Cosmology at the Turning Point of Relativity Revolution. the Debates During the 1920’S on the “De Sitter Effect”

UNIVERSITA` DEGLI STUDI DI PADOVA Dipartimento di Astronomia

Scuola di Dottorato di Ricerca in Astronomia XXI Ciclo (2006 - 2008) Tesi di Dottorato

Cosmology at the turning point of relativity revolution. The debates during the 1920’s on the “de Sitter Effect”

Direttore della Scuola: Prof. Giampaolo PIOTTO Dipartimento di Astronomia - Universit`adi Padova Supervisore: Prof. Giulio PERUZZI Dipartimento di Fisica - Universit`adi Padova Co-supervisore: Prof. Luigi SECCO Dipartimento di Astronomia - Universit`adi Padova

Dottorando: Matteo REALDI

UNIVERSITY OF PADOVA Department of Astronomy

Ph.D. in Astronomy XXI Cycle (2006 - 2008) Dissertation Thesis

Cosmology at the turning point of relativity revolution. The debates during the 1920’s on the “de Sitter Effect”

Ph.D. Coordinator: Prof. Giampaolo PIOTTO Department of Astronomy - University of Padova Supervisor: Prof. Giulio PERUZZI Department of Physics - University of Padova Co-supervisor: Prof. Luigi SECCO Department of Astronomy - University of Padova

Candidate: Matteo REALDI

A mia mamma e mio pap`a. A Gi´o,Anna, Luca, Vale, Albe, Matteino, Bettax. Alla zia Gisa e alla nonna Laura. A Giuliano e Marinella.

Soprattutto, a Michela.

La chair est triste, h´elas!Et j’ai lu tous les livres. (Mallarm´e,Brise marine)

Abstract

This thesis is devoted to a critical analysis of the cosmological debates which took place during the 1920’s about the so-called “de Sitter effect”, which represents the linking thread between the 1917 beginning of the- oretical relativistic cosmology and the 1930 first diffusion and general acceptance of the model of the expanding universe. The de Sitter effect is a theoretical redshift-distance relation which can be derived from the cosmological solution of field equations proposed by de Sitter. This solution and the solution proposed by Einstein repre- sent the first theoretical relativistic cosmological models. They appeared in 1917, when stars, not yet galaxies, where considered the fundamental pieces filling the universe, and the expanding universe still had to enter modern cosmology. During the 1920’s it was just the de Sitter effect which played a fundamental role in the first pioneering attempts to re- late the theoretical relativistic description of the universe to astronomical observations. The models of the universe proposed by Einstein and de Sitter, both based on general relativity, soon appeared as revolutionary tools in order to investigate the properties of the universe as a whole and the connec- tion among space, time and gravitation. In his own spherical model, Einstein proposed a static, finite and unbounded universe. In this uni- verse, according to Machian inspiration, inertia was fully determined by all masses. Dealing with the universe as a whole and its properties, Einstein took into account a hypothetical density of matter which was uniformly and homogeneously distributed through space, foreshadowing

VII VIII Abstract what became later known as the Cosmological Principle. Einstein modi- fied his field equations and introduced a new term with the cosmological constant λ, which in Einstein’s intentions acted like an anti-gravity, in order to express in general relativity his model of a static universe. On the contrary, de Sitter found a suitable solution of field equations which corresponded to a completely empty and static world. In de Sit- ter’s static universe, a spectral displacement was expected from a mass test for the form of the metric and the geodesic equations. This property of de Sitter’s universe became known as the de Sitter effect. Already in 1917 de Sitter related spectral shifts to velocity and distance of as- tronomical objects through his own relativistic solution. He proposed that spectral displacements which were observed in some stars and neb- ulæ could be interpreted in his static and empty world as an apparent (spurious) velocity of test particles due to the peculiar line element, su- perimposed to a relative (Doppler) velocity which resulted from geodesic equations. The first contribution led to a quadratic redshift-distance (or equivalently velocity-distance) relation, while the latter involved a linear dependence. This first suggestion did not pass unnoticed, and during the 1920’s several scientists dealt with the properties of de Sitter’s universe and pro- posed different formulations of the redshift-distance effect which resulted by the metric of such a model. Despite its lack of matter, de Sitter’s universe attracted the attention of scientists because it offered more ad- vantages than Einstein’s one with regard to astronomical consequences and observations. According to Eddington, a general cosmic recession was expected in de Sitter’s universe just because of the presence of the cosmological con- stant. Such a tendency of particles to scatter, which Eddington pro- posed in 1923, could roughly account for the astonishing radial velocities measured by Slipher in spiral nebulæ, the most part of which revealed receding motions from the observer. Moreover, the geometry of de Sitter’s world-model was not uniquely Abstract IX determined, and non-static pictures emerged by appropriate coordinate changes, as done in the 1920’s by Weyl, Lanczos, Lemaˆıtreand Robert- son. Their contributions marked the actual departure from the metric of a static universe, by using a stationary frame of de Sitter’s model. All of them took into account the empirical evidence of relevant veloci- ties in spiral nebulæ, and each of them proposed an own version of the redshift-distance relation in de Sitter’s world. In 1924 Wirtz realized that the universe of de Sitter represented a suitable model accounting for redshift and apparent diameter measured in spiral nebulæ. In the same year, on the contrary, Silberstein criticized the possibility of a general cosmic recession, and considered the distances of globular clusters in order to verify the de Sitter effect. However, the correctness of the method and the results proposed by Silberstein were shortly after denied by Lundmark and Str¨omberg. The de Sitter effect could offer an answer to the question of relevant redshift measurements in nebulæ, however there was an ambiguous for- mulation of such a theoretical relation between velocities and distances. Nevertheless, up to 1930 such an effect was the only possible, however puzzling, explanation of the redshift problem. A suitable test of redshift relations was possible only with reliable determinations of distance of spiral nebulæ. In this controversial picture, the contributions of Hubble marked a turning point in the comprehension of the structure of the universe. Thanks to the revolutionary observations which Hubble furnished dur- ing the 1920’s, spiral nebulæ were finally accepted in 1925 as ‘island universes’, i.e. as true extra-galactic stellar systems. In 1929 Hubble confirmed that such systems receded relatively to one another, and that their radial velocities linearly increased with distances. The puzzling question of the interpretation of the de Sitter effect and the meaning of redshift was solved in 1930, when the cosmology of Lemaˆıtrewas reconsidered in order to explain the cosmic recession of galaxies revealed by Hubble. The model of a non-empty expanding uni- X Abstract verse which Lemaˆıtrehad already proposed in 1927 provided the proper cosmological interpretation of redshift: the displacement of spectral lines was due to the expansion of the universe. The cosmological solution of Lemaˆıtrecorresponded to a “third way” between the Einstein’s model, which had matter but not motion, and the de Sitter’s model, which had motion but not matter. Since 1930, also the cosmological consequences of similar solutions proposed by Friedmann in 1922 and 1924 were fully acknowledged. In their analysis, Friedmann and Lemaˆıtre took into account dynamical models of the universe, i.e. they considered the possibility of a not empty, homogeneous and isotropic universe which world-radius increased in time. Static and stationary models were eventually seen as limiting cases of solutions of field equations describing an expanding universe. As from 1930, the de Sitter effect, which during the 1920’s represented the first hint in the intersection between the new theory of gravitation and observed facts, was seen as an effect of minor importance, and the expanding universe inaugurated another chapter of modern cosmology. The historical analysis which is below proposed is useful to highlight the richness of contributions, attempts and controversies which appeared in the early connections between astronomical observations and predic- tions offered by relativistic cosmology. In particular, scientists involved in the 1920’s debates about the de Sitter effect approached and thoroughly analyzed some fundamental questions in the framework of relativistic cosmology, such as the nature of redshift measurements, the geometry of space, the assumption of a homogeneous and isotropic universe. Several passages from primary literature, original manuscripts, un- published sources and correspondence among scientists have often been quoted, in order to highlight the very contributions by actors involved in those debates. From the present thesis it emerges the fundamental role played by the de Sitter effect in the 1920’s debates. It was a very fruitful phase for the introduction of new ideas, discoveries and changes, and the history Abstract XI of that period permits to understand how cosmology developed passing from a sphere of theoretical speculations to a truly empirical science.

Riassunto

Ci si riferisce alla cosmologia moderna, o scientifica, come alla scienza che studia l’origine e l’evoluzione dell’universo, interpretando il quadro che ne risulta sulla base delle leggi della fisica. In questa tesi viene proposta una ricostruzione storica del ruolo svolto dal cosiddetto “effetto de Sitter” durante le prime fasi della cosmologia moderna. Vengono ricostruiti in particolare i dibattiti cosmologici cen- trati su tale effetto che ebbero luogo negli anni Venti, prima cio`eche il concetto di universo in espansione facesse la sua comparsa ufficiale nella storia della cosmologia moderna. Lo studio `ebasato prima di tutto sulla documentazione originale dell’epoca, sulle corrispondenze tra scien- ziati e su manoscritti inediti. L’indagine storica permette di evidenziare l’importanza fondamentale che l’effetto de Sitter ebbe in quei dibattiti. Proprio attorno a tale effetto, infatti, ruotarono i primi confronti tra le previsioni teoriche relativistiche e le osservazioni su scala cosmologica, fa- vorendo quindi il passaggio della cosmologia da pura discussione teorica a scienza empirica. L’effetto de Sitter si riferisce ad una relazione teorica tra redshift e dis- tanza, e deriva dalla scelta della metrica e dalle equazioni delle geodetiche nell’universo vuoto proposto da de Sitter. Questo modello di universo e il modello di Einstein rappresentano i primi due approcci teorici relativis- tici al problema cosmologico, e furono entrambi proposti nel 1917. Tali modelli sono, appunto, basati sulla relativit`agenerale, la cui diffusione implic`ouna svolta nella comprensione dell’intreccio tra spazio, tempo e gravitazione e, in particolare, nello studio scientifico dell’universo come

XIII XIV Riassunto un tutto. All’inizio del secolo scorso si considerava ancora l’universo come statico e costituito essenzialmente da stelle e nebulæ. Bisogner`aattendere la met`adegli anni Venti perch´evenga accettata l’idea di un universo cos- tituito da galassie, ritenute da allora i “mattoni” fondamentali nella de- scrizione del cosmo. Nel 1917 Einstein propose un modello di universo sferico in cui la ma- teria era distribuita in maniera uniforme ed omogenea. La gravitazione veniva interpretata come curvatura dello spazio e la materia era intera- mente responsabile dell’origine dell’inerzia, soddisfacendo quello che Ein- stein introdusse pi`utardi come “principio di Mach”. Per rendere coerenti i risultati della relativit`agenerale con la supposta staticit`adell’universo, Einstein aveva introdotto nelle equazioni di campo un termine aggiuntivo contenente la costante cosmologica λ. Questa nuova costante era inter- pretabile come una sorta di repulsione cosmica, capace di controbilanciare a grandi distanze l’effetto della gravit`ae di rendere statico l’universo. Sempre nel 1917 apparve, ad opera di de Sitter, un altro lavoro sulla cosmologia, impostato anch’esso sulla teoria della relativit`agenerale. De Sitter mostrava che se la densit`amedia della materia nell’universo poteva essere considerata nulla, la metrica dello spazio-tempo che ne risultava comportava un mondo statico con condizioni fisiche coerenti. A partire dal 1917 e nel corso degli anni Venti la discussione cosmo- logica fu centrata essenzialmente su quale di questi due modelli potesse rappresentare al meglio il cosmo, se il modello statico di Einstein o quello “vuoto” di de Sitter. Tra i due modelli, quello di de Sitter forniva la pos- sibilit`adi interpretare gli spostamenti osservati nelle righe spettrali di stelle e nebulæ. Gi`ade Sitter aveva infatti notato che nel suo modello di universo una particella di prova non poteva rimanere in uno stato di quiete, ma avrebbe mostrato un moto rispetto all’osservatore. Tale re- lazione teorica tra velocit`a(redshift) e distanza divenne in seguito nota come effetto de Sitter. La causa di questo spostamento spettrale predetto dall’effetto de Sitter poteva essere ricondotta sia ad un contributo grav- Riassunto XV itazionale che ad un effettivo moto relativo tra l’osservatore e l’oggetto osservato (effetto Doppler). Numerosi scienziati presero in considerazione le propriet`adell’universo di de Sitter, e proposero differenti interpretazioni e formulazioni dell’effetto de Sitter. Nella sua analisi del 1923 sui modelli cosmologici relativistici, Ed- dington riteneva che dalle propriet`adell’universo di de Sitter si potesse dedurre una recessione cosmica estendibile a tutto l’universo. Questa tendenza delle particelle ad allontanarsi le une dalle altre era dovuta, secondo Eddington, alla presenza della costante cosmologica, e poteva rendere conto delle sorprendenti velocit`aradiali di alcune nebulæ mis- urate proprio in quegli anni da Slipher. La geometria dell’universo di de Sitter non era univocamente deter- minata, e alcuni protagonisti del dibattito cosmologico degli anni Venti proposero modelli stazionari dell’universo di de Sitter. In questo ambito, i contributi di Weyl, Lanczos, Lemaˆıtree Robertson segnarono effettiva- mente il distacco dallo studio di elementi di linea di universi puramente statici. Ognuno di loro tent`oquindi di interpretare l’evidenza osservativa delle elevate velocit`aradiali misurate nelle nebulæ, proponendo versioni differenti della relazione redshift-distanza nell’universo di de Sitter. Nel 1924 Wirtz propose una relazione tra la velocit`adi allontana- mento e il diametro apparente misurato nelle nebulæ. Secondo Wirtz, dunque, l’universo di de Sitter risultava idoneo per spiegare una reces- sione cosmica suggerita dalle osservazioni. Nello stesso anno, al con- trario, Silberstein critic`otale tendenza a recedere, e tent`odi confermare l’effetto de Sitter attraverso le distanze degli ammassi globulari. Di l´ıa poco, Lundmark e Str¨omberg mostrarono che il metodo e i risultati di Silberstein non erano corretti. Le controversie legate all’effetto de Sitter e all’interpretazione del red- shift osservato nelle nebulæ potevano trovare una soluzione solo tramite una corretta stima delle distanze di tali oggetti. A tal proposito i con- tributi di Hubble segnarono una svolta nella comprensione del cosmo. XVI Riassunto

Grazie alle fondamentali osservazioni di Hubble, a partire dal 1925 le nebulose a spirale vennero finalmente considerate come veri e propri sis- temi extragalattici del tutto simili alla nostra galassia, la Via Lattea. Nel 1929, inoltre, Hubble conferm`oche esisteva una recessione cosmica, e che ogni galassia si allontanava dalle altre con una dipendenza lineare tra la velocit`ae la distanza. Nel 1930 l’enigma legato all’effettiva causa del redshift trov`ofinal- mente una soluzione. In quell’anno, infatti, Eddington e de Sitter col- legarono le osservazioni della recessione cosmica contenuta nelle osser- vazioni di Hubble con le predizioni teoriche di nuovi modelli d’universo basati sulla relativit`agenerale. Essi interpretarono correttamente l’espan- sione del cosmo evidenziata dalle osservazioni rivalutando un modello di universo proposto ancora nel 1927 da Lemaˆıtre,ma passato inosservato. L’idea di Lemaˆıtreera basata sulla formulazione di un modello interme- dio tra quello di Einstein, che conteneva materia ma non movimento, e quello di de Sitter, che era vuoto ma al contrario prediceva un movimento delle particelle di prova. Nella proposta di Lemaˆıtreera contenuta la spiegazione cosmolog- ica del redshift: la causa dello spostamento delle righe spettrali andava ricondotta proprio all’espansione dell’universo. Solo a partire da allora venne interamente apprezzato anche il la- voro svolto gi`anel 1922 e nel 1924 da Friedmann. Questi aveva ottenuto soluzioni coerenti delle equazioni di campo ammettendo una densit`ame- dia di materia non nulla e una dipendenza della metrica dal tempo. Le soluzioni di Friedmann prevedevano un’evoluzione dell’universo indipen- dentemente dalla presenza o meno della costante cosmologica. A partire dal 1930, dunque, il concetto dell’universo in espansione inaugur`oun nuovo capitolo della cosmologia moderna. L’effetto de Sit- ter venne da allora visto come un effetto di importanza secondaria, e l’interesse in questa relazione tra distanza e redshift and`ovia via dimin- uendo. Come evidenziato nella presente tesi, fu proprio l’effetto de Sitter che Riassunto XVII caratterizz`oil tortuoso percorso della cosmologia moderna relativistica dalla concezione di un universo statico a quella di un universo in espan- sione. I dibattiti cosmologici degli anni Venti, come avviene in tutte le fasi di scoperta, furono quanto mai fecondi per la produzione di argomen- tazioni e di idee innovative. I protagonisti di quei dibattiti affrontarono in quegli anni questioni fondamentali quali il significato del redshift, la metrica dell’universo, il confronto tra predizioni teoriche ed evidenze os- servative, inaugurando in questa maniera l’approccio moderno alla com- prensione dell’universo come un tutto. Anche se oggi pochi conoscono quel periodo, `epur certo, tanto pi`ualla luce degli attuali sviluppi, il ruolo fondamentale che questi dibattiti svolsero. La storia di quel periodo `eun significativo esempio di come procede l’impresa scientifica.

Contents

Abstract VII

Riassunto XIII

1 Introduction 1 1.1 Summary ...... 5

2 Some issues of present cosmology 7 2.1 The Cosmological Principle ...... 9 2.2 The Robertson-Walker metric ...... 11 2.3 Redshifts in cosmology ...... 16 2.3.1 Velocities and distances ...... 18 2.4 Dark matter and dark energy ...... 22

3 Cosmology at the beginning of XX Century 27 3.1 The sidereal universe and the nebulæ ...... 27 3.2 Cosmological difficulties with Newtonian theory . . . . . 35 3.2.1 Olbers’s paradox ...... 37

4 1917: the universes of general relativity 39 4.1 Einstein, the universe and the relativity of inertia . . . . 39 4.1.1 The debate with Willem de Sitter ...... 42 4.1.2 Towards the solution ...... 49 4.1.3 “A ‘finite’ and yet ‘unbounded’ universe” . . . . . 52 4.1.4 The cosmological constant ...... 55

XIX XX CONTENTS

4.2 The universe of de Sitter ...... 58 4.2.1 The “mathematical postulate of relativity of inertia” 59 4.2.2 A universe without “world matter” ...... 61 4.2.3 Einstein’s criticism ...... 63

5 The “de Sitter Effect” 71 5.1 De Sitter’s first suggestion ...... 74 5.1.1 Redshifts in de Sitter’s universe ...... 79 5.2 Matter or motion? Eddington’s analysis ...... 90 5.3 Weyl, Lanczos and the redshift-distance law ...... 97 5.4 Silberstein’s contributions ...... 105 5.5 Lemaˆıtre’s1925 notes ...... 112 5.6 Shifts in de Sitter’s universe according to Robertson and Tolman ...... 117

6 Observational investigations of redshift relations 129 6.1 The nature of the nebulæ around 1920 ...... 129 6.2 Slipher and the radial velocities of spirals ...... 134 6.3 The K term and the solar motion ...... 136 6.3.1 Wirtz and de Sitter’s cosmology ...... 139 6.4 Astronomers at work: Lundmark and Str¨omberg . . . . . 142 6.5 Hubble and the universe of galaxies ...... 150 6.5.1 The contributions by Humason ...... 153 6.5.2 Hubble’s 1929 relation ...... 154

7 The “third way” between Einstein’s and de Sitter’s solu- tions 159 7.1 1930: Eddington, de Sitter and the expanding universe . 160 7.2 The importance of Lemaˆıtre’s1927 proposal ...... 163 7.2.1 Rediscovering the models of Friedmann ...... 169 7.3 The decline of the interest in the de Sitter effect . . . . . 172

Conclusion 177 CONTENTS XXI

Bibliography 183 Primary literature ...... 183 Secondary literature ...... 201

Acknowledgements 211

Chapter 1

Introduction

Cosmology is the study of the origin, structure and evolution of the universe as a whole, based on interpretations of astronomical observations at different wave-lengths through laws of physics. This thesis is devoted to a historical reconstruction of some debates which happened less than a century ago, during the early developments of modern scientific cosmology. Emphasis is placed on the first descrip- tions of the universe as a whole at the turning point of general relativity revolution, focusing on the pioneering attempts to relate theoretical pre- dictions of relativistic cosmology with astronomical data. Particular attention is given to the debates which took place during the 1920’s about the so-called “de Sitter effect”, a redshift-distance rela- tion which resulted from the model of the universe proposed by Willem de Sitter (1872-1934). This cosmological solution and the solution pro- posed by Albert Einstein (1879-1955), both based on relativistic theory of gravitation, represent the first models of modern cosmology. They were proposed by Einstein and de Sitter in 1917, when stars, and not yet galaxies, were considered the fundamental pieces filling the universe and the concept of the expanding universe still had to enter the history of modern cosmology1.

