to foresee the 1998 discovery of cos-

mic acceleration, Einstein no longer saw anyneed for a cosmological constant. He thoughtFriedmann’s expanding solution with λ = 0 might

be the answer. Starting with the 1946 edition ofhis popular exposition The Meaning of Relativity ,

Einstein interjected that “the mathematicianFriedmann found a way out of this [cosmologicalconstant] dilemma. His result then found a sur-prising confirmation in Hubble’s discovery of theexpansion of the stellar system. . . . The following[15-page exposition] is essentially nothing but anexposition of Friedmann’s idea.”12 Unfortunately,Einstein attributed to Hubble alone what properly

belongs to several people, among them de Sitterand Vesto Slipher.

Later generations have bestowed the title “fatherof modern cosmology” primarily on Lemaître orHubble.13 The debate among historians has focusedon the portion of Lemaître’s 1927 paper7 that con-tains his introduction of the Hubble constant.Strangely, that portion was omitted in the 1931English translation.14 Still, the consensus is that the“Hubble” constant was solely Lemaître’s idea.

Although Lemaître was unaware of Fried-mann’s 1922 and 1924 papers, he appeared on thescene just when the shortcomings of the static solu-tions A and B were becoming clear in the light of thedata coming from the new 100-inch telescope onMount Wilson near Los Angeles. Unlike Friedmann,Lemaître was in possession of Hubble’s 1926 galaxy-distance data and Eddington’s 1923 book on GR andcosmology.15 That book brought Lemaître’s atten-tion to the spectral redshifts of 41 spiral galaxiesmeasured by Slipher (see figure 4).

Plotting the line-of-sight velocity component de-duced from each galaxy’s redshift against Hubble’sestimate of its distance d , Lemaître postulated thatthey were proportional to each other, and he founda best fit for the proportionality constant H . Hismajor contribution was to connect H to the evolvingnonstatic cosmic radius via rH = dr/dt. Thus, meas-uring the Hubble constant yields an estimate of theage of the universe.7 Certainly a great achievement,

but not, I would argue, meriting the paternity of BigBang cosmology.

In fact, Lemaître missed the Big Bang solution inhis 1927 paper. Having rediscovered the Friedmannequations, he failed to consider all classes of solu-tions. Instead, he considered only the limiting casediscussed above, in which C(x) has a double positiveroot. He identified that root with ri , a finite initialcosmic radius like that of Friedmann’s M2 scenario.But unlike the M2 scenario, Lemaître’s solution re-quired a critical value of λ specified by the totalmass of the cosmos.

Lemaître stuck with the finite-initial-radiuslimiting case for years after Einstein (shown withLemaître in figure 5) introduced him to Fried-mann’s papers in 1927. Only in 1931 did he beginto consider the Big Bang scenario.11 So it’s puzzlingthat historians Harry Nussbaumer and Lydia Bieri

42 October 2012 Physics Today www.physicstoday.org

Alexander Friedmann

Figure 5. At Caltech in 1933 are Robert Millikan (chair of Caltech’sexecutive council), Georges Lemaître, and Albert Einstein. Lemaître,an ordained priest, was a physics professor at the Catholic Universityof Louvain, in Belgium.

C A L T E C

H

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recently concluded that “Lemaître owes nothingto Friedmann.”5 Indeed, “nothing” except for theidea that the cosmological constant is a fully inde-pendent parameter and that the universe was bornin a singularity.

Ironically, the idea of an initial singularity wassubverted for decades by early attempts to measureH . Greatly underestimating the distances to remotegalaxies, Hubble understated the age of the uni-

verse by an order of magnitude. Einstein, in his lastyears, despaired of finding a way out of the paradoxof a cosmological age of less than 2 billion years anda geological age that exceeded 4 billion! Only afterEinstein’s death in 1955 did the magnitude ofHubble’s error become clear.

Confirmation and legacyThe early favorite among Friedmann’s three princi-pal scenarios was the periodic world (P in figure 3).It allowed multiple cosmic births and deaths—reminiscent of Greek and Asian philosophies ofreincarnation. But by the early 1990s, cosmologistsgenerally assumed that the cosmos was flat, andthat its expansion rate was asymptotically slowingto zero, with no cosmological constant to resist thepull of gravity. So the 1998 results that revealed anaccelerating expansion came as a great surprise,deemed worthy of the 2011 Nobel Prize.

Teams led by the three laureates—SaulPerlmutter, Adam Riess, and Brian Schmidt—discovered the acceleration by exploiting distanttype 1a supernovae as standard candles. But the 1998results, by themselves, could not discriminate be-tween Friedmann’s M1 and M2 monotonic worlds.