1The formulation of the first two relativistic models of the universe represents a milestone in modern cosmology. Its history has been faced by several authors; see for

1 2 Introduction

These models inaugurated a new phase in cosmological researches: they marked a revolutionary step beyond Newtonian cosmology in the investigation of connection among space, time and gravitation. The the- ory of general relativity soon appeared as a suitable tool to study the physics and the geometry of the universe. Since 1917, when both Einstein and de Sitter proposed a spatially finite world2, appropriate theoretical extrapolations of laws of physics and gravitational properties of matter to the universe as a whole entered cosmology just through “the remark- able rationality and inner logicality of the theory of general relativity” [Tolman 1934, p. 331]. It was during the 1920’s that the two alternative solutions proposed by Einstein and de Sitter were related to observational cosmology. In this perspective, the history of the de Sitter effect and its related de- bates represents the linking thread in order to understand how cosmol- ogy developed from a sphere of theoretical speculations into an empirical science. In Einstein’s spherical world, according to Machian inspiration, the whole of matter was responsible for the origin of inertia. Einstein intro- duced in his field equations a new term, with the cosmological constant λ, example the Editorial Note about the Einstein - de Sitter - Weyl - Klein debate in the Collected Papers of Albert Einstein [CPAE 1998 Editorial], and also [Kerszberg 1989, North 1965, Ellis 1989 ]. The author of present thesis dealt with this topic in his Laurea degree thesis about the renewal of cosmology in XX Century (“La rinascita della cosmologia nel XX secolo”, in Italian; University of Padova) and subsequently in [Realdi-Peruzzi 2009 ]. 2As we shall see in Chapter 4, both Einstein’s and de Sitter’s universes were closed respect to their spatial dimensions. According to de Sitter “infinity is not a physical but a mathematical concept, introduced to make our equations more symmetrical and elegant” [de Sitter 1933, p. 154]. It is worth noting that, mutatis mutandis, in origin the fact that infinity is a foreign notion in physical realm was pointed out by Galileo Galilei (1564-1642): “essendo il moto retto di sua natura infinito, perch´einfinita e indeterminata `ela linea retta, `eimpossibile che mobile alcuno abbia da natura principio di muoversi per linea retta, cio`everso dove vi `eimpossibile di arrivare, non vi essendo termine prefinito” [Galilei 1632, p. 43]. Introduction 3 in order to express in general relativity the observed evidence of a static equilibrium of the universe. De Sitter proposed an alternative solution, i.e. a model of an empty world. In de Sitter’s universe a mass test would have escaped far away from an observer because of the presence of the cosmological constant, so that de Sitter’s universe appeared not really static. This property was later called “de Sitter effect”, and during the 1920’s some discussions arose about its form and its actual astronomical consequences. Indeed the de Sitter effect related distances and velocities, even better redshifts, and for this reason it seemed to be connected by some means with the first relevant redshift measurements in extragalac- tic nebulæ. Thus de Sitter’s model began to appear in a favorable light compared to Einstein’s solution. The interest in the de Sitter effect survived until 1930, when “truly” expanding models of the universe, i.e. solutions already discovered in 1922 by Alexander Friedmann (1888-1925) and independently in 1927 by Georges Lemaˆıtre(1894-1966), were proposed in order to properly explain the evidences of a general cosmic recession confirmed in 1929 by Edwin Hubble (1889-1953). It was Arthur Eddington (1882-1944), together with de Sitter, who clarified in 1930 the relation between the non-static and non-empty Friedmann-Lemaˆıtremodels and observational discoveries of receding galaxies. Before the first diffusion and general acceptance of the model of the expanding universe, the solution of de Sitter played a relevant role in the early phases of modern cosmology: just because of its mentioned prop- erties, it was the precursor of the other non-static solutions to which attention was directed since the 1930’s. “Statique sans l’ˆetre,vide mais non neutre, virtuellement actif sur toute mati`erequ’on voudrait y mettre, r´esultatd’une sym´etrieen trompe-l’œil, solution batˆarded’une ´equation bˆatarde,l’univers de de Sitter - Merleau Ponty wrote in 1965 - ´etait donc un curieux complexe d’´equivoques, qui cependant portait l’avenir de la pense´ecosmologiques” [Merleau Ponty 1965, p. 61]. In particu- lar, the de Sitter effect, which was obtained through the metric of de 4 Introduction

Sitter’s universe, played a fundamental influence in contributions that several scientists offered during the 1920’s both in theoretical and in observational cosmology. Predictions and confirmations of an appropri- ate redshift-distance relation marked the tortuous process towards the change of viewpoint from 1917 paradigm of a static universe to 1930 picture of a universe evolving both in space and in time. In this perspective, such a historical research centered on the rise and decline of the interest in the de Sitter effect aims to highlight the va- riety of ideas and contributions addressed to the cosmological question during the 1920’s debates, which inaugurated the modern approach of cosmologists to the universe as a whole and its properties. Indeed, deal- ing with the geometry of curved spaces in astronomical context, some important topics, which are still present in cosmological debates, entered in that period the description of the universe through applications of general relativity theory. In particular, concepts as the metric of the uni- verse, the homogeneity and isotropy of space, as well the interpretation of extragalactic redshift, were first faced in those debates. In this framework, such a history of the debates about the de Sitter effect reveals its utility. Individual attempts to explain the de Sitter effect and interpretations of redshift measurements through de Sitter’s line element can be viewed as the leading questions in the history of early intersections between observations of distant astronomical objects and a serious and coherent physical theory of the universe given by cosmological solutions of relativistic field equations. Therefore, from this point of view, the issue faced in this thesis rep- resents a remarkable passage in the broader history of the encounter between astronomical observations and the laws of physics, inaugurated through the revolutionary scientific contributions by Galilei, who first “did not separate the two sciences by an impassable barrier” [Drake 1993, p. 237]. The convergence of astronomy and physics into cosmology has greatly developed towards the completely new present cosmological picture. In Summary 5 this picture, both general relativity and quantum physics today share a fundamental role investigating the nature of space, time and gravita- tion. Indeed at present the knowledge of outer space, i.e. of cosmological scales, is deeply connected to that of inner space, i.e. sub-atomic scales, looking for the formulation of a quantum theory of gravitation by inves- tigating unification models of fundamental interactions.

1.1 Summary

An overview of some topics of present cosmology, as the Robertson- Walker metric and the meanings of redshift in cosmology, is reported in Chapter 2. Then the state of cosmological knowledge at the beginning of XX Century, when the universe was still considered static and filled by stars and nebulæ, is illustrated in Chapter 3. Chapter 4 is devoted to the beginning of relativistic cosmology. A critical analysis is furnished about the ideas and attempts which led Einstein and de Sitter to the formulation of the first two rival models of the universe as a whole through general relativity. The main topic of the thesis is then developed in Chapters 5 and 6. The former, Chapter 5, is focused on the de Sitter effect, its math- ematical formulation, and the several formulations of such a theoreti- cal redshift-distance relation which were proposed by the actors of the cosmological debates during the 1920’s: de Sitter, Eddington, Lemaˆıtre, Hermann Weyl (1885-1955), Kornel Lanczos (1893-1974), Ludwik Silber- stein (1872-1948), Howard Robertson (1903-1961), and Richard Tolman (1881-1948). The following chapter, Chapter 6, unfolds the various at- tempts to relate the de Sitter effect to observational data, as done by Carl Wirtz (1874-1939), Knut Lundmark (1889-1958), and Gustav Str¨omberg (1882-1962), who based their conclusions on the astonishing radial ve- locities of nebulæ measured by Vesto Slipher (1875-1969). During the early phases of observational cosmology, the attempts to deduce a re- liable empirical redshift relation successfully culminated in 1929, when 6 Introduction

Hubble confirmed that a linear redshift-distance relation among galaxies existed, and therefore that a systematic recession was actually revealed by observations. In Chapter 7, the 1930 first diffusion of the model of the expanding universe is discussed, paying attention to Lemaˆıtre’scosmology as the “third way” between Einstein’s and de Sitter’s cosmological models which solved the puzzle generated by the de Sitter effect. Chapter 2

Some issues of present cosmology

In this chapter an overview of some issues of present standard cos- mology is given. The starting points on which the following topics are based are rela- tivistic solutions of field equations for an expanding universe. However, it is important to note that the first hints and discussions about some of these issues can be found in the interpretation of properties of static and stationary cosmological models since 1917, i.e. before the general acceptance of the expanding universe. In particular, the assumption of homogeneity and isotropy of space entered modern cosmology by Ein- stein’s model, and the meaning, even better the meanings, of redshift (and also the existence of visual horizons) were faced in 1920’s debates dealing with de Sitter’s cosmological solution, as we shall see in next chapters. Astronomy and cosmology are based on observations at several wave- lengths, which are limited to regions of the universe accessible to tele- scopes. The causal connection between observers and observed objects is given by light: the fact that such a “sidereal messenger” travels through space with a finite speed involves that the farthest we observe in space, the more backward we observe in time.

7 8 Some issues of present cosmology

Among the four fundamental forces in nature, which are gravity, elec- tromagnetic force, strong interaction and weak interaction, only gravity and electromagnetism act on long range. Assuming a neutral charge state of the universe, the fundamental role in the evolution of structures in the universe is thus played only by gravitation. The fundamental theory describing gravity is the general theory of relativity, published by Einstein in its final form in 19161. In Einstein’s theory, space-time can be described as a 4-dimensional Riemannian manifold. Through non-Euclidean geometry, Einstein introduced the concept of curvature of space-time, meaning that metrical properties of space-time are entirely described by tensor quantities gµν’s, which represent the gravitational field. The space-time interval, often called the metric, is given by:

2 µ ν ds = gµνdx dx , (2.1) where µ, ν = 1, 2, 3, 4. Following initial notation used by Einstein, the first three indexes refer to spatial coordinates, while dx4 refers to time co- ordinate; the signature is (–,–,–,+). The summation convention is used, i.e. identical upper and lower indices are implicitly summed. In general relativity an important aspect is that gravity, which is rep- resented through space-time curvature, is included in the line element. Thus a particle in a gravitational field can be considered as moving along geodesics of space-time. Field equations, or “Einstein’s equations”, correspond to the relativis-

1The questions of a general covariant formulation of laws of physics, in particular gravitation, and the “defect” of the existence of privileged observers also in special relativity, i.e. the existence of inertial reference frames, led Einstein to the formulation of his new theory of gravitation, based upon the principle of relativity and the principle of equivalence. See [Janssen 2005 ] for a historical reconstruction of how Einstein formulated general relativity. See also [Renn-Schemmel 2007 ] for a historical analysis of different approaches of the problem of gravitation around the turn of the last century. The Cosmological Principle 9 tic generalization of Poisson’s equation2, and relate space-time geometry

(through Riemann curvature tensor Rµνσρ) to energy-momentum tensor

Tµν, which describes the matter and energy contribution:

1 R − g R = −κT (2.2) µν 2 µν µν or equivalently µ ¶ 1 R = −κ T − g T . (2.3) µν µν µν 2

Here Rµν is the Ricci tensor, i.e. the contracted Riemann tensor, R is µν the Ricci scalar, i.e. a scalar curvature obtained from g Rµν, and T is µν 8πG obtained from g Tµν; κ is a constant equal to c4 , where c is the speed of light and G the gravitational constant. The solution of field equations with regard to the universe as a whole permits to determine the metric of the universe, i.e. all information about the geometry of space-time.

2.1 The Cosmological Principle

From astronomical observations on large scale and from theoretical models interpretation, the present picture is that of an expanding uni- verse, which evolved from an initial singularity, i.e. from an extremely dense and hot phase: this is the standard model of the universe, the so- called “hot Big-Bang model”. There are some fundamental results which

2Such a scalar equation was formulated in 1813 by Sim´eonDenis Poisson (1781- 1840) and describes how gravitational potential φ (determined by density of matter ρ) behaves: ∂2φ ∂2φ ∂2φ ∇2φ ≡ ∆φ = + + = 4πGρ. ∂x2 ∂y2 ∂z2 The gravitational potential φ is defined as Z ρ φ(r) = −G dV, r0 where G is the universal gravitational constant. Thus gravitational force for a point mass at a distance r is F = −∇φ. Laplace’s equation is obtained from Poisson’s equation when zero density of matter is considered: ∇2φ = 0. 10 Some issues of present cosmology support this picture, coming both from pure astronomical evidences and from data comparison with the Standard Model of particle physics:

• the recession of galaxies, which was interpreted since the 1930’s as a feature of an expanding universe

• the light elements abundance, which observed values agree with predicted abundances from the primordial nucleosynthesis during the early phases after the Big-Bang

• the cosmic background radiation, which is at present observed at microwave lengths (CMB), and is interpreted as the relic of radia- tion originated at the last scattering surface from proton-electron ionized plasma at the time of decoupling, at a temperature of about T ' 3000 K, when the universe was 380’000 years old, about 13.7 · 109 years ago. Such a radiation was first observed in 1965 by Arno Penzias (1933- ) and Robert Wilson (1936- )3, giving a very proof that the universe expanded from a hot early phase. The measured radiation, according to latest results4, corresponds to the radiation of a black-body at a temperature of T ' 2.725 K.

Moreover, present results from cosmological surveys permit to assume that the universe we observe is homogeneous and isotropic on scales larger than approximately5 200 h−1 Mpc. Matter and radiation are assumed to be uniformly distributed through space on the very largest scales, with

3Actually, Penzias and Wilson measured a background radiation since 1963, then in 1965 Robert Dicke (1916-1997) and his colleagues at Princeton rightly interpreted the meaning of such a radiation as a cosmic radiation relic. See [Penzias-Wilson 1965 ] and [Dicke et al. 1965 ]. 4See [Komatsu et al. 2008 ] for a present cosmological interpretation of observations of CMB and its anisotropies. 5 H Here h correspond to 100 , where H is the Hubble parameter defined in Section 2.2. Mpc means Megaparsec, one of the astronomical quantities which are used for distances. It corresponds to 106 pc, i.e. to 3.09 · 1022 m. Other useful units of length are the light year, equal to 9.46· 1015 m (1 pc ' 3.26 light years), and the astronomical unit (AU), corresponding to 1.49 · 1011 m. The Robertson-Walker metric 11 neither privileged directions nor privileged positions. As already men- tioned, it was Einstein who first assumed in his 1917 static cosmological model the homogeneity and isotropy of matter (see Chapter 4). Such a fundamental condition is referred in the literature as the “Cosmolog- ical Principle”. In particular, in the picture of an expanding universe, the cosmos exhibits the same properties at a fixed time t, i.e. physical properties of space depend only on time coordinate, which is called for this reason “cosmic time”. The cosmological principle permits to identify 3-dimensional spatial surfaces at constant time which are homogeneous and isotropic, thus maximally symmetric. Since the mentioned 1965 observation of the cosmic microwave back- ground radiation, Big-Bang model was clearly preferred to a rival theory, the Steady State cosmology. Such a theory was proposed in 1948 by Thomas Gold (1920-2004), Hermann Bondi (1919-2005) and Fred Hoyle (1915-2001), and does not predict an initial singularity of the universe6. The steady state model is based on a continuous creation of matter, in order to satisfy the “Perfect Cosmological Principle”, meaning that the universe looks the same both in every direction and at every time.

2.2 The Robertson-Walker metric

The only solution which satisfies the cosmological principle for an expanding universe is the Robertson-Walker (RW) metric7: · ¸ dr2 ds2 = −a2(t) + r2dθ2 + r2 sin2 θ dφ2 + c2dt2. (2.4) 1 − kr2 Here the signature is (–,–,–,+). The coordinate t is the cosmic time of comoving objects. Polar coordinates (r, θ, φ) refer to a comoving reference frame, in the sense that they are constant for each particle of the “perfect fluid” (to which the content of the universe is approximated) at rest

6See [Bondi-Gold 1948 ], [Hoyle 1948 ]. 7Such a line element was independently proposed in 1935 by Robertson in [Robert- son 1935 ] and in 1936 by Arthur Walker (1909-2001) in [Walker 1936 ]. 12 Some issues of present cosmology with respect to this reference frame. The function a(t), historically first denoted as R(t), is the cosmic scale factor, or expansion parameter, which depends only on time and varies according to expansion8. The parameter k determines the constant curvature of spatial sections. It can be negative (k = −1), null (k = 0), or positive (k = +1), yielding respectively an open universe (3-dimensional hyperbolical space, or 3-hyperboloid), a flat universe (Euclidean space) or a closed universe (3-dimensional spherical space). In fact, the curvature parameter can be scaled in such a way to assume only values k = (1, 0, -1). The expansion parameter a(t) is related to the curvature of space. Indeed closed and open spaces have positive k and negative Gaussian curvature CG = a2 , respectively. The parameter a(t) thus represents the radius of spatial curvature, which in cosmology −1/2 a describes the modulus of Gaussian curvature radius RG = C = √ G |k| [Coles-Lucchin 2002, pp. 9-13]. In the next chapters of present thesis, following notation in primary sources of the period 1917-1930, the symbol R will be used in order to represent the radius of the universe9. Indeed, before the 1930 diffusion of the model of the expanding universe, such a notation was used to represent the constant world-radius in early relativistic static and finite models of the universe proposed by Einstein and de Sitter. It was Ed- dington who introduced in 1930 the symbol a(t) for the world-radius in an expanding universe, R(t) ≡ a(t)[Eddington 1930a]. Friedmann and Lemaˆıtre,as we shall see, were the first who independently considered a world-radius depending on time for a not empty universe.

8Following conventions in [Coles-Lucchin 2002 ], scale factor a(t) has the dimen- sions of a length, and the comoving coordinate r is dimensionless. 9The mentioned world-radius R is not the Ricci curvature scalar obtained from µν g Rµν , i.e. from the contraction of Ricci tensor which appears in Einstein’s field equations. The Ricci scalar is related to the expansion parameter a(t) and the spatial curvature k through [Carroll 1997, p. 220]:

6 R = (aa¨ +a ˙ 2 + kc2). (2.5) a2 . The Robertson-Walker metric 13

During the 1920’s debates, several authors dealt with the properties of de Sitter’s universe, and proposed non-static interpretations of the line element of such an empty world. In this framework, the theoretical contributions by Robertson, Lemaˆıtre,Weyl and Lanczos, which will be analyzed in Chapter 5, marked the departure from the static picture of the universe. In particular, they considered the universe of de Sitter in a stationary frame. As we will see, each of the mentioned authors interpreted in a different way the notion of a stationary world for their own versions of de Sitter’s universe. In the retrospect, following [Ellis 1990, pp. 100-101], their contributions can be actually viewed as true expanding versions of the universe of de Sitter. It is useful to mention the present meaning in general relativity of the concepts of static and stationary gravitational fields. Following [Landau- Lifˇsitz1960, Ita. tr. p. 325], a static gravitational field corresponds to the constant field which is generated by a unique body, where in the metric the gµν terms are not dependent on time coordinate x4 (i.e. the body is at rest and both two time directions are equivalent), and all gα 4 are equal to zero10. On the contrary, in a constant gravitational field which is generated by a body which has axial symmetry, as for example by a body rotating along one of its symmetry axis, the two directions of time are not equivalent. Such a gravitational field, where in general gα 4 6= 0, is denoted as stationary. In other words, a stationary space- time admits a time-like Killing vector field. Such a space-time is also static if this Killing vector field is orthogonal to space-like hyper-surfaces at constant time. By using another radial comoving coordinate,r ˜, an equivalent expres- sion of the RW metric is: £ ¤ ds2 = −a2(t) dr˜2 + f(˜r)2(dθ2 + sin2 θ dφ2) + c2dt2. (2.6)

Here the geometry of spatial sections is determined by the function f(˜r),

10The symbol α (and, in following pages, also the symbol β) refers to the spatial coordinates of the metric: α, β = 1, 2, 3. 14 Some issues of present cosmology the comoving angular diameter distance, which is equivalent to r in the metric 2.4, and is equal to (sinr ˜,r ˜, sinhr ˜), for k = ( 1, 0, −1), respec- tively. The expansion rate of the universe is measured through the Hubble parameter H(t): da 1 a˙(t) H(t) ≡ = . (2.7) dt a(t) a(t)

In particular, H0 ≡ H(t0) is the value of Hubble parameter at present time t0. Dimensionally, the Hubble parameter is the inverse of a time −1 and its present value is H0 ' 70.1 ± 1.3 Km/sec Mpc [Komatsu et al. 2008 ]. The rate of change of the expansion rate is measured by the deceleration parameter q: a a¨ q(t) = − . (2.8) a˙ 2 The geometry of space depends on its content: the spatial sections of the universe can be closed, flat or open provided that the density parameter

ρ(t) k Ω(t) ≡ = 1 + 2 (2.9) ρc(t) a˙ is respectively greater, equal or less than 1. Critical density ρc is the density of a flat universe: µ ¶ 3 a˙ 2 3 ρ (t) = = H2(t). (2.10) c 8π G a 8π G

Since the material content of the universe is diluted during the expansion over an increasing volume, matter (and also radiation) density decreases with time. As already mentioned, the matter content is compared to a perfect fluid with neither viscosity nor heat flow. Thus the energy- momentum tensor is

2 Tµν = −p gµν + (p + ρ c )UµUν, (2.11) where U is the 4-velocity of the perfect fluid. The 4-velocity vector represents the average motion of matter, and in comoving coordinates is The Robertson-Walker metric 15 written as U µ = (0, 0, 0, 1). With these conditions and by considering the Robertson-Walker metric, Einstein’s equations reduce to 2 equations, the well known Friedmann- Lemaˆıtre(FL) solutions11, which describe the evolution of an expanding universe: 4 ³ p ´ a¨ = − π G ρ + 3 a, (2.12) 3 c2 8 a˙ 2 + kc2 = π Gρ a2. (2.13) 3 The first equation corresponds to the time-time component of Einstein’s equations. The second equation can be obtained from the first one, by considering Birkhoff theorem12 and the postulate of adiabatic expansion of the universe: d(ρ c2a3) = −p da3. (2.15)

The behavior of scale factor, a(t), depends thus on the total density ρ, which is related to the pressure by the state equation of a perfect fluid,

p = ω ρ c2. (2.16)

If 0 ≤ ω ≤ 1 (the so-called Zel’dovich interval), models of a homogeneous and isotropic universe have an initial singularity at a = 0. There is an 1 unavoidable singularity even for − 3 < ω ≤ 0 [Coles-Lucchin 2002, p. 11Friedmann’s and Lemaˆıtre’sproposals will be described in Chapter 7. 12Such a theorem was proposed in 1923 by George Birkhoff (1884-1944). It states that a spherically symmetric gravitational field in an empty space is static and can be described through the Schwarzschild metric, i.e. by the solution of Einstein’s field equations considering a mass-energy source in empty space. Such a metric was discovered in 1916 by Karl Schwarzschild (1873-1916): µ ¶ dr2 ¡ ¢ 2GM ds2 = −¡ ¢ − r2 dθ2 + sen2θdφ2 + 1 − c2dt2. (2.14) 2GM c2r 1 − c2r

Birkhoff’s relativistic result is analogous to the classic result obtained by Newton, based on Gauss theorem for a gravitational field, which states that the gravitational field outside a spherical object is the same as the whole mass of such an object is concentrated at its center [Coles-Lucchin 2002, p. 24]. 16 Some issues of present cosmology

1 36]. The cases ω = 0, ω = 3 , ω = 1 correspond to a dust, radiation, stiff matter universe, respectively.

2.3 Redshifts in cosmology

Redshift or blueshift measurements and their interpretations play a fundamental role in astronomy and in cosmology. The redshift (or blueshift) z is due to an increasing (or decreasing) of wavelength λ (not to confuse with the cosmological constant!), and corresponds to:

λ − λ λ z = 0 e ⇒ 1 + z = 0 , (2.17) λe λe where λ0 and λe denote, respectively, the wavelength measured by the observer and the original emitted wavelength. There are three kinds of shift to take into account in astronomical obser- vations [Harrison 1981, p. 235]:

• the gravitational shift, which can be either red or blue, and is due to light traveling close to massive bodies: 1 zG = q − 1. (2.18) Rs (1 − R )

2GM Here Rs = c2 is the Schwarzschild radius of the source of gravi- tational shift approximated to a sphere of radius R and mass M.

• the Doppler shift, either red or blue, originated by relative motions between observer and observed object through space13: v z = , (2.19) D c 13The interpretation of the change in the frequency of sound-waves which is heard from a moving source of sound was proposed in 1842 by Christian Doppler (1803- 1853). It was Armand Fizeau (1819-1896) who in 1848 rightly predicted the displace- ment of lines for light coming from stars. For this reason such an effect is also known as the Fizeau-Doppler effect. Redshifts in cosmology 17 µ ¶ c + v 1/2 z = − 1. (2.20) D c − v

The first is the classic Doppler formula, to be used when the velocity of the object v is small when compared to speed of light, and the second is the special relativistic Doppler formula. This latter gives:

(z2 + 2z) v(z) = c . (2.21) (z2 + 2z + 2)

• the expansion (or cosmological) redshift, which formulation was proposed by Lemaˆıtrein 1927, as we shall see later. Such an effect is due to waves stretched by the expansion of the universe propor- tionally to the scale factor:

a0 zC = − 1, (2.22) ae

where a0 ≡ a(t0) is the scale factor at reception time (i.e. the

present scale factor), and ae ≡ a(te) is the scale factor at signal emission.

Following [Ellis 1989, p. 374], several contributions to redshift mentioned above can be generally resumed as:

(1 + ztot) = (1 + zDS )(1 + zGS )(1 + zC + zGC )(1 + zDO )(1 + zGO ). (2.23)

The terms zGS , zGO , zGC correspond to gravitational shift contributions due to inhomogeneity of matter distribution at the source, near the ob- server, and on large scale, respectively. The shifts zDS and zDO are origi- nated by the relative (Doppler) motion of the source and of the observer respectively. Finally, zC is the expansion redshift due to increasing scale factor in the universe. Such a summary will be useful in Chapter 5 de- scribing the de Sitter effect and several interpretations of redshifts in de Sitter’s universe. 18 Some issues of present cosmology

2.3.1 Velocities and distances

Cosmological redshift, by equation 2.22, gives informations about how much the universe increased in size (i.e. expanded) from the emission time of an observed object. However a direct information about distances and recession velocities can not be obtained. It is indeed necessary to know also the geometry of the universe and the expansion rate of scale factor. Equation 2.21 can be applied only in (local) inertial frames. Neglect- ing contributions by gravitational and Doppler redshifts, the general relativistic relation between recession velocity and cosmological redshift [Davis-Lineweaver 2004, p. 99] is given by: Z a˙(t) z dz0 vrec(t, z) = c 0 . (2.24) a0 0 H(z )

Setting t = t0, we obtain the recession velocity that the object with the measured redshift has today: Z z dz0 vrec(t0, z) = c H0 0 . (2.25) 0 H(z ) When we speak about distances, we have to distinguish between the- oretical concepts of proper and comoving distances. The proper distance is the radial distance at a fixed time; from equation 2.6, being dt = 0 because of the fixed time and dφ = 0, dθ = 0 because of the radial direction, the integral of ds gives:

Dpr(t) = a(t)˜r. (2.26)

Thus proper distance is obtained from the comoving distancer ˜ multiplied to scale factor. Differentiating this formula with respect to time we obtain the theoretical velocity-distance law [Harrison 1993, p. 30]: dD (t) da(t) pr = r˜ ⇒ v (t) = H(t)D (t). (2.27) dt dt rec pr Such a linear relation is valid for all distances in an expanding RW uni- verse, and for example predicts also recession velocities exceeding speed Redshifts in cosmology 19 of light. In particular, the so-called Hubble radius measured at present time c DH0 = (2.28) H0 is the distance at which the recession velocity is equal to the speed of light, and beyond the sphere of that radius, the Hubble sphere, recession velocities exceed speed of light. As we will see in Chapter 6, Hubble in 1929 confirmed that a linear relation existed between distances and radial velocities, thus relation 2.27 is usually referred to as the Hubble law14. However, as pointed out in [Harrison 1993, p. 31], the true Hubble law should be considered the empirical redshift-distance relation

zc = H(t)DL. (2.29)

Here DL is the luminosity distance, an observable quantity which will be described later. This relation is due to the direct estimate of velocity from redshift through equation 2.19, as a habit in early observational cosmology15. Such a relation is linear, i.e. coincides with the general theoretical velocity-distance law (equation 2.27), only for small redshifts and small distances compared to Hubble’s radius. The comoving distance of an object can be calculated through the fact that photons travel along null geodesic, ds = 0. In particular, at present time t0, the comoving distance of an object which emitted light at t is: e Z t0 dt0 r˜(te) = c 0 , (2.30) te a(t )

14Hereafter it comes also the notation of H for the “constant” of proportionality between velocities and distances. 15It is important to note that also Lemaˆıtreproposed such a relation already in 1927, as we shall see. Two years later, Hubble, unaware of Lemaˆıtre’sresult, confirmed this linear relation from new estimates of distance and radial velocity of galaxies which Hubble himself and Humason had obtained by using the most powerful telescope operating at that time, the 100-inch Hooker reflector at Mt. Wilson. 20 Some issues of present cosmology

The proper distance to such a comoving object is thus obtained (from the above equation and equation 2.26) by multiplying to a(t0). As mentioned, an important observable quantity is the luminosity distance DL, which is useful to approximately estimate cosmological pa- rameters through measurements of redshift. It is defined as:

L D2 = , (2.31) L 4π l where l and L are the apparent and absolute luminosity of a distant source, respectively. The luminosity distance varies with redshift, be- cause the flux of photons is diluted traveling in an expanding RW uni- verse [Coles-Lucchin 2002, p. 20]. It can be written as [Davis-Lineweaver 2004, p. 105]:

DL(z) = a(t)f(˜r)(1 + z). (2.32)

Thus, for small redshift and by a Taylor series expansion of scale fac- tor, measurements of luminosity distances allow to determine H0 and q0 through the approximate formula [Coles-Lucchin 2002, p. 20]: · ¸ c 1 2 DL(z) = z + (1 − q0)z + ... . (2.33) H0 2

The luminosity distance of a source is related to observable quantities (with which astronomers usually work) as the absolute magnitude M and apparent magnitude m through the distance modulus equation. Such an equation, neglecting absorptions, takes the form:

m − M = −5 + 5 log DL. (2.34)

Therefore, the luminosity distance can be obtained for some astronom- ical objects which absolute magnitude is known. For example, Cepheid variable stars are important distance indicators, since they have a firm relation between their period of variation and their absolute luminos- ity. As we shall see in next chapters, such standard candles played a fundamental role in the rise of observational cosmology. Redshifts in cosmology 21

Another useful observable quantity is the angular diameter distance y DA(z) = θ , i.e. the distance of an object of physical size y observed with apparent angular size θ. In an expanding universe it can be expressed as [Davis-Lineweaver 2004, p. 105]:

−1 DA = a(t)f(˜r)(1 + z) , (2.35) the meaning of f(˜r) being described in equation 2.6. The angular diam- eter distance is related to the luminosity distance through:

2 DL = (1 + z) DA. (2.36)

The observable part of the universe is defined through the particle horizon. Our particle horizon is the frontier which divides, at the instant of observation t0, world-lines which intersect our past light cone, i.e. world-lines which can be observed, from world-lines which lie outside our past light cone [Harrison 1991, p. 61]: Z t0 dt0 r˜ph(t0) = c 0 . (2.37) 0 a(t ) From equation 2.26, the corresponding proper distance to the particle horizon is:

Dph(t0) = a(t0)˜rph(t0). (2.38)

Our particle horizon is thus the distance light we receive or can receive has traveled from t = 0 to t0. Another important horizon, which is exemplified by de Sitter’s uni- verse16, is the event horizon: Z +∞ dt0 r˜eh(t) = c 0 . (2.39) t a(t )

16Actually, through the “exponential” representation of de Sitter’s line element (where a(t) ∝ eH t), which is at present used to describe a vacuum energy dominated universe, it can be shown that de Sitter’s universe has an event horizon at distance c Deh = H , i.e. at the constant proper Hubble radius [Rindler 1956, p. 139]. 22 Some issues of present cosmology

The event horizon, following [Rindler 1956, p. 134], is the frontier which divides events which have been, are, will be observable by us, from eter- nally unobservable events. The superior limit in the above integral is t → +∞ by supposing an eternally expanding universe17.

2.4 Dark matter and dark energy

With regard to the material and energy content of the universe, cur- rently the relevant contributions are considered to be given not by ordi- nary matter, but from dark matter and dark energy. Measurements of rotation curves of galaxies permit to estimate the mass of these structure. However, a general discrepancy exists between observed and predicted mass. A possible solution seems that to consider

17The existence of cosmological horizons is related to the problem of causal con- nection among different regions of the universe. For example, from considerations on the CMB, it can be shown that regions which now are separated by more than ∼ 2◦ are causally disconnected. Thus, the fair isotropy of CMB can not be explained: this is the so-called horizon problem. A possible solution to such a problem is given by the mechanism of inflation, first proposed by Alan Guth (1947- ) in 1981. The infla- tionary theory is based on a spontaneous broken symmetry as predicted in Standard Model of particle physics. Such a theory takes into account an accelerated phase of the universe right after the Big-Bang, which took place between 10−34 sec and 10−32 sec, due to the repulsion effect of vacuum energy with negative pressure. During such 1 a inflationary phase the expansion is accelerated:a ¨ > 0 (ω < − 3 ). Thus the size of the comoving effective horizon, which during inflation is not the particle horizon

(˜rph), but is the Hubble radius (˜rH ), decreases with time: r˜˙H ∝ −a¨. During inflation the proper distance to the Hubble sphere remains constant and is coincident with the event horizon [Davis-Lineweaver 2004, p. 100]. Therefore a region of the universe which entered such a “horizon” before the beginning of inflation, i.e. which became causally connected with regions within the Hubble sphere, can “escape” such an effec- tive horizon during inflation (because the comoving effective horizon decreases with time during such a phase) and eventually reenter the horizon once more, after the end of inflation [Coles-Lucchin 2002, pp. 149-151]. We refer to [Harrison 2000, Chap- ters 21-22] for further readings about the existence and description of cosmological horizons, which represent an interesting and complex issue. Dark matter and dark energy 23 the existence of a relevant fraction of non-luminous and non-ordinary matter (thus dark matter), which is dominant in the total matter contri- bution in the universe over the baryonic (ordinary) matter18. Even if a small fraction of baryonic dark matter may also exist, the most fraction of dark matter is characterized by a non-baryonic nature (the candidate particles of dark matter are considered among exotic elementary parti- cles predicted in supersymmetry theories). Its interaction with ordinary matter occurs only by gravity. In the common accepted cosmological sce- nario dark matter seems to be not relativistic (cold dark matter, CDM). This kind of matter forms a (dark) halo around galaxies, so that the rotational curves are altered by its presence. In recent years, some fundamental observations of luminosity of dis- tance indicators, type Ia supernovæ, revealed that the universe is accel- erating, i.e. its rate of expansion is increasing [Riess et al. 1998, Perl- mutter et al. 1999 ]. Accounting for this newly observed acceleration, an unknown kind of energy, called dark energy, has been postulated. Such a dark energy permeates the universe: its contribution is now dominant and determines the accelerated expansion acting as a negative pressure. One of the possible explanations of dark energy is vacuum energy. The simplest form of a cosmic repulsion which accelerates the expansion of the universe is a cosmological term in field equations. As we shall see in Chapter 4, in 1917 Einstein introduced a new term, λ gµν, in field equations 2.2 in order to obtain a suitable relativistic solution describing a static universe. This new term acted like an anti-gravity term, and, in Einstein’s intentions, could balance gravity effects on large scales. In the current picture, accounting for an accelerated universe, a similar cosmo- logical term (now denoted with Λ) can be inserted in field equations: 1 R − g R − Λg = −κ T . (2.40) µν 2 µν µν µν Equivalently, a new contribution can be considered in the right-hand side

18Already in 1933 Fritz Zwicky (1898-1974) suggested the existence of some kind of invisible matter studying rotational curves of galaxies in Coma cluster [Zwicky 1933 ]. 24 Some issues of present cosmology of field equations, among the energy-matter contributions which curve space-time:

1 R − g R = −κ T˜ = −κ(T + ρ g ). (2.41) µν 2 µν µν µν Λ µν

The present role of the cosmological constant is inspired by quantum mechanics. It can indeed be interpreted not only as an intrinsic property of space, but also as a kind of energy, in particular vacuum energy:

8π G Λ = ρ . (2.42) c4 Λ

Vacuum energy has negative pressure (i.e. ω = −1 in equation 2.16), and its contribution has to be taken into account in the total density parameter Ω(t) as defined in 2.9. It is supposed to be constant on time, so that its accelerating effect became dominant over time-decreasing matter density about 5 · 109 years ago. From measurements of anisotropies of CMB (which are of order of 10−5 and correspond to the “seeds” of structures in the universe), the cur- rent picture is that of a flat universe (ktot ' 0 or equivalently Ωtot(t0) ' 1). Furthermore, investigations of clusters abundances and results from gravitational lensing permit to estimate the present total matter (dark and baryonic) contributions at about 30% in the density parameter. Thus the remaining contribution is supposed to be done by dark energy. Com- bining the results from type Ia supenovæ, CMB anisotropies, and acoustic oscillations of baryons, the total energy content is assumed to be com- posed by baryons ' 4.6%, dark matter ' 23%, dark energy ' 72% [Komatsu et al 2008 ]. A ΛCDM model seems consistent with present observations. How- ever, this issue has still to be clarified19, and other possible explanations

19For example, some problems are the present value of vacuum energy density compared to the expected value from theoretical results of supersymmetry theories and its fine-tuned coincident value with present matter density. See [Carroll 2001, Peebles-Ratra 2003 ] for further readings on this subject. Dark matter and dark energy 25 for the accelerated expansion have been proposed, as a quintessence dy- namical field, or modified gravity. A suitable explanation of the accel- erated universe represents one of the important challenges in cosmology and theoretical physics, as well in search of quantum gravity theory.

Chapter 3

Cosmology at the beginning of XX Century

In order to better understand the rise and early convergence of theo- retical and observational cosmology, it is worthwhile to briefly illustrate in the following pages the general scientific knowledge on the universe at about a century ago.

3.1 The sidereal universe and the nebulæ

The structure and duration of the universe, quoting Simon Newcomb (1835-1909), “is the most far-reaching problem with which the mind has to deal” [Newcomb 1902, p. 226]. According to Newcomb, at the very beginning of last Century first steps were made to attack such a problem by scientific methods, not merely from a speculative point of view. Some questions were involved, as the extent of the universe of stars through an infinite or finite space, as well the arrangement of stars in space and the duration of the universe in time. However, no certain answers could yet be furnished by observations. Moreover, as pointed out in 1911 by French mathematician Henri Poincar´e(1854-1912) in the introduction to the “Le¸cons sur les Hy-

27 28 Cosmology at the beginning of XX Century poth`esesCosmogoniques”, the cosmogonic problem of the origin of the universe was still far to be clarified. Science could not yet explain some questions about the evolution of the cosmos already inspired by works of Pierre-Simon Laplace (1749-1827). How the cosmos had evolved towards the present and ordered state, starting from a hypothetical primeval distribution of matter uniformly spread across space? According to Poincar´e,“nous ne pouvons donc terminer que par un point d’interrogati- on” [Poincar´e1911, p. XXV]. Observations could be helpful to eventu- ally interpret cosmogony as an experimental science, however the relation between the Milky Way and other celestial objects, in particular spiral nebulæ, still represented to Poincar´ea challenge without definite results. Scientific investigations of heavens were firmly linked up with Galilean experimental method and theoretical synthesis. From this point of view, astronomy succeeded with positive results by applications of Newtonian gravitation law and classical mechanics on scales referred to the solar system. Some problems of celestial mechanics were solved, so to confirm the law of motion proposed times before by Isaac Newton (1642-1727). A long-standing question was the precession of the perihelion of Mercury. The amount of the measured precession of the most inner planet con- siderably differed from expected values of planetary perturbation, rep- resenting an exception with regard to predictions of Newton’s theory. Works by Urbain Le Verrier (1811-1877) and subsequently by Newcomb settled, in 1882, a value for such a discrepancy to 43” per century. It is well known that general relativity offers a solution for this dilemma: the difference between the value of the precession of Mercury perihelion predicted by general relativity and the value measured by observations is negligible, even better, such a theoretical prediction represents one of the crucial tests giving validity to Einstein’s theory of gravitation. Astronomical observations were all the more detailed and careful: astronomy quickly developed into astrophysics, thanks to advantages of photometry and spectroscopy. However, despite technical progresses and advancements of data interpretations through theoretical models, the The sidereal universe and the nebulæ 29 source of stellar evolution was still unknown, and, moreover, no large- scale systematic velocity fields were pointed out in stellar motions by observed data. Thus at the beginning of last Century, during the rise of special and general relativity, it was well accepted by astronomical community the paradigm of a static universe, which fundamental filling pieces were considered stars, not yet galaxies. In fact, during the 1850’s there was a controversy about the concept of time from the point of view of thermodynamics laws1. As postulated by William Thomson (Lord Kelvin, 1824-1907) and Rudolph Clausius (1822-1888), the second law of thermodynamics involved the irreversibil- ity of physical processes and the concept of entropy. Such a state func- tion tends to increase in isolated systems. Extrapolating the property of increasing entropy to the universe as a whole, now considered as an isolated system, the result of continuous energy dissipation, quoting Lord Kelvin, “would inevitably be a state of universal rest and death, if the universe were finite and left to obey existing laws” [Thomson 1862, p. 289]. Thus the universe would have ended in a “heat death”, correspond- ing to the end of physical transformations and to an unchanging thermal state of the universe. Therefore, according to Clausius, it could be pos- sible to postulate in these terms both a beginning of the universe, i.e. an initial state of the universe as a whole with zero entropy, and a final state characterized by the maximum entropy. It was through the in- terpretation of laws of thermodynamics proposed by Ludwig Boltzmann (1844-1906) that the question about time duration of the universe was clarified. Indeed, according to Boltzmann’s view, the direction of time

1It is interesting to note that, during the early phases of relativistic cosmology, considerations on the universe as a whole from the thermodynamic point of view were not essentially taken into account. Except for Tolman, who investigated in late 1920’s how structures could evolve in a static universe [Tolman 1934 ], scientists involved in cosmological debates mainly dealt with the geometry of the universe and with the relation between the curvature of space-time and astronomical observations [Ellis 1993, p. 316]. 30 Cosmology at the beginning of XX Century could be appreciated in local, not global, thermodynamic phenomena. Thus a definite time direction could not be given to the universe as a whole, but only to some of its parts, as for example to the observable portion of the universe [Boltzmann 1897 ]. This controversy apart, the universe was in general considered to be unbounded in space and infinite in time, and, recalling the ancient Greek conception, basically filled by “fixed” stars, meaning that stars were aimed both by small velocities (compared to speed of light) and small proper motions. Astronomical investigations during those years were directed at the comprehension of distances, positions and displacements of celestial bod- ies, as a sort of continuation of pioneering works inaugurated by William Herschel (1738-1822). Over a century earlier indeed Herschel tried to de- termine through stars counts the structure of our Galaxy, the Milky Way. Those observations were continued by Herschel’s son, John (1792-1871), and then resumed by Jacob Cornelius Kapteyn (1851-1922). Dealing with the problem of sidereal distances by a statistical approach, Kapteyn pro- posed a model of the stellar system as a flat and static structure, the so-called “Kapteyn universe”. In 1920 he assumed the Sun to be at the center of such a system [Kapteyn 1920 ]. However, in order to better ex- plain from a dynamical point of view his own discovery of star-streaming, Kapteyn shortly after proposed that the Sun should have been at about 650 pc from the center of the Galaxy, then considered as a flat rotating disk [Kapteyn 1922 ]. Detailed investigations of globular clusters marked a turning point in the comprehension of the structure of the Milky Way. It was Harlow Shapley (1885-1972) who gave a fundamental contribution in this field. Shapley used statistical parallaxes in order to determine absolute mag- nitude of RR Lyrae stars, i.e. pulsating variable stars (like Cepheids) which change in brightness with a regular period. In this way Shapley could estimate distances of these objects in globular clusters, by using the period-luminosity relation discovered in 1912 by Hernrietta Leavitt The sidereal universe and the nebulæ 31

(1868-1921). In 1918 Shapley suggested a picture of our Galaxy as a flat rotating disk with a diameter which he set in 1919 of about 300’000 light years, filled by stars and nebulæ and surrounded by a spherical halo of globular clusters. The center of the Galaxy was thousands of light years far from the Sun (65’000 light years), in the direction of Sagittarius [Shapley 1918, Shapley 1919, Paul 1993 ]. Doubts remained about the real extent of the Galaxy, and mainly about the nature and distances of spiral nebulæ. During the 1880’s, J. L. Emil Dreyer (1852-1926) compiled a vast catalogue of stars clusters and nebulæ, the “New General Catalogue” (NGC), which contained, together with following supplements, nearly 8’000 objects. Spectroscopy revealed for some of these objects the same features of stellar systems: were spiral nebulæ thus to be regarded as extragalactic systems? Could them be compared to the Milky Way? On the contrary, did the whole of stars and nebulæ belong to the Galaxy, considering it as a unique universal system? It seemed to reappear, now supported by observations, Kantian suggestion of “island universes”, one of the speculative cosmologies of XVIII Century. Indeed, some insights on philosophical grounds into the universe as a whole date back to XVIII Century. Emanuel Swedenborg (1688-1772), Thomas Wright (1711-1786), Johann Lambert (1728-1777) and Immanuel Kant (1724-1804) tried to understand how the cosmos was arranged. However, their remarks did not much involve scientific community. In- deed, up to Herschel’s investigations, astronomers were usually interested in celestial mechanics referred to the solar system. Kant put forward the idea that nebulæ formed part of an infinite cosmos: in his 1755 “All- gemeine Naturgeschichte und Theorie des Himmels” (Universal Natural History and the Theory of Heavens), the philosopher from K¨onigsberg suggested the existence of an unlimited number of flat shaped island universes spread over an infinite space. On the contrary, Lambert hy- pothesized that celestial bodies formed part of a hierarchical universe, culminating with a unique structure. 32 Cosmology at the beginning of XX Century

Recalling Lambert’s idea, in 1908 and subsequently in 1922 Swedish astronomer Carl Charlier (1862-1934) proposed a static and hierarchical model of the universe. Charlier discarded the hypothesis of a uniform distribution of matter through space, which, on the contrary, played an important role in the formulation of first relativistic models of the uni- verse. He supposed that celestial bodies formed gradually increasing spherical systems, which Charlier explicitly called galaxies, of order 0, 1,

2, ... and radii R0, R1, R2, ... , respectively. According to Charlier:

• N1 stars were arranged as system G1, of order 1 and radius R1

• N2 systems G1 formed system G2, of order 2 and radius R2

• N3 systems G2 formed system G3, of order 3 and radius R3...