The litmus test invokes Friedmann’s formulafor the cosmic radius rf at the inflection point in M1.In modern notation, it reads

rf = r0 (ΩM/2ΩΛ)1/3 , (5)

where ΩM and ΩΛ are the present mean cosmic en-ergy densities due, respectively, to matter and λ ,

both normalized to the critical total energy densityrequired by inflationary cosmology. With the ap-proximate values—ΩM = 0.3, ΩΛ = 0.7, t0 = 14 billionyears—determined by a reassuring convergence ofcosmological data, equation 5 says that the inflec-tion from deceleration to acceleration should havehappened about 5.6 billion years ago. Only in 2004was the issue resolved, when Riess and coworkersconfirmed the M1 prediction by measuringultrahigh-redshift supernovae from the epoch ofcosmic deceleration (see PHYSICS TODAY , June2004, page 19).

There has been a tendency to present Fried-mann as merely a mathematician, unconcernedwith the physical implications of his discovery.13

But such a view is belied even by Friedmann’s con-siderable achievements in meteorology and aero-dynamics. The wide spectrum of the problems hesolved, as seen in his collected works,16 leaves nodoubt that he cared about verification of his theo-ries. His death at age 37 prevented him from see-ing any of the observational triumphs of his pio-neering cosmological ideas. I contend that hisearly death has contributed to the undervaluation

and misrepresentation of his contributions tomodern cosmology.

It’s clear that in adumbrating Big Bang cosmol-ogy, Friedmann went much further than his prede-cessors or early successors like Lemaître. He likedto quote Dante’s line: “L’acqua ch’io prendo già mainon si corse” (The sea I am entering has never yet

been crossed). His approach, the first correct appli-cation of GR to cosmology, introduced the idea of an

expanding universe, possibly born from a singular-ity. Moreover, realizing that GR admits a variety ofcosmological metrics, Friedmann first alerted physi-cists to the possibility that the cosmos might be neg-atively curved and infinite in size.

Still, after the 1930s, Lemaître received almostall the credit for the Big Bang theory. But the voicesof Russian physicists speaking out on behalf ofFriedmann’s achievements were ultimately heard.One of them, Yakov Zeldovich, wrote that

Friedmann published his works in1922–1924, a time of great hardships.In the issue of the 1922 journal that

carried Friedmann’s paper, there was anappeal to German scientists to donatescientific literature to their Sovietcolleagues, who were separated from itduring the revolution and the war.Friedmann’s discovery under thoseconditions was not only a scientific butalso a human feat!16

I thank Larry Horwitz, Alexei Kojevnikov, ZinovyReichstein, Robert Schmidt, Reinhold Bien, Dierck Lieb-scher, Leos Ondra, Evgeny Shapiro, and Eduardo VilaEchagüe for helpful discussions.

References

1. A. Friedmann, Z. Phys. 10 , 377 (1922).2. A. Friedmann, Z. Phys. 21 , 326 (1924).3. R. Swedish Acad. Sci., “The Accelerating Universe”

(2011), available at http://www.nobelprize.org/nobel_prizes/physics/laureates/2011/advanced-physicsprize2011.pdf.

4. Biographical details can be found in E. A. Tropp,V. Y. Frenkel, A. D. Chernin, Alexander A. Friedmann:The Man Who Made the Universe Expand , Cambridge U.Press, London (1993).

5. References to general-relativity papers not individu-ally cited can be found in H. Nussbaumer, L. Bieri,Discovering the Expanding Universe, Cambridge U.Press, New York (2009).

6. W. de Sitter, Mon. Not. R. Astron. Soc. 78 , 3 (1917).7. G. Lemaître, Ann. Soc. Sci. Brux. 47A , 49 (1927).

8. A. Einstein, W. de Sitter, Proc. Natl. Acad. Sci. USA 18 ,213 (1932).

9. A. Friedmann, Mir kak prostranstvo i vremia , Academia,Petrograd (1923); German trans., Die Welt als Raumund Zeit , G. Singer, trans., Harri Deutsch, Frankfurt(2006).

10. E. Hubble, Astrophys. J. 64 , 321 (1926).11. G. Lemaître, Nature 127 , 706 (1931).12. A. Einstein, The Meaning of Relativity , 5th ed.,

Princeton U. Press, Princeton, NJ (1951).13. H. Kragh, R. W. Smith, Hist. Sci. 41 , 141 (2003).14. M. Livio, Nature 479 , 171 (2011).15. A. S. Eddington, The Mathematical Theory of Relativity ,

Cambridge U. Press, London (1923).16. A. A. Friedmann, Collected Works , Nauka, Moscow

(1966).

www.physicstoday.org October 2012 Physics Today 43