And so forth. Charlier proposed that this scheme could also be inverted. Indeed a hierarchical structure of spherical systems could be found both in the infinitely great and in the infinitely small2:

2Charlier’s speculative attempt was not an isolated hint in the history of connec- tions between macro-systems and micro-systems during those years. For example, Eddington attempted to relate the cosmological problem to the atomic one. In 1931 Eddington suggested that, starting from the solution of wave-equation, it could be possible to obtain a value of the cosmological constant in agreement with expected value of λ measured from galaxies recession. Indeed, according to Eddington, the cos- mological constant, which Einstein introduced in 1917 and “abandoned” just in 1931 (see later, Section 7.2.1), had a non-zero value and gave the expansion of the universe [Eddington 1931 ]. Eddington considered λ as one of the fundamental entities in na- ture, together with α (which is the fine structure constant), the number of particles expected in an expanding universe, and the ratio of electrostatic and gravitational forces. He firmly attempted to develop a fundamental theory which could relate cos- mology to laboratory physics [Eddington 1946 ]. However, his considerations did not much appeal to physics community (see [Longair 2004 ] for further readings on Ed- dington’s approach). Also Paul A. M. Dirac (1902-1984) pointed out the importance of numerical coincidences between physical and cosmological quantities. Accounting for identifications of large numbers, Dirac proposed in 1938 that the gravitational constant could vary with time [Dirac 1938 ]. The sidereal universe and the nebulæ 33

• N0 meteorites formed star G0

• N1 molecules formed meteorite G1

• N2 electrons formed molecule G2

• N3 sub-electrons formed electron G3

According to stars and nebulæ counts, and measuring their apparent dimensions, Charlier proposed that all spherical systems satisfied the relation: Ri p > Ni. (3.1) Ri−1 This relation was useful, for example, to estimate extent of the second order system G2 which contained our Galaxy:

R2 p > N2. (3.2) R1

6 By supposing that the galaxy G2 contained N2 ≈ 10 nebulæ, Charlier proposed a radius R2 > 1000 R1, being R2 the “metagalaxy” radius and

R1 the Galaxy radius respectively. In this detailed description of an infinite world, Charlier supported the hypothesis that collisions among nebulæ caused the formation of spiral structures in each system [Charlier 1925 ]. The status of spiral nebulæ, together with the dimension of the Milky Way and other astronomical issues as the role of distance indicators, the interpretation of star counts and stellar evolution theory, were faced in a famous discussion, “the Scale of the universe”, between Shapley and Heber D. Curtis (1872-1942). It took place on April 26, 1920 in Washington and passed into the literature as “the Great Debate”. During that meeting, Shapley and Curtis presented opposite proposals about the nature of celestial systems3.

3The great importance of the discussion between Shapley and Curtis was to stim- ulate further researches in that topic. From this point of view, the history of modern cosmology is marked by several debates highlighting important astronomical issues, 34 Cosmology at the beginning of XX Century

Shapley asserted that spiral nebulæ were not comparable in size and in constitution with our Galaxy: “I prefer to believe - Shapley said - that they are not composed of stars at all, but are truly nebulous ob- jects” [quoted in Hoskin 1976, p. 177]. On the contrary, Curtis, holding the same point of view of William W. Campbell (1862-1938), supported theory of spirals as island universes of the order of size of our Galaxy, as suggested long before by Kant. In Curtis’ picture, the Milky Way was counted as a spiral-arms system, as already given in 1900 by Cornelius Easton (1864-1929). In the retrospect, each of the two protagonists pre- sented right arguments with regard to different topics they faced: Shap- ley was correct in his estimates of globular clusters distance, and Curtis rightly interpreted extragalactic position of spirals4. The controversy on the existence of extragalactic systems was shortly after solved, thanks to high resolution images produced by 100-inch (2.5 m) Hooker reflector telescope at Mt. Wilson. The 1917 “first light” of this telescope represents a milestone in the history of the comprehension of galactic structures. As we will see later, just through Mt. Wilson observations, Hubble, whose name is closely connected to the rise of observational cosmology, located Cepheid stars in M31 and M33 nebulæ. Thus Hubble could estimate the distance of these objects. They were placed, according to Hubble, at a distance of about 930’000 light years which are also useful to illustrate how definite interpretations of astronomical ob- servations are not easy to be established. One of the most important controversy in cosmology focused on the rival theories of Big-Bang model and steady state cosmology (we refer to [Kragh 1996 ] for further readings about it). With regard to other topics, in 1972 Arp and Bahcall discussed about the conventional interpretation of redshift measurements [Field 1973 ]. During the 1990’s other debates took place about the distance scale to gamma ray bursts and the nature and scale of the universe [Debates 1998 ]. 4See [Hoskin 1976 ] and [Trimble 1995 ] for some review articles on the Great De- bate. The nature of the nebulæ and relevant topics related to the astronomical obser- vations of spirals which were faced during the 1920’s will be reconsidered in Chapter 6 of present thesis. Cosmological difficulties with Newtonian theory 35

(285’000 pc), undoubtedly outside boundaries proposed by Shapley for galactic globular clusters [Hubble 1925b]. Since 1925 spiral nebulæ were eventually considered as true extragalactic systems similar to our Galaxy: the realm of the stars was replaced in 1925 by the realm of the galaxies.

3.2 Cosmological difficulties with Newto- nian theory

As seen, at the beginning of last century the most accepted view was that of a sidereal and infinite universe. In fact the cosmological issue was not truly faced: there were debates about the nature and position of nebulæ, and questions about the finiteness or infiniteness of space were investigated only from a speculative point of view. Spectroscopical mea- surements permitted to determine radial velocities and proper motions of many stars. Thanks to observations by Vesto M. Slipher (1875-1969), first relevant redshifts were measured in nebulæ and interpreted as ra- dial velocities5 (through 2.19 formula). However, any departure from the picture of a static universe was not debated. Newtonian theory of gravitation, based on usual Euclidean geometry, involved some difficulties and contradictions when extrapolated to an infinite universe. If gravitational force is admitted to be universally valid, how the gravitational effect of an infinite number of stars spread over an unbounded space can be balanced towards the observed equilibrium? In other words, the integral of the contributions by all masses to the gravitational force of a mass test does not converge. Among several attempts to solve such cosmological difficulty in New- tonian theory6, at the end of XIX Century both Hugo von Seeliger (1849- 1924) and Carl Neumann (1832-1925) proposed a modification in the gravitational potential φ. In 1885 Seeliger highlighted that both gravita-

5See Chapter 6 for a description of early measurements of radial velocities in spirals. 6See [Norton 1999 ] and [North 1965, Chapter 2], for further readings on this issue. 36 Cosmology at the beginning of XX Century tional force F and potential φ diverged when a uniform density of matter ρ was spread over an infinite volume V . Neumann, dealing with these issues which he already faced in 1874, suggested that Poisson’s equa- tion should have been modified in order to obtain a uniform and static distribution of matter through space. As a possible solution, both of them independently proposed in 1895 (von Seeliger) and 1896 (Neumann) a similar modification in gravita- tional law. They introduced an exponential term in the expression of φ. Their modifications can be resumed as: Z ρ 0 φ(r) = −G e−λ r dV. (3.3) r0 This new exponential term was crucial at great distances. In such a way the gravitational force diminished on large scale more rapidly than usual r−2 in Newton’s law. The new term balanced gravitational attraction of matter and could be interpreted as a sort of “cosmic repulsion”, so that an average density of matter that was constant everywhere was well allowed. In 1917, as we will see, Einstein used the same analogy and modi- fied his relativistic field equations by the so-called “cosmological term” to obtain a static model of the universe (which, however, Eddington demonstrated in 1930 to be unstable). In order to suitably deal with gravitation on large scale, another possi- bility was to renounce Euclidean geometry and to consider curved spaces. It is well known that Einstein followed this road towards his cosmolog- ical solution of field equations. However, between the end of XIX Cen- tury and the beginning of XX Century, just a few astronomers ventured this path. Among them, Newcomb in 1877 gave attention to the con- sequences of elliptical curved spaces in parallax measurements. In 1900 Karl Schwarzschild (1873-1916) investigated the effects of hyperbolical and elliptical geometries of space in astronomical context7 [Schwarzschild 1900 ].

7This pioneering pre-relativistic work by Schwarzschild, as we will see in next chap- Cosmological difficulties with Newtonian theory 37

3.2.1 Olbers’s paradox

Besides contradictions related to gravity effects on large scale, there were also objections to Newtonian cosmology related to luminosity ef- fects. Recalling the suggestion by William Stukeley (1687-1765), already in XVIII Century Jean Philippe de Ch´eseaux(1718-1751) dealt with the problem of the darkness of night sky. Indeed in an infinite and Euclidean universe, accounting for an infinite number of uniformly distributed stars with the same luminosity, it is expected an equal contribution to the to- tal luminosity both from stars closest to the Earth and from stars placed on a spherical surface at a greater distance than the formers. Indeed the number of stars in latter shell increases by the square of the distance, while the luminosity flux diminishes by the inverse square of the dis- tance. Thus, considering the equal contribution from an infinite number of shells, one could expect to see always a bright sky. In other words, the total luminosity flux diverges. Indeed apparent luminosity l at distance L dL = r is related to absolute luminosity L by l = 4π r2 (see equation 2.31). Being n the constant stars density and 4π nr2dr the amount of stars between r and r + dr, the total flux luminosity U is

Z ∞ µ ¶ Z ∞ L 2 U = 2 4πnr dr = Ln dr. 0 4πr 0 Thus such an integral diverges. In order to explain the absence of such a bright sky, de Ch´eseaux postulated a loss of light traveling across space, so that the contribution by farthest stars was negligible. In 1823 Heinrich Olbers (1758-1840) took into account the same issue. Following de Ch´eseaux’spossible explanation, Olbers pointed out that a loss of the order of 1/800 in stars luminosity was sufficient to explain the evidence of dark sky. Such a cosmological enigma passed into the literature as the “Olber’s paradox”, and could not be suitably solved by ter, played a significant role in Einstein’s and de Sitter’s approaches to the question of the geometry of the universe. 38 Cosmology at the beginning of XX Century admitting an Euclidean universe which was infinite in space and time. The concept of an initial singularity, which is a feature of Big-Bang standard cosmology, offers a solution to this problem8. In an expanding universe, being a(t) the parameter expansion (scale factor), the apparent luminosity l is: 2 L a (te) l = 4 2 , (3.4) 4π a (t0) re where re is star comoving distance, t0 is time of received signal and te is time of emission signal. The amount of stars with luminosity between L and L + dL, which light is emitted between te − dte and te and received at time t0, is 2 2 dN = 4π a (te) re n(te,L)dtedL, (3.5) where n(te,L)dL is stars density at te. Thus the total density of stars observed at t0 is ZZ Z Z · ¸ t0 4 a(te) U0 = l dN = n(te,L) dLdte. (3.6) −∞ a(t0) If a initial singularity is postulated, as in Big-Bang models (but not for example in steady-state cosmology), such a integral does not diverge, being the inferior limit of integration t = 0, not t → −∞.

8The solution here presented is taken from [Weinberg 1972, p. 612]. Chapter 4

1917: the universes of general relativity

This chapter is devoted to the historical reconstruction of significant steps that Einstein and de Sitter took towards the formulation of the first two relativistic models of the universe1. Such solutions of field equations were proposed during the famous 1916-1918 debate between Einstein and de Sitter. In this debate the origin of inertia represents one of the main issues, and can be viewed as the leading thread which led to the first world models.

4.1 Einstein, the universe and the relativity of inertia

Einstein proposed his cosmological solution in the fundamental pa- per “Kosmologische Betrachtungen zur allgemeinen Relativit¨atstheorie” (Cosmological Considerations in the General Theory of Relativity), first published in February 1917. In this paper Einstein illustrated the “rough

1Relevant parts of this chapter are taken from [Realdi-Peruzzi 2009 ]. With re- gard to Einstein’s papers and correspondence, references are made to the English translation of the Collected Papers of Albert Einstein, hereafter [CPAE].

39 40 1917: the universes of general relativity and winding road” [Einstein 1917b, p. 423] that he had to follow to obtain the whole material origin of inertia in the framework of General Rela- tivity. Einstein considered his cosmological result as the achievement of what he called the “relativity of inertia” [Einstein 1913b, p. 197]. Indeed, accounting for Mach’s ideas, Einstein pointed out that “inertia is simply an interaction between masses, not an effect in which ‘space’ of itself were involved, separate from the observed mass” [CPAE 1998E doc. 181, p. 176]. The interest which Einstein addressed to the universe as a whole came from the necessity of a global explanation of the relations among space, time and gravitation through the principle of relativity. In partic- ular, the universe as a whole represented to Einstein an ideal setting in which the relativity of inertia could be verified. This idea can be inferred from a letter to Michele Besso (1873-1955) that Einstein wrote in May 1916. In this letter Einstein illustrated his interest in the implications of a finite universe, “that is, a world of naturally measured finite extension, in which all inertia is truly relative” [CPAE 1998E doc. 219, p. 213]. It is well known that some ideas of the Austrian scientist and philoso- pher Ernst Mach (1838-1916) had a fundamental influence on Einstein’s concept of inertia and on the formulation of general relativity2. In 1916, a few days before the publication of his review paper containing the new theory of gravitation [Einstein 1916b], Einstein wrote an obituary of Mach [Einstein 1916a]. In these pages Einstein acknowledged the philosopher as a true precursor of General Relativity, for having “clearly recognized the weak points of classical mechanics, and thus came close to demand a general theory of relativity” [Einstein 1916a, p. 144]. In Mach’s critical analysis, the Newtonian absolute space and absolute time were considered as pure imaginary entities. By way of experiment it was not possible to know anything about them. Mechanics should be founded, according to Mach, on experimental knowledge about the rela- tive motions and positions of bodies. No absolute space was necessary to

2See [Renn 1994 ] for a comprehensive study about Mach’s influence on Einstein’s formulation of general relativity. Einstein, the universe and the relativity of inertia 41 define inertial frames. Only the relative motion existed, and there was not any difference between rotation and translation. The very existence of relative motions was the basis of Mach’s interpretation of physical measurements. Quoting Mach, “when we say that a body preserves un- changed its direction and velocity in space, our assertion is nothing more or less than an abbreviated reference to the entire universe”[Mach 1883, Engl. tr. p. 286]. Motions had to be referred directly to all masses in the universe, and not to an absolute space. In this way the average motions referred to closest celestial bodies could be considered null, and in this approximation the fixed stars became the reference frame. These criticisms played an important role in Einstein’s concept of in- ertia, which led Einstein to cosmology. Inertia could not be interpreted as an absolute intrinsic property: according to Einstein it had its origin in the interactions among bodies. Since 1912 Einstein dealt with this statement in some papers3. After publishing in 1916 the general the- ory of relativity in its final form, the considerations on the universe as a whole, i.e. on the whole of matter, were for Einstein the natural and yet necessary investigation both in the relationships among space, time and gravitation, and in the derivation of local Dynamics by the effects of the total world content [Barbour 1990, p. 57]. General relativity, indeed, should have expressed that there was neither locally nor glob- ally any independent property of space. Thus the relativity of inertia, i.e. the requirement that the metric should be fully determined by mat- ter, was a fundamental question. Einstein finally achieved this result in 1917, through a suitable relativistic model of the universe. According to Einstein, this model, which he proposed in his 1917 “Cosmological considerations”, was the very proof that his new theory of gravitation could “lead to a system free of contradictions” [CPAE 1998E, doc. 306, p. 293].

3See [Einstein 1995 ] for some selected quotations revealing Mach’s influence on Einstein’s view of inertia. 42 1917: the universes of general relativity

4.1.1 The debate with Willem de Sitter

The road to Cosmology passed through The Netherlands. The initial investigations about different effects of the masses of the universe can be inferred in a few letters between Einstein and Hendrik A. Lorentz (1853-1928) [CPAE 1993 E, doc. 467; CPAE 1998E, doc. 43, doc. 47, doc. 225, doc. 226]. They discussed in particular about the relativity of rotation, and about the relationships among fixed stars, centrifugal forces and Coriolis forces4. However, it was Willem de Sitter who gave important contributions to the cosmological consequences of general relativity5. Einstein and de Sitter met in Leiden in the fall of 19166. They held a fruitful corre- spondence focused in particular on the boundary conditions at spatial infinity. According to Machian view, “in a consistent theory of relativity - Einstein wrote - there can be no inertia relatively to ‘space,’ but only an inertia of masses relatively to one another”[Einstein 1917b, p. 424]. This statement required that at very large distances from all masses the sources of inertia could not influence a mass test: inertia of this body had to be zero. Such a condition was represented by a space-time that at infinity was pseudo-Euclidean. In the absolute reference frame of Newtonian theory this condition could be expressed, in relativistic notation, by requiring that at infin- ity the potentials (gµν’s) assumed the diagonal values of Minkowski flat

4See [Janssen 1999 ] for a detailed study about the rotation in Einstein’s theory. 5It is important to note that just through the fundamental articles by de Sitter about the astronomical consequences of general relativity, which were published in the Monthly Notices of the Royal Astronomical Society in 1916 and 1917 [de Sitter 1916a, de Sitter 1916c, de Sitter 1916d, de Sitter 1917b], the scientific community in Great Britain (thus in particular Eddington) and in United States became aware of Einstein’s new theory of gravitation. 6During a stay in Leiden, from September 27 to October 12, as inferred from [CPAE 1998E, doc. 260, doc. 263]. Einstein, the universe and the relativity of inertia 43 space-time:   −1 0 0 0      0 −1 0 0     0 0 −1 0  0 0 0 +1 The presence of some material sources did not influence these values, except for the g44 (time-dependent) term, which became in this case: 2φ g = 1 + . (4.1) 44 c2 The gravitational potential φ was produced by these sources, and could be calculated by Poisson’s equation. How to formulate this condition inside the framework of the new the- ory of gravitation soon became for Einstein a “fundamentally important question” [Einstein 1917b, p. 421]. In de Sitter’s words, general relativ- ity “has no room for anything whatever that would be independent of the system of reference” [de Sitter 1916b, p. 527]. There was not any absolute property, and relativistic field equations were the “fundamental ones” [de Sitter 1916b, p. 529]. The energy-momentum tensor appeared as the source of the potentials, i.e. as the source of the geometry of space-time. However, this statement was not sufficient to assert that the whole of the gµν’s was of material origin. It was necessary to assign the constants of integration, namely the boundary conditions. As de Sitter wrote, “we must be prepared to have different constants of integration in different systems of reference” [de Sitter 1916b, p. 531]. Above all, in order to preserve the principle of relativity, the values for the potentials at infinity had to be invariant for all transformations [de Sitter 1916c, p. 181]. The original manuscripts of de Sitter which are stored at Leiden Ob- servatory are helpful to reconstruct the first attempts to solve the ques- tion of boundary conditions at infinity. De Sitter reported in a notebook some topics faced about the relativity of rotation during conversations in Leiden among Einstein, de Sitter himself, Paul Ehrenfest (1830-1933) 44 1917: the universes of general relativity and Gunnar Nordstr¨om(1881-1923) on September 28-29, 1916. “Ein- stein - de Sitter wrote - wants the hypothesis of the closeness of the world. He means by that that he makes the hypothesis (conscious that it is a hypothesis which cannot be proven) that at infinity (that is at very large, mathematically finite, distance, but further than any observable material object (...)) there are such masses (...) that the gµν assume certain degenerate values (these have not to be 0, that is a priori not to be said), the same in all systems. (...) He is even prepared to give up the complete freedom of transformation (...). If it is possible to find a set of degenerate values of the gµν that is invariant for a not too restricted group of transformations, is a question that can be solved mathemati- cally. Is the answer no (what Ehrenfest and I expect), then Einstein’s hypothesis of the closeness is untrue. Is the answer yes, then the hypoth- esis is not incompatible with the relativity theory. However, I even then maintain my opinion that it is incompatible with the spirit of the princi- ple of relativity. And Einstein admits that I have the right to do so. Also the rejection of the hypothesis is completely admissible in the relativity theory” [de Sitter Archive, Box S12. Engl. tr. by Jan Guichelaar].

Einstein proposed a set of degenerate values for the gµν’s which were invariant at infinity [de Sitter Archive, Box S12; de Sitter 1916b, p. 532; de Sitter 1916c, p. 181]:

  0 0 0 ∞      0 0 0 ∞     0 0 0 ∞  ∞ ∞ ∞ ∞2

Einstein thought to achieve in this way the relativity of inertia, since these values could satisfy the condition of vanishing inertia of a test mass at infinity. He considered a reference frame in which the gravitational field Einstein, the universe and the relativity of inertia 45

Figure 4.1: Detail from the notebook of de Sitter. The Dutch astronomer reported the set of degenerate values which Einstein suggested during conver- sations in Leiden with de Sitter himself, Ehrenfest and Nordstr¨om,September 28-29, 1916. According to Einstein, such values could satisfy the requirement of the relativity of inertia [from de Sitter Archive, Box S12]. was spatially isotropic. In this approximation the space-time interval7

2 µ ν ds = gµνdx dx (4.2) assumed the simpler form:

2 2 2 2 2 ds = −A[(dx1) + (dx2) + (dx3) ] + B(dx4) . (4.3) √ In a Galilean reference frame, by using the Minkowski metric ( −g = √ 1 = A3B), the momentum terms

√ dxα m −g g (4.4) µα ds became (to a first approximation for small velocities):

A dx m√ α , (4.5) B dx4

7Hereafter the notation which appears in equation 2.1, i.e. the upper-lower indices summation, is used. However, in Einstein’s and de Sitter’s original papers the metric 2 was written as ds = gµν dxµdxν . 46 1917: the universes of general relativity where α, β = 1, 2, 3. In this static case the potential energy was equal √ to m B. By assuming that far from heavenly bodies space-time was pseudo-Euclidean, the condition of vanishing inertia could be mathemat- ically obtained with A diminishing to zero or equivalently with B tending to infinity. The degenerate values for gµν’s proposed by Einstein could satisfy this requirement. Thereby the potential energy became infinite at very great distances, so that any material point could not leave the system [Einstein 1917b, p. 425].

According to de Sitter, these values for the gµν’s were the “natural” [de Sitter 1916c, p. 182] ones. Any other set of different values, i.e. any variation from them, should be determined by some material sources. The meaning of this set of values was that “at infinity the 4-dimensional time-space is dissolved into a 3-dimensional space and a one-dimensional time” [de Sitter 1916c, p. 182]: de Sitter pointed out that, by this ref- erence frame, the time coordinate recalled some properties of Newtonian absolute time. Some portions of the observable universe, placed at very great dis- tances from material sources, could be well approximated by diagonal gµν’s of Minkowskian flat space. Since these values were different from the “natural” ones, an explanation of this variation was thus necessary. Ein- stein considered the existence of some hypothetical and unknown masses. These “supernatural masses” [de Sitter 1916c, p. 183] were placed at fi- nite but very large distances from all observable heavenly bodies, and were the source of degenerate values at infinity. Quoting de Sitter, this hypothesis “implies the finiteness of the physical world, it assigns to it a priori a limit, however large, beyond which there’s nothing but the

field of gµν’s which at infinity degenerated into the natural values” [de Sitter1916b, p. 532]. De Sitter criticized this attempt to explain the origin of inertia. He did not accept this “envelope” as a physical reality, because it “will al- ways remain hypothetical and will never be observed” [CPAE 1998E, doc. 272, p. 260]. Observable stars and nebulæ were not part of this Einstein, the universe and the relativity of inertia 47 boundary. Light received from those bodies had approximately the same wave-length as the light from terrestrial bodies, i.e. there was roughly the same deviation from the diagonal Minkowskian values for the former as for the latter. Thereby those stars and nebulæ should have been in- side the envelope [de Sitter 1916c, p. 182]. According to de Sitter, such unknown and “invisible” masses took the same role of absolute space. This explanation of the origin of inertia was not more satisfactory than Newtonian explanation, and moreover was “practically equivalent to no explanation at all, or to a confession of our ignorance” [de Sitter 1916c, p. 183]. De Sitter acknowledged general relativity to represent “an enor- mous progress over the Physics of yesterday” [de Sitter 1916c, p. 183]. He found this ad hoc envelope very hard to accept. “It is not possible - de Sitter wrote to Einstein - that, in the end, the explanation for inertia must be sought in the infinitely small rather than in the infinitely large?” [CPAE 1998E, doc. 272, p. 261]. De Sitter preferred to doubt about the origin of inertia rather than accept this solution8. De Sitter’s remarks persuaded Einstein to modify his own hypothesis. Since the boundary conditions problem was “a purely matter of taste,

8It is worthwhile to note that G. F. Bernhard Riemann (1826-1866) used the same words in 1854, with reference to the relations between geometrical assumptions and empirical determinations. In his fundamental lecture “On the hypotheses which lie at the bases of geometry” Riemann said: “The questions about the infinitely great are for the interpretation of nature useless questions. But this is not the case with the questions about the infinitely small. It is upon the exactness with which we fol- low phenomena into the infinitely small that our knowledge of their causal relations essentially depends. The progress of recent centuries in the knowledge of Mechan- ics depends almost entirely on the exactness of the construction which has become possible through the invention of the infinitesimal calculus, and through the sim- ple principles discovered by Archimedes, Galileo, and Newton, and used by modern Physics. But in the natural sciences which are still in want of simple principles for such constructions, we seek to discover the causal relations by following the phe- nomena into great minuteness, so far as the microscope permits. Questions about the measure-relations of space in the infinitely small are not therefore superfluous questions” [Riemann 1854, Engl. tr. p. 150]. 48 1917: the universes of general relativity which will never gain scientific significance” [CPAE 1998E, doc. 273, p. 261], Einstein abandoned the idea of the distant masses, and gave a different interpretation. Observable regions of the universe contained a small mass when compared to the total mass of the universe. Inside those portions of space “the inertia is determined by the masses there, and only by these masses” [CPAE 1998E, doc. 273, p. 261]. In such a way any envelope was not necessary. In different regions of the universe, the gµν’s and so the inertia were determined both by masses inside those regions and by boundary conditions at infinity. According to Einstein, “no motive remains to place such great weight on the total relativity of inertia” [CPAE 1998E, doc. 273, p. 262]. However, also this new interpretation soon became “intolerable” [CPAE

1998E, doc. 308, p. 296] for Einstein. The problem of the gµν’s at in- finity was still an open issue, and Einstein abandoned the hypothesis of degenerate values. Velocities measured on stars were small when com- pared with the speed of light. For this reason it was possible to consider √ only the contribution of the g44 dx4 term in the expression of the space- time interval. Also because of the same reason, the energy-momentum tensor could be approximated to the T 44 term. However, these approx- imations did not agree with degenerate boundary conditions. As Ein- stein noted, “in the retrospect this result does not appear astonishing” [Einstein 1917b, p. 426]. The small velocities of stars indeed allowed to consider a quasi-static stellar system. The gravitational potential of such a system could not assume extremely large (or infinite) values, nor values much greater than those on Earth. The possibility to assign values that were not invariant at infinity appeared to Einstein as a renounce, and not as a true solution to the problem of boundary conditions: “I must confess - Einstein wrote - that such a complete resignation in this fundamental question is for me a difficult thing” [Einstein 1917b, p. 426]. Another possibility was to assign at infinity the diagonal values of flat space, i.e. to impose at very large distances the scheme of Special Rel- Einstein, the universe and the relativity of inertia 49 ativity. Also this choice was not satisfactory for many reasons. Firstly, such a defined reference frame contradicted the Principle of Relativity [Einstein 1917b, p. 427]. Secondly, this hypothesis required null values for the Riemann tensor Rµνσρ at infinity. This requirement corresponded to assign 20 independent conditions, while only 10 terms of the curvature tensor Rµν were considered in field equations. There was not any phys- ical foundation for this remarkable statement [Einstein 1922a, p. 359]. Moreover, the hypothesis of a universe that was pseudo-Euclidean at in- finity was not satisfactory at all because it involved a sort of spatial origin of inertia. If the gµν’s were constant at infinity, then on large scale the physical properties of space would have not been dependent on matter [Einstein 1922a, p. 359]. As Einstein pointed out, “inertia would indeed be influenced, but not would be conditioned by matter (present in finite space)” [Einstein 1917b, p. 427]. On the contrary, quoting Einstein, “the essence of my theory is precisely that no independent properties are attributed to space on its own” [CPAE 1998E, doc. 181, p. 176]. Einstein did not succeed in assigning suitable boundary conditions at infinity, in order to satisfy both the Machian view on inertia, and the Principle of Relativity. By observed data, the isotropy of space could be roughly considered, in the sense that a uniform distribution of the stars appeared as the “natural” [Einstein 1922a, p. 363] one.

4.1.2 Towards the solution

It was well known that classical celestial Mechanics involved some dif- ficulties in the cosmological framework. According to Newton’s theory, matter should have concentrated in a finite region of the infinite space, the universe being in such a way “a finite island in the infinite ocean of space” [Einstein 1917a, p. 362]. Considering at great distances the solu- tion of Poisson’s equation, the density of stars should have diminished to zero, and there the gravitational potential should have tended to a fixed (or null) limiting value. This world was empty at infinity and was not 50 1917: the universes of general relativity isotropic: it was neither possible to consider all points on average equiv- alent, nor everywhere assign the same mean density of matter [CPAE 1998E, doc. 604, p. 630]. In Newtonian theory, as Einstein pointed out, it was possible to imag- ine the universe gradually losing its content. As the radiation was able to radially pass away, in the same way heavenly bodies with enough kinetic energy could leave the system of stars and “lost in the infinite” [Ein- stein 1917b, p. 422]. Furthermore, this world built on Newtonian laws could not exist from the statistical point of view. Comparing stars to gas molecules, the application of Boltzmann’s law required a finite den- sity ratio among different points. This condition corresponded to a finite difference of the gravitational potential between the center of the system and the infinity. Thus a null value for the density at infinity would have required a null value for the density at the center9 [Einstein 1917b, p. 422]. As seen in Chapter 3, at the end of XIX Century both von Seel- iger and Neumann introduced an exponential term in the expression of gravitational potential in order to solve the cosmological difficulties in Newtonian theory. At the time of writing his “Cosmological consider- ations”, Einstein was not aware of this attempt [Einstein 1919, p. 89], and proposed a similar modification. Einstein pointed out that his own method “does not in itself claim to be taken seriously” [Einstein 1917b, p. 423], but was useful to suggest a possible solution. According to Einstein, also the solution proposed by von Seeliger, with which Einstein became acquainted some months later, could “free ourselves from the distasteful conception that the material universe ought to possess something of the nature of a center”10 [Einstein 1917a, p. 362], but at the same time was “a complication of Newton’s law which has neither empirical nor theoretical foundation” [Einstein 1917a, p. 363].

9See [Kerszberg 1989, pp. 145-152], for some aspects on this subject. 10This statement is taken from the “Part III: Considerations on the Universe as a Whole” of [Einstein 1917a], which Einstein added in the third edition of 1918. Einstein, the universe and the relativity of inertia 51

Einstein considered an extension in Poisson’s equation:

∇2φ − λφ = 4πGρ. (4.6)

The solution of such a modified expression was: 4πG φ = − ρ (4.7) λ 0

This solution was dynamically correct. In this relation, λ and ρ0 re- spectively denoted a universal constant and a uniform density of mat- ter. Considering the universe as a whole, stars and planets represented some local not homogeneous distributions of matter. Therefore ρ0 cor- responded to the hypothetical mean density of fixed stars, if the latter were assumed to be uniformly distributed through space. “In order to learn something of the geometrical properties of the universe as a whole” [Einstein 1922a, p. 363], it was indeed convenient to consider a continue distribution of matter with a uniform and extremely small (but not zero) density. Being ρ0 constant, φ was constant too, so the universe did not have a center. There was not any preferred position nor any preferred direction. Some local non-uniform distributions of matter were related to some variations in the potential φ. In this case, the latter was roughly equal to the Newtonian field, because λφ could be chosen very small when compared with 4πGρ [Einstein 1917b, p. 423]. Einstein extended this modification in the framework of his new the- ory of gravitation. He expressed in general relativity that both the po- tential and the mean density of matter remained constant in space and in time. In order to satisfy these conditions, Einstein respectively eliminated spatial infinity and introduced in his field equations the so-called cosmo- logical term, namely the fundamental tensor gµν multiplied by −λ, a uni- versal but unknown constant. The condition of spatial closure ensured that both the gravitational potential and the mean density of ponderable matter remained constant in space. He introduced the cosmological con- stant accounting for the supposed static nature of the universe, i.e. to preserve the gravitational potential and the density of matter constant 52 1917: the universes of general relativity in time. In this way field equations expressed the observational evidence of the static equilibrium of the universe. His fundamental paper “Cosmological considerations in the general theory of relativity” was published in 1917, February 15. It represented “the first serious proposal for a novel topology of the world as a large” [Pais 1982, p. 286]. Einstein was aware that this solution could appear “outlandish” [CPAE 1998E, doc. 293] or “adventurous” [CPAE 1998E, doc. 298] to his colleagues, and thought “to have erected, from the standpoint of astronomy, but a lofty castle in the air” [CPAE 1998E, doc. 311, p. 301]. However, Einstein was satisfied because, by this model of the universe, “nothing new can be found from general relativity anymore: identity of inertia and gravity; the metric behavior of matter is determined by the interaction of the bodies; independent properties of space do not exist. With this, in principle, all is said” [CPAE 1998E, doc. 453, p. 459].

4.1.3 “A ‘finite’ and yet ‘unbounded’ universe”

As already mentioned in Chapter 3, Schwarzschild studied in 1900 the astronomical consequences of an elliptic geometry of space. It is interesting to note, as pointed out in [Schemmel 2005, p. 470], that Schwarzschild dealt with the possibility of a closed universe with elliptic geometry as a solution of field equations in some correspondence with Einstein already in early 1916 [CPAE 1998E, doc. 188]. Einstein avoided the difficulty to obtain boundary conditions at spa- tial infinity by using a very similar solution, i.e. by considering the uni- verse “as a continuum which is finite (closed) with respect to its spatial dimensions”[Einstein 1917b, p. 427]. There were two possibilities: there could be an infinite extension of the universe or a finite one. Non-Euclidean geometry allowed to right- fully consider the finiteness of space without contradictions respect to experimental results [Einstein 1917a, p. 364]. A spatially infinite uni- Einstein, the universe and the relativity of inertia 53 verse was possible only with a vanishing density of matter, in the sense that “the ratio of the total mass of the stars to the volume of space (...) approaches zero as greater and greater volumes are considered” [Einstein 1921a, p. 215]. Such a hypothesis of vanishing density was admissible from a logical point of view, but was less probable than a finite aver- age density of matter in the universe [Einstein 1922a, p. 368]. The possibility to assign this non-zero density was useful to eliminate some difficulties due to local non-homogeneous distributions of masses. This method emulated the way of the geodesists, who, through an ellipsoid, “approximate to the shape of the earth’s surface, which on a small scale is extremely complicated” [Einstein 1917b, p. 428]. Einstein based his considerations both on the hypothetical density of matter, and on the supposed static nature of the world-system, i.e. on a “magnitude (‘radius’) of space independent of time” [Einstein 1917a, p. 392]. At the beginning of XX Century, as seen, no large-scale sys- tematic velocity fields were indeed pointed out in stellar motions by ob- servations. Velocities measured on stars were very small when compared with the speed of light, so a reference frame existed in which the mat- ter approximately was permanently at rest. In this approximation the energy-momentum tensor became:   0 0 0 0     0 0 0 0    0 0 0 0  0 0 0 ρ c2

The hypothesis of a constant gravitational field corresponded to the gµν’s independence of the time-coordinate x4 in the equation of motion of a material point, d2x dx dx µ + Γµ ν σ = 0. (4.8) ds2 νσ ds ds Thus g44 = 1 for every xµ. Moreover, the static case corresponded to g0α = 0. In this approximation the space-time interval resulted:

2 2 α β ds = dx4 − gαβdx dx . (4.9) 54 1917: the universes of general relativity

The odd consequence of this approximation, as Einstein pointed out, was that “now a quasi-absolute time and a preferred coordinate system do reappear at the end, while fully complying all the requirements of relativity” [CPAE 1998E, doc. 298]. According to Einstein, even if “the ‘spatial’ or ‘temporal’ nature is real, it is not ‘natural’ for one coordinate to be temporal and the others spatial” [CPAE 1998E, doc. 270, p. 257].

It was necessary to give an expression for the gαβ’s, i.e. for the poten- tial terms independent of time. Einstein considered a spherical spatial continuum, as the geodesist way11. Quoting Einstein, “the curvature of space is variable in time and place, according to the distribution of matter, but we may roughly approximate to it by means of a spherical space”12 [Einstein 1917b, p. 432]. The hypothetical uniform distribution of matter allowed a constant curvature of space. On a two-dimensional spherical surface, which is defined in a 3-dimensional Euclidean space, all points are likewise equivalent. Einstein used the 3-dimensional analogy: the universe could be well represented through a 3-dimensional spherical space with constant curvature. As noted by de Sitter in March, 1917, “the idea to make the 4-dimensional world spherical, in order to avoid the necessity of assigning boundary conditions, was suggested several months ago by Prof. Ehrenfest, in a conversation with the writer. It was, how- ever, at that time not further developed” [de Sitter 1917a, p. 1219]. This universe had no preferred points; it had a finite volume equal to 2πR3 (dependent on the radius R), and its surface was unbounded [Einstein 1917a, p. 367]. In this framework, the hypothesis of an isotropic and homogeneous space can be seen as the first suggestion to what became some years later

11Einstein’s suggestion to assume global average property of matter when consid- ering extragalactic scales (as the “geodesist way”) turned out to be one of the typical features in modern approach to cosmology. 12In order to better describe the universe as a whole, right after publishing the “Cos- mological considerations”, Einstein acknowledged the elliptical geometry as “more obvious” than the spherical one [CPAE 1998E, doc. 319, doc. 359]. Einstein, the universe and the relativity of inertia 55 the “Cosmological Principle”. Introducing (as a mathematical tool) a 4-dimensional Euclidean space, such a hyper-sphere was defined by:

2 2 2 2 2 R = ξ1 + ξ2 + ξ3 + ξ4 . (4.10)

All points of this hyper-surface filled the 3-dimensional continuum of the spherical space with constant curvature R. The line element of the 4-dimensional hyper-surface was

2 2 2 2 2 dσ = dξ1 + dξ2 + dξ3 + dξ4 . (4.11)

The projection on the hyper-plane ξ4 = 0 could be used to obtain the line element of the spherical space:

2 α β dσ = γαβdξ dξ , (4.12)

ξαξβ γαβ = δαβ + 2 2 2 2 , (4.13) R − (ξ1 + ξ2 + ξ3 ) where α, β = 1, 2, 3 and δαβ = 1 if α = β; δαβ = 0 if α 6= β. In such a way Einstein found for his static model of the universe this expression for the spatial potentials: · ¸ xαxβ gαβ = − δαβ + 2 2 2 2 . (4.14) R − (x1 + x2 + x3) This model fully achieved the relativity of inertia. The condition of closure of the universe replaced boundary conditions at infinity. There was not any independent property of space which claimed to the origin of inertia, so the latter was entirely produced by masses in the universe.

4.1.4 The cosmological constant

As seen, the requirement of a gravitational field independent of time led to the formulation of the metric of the universe. However, there was not any agreement between these values and field equations. For this 56 1917: the universes of general relativity

Figure 4.2: Einstein’s static model of the universe. One of the three spatial dimensions is shown, and Einstein’s spherical world corresponds to the surface of the cylinder. The time coordinate is along the vertical axis [from Robertson 1933, p. 70].

reason Einstein modified the latter. In fact a spatially finite universe ap- peared to be a good solution to the cosmological problem, i.e. to achieve the relativity of inertia. “From the equations of the general theory of relativity - Einstein wrote - it can be deduced that this total reduction of inertia to interaction between masses, as demanded by E. Mach, for ex- ample, is possible only if the universe is spatially finite” [Einstein 1921a, p. 215]. It was thus necessary to find a suitable formulation of relativistic field equations to express also in general relativity the observed evidence Einstein, the universe and the relativity of inertia 57 of a static equilibrium of the universe13. Einstein introduced a new term on the left-hand side of field equa- tions: 1 R − g R − λg = −κT . (4.15) µν 2 µν µν µν

He added the fundamental tensor gµν multiplied by −λ, an unknown constant. This constant was sufficiently small, so that the modified field equations were “compatible with the facts of experience derived from the solar system” [Einstein 1917b, p. 430]. Both the general covariance and the laws of conservation of momentum and energy were still satisfied. The solution of field equations for the simplest case of one of the two world points with coordinates x1 = x2 = x3 = x4 = 0 led to the connection among the new universal constant, the radius of the universe and the mean density of matter: κρc2 1 λ = = ; (4.16) 2 R2 13It is useful to note an interesting analogy about the idea of the quasi-static equilib- rium of the universe. “I do not seriously consider believing - Einstein wrote to Besso - that the universe is statistically and mechanically at equilibrium, even though I argue as I do. The stars would all have to conglomerate, of course (if the available volume is finite)” [CPAE 1998E, doc. 308, p. 296]. In 1897 Boltzmann illustrated some analogue considerations on the equilibrium of the universe. According to Boltzmann, “the universe as a whole is in thermal equilibrium, and therefore dead.”. However, “there must be here and there relatively small regions of the size of our galaxy (which we call worlds), which (...) deviate significantly from thermal equilibrium. Among these worlds the state probability increases as often as it decreases. For the universe as a whole the two direction of time are indistinguishable, just as in space there is no up or down” [Boltzmann 1897, Engl. tr. p. 242]. The question of the arrow of time has been faced in a famous debate between Einstein and Walter Ritz (1878-1909) in 1909. In Einstein words, the “irreversibility of electromagnetic elementary processes”, as well the “irreversibility of the elementary processes of atomic motion” [Einstein 1909a, p. 359], “is exclusively due to reasons of probability” [Einstein 1909b]. Even if we have not found explicit references to the paper of Boltzmann in Einstein’s papers and correspondence, the Boltzmann’s proposal for the universe as a whole in thermal equilibrium could have been of some importance for Einstein’s original idea of the universe as a whole in static equilibrium. 58 1917: the universes of general relativity

4π2 M = ρ 2π2R3 = R. (4.17) κc2 −22 By using the spatial density of matter from star counts (ρ0 ' 10 g/cm3), the previous relation led to a world-radius R = 107 light-years, whereas the farthest visible stars were estimated at 104 light-years [CPAE 1998E, doc. 308, p. 297]. As Einstein pointed out, the new λ-term “is not justified by our ac- tual knowledge of gravitation. (...) That term is necessary only for the purpose of making possible a quasi-static distribution of matter” [Ein- stein 1917b, p. 432]. The modified field equations were consistent with the metric of the static and closed universe. According to Einstein, “the burning question whether the relativity concept can be followed through the finish or whether it leads to contradictions” was thus solved. Actu- ally, through this extension of field equations, Einstein was “no longer plagued with the problem, while previously it gave no peace” [CPAE 1998E, doc. 311, p. 301].

4.2 The universe of de Sitter

Right after Einstein’s model appeared, de Sitter proposed his own solution of field equations. The Dutch astronomer admired Einstein’s conception of the universe “as a contradiction-free chain of reasoning” [CPAE 1998E, doc. 312], and gave a different solution also maintaining the λ-term. However, de Sitter preferred the original relativistic theory of gravi- tation, “without the undeterminable λ, which is just philosophically and not physically desirable” [CPAE 1998E, doc. 313, p. 305]. According to de Sitter, indeed, the cosmological constant “is a name without any meaning, which (...) appeared to have something to do with the consti- tution of the universe; but it must not be inferred that, since we have given it a name, we know what it means. (...) It is put in the equa- tions in order to give them the greatest possible degree of mathematical The universe of de Sitter 59 generality” [de Sitter 1932b, p. 121].

4.2.1 The “mathematical postulate of relativity of inertia”

De Sitter approached the cosmological problem in a different way. He proposed, as a kind of reply to Einstein’s model, a finite and empty universe, carrying on a mathematical conception of inertia. The corre- spondence of de Sitter reveals that, in this framework, some important suggestions can be attributed to Ehrenfest, colleague of de Sitter in Lei- den during those years [CPAE 1998E, doc. 321]. De Sitter was “sceptical” [CPAE 1998E, doc. 327] about Einstein’s model, and about the assumption that the world was mechanically quasi- stationary. According to de Sitter, “all extrapolation is uncertain. (...) We only have a snapshot of the world, and we cannot and must not con- clude (...) that everything will always remain as at that instant when the picture was taken” [CPAE 1998E, doc. 321]. Otherwise “the extrap- olation leads us - de Sitter wrote to Einstein - to assume that we have solved a puzzle, when we have just clothed it in other words” [CPAE 1998E, doc. 321, p. 312]. Moreover, the hypothetical average density of “world matter” ρ was objectionable, because the distribution of stars in the observable portion of the universe was extremely not homogeneous. De Sitter proposed a distinction between the “world matter” and the “ordinary matter”. The former was hypothetically distributed through space with density ρ0. The latter corresponded to observable objects as planets and stars, i.e. to locally condensed world matter with density ρ1. By this assumption, de Sitter pointed out that “inertia is produced by the whole of world matter, and gravitation by its local deviations from homogeneity” [de Sitter 1917b, p. 5]. Neglecting all pressures and internal forces, and supposing all matter to be at rest, the energy-momentum tensor became:

2 T44 = (ρ0 + ρ1) c g44. (4.18) 60 1917: the universes of general relativity

De Sitter made the hypothesis to neglect gravitation on large-scale, and to take ρ0 constant. The 3-dimensional finite world proposed by Einstein satisfied what de Sitter called the “material relativity requirement” [CPAE 1998E, doc. 321, p. 312], or equivalently the “material postulate of relativity of in- ertia” [de Sitter 1917b, p. 5]. This requirement denied the possibility of the existence of a world without matter: “if all matter - de Sitter wrote - is supposed not to exist, with the exception of one material point which is used as a test-body, has then this test-body inertia or not? The school of Mach requires the answer No. (...) This world matter, however, serves no other purpose than to enable us to suppose it not to exist” [de Sitter 1917a, p. 1222].

According to de Sitter, the postulate that at infinity all gµν’s were invariant for all transformations, i.e. the requirement that the metric satisfied general covariance, was more important than the Machian pos- tulate of relativity of inertia introduced by Einstein. Quoting de Sitter, the relativity of inertia was only “a somewhat vague phrase to which var- ious meaning were attached” [de Sitter 1933, p. 158]. This interpretation is the main difference between Einstein’s and de Sitter’s approaches. As de Sitter pointed out, in Einstein’s model, for the hypothetical value R → ∞, the whole of gµν’s degenerated to   0 0 0 0     0 0 0 0   0 0 0 0 0 0 0 1

This set of values was invariant for all transformations for which, at infinity, t0 = t. In other words, in Einstein’s world it was possible to

find systems of reference in which the gµν’s only depended on space- variables, and not on “time”. However, the “time” of such a systems had “a separate position” [de Sitter 1917a, p. 1223], because it was “the same always and everywhere” [de Sitter 1917b, p. 11]. For such a The universe of de Sitter 61 reason, according to de Sitter, the time coordinate in Einstein’s model was nothing else than an absolute time, and there the world matter took “the place of the absolute space in Newton’s theory, or of the inertial system” [de Sitter 1917b, p. 9]. De Sitter proposed that the potentials should have degenerated at infinity to the values:   0 0 0 0     0 0 0 0   0 0 0 0 0 0 0 0

“If at infinity - de Sitter claimed - all gµν’s were zero, then we could truly say that the whole of inertia, as well as gravitation, is thus produced. This is the reasoning which has led to the postulate that at infinity all gµν’s shall be zero” [de Sitter 1917b, p. 4]. De Sitter called this requirement the “mathematical relativity condition” [CPAE 1998E, doc. 321, p. 312], or the “mathematical postulate of relativity of inertia” [de Sitter 1917b, p. 5]. Indeed, such a condition corresponded to the possibility that “the world as a whole can perform random motions without us (within the world) being able to observe it” [CPAE 1998E, doc. 321, p. 312]. “The postulate of the invariance of the gµν’s at infinity - de Sitter concluded - has no real physical meaning. It is purely mathematical” [de Sitter 1917a, p. 1223].

4.2.2 A universe without “world matter”

In a letter to Einstein [CPAE 1998E, doc. 313] de Sitter proposed his own solution for the metric of the universe as a whole, actually the second relativistic model in modern Cosmology. The Dutch astronomer considered field equations with the λ-term and without matter, i.e. he assumed ρ0 = 0: 1 R − g R − λg = 0. (4.19) µν 2 µν µν 62 1917: the universes of general relativity

These equations were satisfied by the gµν’s given by the space-time in- terval: −dx2 − dy2 − dz2 + c2dt2 ds2 = . (4.20) λ 2 2 2 2 2 2 [1 − 12 (c t − x − y − z )]

The coordinates (x, y, z, t) could have infinite values, provided that gµν’s were null at infinity. Such a condition was equivalent to the finiteness of the world in natural (proper) measure. In fact the length of any semi-axis in natural measure was: Z ∞ √ Lα = −gαα dxα. (4.21) 0

A finite world (i.e. a finite value of Lα) necessary implied gαα = 0 for xα → ∞, and vice versa [de Sitter 1917b, p. 5]. De Sitter pointed out that in his model no world matter was necessary, and the insertion of the λ-term satisfied the mathematical postulate of relativity of inertia. In this system there was not any universal time, nor any difference between the “time” and the other coordinates: none of these coordinates had any physical meaning [de Sitter 1917b, p. 11]. The cosmological constant determined the value of the curvature radius R: 3 λ = . (4.22) R2

By using an imaginary “time”-coordinate ξ4 = ict, the geometry of de Sitter’s world was that of a 4-dimensional hyper-sphere which could be described in a 5-dimensional Euclidean space:

2 2 2 2 2 2 R = ξ1 + ξ2 + ξ3 + ξ4 + ξ5 . (4.23)

In hyper-spherical coordinates the line element of such a 4-dimensional world resulted:

ds2 = −R2{dω2 + sin2 ω[dζ2 + sin2 ζ(dψ2 + sin2 ψ dθ2)]}, (4.24) where 0 ≤ θ ≤ 2π; 0 ≤ ψ, ζ, ω ≤ π. Equivalently, by replacing the imaginary “time”-coordinate ξ4 with a real time-coordinate (ξ4 → iξ4), The universe of de Sitter 63 the geometry of de Sitter’s world corresponded to a 4-dimensional hyper- boloid in a 4+1-dimensional Minkowski space-time:

2 2 2 2 2 2 R = ξ1 + ξ2 + ξ3 − ξ4 + ξ5 . (4.25)

By pseudo-spherical coordinates (with iω0 = ω), the interval of space- time resulted:

ds2 = R2{dω02 − sinh2 ω0[dζ2 + sin2 ζ(dψ2 + sin2 ψ dθ2)]}, (4.26) where 0 ≤ θ ≤ 2π; 0 ≤ ψ, ζ ≤ π; −∞ < ω0 < +∞. The potentials in the hyper-spherical coordinate system were: · ¸ xµxν gµν = − δµν + 2 2 2 2 2 . (4.27) R − (x1 + x2 + x3 + x4) Thus the solution proposed by de Sitter, −dx2 − dy2 − dz2 + c2dt2 ds2 = , (4.28) λ 2 2 2 2 2 2 [1 − 12 (c t − x − y − z )] could be obtained by the stereographic projection of the 4-dimensional hyper-sphere on the Euclidean space, or equivalently by the projection of the hyperboloid on a 3+1-dimensional Minkowski space-time14 [CPAE 1998 Editorial, p. 353]. “If a single test particle - de Sitter wrote to Einstein - existed in the world, that is, there were no sun and stars, etc., it would have inertia” [CPAE 1998E, doc. 313, p. 303]. Actually in the universe proposed by de Sitter a suitable metric was obtained without any physical masses. Such forms of matter as stars and nebulæ were to be regarded as “test particles” in a fixed background metric, which curvature was determined by the cosmological constant [Bernstein-Feinberg 1986, p. 10].

4.2.3 Einstein’s criticism

Einstein acknowledged de Sitter’s solution to be “very interesting” [CPAE 1998E, doc. 317, p. 308], but “must have been disappointed”

14See later, Chapter 5, for further descriptions of the geometry of de Sitter’s world. 64 1917: the universes of general relativity

Figure 4.3: Example of the stereographic projection of a spherical surface on the Euclidean space [from Harrison 2000, p. 376].

[Pais 1982, p. 287], and tried to discard this anti-Machian solution: “I cannot grant - Einstein wrote to de Sitter - your solution any physical possibility” [CPAE 1998E, doc. 366]. Actually, the cosmological term took a fundamental role in de Sitter’s model in order to involve a sort of spatial (and not material) origin of inertia. “The gµν field - Einstein replied to de Sitter - should be fully determined by matter, and not be able to exist without the latter”[CPAE 1998E, doc. 317, p. 309]. First of all Einstein objected that the hyperboloid surface λ 1 − (c2t2 − x2 − y2 − z2) = 0 (4.29) 12 was a singularity. On this surface there was a discontinuity, because the g44 term “jumped” [CPAE 1998E, doc. 317, p. 309] from +∞ to

−∞, and gαα’s from −∞ to +∞. Such a surface lied in the physically finite, but it was not possible to assume infinite values for the poten- tials, because of the supposed static nature of the universe and the small velocities measured on stars. Moreover, the 4-dimensional continuum proposed by de Sitter did not have the property that all its points were equivalent [CPAE 1998E, doc. 351]. It had indeed a preferred point, i.e. the center of the conic section

1 + (c2t2 − x2 − y2 − z2) = 0. (4.30) The universe of de Sitter 65

De Sitter replied that the hyper-surface involved a finite natural spatial distance and an infinite natural temporal distance. Thus the disconti- nuity was only apparent, and this problem was “not interesting” [CPAE 1998E, doc. 327]. Also the supposed preferred point was later shown to be a geometrical consequence of that choice of coordinates, and not a true physical aspect. “My 4-dimensional world - de Sitter remarked to Einstein - also has the λ-term, but no world matter”[CPAE 1998E, doc. 363]. In order to better compare his own model with Einstein’s solution, de Sitter proposed another expression of the metric [CPAE 1998E, doc. 355]. By using spherical polar coordinates, he represented the hyper- boloid universe (system B) as the Einstein’s universe (system A), i.e. as 3-dimensional spheres embedded in a 4-dimensional Euclidean space: r ds2 = −dr2 − R2 sin2 (dψ2 + sin2 ψdθ2) + c2dt2, (4.31) A R r r ds2 = −dr2 − R2 sin2 (dψ2 + sin2 ψdθ2) + cos2 c2dt2. (4.32) B R R De Sitter pointed out that, between the possible forms of space with constant curvature, the elliptical space was more preferable than the spherical one. He acknowledged the importance of the already mentioned 1900 work by Schwarzschild [CPAE 1998E, doc. 355]. In the elliptical space, which also was closed respect to its dimensions, any two straight lines could not have more than one point in common, so there were not the “antipodal” points which on the contrary were present in spherical space [de Sitter 1917b, p. 8]. Einstein agreed with de Sitter on the choice of elliptical space [CPAE 1998E, doc. 359], but he noticed that the spherical geometry he used in the “Cosmological considerations in the general theory of relativity” was just an approximation. According to Einstein, it was useful to show “through an idealization, that a spatially closed (finite) system is possible. (...) The system could actually be quite irregularly curved, also on a large scale, that is, it could relate to the spherical world like a potato’s surface to a sphere’s surface” [CPAE 1998E, doc. 356, p. 346]. 66 1917: the universes of general relativity

Figure 4.4: Example of the elliptical space, which does not have antipodal points and covers half of the surface of a sphere [from Harrison 2000, p. 202].

By the new expression of the line element (the so-called “static form”) it was clear that all the points in de Sitter’s world were equivalent. How- ever, as Einstein pointed out, the g44 coefficient of the temporal term in 2 r system B depended on position. Being g44 = cos ( R ), such a potential π changed its value from 1 (for r = 0) to 0 (for r = 2 R). According to π Einstein, time clocks slowed down approaching r = 2 R: this null value of the potential involved that all masses had the tendency to aggregate at this “equator” [CPAE 1998E, doc. 363]. “It seems - Einstein wrote in 1918 - that no choice of coordinates can remove this discontinuity. (...) We have to assume that de Sitter solution has a genuine singularity on π the surface r = 2 R in the finite domain. (...) The de Sitter system does not look at all like a world free of matter, but rather like a world whose π matter is concentrated entirely on the surface r = 2 R”[Einstein 1918b, p. 37]. According to Einstein, a free of matter solution of field equations was inconceivable. Through his critical comment to de Sitter’s solution, Ein- stein advocated the belief that the cosmological constant did not involve any sort of spatial origin of inertia. The idea of the material origin of inertia inspired by Mach was so important that Einstein elevated the The universe of de Sitter 67 relativity of inertia to one of the fundamental principles of his new the- ory of gravitation. Einstein indeed at almost the same time proposed both such a critical comment on de Sitter’s solution and a new paper on the foundations of General Relativity. In the latter, Einstein wanted to underline the importance of three independent aspects upon which the theory was based. Together with the Principle of Relativity and the Principle of Equivalence, he took into account a third aspect, which he called “Mach’s Principle”15: “The G-field is completely determined by the masses of the bodies. Since mass and energy (...) are the same, and since energy is formally described by the symmetric energy tensor (Tµν), it follows that the G-field is caused and determined by the energy tensor of matter” [Einstein 1918a, p. 33]. De Sitter acknowledged Einstein’s remark about solution B to be correct, but gave a different interpretation. According to the Dutch as- tronomer, such a remark involved a philosophical, and not a physical requirement [CPAE 1998E, doc. 501, p. 523]. In fact, the “equator” at π r = 2 R was at a finite distance in space, but was physically inaccessible [de Sitter 1918, p. 1309]. It was a sort of “mass-horizon”. The velocity π of a material particle became zero for r = 2 R. Thus a material particle which was on the polar line on the origin could have no velocity, nor energy. “All these results - de Sitter stated - sound very strange and paradoxical. They are, of course, all due to the fact that g44 becomes π zero for r = 2 R. We can say that on the polar line the 4-dimensional time-space is reduced to the 3-dimensional space: there is no time, and consequently no motion” [de Sitter 1917b, p. 17]. The time needed by a ray of light, or by a material particle, to travel by any point to the π equator was infinite. Thus the singularity at r = 2 R could never affect any physical experiment [de Sitter 1918, p. 1309]. The debate between Einstein and de Sitter finished with these differ-

15The Einstein-Mach question and its related topics are fundamental issues in the history of physics. We refer to [Barbour 1995 ] for further readings on these still open matters. 68 1917: the universes of general relativity

Figure 4.5: The static form of de Sitter’s universe in pseudo-spherical coor- dinates. Such a universe corresponds to the surface of the hyperboloid (one spatial dimension and the time coordinate). Static coordinates cover only the part of the hyperboloid bounded by the generators. Lines at constant time intersect each other at the ‘equator’ [adapted from Lord 1974, p. 121]. ent interpretations of such a property of de Sitter’s empty universe. The issue about the correctness of de Sitter’s solution was solved by Felix Klein (1849-1925). Einstein, indeed, faced this topic with the au- thoritative mathematician and with Weyl. In some correspondence with Einstein, Klein showed that the singularity at the equator in de Sitter’s universe could be eliminated. Indeed, by using pseudo-spherical coordi- nates of the line element of the hyperboloid form, it could be possible to write the line element in the static form. Thus such a singularity could “simply be transformed away” [CPAE 1998E, doc. 566, p. 593]: the pres- The universe of de Sitter 69 ence of such a “mass-horizon” was only a geometrical consequence of the choice of coordinates, i.e. it was a coordinate singularity, not an intrinsic one16. The matter-free model proposed by de Sitter was free of singular- ities, and its space-time points were all equivalent. Indeed, as shown in [Schr¨oedinger 1957, pp. 14-17] and in [Rindler 1977, p. 185], the appar- ent singularity derives from the fact the coordinates of the static form of de Sitter’s line element cover only a portion of de Sitter hyperboloid. In front of the explanation proposed by Klein, Einstein admitted that de Sitter’s solution existed. However, he still believed that this anti- Machian universe was not a physical possibility [CPAE 1998E, doc. 567]. In particular, Einstein objected that, from Klein’s result, it emerged a non-static interpretation of de Sitter’s model. “For in this world - Ein- stein wrote - time t cannot be defined in such a way that the three- dimensional slices t = const. do not intersect one another and so that these slices are equal to one another (metrically)” [CPAE 1998E, doc. 567, p. 594]. In the hyperboloid representation, hyper-surfaces at differ- ent times intersected each other at the equator, therefore time coordinate could not be uniquely defined. According to Einstein, this singularity- free solution of field equations without matter existed, however such a solution was not static17. Actually, the non-static character of de Sitter’s world which Einstein pointed out from the geometrical interpretation of Klein can be con- sidered as the very first hint in the departure from static cosmological solutions of field equations in modern cosmology. Before the expanding universe entered modern cosmology, however, it was the interest in the

16See [CPAE 1998 Editorial, Earman-Eisenstaedt 1999 ] for detailed reports on the Einstein - Klein - Weyl discussions about such a singularity in de Sitter’s cosmological model. A useful reading about Weyl’s contributions to cosmology, highlighting Weyl’s interpretations of de Sitter’s solution, is [Goenner 2001 ]. 17As we shall see, before the 1930 general acceptance of the expanding universe, Einstein objected both to the non-static universe which Friedmann proposed in 1922 and to the similar solution which Lemaˆıtregave in 1927 that they did not correspond to a physical possibility. 70 1917: the universes of general relativity

“de Sitter effect” which drew the attention in cosmological discussions to- wards the connection between relativistic world models and astronomical data by non-static interpretations of the line element. Chapter 5

The “de Sitter Effect”

The so-called “de Sitter effect” is a theoretical redshift-distance rela- tion which can be obtained through the metric of de Sitter’s universe1. In this chapter the variety of theoretical predictions of such an effect which appeared during the 1920’s is analyzed2. Already in 1917 de Sitter related spectral shifts to velocity and dis- tance of astronomical objects by his relativistic solution. He proposed that spectral displacements which were observed in some stars and neb- ulæ could be interpreted in his static and empty world as an appar- ent (spurious) velocity of test particles due to the peculiar g44 term in de Sitter’s line element, superimposed to a relative velocity which re-

1In present thesis we will generally refer to the de Sitter effect as a redshift-distance relation. However, the strict interpretation would refer to a shift-distance relation, since, as we shall see, the de Sitter effect predicted also approaching motions. 2In Chapter 2 we mentioned different kinds of shift which can be related to astro- nomical observations, i.e. the gravitational, Doppler, and cosmological (expansion) shift. During the historical period which is considered in the present thesis, redshift and blueshift measurements corresponded to velocity measurements, because of the habit in the early phases of relativistic cosmology to directly estimate velocities from spectral displacements by classic or special relativistic Doppler formula. In this sense, the de Sitter effect was considered during the 1920’s, i.e. before the acceptance of the expanding universe, as a redshift-distance relation or equivalently as a velocity- distance relation.

71 72 The “de Sitter Effect” sulted from geodesic equations. The first contribution led to a quadratic velocity-distance relation, while the latter involved a linear dependence. These suggestions did not pass unnoticed. During the 1920’s several scientists dealt with the properties of de Sitter’s universe and proposed different formulations of the redshift-distance effect which resulted by the metric of such a model. Despite its lack of matter, de Sitter’s uni- verse attracted the attention of scientists for several features, such as the not univocal geometrical description, the presence of the singular mass-horizon, and the possibility to explain spectral displacements in as- tronomical objects. In particular, as we shall see, some analysis about the de Sitter effect roughly admitted a linear relation between veloci- ties and distances. Although there was an ambiguous formulation of the theoretical relation between velocities and distances, the de Sitter effect seemed to offer an answer to the question of relevant redshift measure- ments in nebulæ, and up to 1930 such an effect was the only possible, however puzzling, explanation of the redshift problem. Moreover, it turned out that the empty universe of de Sitter could be represented also by stationary frames (with a closed or a flat geometry of spatial sections). Weyl, Lanczos, Lemaˆıtreand Robertson interpreted de Sitter’s universe as a stationary world, by using different definitions of a stationary space-time and different coordinates. Their contributions marked the theoretical departure from a static picture of the universe3. Because of the properties of this model, during the 1920’s de Sitter’s solution was preferred to the “rival” static solution proposed by Einstein, which did not allow to such an interpretation of redshift. The interest in the de Sitter effect raised in 1917 and faded away in 1930, when the observed redshifts were eventually interpreted as due to

3As already mentioned (see Chapter 2), in the retrospect these authors actually used an expanding FLRW frame. The possibility to write the line element of de Sitter’s universe in many ways comes from the fact that such a space-time is a space- time of constant curvature, and there is not a unique choice to specify the 4-velocity which represents the average motion of matter [Ellis 1990, p. 100]. 73 a motion of recession in an expanding universe. The history of the first diffusion of the model of the expanding universe is strictly connected to the history of the investigations of a suitable relation between velocity and distance of spiral nebulæ. Even though de Sitter effect turned out to be of minor importance and, compared to the genuine receding motion of distant nebulæ, was regarded since 1930 as an “imitation recession” [Ed- dington 1933, p. 2], several attempts to predict and to confirm through observations the de Sitter effect foreshadowed the tortuous transition to the expanding universe. After the 1925 acceptance that spiral nebulæ were “island universes”, i.e. really extragalactic systems, in 1929 a linear relation between velocity and distance of these objects was confirmed by Hubble’s observations4. In 1930 the expanding models of the universe independently proposed by Friedmann and Lemaˆıtrealready in, respec- tively, 1922 and 1927 were accepted by the scientific community in order to describe the universe as a whole. Friedmann and Lemaˆıtretook into account in their own dynamical models the possibility of a not empty, ho- mogeneous and isotropic universe with a world-radius increasing in time. Static and stationary models were eventually seen as limiting cases of solutions of field equations describing an expanding universe. Before this second renewal of cosmology, i.e. before the general accep- tance of the expanding universe, investigations about the properties of the model proposed in 1917 by de Sitter played a remarkable role in the first connections between theoretical and observational cosmology which took place during the 1920’s. Between the two rival relativistic models of the universe, de Sitter’s solution offered more advantages than Einstein’s one with regard to astronomical consequences and observations. In this perspective, the history of the de Sitter effect which is below

4We recall that in the picture of present cosmology, the relation which Hubble pro- posed in 1929, the so-called Hubble law, should be written as zc = Hr, i.e. strictly as an empirical redshift-distance relation. Such a law coincides with the general the- oretical velocity-distance relation v = Hr only for small distances and small redshifts compared to Hubble radius. 74 The “de Sitter Effect” reconstructed is useful to highlight the richness of contributions which appeared in the early intersection of predictions offered by relativistic cosmology and confirmations through astronomical observations. In par- ticular, the actors during the 1920’s discussions about the de Sitter effect approached and thoroughly analyzed some fundamental questions in the framework of relativistic cosmology, such as the nature of redshift, the geometry of space, the assumption of a homogeneous and isotropic uni- verse.

5.1 De Sitter’s first suggestion

Einstein and de Sitter, whose insights into the relativity of inertia inaugurated the rise of relativistic cosmology, followed different paths comparing their own theoretical world models to astronomical measure- ments. As seen in Section 4.1.4, Einstein proposed in 1917 a possible value of the radius of his own spherical universe by using star count estimates for the density of matter5. On the contrary, dealing with the features and the geometry of his own empty world, de Sitter wrote the line element of this model in dif- ferent ways, and studied properties of test particles and light rays in it. Investigating the astronomical consequences of general relativity, de Sitter compared in his papers results obtained through solutions A and

5Einstein, in collaboration with Erwin Freundlich (1885-1964), pursued a possible confirmation that a non-zero cosmological constant could be revealed by observations. In particular, in the framework of Newtonian approximation of general relativity, Ein- stein investigated the application of Newtonian law of gravitation to globular clusters. Being globular clusters in stationary equilibrium, through some assumptions on the average masses and luminosity of stars the virial theorem could be used to obtain an average theoretical velocity of stars. Therefore the comparison with observed veloci- ties would have revealed the presence of a non-zero cosmological constant, which was necessary to prevent the collapse of such stellar systems [Einstein 1921b]. De Sitter’s first suggestion 75

B6. It was just the attention de Sitter drew on redshift interpretation which inaugurated the connection of observational astronomy on large scale with theoretical predictions given by models of the universe based on general relativity. In particular, as we shall see, de Sitter was the first to take into account both redshifts measured on stars and relevant re- cession velocities obtained from spectroscopic analysis of nebulæ, clearly attempting to relate these observational evidences to the geometry of the universe [Ellis 1989, p. 372]. De Sitter proposed his own cosmological solution of field equations right after Einstein’s “Cosmological considerations” appeared in 1917 [CPAE 1998E, doc 313; de Sitter 1917a; de Sitter 1917b]. Despite this model, i.e. solution B, was purely obtained from a “mathematical pos- tulate” and corresponded to an empty world, de Sitter attached great importance to its astronomical consequences. In this framework, the correspondence with Kapteyn, whom de Sit- ter asked for advices just during the debate with Einstein, reveals that de Sitter was interested in the suitability of his own model with regard to parallax measurements and to estimates of the mass of our sidereal system. Most of all, de Sitter dealt with the possibility of a systematic redshift from observations of stars and nebulæ. Actually, he was very interested whether a general average redshift could be revealed by obser- vational evidences. “These are hard nuts - Kapteyn replied to de Sitter in June 1917 - you are giving me to crack” [van der Kruit-van Berkel 2000, p. 96]. System B was the 4-dimensional analogy of the 3-dimensional spher- ical space of system A. The line element of system B, following de Sitter, could be written as:

ds2 = −R2{dω2 + sin2 ω[dζ2 + sin2 ζ(dψ2 + sin2 ψ dθ2)]}, (5.1)

6It is important to mention that, beside solutions A and B, de Sitter took into account for the sake of comparison the solution of field equations without λ, i.e. the line element of the special theory of relativity, which he denoted as solution C. 76 The “de Sitter Effect” where ψ and θ were real angles, and ω and ζ were imaginary angles [de Sitter 1917c, p. 229]. Equivalently, avoiding imaginary quantities by ω = iω0 and ζ = iζ0, the metric was:

ds2 = R2{dω02 − sinh2 ω0[dζ02 + sinh2 ζ0(dψ2 + sin2 ψ dθ2)]}. (5.2)

Furthermore, by the new real coordinates χ and η defined as

sin ω sin ζ = sin χ, (5.3) r = R χ, (5.4) tan ω cos ζ = tan iη, (5.5) t = R η, (5.6) the line element in the form 5.1 could be written in the “static” form: r r ds2 = −dr2 − R2 sin2 (dψ2 + sin2 ψdθ2) + cos2 c2dt2. (5.7) R R According to de Sitter, elliptical geometry was more suitable than the spherical one for the absence of antipodal points. In order to describe spaces with constant positive curvature, elliptical space was “really the simpler case, and - de Sitter wrote - it is preferable to adopt this for the physical world” [de Sitter 1917b, p. 8]. Spherical space filled elliptical space twice, and for small values of r compared to R these two spaces could be approximated by the Euclidean space [de Sitter 1917c, p. 231]. Such an elliptical space, as the spherical one, could be projected on an Euclidean space or on a hyperbolical space. In the first case, elliptical space was projected on the whole of Euclidean space through the trans- formation7: r = R tan χ. (5.8)

The line element was: 2 2 2 2 2 2 2 2 d r r [dψ + sin ψdθ ] c dt ds = −¡ ¢ − 2 + 2 . (5.9) r2 2 1 + r 1 + r 1 + R2 R2 R2 7The symbol r was introduced by de Sitter. It represents a spatial coordinate, not a vector. De Sitter’s first suggestion 77

With regard to hyperbolical space, de Sitter considered a new coordinate h, so that: cos χ dh = dr. (5.10) The integral of this expression was: h r r sinh = tan = . (5.11) R R R Thus spatial sections could be described also through hyperbolical space (or space of Lobatschewsky) with constant negative curvature [de Sitter 1917b, p. 13]. The line element could be written:

2 2 h 2 2 2 2 2 2 −dh − sinh R [dψ + sin ψdθ ] + c dt ds = 2 h . (5.12) cosh R Also in this case, the elliptical space was projected on the whole of hyper- bolical space. On the contrary, as de Sitter pointed out, the projection of spherical space of system A, i.e. of Einstein’s model, filled both Euclidean space and hyperbolical space twice [de Sitter 1917c, p. 233]. Therefore de Sitter showed that the spatial geometry of his static and empty model of the universe, with positive curvature proportional 3 to the cosmological constant (ρ0 = 0; λ = R2 ), could be represented by an elliptical geometry, and equivalently described through the projection on Euclidean and hyperbolical spaces. Hyperbolical geometry was useful to derive a formula for parallax. Indeed, as de Sitter pointed out, the rays of light were not geodetic lines in 3-dimensional space (r, ψ, θ), where the speed of light was equal to v = c cos θ. They were not geodesics also in space (r, ψ, θ). Light rays were straight (i.e. geodetic) lines in hyperbolical space (h, ψ, θ), where speed of light was constant in all directions [de Sitter 1917b, p. 13]. The parallax p of a star at a distance r from the Sun could be obtained in such a space from: a h tan p = sinh coth , (5.13) R R being a the average distance between the Sun and the Earth. Thus: a h p = coth . (5.14) R R 78 The “de Sitter Effect”

From equation 5.11 it followed: r a a r2 p = = 1 + . (5.15) R sin χ r R2

In system B, the parallax p of a star was never zero, and reached its r π minimum value at h → ∞, i.e. at the “mass horizon” χ ≡ R = 2 [de Sitter 1917b, p. 13]. On the contrary, in system A, being for spherical a π space p ' r , parallax p was zero for r = 2 R, i.e. at the largest distance which was possible in elliptical space [de Sitter 1917c, p. 233]. A lower limit of the curvature radius, R > 4 · 106 AU, was found in 1900 by Schwarzschild by using hyperbolical spaces and parallaxes of the order of 000.05. This value, according to de Sitter, could be applied in system B, while measured parallaxes did not give a limit of R in system A[de Sitter 1917c, p. 234]. De Sitter proposed different ways to estimate the curvature radius in system A. First, he considered the angular diameter δ of an object of linear diameter d at the distance r from the Earth. In elliptical space the relation with R was given by: d δ = . (5.16) R sin χ

“It is very probable - de Sitter wrote in 1917 - that at least some of the spiral nebulæ or globular clusters are galactic systems comparable with our own in size” [de Sitter 1917b, p. 24]. Taking for example for one of these systems d = 109 and δ = 50, it followed R ≥ 1012 AU in system A. Another estimate was obtained by considering for the star density of the universe (ρ0) the same density at the center of the galactic system (80 stars/1000 pc3); in this case, the curvature radius in system A was R = 9 · 1011 AU. In addition, being space finite and straight lines closed, one should expect in system A to see the antipodal image of the Sun. Since this was not the observed case, light should have been absorbed traveling round the world. Taking an absorption of 40 magnitudes, which had De Sitter’s first suggestion 79 been already proposed in 1900 by Schwarzschild, de Sitter obtained a 1 12 value R > 4 · 10 AU of the radius in Einstein’s universe [de Sitter 1917c, p. 234]. However, de Sitter remarked that “all this of course is very vague and hypothetical. Observation only gives us certainty about the existence of our own galactic system, and probability about some hundreds more. All beyond this is extrapolation” [de Sitter 1917c, p. 237]. Estimates of R in system B could not be obtained by using the fact that the “back of the Sun” was not observed, since in such a system “light requires an infinite time for the voyage round the world” [de Sitter 1917b, p. 26]. Also the relation between apparent and linear diameter of spirals could not be applied. Such a relation was in system B: d d δ = = . (5.17) h R tan χ R sinh R

π Thus δ was zero at r = 2 R.

However, it was the g44 term which was useful to estimate the world radius in system B, leading to the interpretation of spectral displacements in de Sitter’s universe.

5.1.1 Redshifts in de Sitter’s universe

Since in the static form of system B the g44 term diminished with increasing r, the frequency of light vibrations diminished with increasing distances from the observer at rest at the origin of coordinate, i.e., as de Sitter wrote, “the lines in the spectra of very distant objects must appear displaced towards the red” [de Sitter 1917c, p. 235]. According to de Sitter, this effect corresponded to a “spurious positive radial velocity” [de Sitter 1917b, p. 26] of very distant stars and nebulæ. Such a displacement was indeed produced by the inertial field, and was superposed on the displacement due to the gravitational field of objects themselves. In order to explain such a spurious velocity, de Sitter took into account the gravitational contribution to redshift in spectral lines produced by stars, 80 The “de Sitter Effect” i.e. one of the “crucial phenomena” [de Sitter 1933, p. 150] which could be described through Einstein’s new theory of gravitation8. For a fixed star in a general static field, i.e. for a fixed point in 3-dimensional space, the line element could be written:

ds2 = f c2dt2, (5.18) where, for small deviations from the diagonal values of Minkowski space- time (denoted by de Sitter as Galilean values), f was obtained by the Newtonian potential φ:

2φ f = 1 + γ ' 1 + . (5.19) c2

Since dt 1 = √ , (5.20) ds c 1 + γ the measure of time was different at different places in the gravitational field. Spectral lines originating in a strong gravitational field, for example on the solar surface, would be displaced towards the red to an observer in a weaker gravitational field. At the surface of the Sun:

dt 1.00000212 = . (5.21) ds c

The ratio between the observed and emitted wavelengths was:

λ 1 1 0 = √ ' 1 − γ. (5.22) λe 1 + γ 2

Therefore, interpreting the observed shifts through classic Doppler’s for- mula (z ≡ λ0−λe = v ), the displacement on the solar surface was the λe c same as produced by a radial velocity of 0.00000212c, or 0.634 km/sec

8It is worth noting that de Sitter considered also the blueshift contribution due to the general gravitational field of a shell of fixed stars. Such a displacement towards 1 the violet, according to de Sitter, was smaller than about 3 km/sec, and its effect was canceled by the displacement towards the red produced by the gravitational field of each single star [de Sitter 1916c, pp. 175-177]. De Sitter’s first suggestion 81

[de Sitter 1916a, p. 719]. For a star of mass M and density ρ (with solar mass and density M¯ = ρ¯ = 1), such a gravitational shift K was:

2 1 K = 0.634 M 3 ρ 3 . (5.23)

Referring to Campbell’s observations9, Helium stars (i.e. B stars), for which K ' 1.4 km/sec, showed a systematic redshift corresponding to a radial velocity of about v = +4.5 km/sec. De Sitter pointed out that apparent (spurious) positive radial velocities were therefore produced by the diminution of g44 in the line element of his own model. Indeed just one third of this observed value could be associated to the gravitational shift at the star surface. The remaining v = +3 km/sec corresponded to the displacement by the inertial field in system B [de Sitter 1917c, p. 235]. De Sitter used such a displacement in order to calculate the radius of his universe. Since the ratio of the observed and emitted wavelengths was related to the velocity v through Doppler’s theory, it followed [de Sitter 1917c, p. 235]: v f = g = 1 + γ ' 1 − 2 = 1 − 2 · 10−5. (5.24) 44 c 2 r In the static form of system B, g44 = cos R . Assuming the average distance of Helium stars at about r = 3 · 107 AU, it resulted a curvature 2 10 radius of system B of R = 3 · 10 AU [de Sitter 1917b, p. 27]. De Sitter took into account the most relevant radial velocities, both positive and negative, which were known in 1917, and attempted to re- late these observational evidences to the geometry of his own universe. This was the first suggestion which inaugurated the intersection between astronomical observations on large scale and an appropriate description of the universe as a whole given by relativistic solutions of field equations. Referring to data from the 1917 Council of the Royal Astronomical Society about spiral nebulæ [Eddington 1917 ], de Sitter pointed out that

9Campbell’s analysis of the K term from observations of velocities of B stars will be described in next chapter. 82 The “de Sitter Effect” the lesser (meaning the Small) Magellanic Cloud was estimated to be at r > 106 AU, with a radial velocity v ' +150 km/sec. These values gave, for system B, a curvature radius R > 2 · 1011 AU [de Sitter 1917b, p. 27]. By several independent observations, three objects (NGC 4594, NGC 1068 and the Andromeda nebula) showed very large radial velocities com- pared with usual velocities of stars in Sun neighborhood [de Sitter 1917c, p. 236]:

NGC 4594 Pease + 1180 km/sec Slipher + 1190 ” NGC 1068 Pease + 765 km/sec Slipher + 1100 ” Moore + 910 ” Andromeda Wright – 304 km/sec Pease – 329 ” Slipher – 300 ”

Their average velocities were [de Sitter 1917b, p. 27]:

NGC 4594 + 1185 km/sec NGC 1068 + 925 ” Andromeda – 311 ”

Taking v = +600 km/sec as the mean of these velocities, and the curva- 2 10 ture radius R = 3 · 10 AU, the distance of these nebulæ was, at least, r = 4 · 108 AU [de Sitter 1917c, p. 236]. Alternatively, by assuming for these objects a distance of about r = 2 · 1010 AU, the curvature radius of system B was found of the order of R = 3 · 1011 AU [de Sitter 1917b, p. 28]. As showed above, the displacement towards the red depending on the g44 term for all spectral lines in de Sitter’s static universe was: λ − λ 1 r z ≡ 0 e = √ − 1 = sec − 1. (5.25) λe g44 R By considering the spatial projection on Euclidean space, through which De Sitter’s first suggestion 83 the line element of solution B became:

2 2 2 2 2 2 2 2 d r r [dψ + sin ψdθ ] c dt ds = −¡ ¢ − 2 + 2 , (5.26) r2 2 1 + r 1 + r 1 + R2 R2 R2 redshift, i.e. velocity, was related to distance by the quadratic form10:

2 λ0 − λe 1 r ' 2 . (5.28) λe 2 R Superimposed to such a gravitational shift, which was responsible for an apparent velocity, there was another effect in de Sitter’s universe, which de Sitter interpreted as a sort of Doppler effect11 due to the veloc- ity of particles in his empty world. From the geodesic equations, indeed, a test particle in de Sitter’s universe showed a velocity contribution de- pending on distances, which was a “velocity due to inertia” [de Sitter 1917b, p. 27]. With regard to the equations of motion of a material particle in the field of pure inertia, by using coordinates (r, ψ, θ, ct), the differential equations of the geodesic were [de Sitter 1917b, p. 15]: "µ ¶ µ ¶ # d2r r dψ 2 dθ 2 = + r + sin2 ψ , (5.29) c2dt2 R2 c dt c dt

10We recall a present summary of several contributions to shifts, which has been mentioned in Chapter 2. Following [Ellis 1989, p. 374], the gravitational, kinematic and cosmological shifts can be resumed as:

(1 + ztot) = (1 + zDS )(1 + zGS )(1 + zC + zGC )(1 + zDO )(1 + zGO ). (5.27)

In the static frame, as in de Sitter’s original proposal, the term zC is equal to zero. The redshift contribution 5.28, obtained by de Sitter as a gravitational shift from the metric of solution B, corresponds to the term zGC , and reflects the space-time curvature [Ellis 1990, p. 101]. 11 Such a contribution is equivalent to an additional term zDS in the shift summary of the previous note. The same consideration can be applied to the analysis of the contributions to redshift in de Sitter’s universe proposed by Eddington in 1923, as we will see later. However, according to Eddington the amount of the velocity (Doppler) effect was roughly the same as the quadratic distance (gravitational) effect. 84 The “de Sitter Effect” µ ¶ µ ¶ µ ¶ µ ¶ d2θ 2 dr dθ dψ dθ = − − 2 cot ψ , (5.30) c2dt2 r c dt c dt c dt c dt µ ¶ µ ¶ µ ¶ d2ψ 2 dr dψ dθ 2 = − + sin ψ cos ψ . (5.31) c2dt2 r c dt c dt c dt

◦ dψ Taking ψ = 90 , dt = 0, the integration of geodesics gave: µ ¶ dθ r2 = c = k , (5.32) dt 1 µ ¶ µ ¶ dr 2 dθ 2 r2 + r2 = + k . (5.33) dt dt R2 2

In order to find the equation³ of the´ orbit of a material particle, de Sitter 2 2 r0 put k1 = (r0 v0) and k2 = v − 2 , where r0 was the minimum distance 0¡ ¢R dθ from the origin, and v0 = r0 dt 0 was the particle velocity at that point. By using a = r0 and b = (R v0), the equation of the orbit was [de Sitter 1933, p. 195]: µ ¶ dr 2 r2(r2 − a2)(r2 + b2) = . (5.34) dθ a2b2 The integration gave an hyperbola: in system B a material particle under the influence of inertia alone did not describe a straight line with constant velocity. The orbit was an hyperbola of which the real axis was r0 and the imaginary axis was Rv0: x2 y2 − = 1, (5.35) a2 b2 with x = r cos(θ − θ0) and y = r sin(θ − θ0). For v0 = 1 the velocity was equal to the speed of light, therefore in system B, in the reference system (r, ψ, θ, ct), also the orbit of rays of light were hyperbolas whose imaginary axis was R [de Sitter 1917b, p. 19]. The radial velocity was in this notation [de Sitter 1933, p. 195]: µ ¶ µ ¶ µ ¶ dr 2 r2 a2 b2 = 1 − 1 + . (5.36) dt R2 r2 r2

According to this relation, the velocities due to inertia had no preference in sign. De Sitter’s first suggestion 85

The total shift in de Sitter’s universe, i.e. what de Sitter interpreted as a real motion due to inertia together with a spurious redshift contribution 12 due to the diminution of g44, took the form : λ − λ r 1 ³ r ´2 0 e ' ± + . (5.38) λe R 2 R Strictly, it was the quadratic relation which became known as the de Sitter effect [Ellis 1990, p.101]. In 1917, at the very beginning of relativistic cosmology, de Sitter re- marked that, with regard to the relevant radial velocities observed in nebulæ, “conclusions drawn from them are liable to be premature” [de Sitter 1917b, p. 27]. If, however, future observations confirmed a system- atical displacement towards the red, this evidence would have been an indication in order to adopt system B in preference to system A. Since these two systems differed in their physical consequences at large dis- tance, according to de Sitter the study of systematic radial motions of spirals was exactly the key method in order to decide between system A and system B: “if continued observations should confirm the fact that the spiral nebulæ have systematically positive radial velocities - de Sitter remarked - this would be certainly an indication to adopt the hypothesis B in preference to A” [de Sitter 1917b, p. 28]. “At the present time - de Sitter wrote in 1920 - the choice between the systems A and B is purely a matter of taste. There is no physical criterion as yet available to decide between them” [de Sitter 1920, p.

12It is useful to mention the general spectral shift in de Sitter’s universe which North derived in his own book about the history of cosmology, “The measure of the universe”. By using the metric of the static form of de Sitter’s universe, North gave the general formula [North 1965, p. 95]: h ³ r ´i ³ r ´ 1 + z = α 1 + v sec sec2 , (5.37) R R where α was a parameter fixing the orbit of the source. As we shall see in next sections, during the 1920’s several scientists proposed different formulations of the de Sitter effect. According to North, Weyl’s 1923 result is not confirmed through the general formula proposed by North himself [North 1965, p. 101]. 86 The “de Sitter Effect”

867]. Referring to data available in 1920, radial velocities of 25 spirals were known. Among them, as de Sitter pointed out, only three were negative, and the mean velocity was v ' +560 km/sec. Thus there was a rough, not (yet) systematic, observed tendency of spirals to recede, which could eventually lead to discriminate between Einstein’s and de Sitter’s universes. However, quoting de Sitter, “the decision between these two systems must, I fear, for a long time be left to personal predilection” [de Sitter 1920, p. 868].

As seen above, first hints about the interpretation of relevant shift displacements through relativistic solutions of field equations were pro- posed in 1917 by de Sitter, investigating the features of his own model of the universe. In the following years, as we shall see in next sections, several authors dealt with different interpretations of the line element of de Sitter’s universe and the connection of the de Sitter effect to shifts measurements.

De Sitter directed Leiden Observatory from 1919 to 1934, and, after 1920, did not consider the cosmological question in other published pa- pers up to the 1930 second renewal of cosmology. As Eddington wrote in 1934 in an obituary for de Sitter, the Dutch astronomer was “the man who discovered a universe and forgot about it” for at least ten years [Eddington 1934, p. 925].

Some 1929 correspondence between de Sitter and Frank Schlesinger (1871-1943), astronomer at Yale University Observatory, are useful to emphasize de Sitter’s point of view on redshift in his own model. In these letters de Sitter resumed the question about the velocity-distance effect at the turning point of Hubble’s 1929 confirmation that a linear velocity- distance relation existed. De Sitter considered the twofold contribution to redshift predicted by his empty model, and tried to explain the absence of observations of approaching objects which on the contrary were predicted through the term corresponding to negative radial velocities. In a letter to Schlesinger dated November 8, 1929, de Sitter pointed out that, being De Sitter’s first suggestion 87 the approximate formula of the velocity-distance effect v r 1 ³ r ´2 = ± + , (5.39) c R 2 R the first term represented a “real velocity due to repulsive force of the origin of the coordinate, which is a consequence of the adopted form of the gravitational potential” [de Sitter Archive, Box 17.7A]. The observed velocities, de Sitter wrote in this letter, were all positive, and could not be represented by a linear formula. “Why - de Sitter inferred - are all the spirals found on the receding branches and none on the approaching? The evident solution (evident once you think about it) is: some of them (say one half) have originally been on the approaching branches, but have long since past their nearest point, and are now receding” [de Sitter Archive, Box 17.7A]. As de Sitter noted, this suggestion could be attributed to Eddington. Indeed, some months earlier de Sitter and Eddington discussed about spiral nebulæ during a trip to South-Africa, where the Meeting of the British Association for the Advancement of Science took place. On July 2, 1929, de Sitter reported that “Eddington is of the opinion that two big negative velocities are too much of an exception. They are however both in Andromeda, in the direction of our movement by the rotation of the galaxy. He had never noticed this, and thinks (...) that the explanation by ‘de Sitter’s world’ has become much more probable. (...) The velocity is ± at + bt2. The term with a must be preponderant for the clear (and big) nebulæ. Why always +? (this is my question). Eddington seems to want to answer: at the creation all the r’s were small” [de Sitter Archive, Box 43.6. Engl. tr. by Jan Guichelaar]. At the end of December, 1929, de Sitter wrote in another letter to Schlesinger that he had derived empirical formulas for distances of spirals as a function of the diameter and the magnitude, which, quoting de Sitter, “still represent what I think is the best we can do at present” [de Sitter Archive, Box 17.7A]. De Sitter later published these results in [de Sitter 1930a, de Sitter 1930b]. The application of such relations to radial 88 The “de Sitter Effect” velocities led to a linear relation between velocity and distance:

∆ = 2000 v, (5.40)

∆ being the distance in units of 1024 cm, and v the velocity in that of light as a unit. The factor 2000 was the radius of the universe when solution B was adopted (R = 2 · 109 light-years) [de Sitter Archive, Box 17.7A].

Figure 5.1: Detail from a letter by de Sitter to Schlesinger (November 8, 1929). In this letter de Sitter, following Eddington’s suggestion, considered the possibility that some spirals “have originally been on the approaching branches, but have long since past their nearest point, and are now receding” [from de Sitter Archive, Box 17.7A].

“We are confronted - de Sitter wrote to Schlesinger - with the math- ematical problem: what becomes of the empty world B if you fill it with matter. I have not yet been able to solve this problem completely, but I have reasons to expect that the solution will be intermediate between the solutions A and B” [de Sitter Archive, Box 17.7A]. As we shall see, also Eddington considered (already in 1923) the possibility that the actual world corresponded to an intermediate state between Einstein’s and de Sitter’s universes. De Sitter’s first suggestion 89

In 1930, having learned about 1927 Lemaˆıtre’scontributions to the cosmological question, de Sitter “accepted with enthusiasm” [Eddington 1934, p. 925] such a new development of the theory and the related interpretation of redshift as due to a motion of recession in an expanding universe. In 1933, in the appendix of a report on the astronomical aspect of relativity theory, de Sitter returned on the twofold source of redshift in his own empty and static model of the universe. He explained that “in 1917 dr (...) it was not realized that the velocity dt would always be positive, and it was thought that this Doppler effect would not be systematic, ¡ ¢ 1 r 2 the redshift 2 R being the only systematic effect” [de Sitter 1933, p. 195]. From late 1920’s, velocities of very distant celestial objects turned out to be all positive; therefore according to de Sitter such an evidence could be explained in his own model by postulating, as already mentioned in his 1929 letter to Schlesinger, that “all observable bodies are on the receding branches of their respective hyperbolas, having passed the apex long ago, so that none remain on the approaching branches” [de Sitter 1933, p. 195]. In such a 1933 report, de Sitter showed that, being the Doppler ef- r fect a first-order effect proportional ro R , a linear relation was actually predicted also by his static model, having corrected the Doppler formula to the second order. For a general velocity q, the correct Doppler effect formula was: λ − λ 1 0 e = q − q2. (5.41) λe 2 Taking dr r q = = , (5.42) dt R the contribution by the quadratic term was canceled, leaving the linear effect: λ − λ r 0 e = . (5.43) λe R This effect, de Sitter concluded, was “in exact agreement with the result from the new theory of the expanding universe” [de Sitter 1933, p. 196]. 90 The “de Sitter Effect” 5.2 Matter or motion? Eddington’s analy- sis

Eddington and the British scientific community became aware of de- tails of Einstein’s new theory of gravitation through de Sitter’s papers which appeared in 1916 and 1917 in the Monthly Notices of the Royal Astronomical Society13. As from 1917, Eddington dealt with the general theory of relativity in many papers and books, giving fundamental contributions in this field14. With regard to the curvature of space and time and the considerations on the universe as a whole, in front of the two rival possibilities proposed by Einstein and de Sitter, Eddington was inclined to prefer the empty solution of de Sitter. Eddington found Einstein’s model objectionable for some aspects. “I feel - Eddington wrote to de Sitter in August, 1917 - a strong objection to system A. It seems to me that the world-matter in that is simply the aether coming back again” [de Sitter Archive, Relativity Box, A2]. Re- ferring to the presence of antipodal points in Einstein’s world, Eddington regretted “being unable to recommend this rather picturesque theory of anti-suns and anti-stars. It suggests that only a certain proportion of the visible stars are material bodies; the remainder are ghosts of stars, haunting the places where stars used to be in a far-off past” [Eddington 1918, p. 87]. Einstein’s hypothesis allowed to a direct relation between the world radius R and the total amount of matter M in it. As Eddington wrote,

13In June 1916, Eddington had written to de Sitter: “Hitherto I had only heard vague rumors of Einstein’s new work. I do not think anyone in England knows the details of his paper” [de Sitter Archive, Relativity Box, A2]. 14We mention, for example, Eddington’s contribution to the fundamental 1919 ex- periments devoted to the confirmation of the amount of light-bending as predicted by general relativity, for which we refer to [Kennefick 2007 ]. A useful paper about Eddington’s works in relativity and Einstein’s reaction to them is [Stachel 1986 ]. Biographical material about Eddington can be found in [Douglas 1956a]. Matter or motion? Eddington’s analysis 91

“there is something rather fascinating in a theory of space by which, the more matter there is, the more room is provided” [Eddington 1918, p. 90]. However, following de Sitter’s remarks, Eddington criticized the introduction of such a vast quantity of undetected world matter15. Also the cosmological constant was objectionable, being a “very artifi- cial adjustment” [Eddington 1918, p. 87], which was “very hard to accept at any rate without some plausible explanation of how the adjustment is brought abroad” [Eddington 1920, p. 163]. However, as mentioned in Chapter 3, in following years the role of the cosmological constant gained great importance in Eddington’s search for a fundamental theory which could relate constants in nature. The alternative solution proposed by de Sitter seemed “much less open to objection” [Eddington 1918, p. 87]. “If the real time is used - Eddington wrote about solution B - the world is spherical in its space dimensions, but open towards plus and minus infinity in its time dimen- sion. (...) It might seem that this kind of fantastic world building can

15Already in 1916, i.e. before Einstein’s and de Sitter’s cosmological solutions appeared, Eddington criticized Einstein’s initial assumption of unobservable super- natural masses which influenced local observable phenomena. In a letter to de Sitter dated October 13, 1916, Eddington criticized the supernatural masses proposed by Einstein which were responsible for the degeneration of potentials at infinity in the form   0 0 0 ∞      0 0 0 ∞     0 0 0 ∞  ∞ ∞ ∞ ∞2

Quoting Eddington, such hypothesis contradicted “the fundamental postulate that observable phenomena are entirely conditioned by other observable phenomena. It is no great advance - Eddington wrote - to be told that instead of being conditioned by a framework of reference (which we can materialize as the aether), they are conditioned by things equally outside observations at infinity. But then (one asks) where is infinity according to the new conceptions? I do not know that is matters, because - Eddington remarked - infinity is necessarily outside observation and that is the main point” [de Sitter Archive, Relativity Box, A2]. 92 The “de Sitter Effect” have little to do with practical problems. But that is not quite certain” [Eddington 1920, pp. 159-160]. The great advantages of this model were the presence of the potentials gµν which at infinity were invariant for all transformations, and the absence of any assumption of the existence of a large amount of “not yet recognized” world matter [Eddington 1918, p. 88]. De Sitter’s model especially offered the possibility to explain the large velocities of spirals, and permitted to estimate the value of the world radius: the theory of de Sitter, according to Eddington, “is of course very speculative, but is the only clue we possess as to the dimensions of space” [Eddington 1929, p. 167]. In a letter to Shapley, December 1918, Eddington explained that “de Sitter’s hypothesis does not attract me very much, but he predicted this (spurious) systematic redshift before it was discovered definitely; and if, as I gather, the more distant spirals show a greater recession that is a further point in its favour” [quoted in Smith 1982, p. 174]. In his 1920 book “Space, time and gravitation” Eddington mentioned the two interpretations of spectral displacement proposed by de Sitter in 1917. With regard to measurements of motion in the line-of-sight of nebulæ, Eddington pointed out that “the data are not so ample as we should like; but there is no doubt that large receding motions greatly preponderate” [Eddington 1920, p. 161]. Therefore this evidence could be interpreted as a genuine phenomenon of recession; however, it could not be discarded the possibility that such an effect was really due to the slowing down of atomic vibrations, allowing to an apparent recession. The de Sitter effect was faced in details in Eddington’s book “The mathematical theory of relativity”. Such a book, which appeared in 1923, was later acknowledged by Einstein as “the finest presentation of the subject in any language” [Douglas 1956b, p. 100]. In the chapter which Eddington addressed to the curvature of space and time, solu- tions A and B were considered by Eddington “as two limiting cases, the circumstance of the actual world being intermediate between them. De Matter or motion? Eddington’s analysis 93

Sitter’s empty world is obviously intended as a limiting case; and the presence of stars and nebulæ must modify it, if only slightly, in the di- rection of Einstein’s solution” [Eddington 1923, p. 160]. This is a very important remark highlighting Eddington’s approach to the search of a suitable model of the universe, because it contained the suggestion to in- vestigate intermediate solutions between a model which had matter but not motion, i.e. Einstein’s world, and a model which was empty but showed motion, i.e. de Sitter’s world. As we shall see, the 1927 model of the universe proposed by Lemaˆıtre would have represented such an intermediate solution. In 1930 Lemaˆıtre drew the attention of Eddington to the solution of an expanding (and not empty) universe which Lemaˆıtrehad already discovered in 1927. There- fore in 1930 Eddington acknowledged that such a dynamical solution discovered by Lemaˆıtrewas “a brilliant and remarkably complete solu- tion of the various questions connected with the Einstein and de Sitter cosmogonies” [Eddington 1930a, p. 668]. Through Lemaˆıtre’ssolution Eddington proved that Einstein’s world was unstable, claiming that “the equilibrium having been disturbed, the universe will progress through a continuous series of intermediate states towards the limit represented by de Sitter’s universe” [Eddington 1930b, p. 850]. Nonetheless, in 1923, the possibility to discriminate whether de Sit- ter’s or Einstein’s form was “the nearer approximation to the truth” [Eddington 1923, p. 160] was strongly challenged by the preponderance of positive velocities of spirals, which, according to Eddington, favoured the model of de Sitter for its interpretation of redshift. In order to describe the properties of de Sitter’s world, Eddington considered the original line element in the form [Eddington 1923, p. 156]:

ds2 = −R2dχ2 − R2 sin2 χ(dθ2 + sin2 θ dφ2) + R2 cos2 χ dt2. (5.44)

Thus, speed of light at the origin was equal to R. The coordinate χ r corresponded to ( R ), being r the same radial coordinate used by de 94 The “de Sitter Effect”

Sitter. Since, for a clock at rest, the metric was:

ds = R cos χ dt, (5.45) there was a displacement of spectral lines towards the red in distant objects at rest “due to the slowing down of atomic vibrations, which would be erroneously interpreted as a motion of recession” [Eddington 1923, p. 161]. With regard to the question of the mass horizon, Eddington pointed out that, because of the symmetry of the original line element proposed by de Sitter, “there can be no actual difference in the natural phenomena at the horizon and at the origin” [Eddington 1923, p. 157]. The singu- larity in ds2 could be removed, or equivalently introduced, by transfor- mations of coordinates: “it is impossible - Eddington noted - to know whether to blame the world-structure or the inappropriateness of the coordinate-system. (...) I believe then that the mass-horizon is merely an illusion of the observer at the origin, and that it continually recedes as we move towards it” [Eddington 1923, pp. 165-166]. By the new coordinater ˜ defined as:

r˜ = R sin χ, (5.46) the line element of system B became: µ ¶ 1 r˜2 ds2 = −¡ ¢dr˜2 − r˜2dθ2 − r˜2 sin2 θ dφ2 + 1 − dt2. (5.47) r˜2 R2 1 − R2 Here the usual unit of t (with c = 1) was restored. From the geodesic dr equations, and by considering a particle at rest ( ds = 0), it followed: d2r˜ 1 = λ r,˜ (5.48) ds2 3 3 where λ = R2 . Therefore, as Eddington pointed out, “a particle at rest will not remain at rest unless it is at the origin” [Eddington 1923, p. 161], and was repelled away with an acceleration increasing with the distance to the origin. Actually, such a tendency to scatter of particles revealed Matter or motion? Eddington’s analysis 95 that de Sitter’s world “becomes non-statical as soon as any matter is inserted in it. But this property - Eddington claimed - is perhaps rather in favour of de Sitter’s theory than against it” [Eddington 1923, p. 161]. Indeed, such an effect could explain the relevant velocities measured in spiral nebulæ, which represented “one of the most perplexing problems of cosmogony” [Eddington 1923, p. 161]. In his book Eddington reported radial velocities measured in 41 nebulæ by Slipher. The table with these observations [Eddington 1923, p. 162] was arranged just by Slipher, and contained many unpublished data related to objects of the New General Catalogue. In addition, Eddington remarked that, beside these data, an additional nebula (NGC 1700) showed a large receding velocity by the observations of Francis Pease (1881-1938). Among these 42 objects, there was a strong preponderance of positive velocities, which suggested a recession, however not entirely systematic.

Figure 5.2: List of radial velocities of spiral nebulæ measured by Slipher, which appeared in Eddington’s 1923 book ‘The mathematical theory of rela- tivity’ [from Eddington 1923, p. 162].

With regard to this motion of recession observed in spirals, Eddington 96 The “de Sitter Effect” followed de Sitter’s double explanation: beside the motion due to inertia, which for Eddington corresponded to a general tendency of particles to scatter (according to equation 5.48), there was a spurious motion due to the singular g44 term in de Sitter’s metric (equation 5.45). 1 In particular, Eddington pointed out that the acceleration 3 λ r˜ (equa- tion 5.48), if continued for the time (in natural unit) Rχ = r, would have caused a change of the velocity of a test particle of the order of: 1 r˜2 λ r˜2 = . (5.49) 3 R2

Since (from the metric in the form 5.47) the shift due to g44 term resulted µ ¶ 1 r˜ 2 z ' , (5.50) 2 R Eddington concluded that “the Doppler effect of this velocity would be roughly the same as the shift to the red caused by the slowing down of atomic vibrations” [Eddington 1923, p. 164]. Eddington interpreted redshift “as an anticipation of the motion of recession which will have been attained before we receive light” [Edding- ton 1923, p. 164]: nebulæ did not have the motion revealed by redshift at the moment of emission of light, but they acquired such a motion during the time light took to travel towards the observer. However, in 1923 observations did not reveal a systematic shift to the red: from Slipher’s data, some nebulæ (NGC 221, NGC 224, NGC 598) showed relevant negative velocities. This approaching motion (due to blueshift measurements) towards the observer at the origin could not be explained through geodesic equations. Therefore, quoting Eddington, “the cosmogonical difficulty is perhaps not entirely removed by de Sitter’s theory” [Eddington 1923, p. 162]. Eddington returned on such an interpretation of the effect foreseen by de Sitter also in his 1933 book “The expanding universe”. He explained that redshift of galaxies corresponded, now in the theory of the expanding universe, to an actual motion of recession, and were no more to be inter- preted as the “rather mysterious slowing down of time at great distances Weyl, Lanczos and the redshift-distance law 97 from the observer” [Eddington 1933, p. 49]. With regard to his own considerations about de Sitter effect which he had proposed in 1923, Ed- dington remarked in 1933 that “it was a question of definition” whether the slowing down of atomic vibrations was to be regarded as a genuine or spurious motion. “During the time that its light is traveling to us - Eddington wrote about de Sitter effect - the nebula is being accelerated by the cosmical repulsion and acquires an additional outward velocity exceeding the amount in dispute” [Eddington 1933, p. 49]. Therefore, according to Eddington, in the de Sitter effect the velocity was spurious at the time of emission, and became genuine at the time of reception. The analysis of Friedmann’s and Lemaˆıtre’sexpanding models, as Ed- dington pointed out in 1933, showed that the slowing down of time had to be regarded as a second-order term, which was small compared to the effect due to the expansion.

5.3 Weyl, Lanczos and the redshift-distance law

In 1923, the same year when Eddington’s book “The mathematical theory of relativity” appeared, explicit formulations of redshift-distance relations in de Sitter’s universe were independently proposed by Weyl and Lanczos. These significant contributions are related to the depar- ture from a static frame which both Weyl and Lanczos proposed in their own considerations about de Sitter’s world. In particular, Weyl derived in 1923 a redshift-distance relation which was linear at small distances. In 1933, in a summary paper about relativistic cosmology in the framework of the expanding universe, Robertson acknowledged that the “appropri- ate linearity of velocity of recession with distance was first predicted by Weyl in 1923, on the basis of the more restricted cosmology of de Sitter” [Robertson 1933, pp. 68-69]. After the 1917 proposal by de Sitter of an empty universe as a so- 98 The “de Sitter Effect” lution of field equations, Weyl participated in discussions with Einstein and Klein about the mass-horizon singularity in de Sitter’s static form, advocating that de Sitter’s universe actually had matter at its horizon [CPAE 1998E, doc. 556]. By the spherical coordinate system in the static form of de Sitter’s line element, system B corresponded to a really static world “that - Weyl remarked - cannot exist without a mass-horizon” [Weyl 1921, Engl. tr. p. 282]. However, as already mentioned at the end of Chapter 4, Klein showed that the singularity at the mass-horizon was due to the choice of coordinates, and was not an intrinsic singularity. In the hyperboloid version, which was only partially covered by static coordinates, such a world was not static on the whole, and, according to Weyl, “was separated from any static solution of the same topology by an abyss” [quoted in Earman-Eisenstaedt 1999, p. 198]. The de Sitter hyperboloid was “an homogeneous state of the world” [Weyl 1930, p. 300]. Despite the static coordinates represented only a sector of such a hyperboloid, Weyl shifted his own attention to this version of the universe of de Sitter. Indeed such a world-description seemed more satisfactory than Einstein’s one for the observed fact that the further distances were considered, the more increasing velocities were involved. Both the analysis of de Sitter (1917) and Eddington (1923) were lack- ing in a suitable assumption about the undisturbed state of stars when the homogeneity and isotropy of space were taken into account. In this framework, Weyl introduced the concept of causal connection of the el- ements of proper time by postulating the existence of a system of null geodesics diverging towards the future, in which all time moments re- mained equivalent. Following [Bergia-Mazzoni 1999, p. 332], the as- sumption that all null geodesics were oriented in one direction towards the infinite future, and in the other direction towards the infinite past, avoided the overlapping of the past and future light cones. “The different points on the world line of a point-like source - Weyl wrote in 1923 - are the origin of (3-dimensional) surfaces of constant phase that form null Weyl, Lanczos and the redshift-distance law 99 cones opening towards the future” [Weyl 1923a, p. 322. Engl. tr. in Har- vey et al., p. 1017]. These null cones opening to the future time-direction filled a portion of the de Sitter’s universe which Weyl denoted as “the range of the influence of a star. (...) There are ∞3 stars or geodesics to which the same range of influence belongs as to the arbitrarily chosen star” [Weyl 1930, p. 302]. De Sitter’s hyperboloid “is distinguished from the well-known Einstein’s solution, which is based on a homogeneous dis- tribution of mass, by the fact that the null cone of future belonging to a world-point does not overlap with itself; in this causal sense, the de Sitter space is open” [Weyl 1930, p. 301]. Such an assumption - Weyl claimed - “means that the stars of the system are able to act upon one another to eternity” [Weyl 1924, p. 476], and “is that, in the undisturbed state, the stars form such a system of common origin” [Weyl 1930, p. 303]. The hypothesis suggested by Weyl was thus that celestial objects could be assumed as uniformly distributed and at rest respect to the spatial coordinates, whatever the origin of coordinates.

Figure 5.3: Weyl’s postulate. World-lines belong to a pencil diverging towards the future [from Harrison 2000, p. 294].

“Weyl 1923 postulate - Ellis notes - was intended to supply a way of uniquely choosing the family of stationary expanding geodesic world 100 The “de Sitter Effect” lines” [Ellis 1989, p. 375]. Weyl’s fundamental assumption became later known as “Weyl Principle” and actually was the first proposal of causal connection in relativistic cosmology16. By quoting Ehlers, “to specify a cosmological model, he [Weyl] realized, it does not suffice to choose a space-time, i.e. a Lorentz manifold. One must also specify a congruence of time-like lines to represent the mean motion of the ‘stars’ (galaxies). Moreover, if one wishes to express that all stars have a common ori- gin, this congruence should be such that all its members share the same domain of action”[Ehlers 1988, p. 95]. Through this hypothesis Weyl introduced a unique definition for spec- tral displacement. According to Weyl, the Doppler effect and what Weyl denoted as the Einstein effect (meaning the gravitational one) were in- dissolubly related [Weyl 1923b, p. 375]. In de Sitter’s hyperboloid, ne- glecting the matter contributions to gravity, world-lines belonged to a ∞3 pencil diverging towards the future. Such a divergence gave proof of a universal tendency of matter to scatter, due to the presence of the cosmological term [Weyl 1923a, p. 322]. By considering the world lines of a source (s) and of an observer (σ), each point of the first corresponded to a point on the latter, i.e. to the intersection of the observer world line with the future null cone emanating from the source: σ = σ(s). Peri- odic processes at the source would have resulted as periodic also by the observer, however with the period increased according to the ratio: dσ α = . (5.51) ds A spectral line measured by the observer with frequency ν corresponded ν to a spectral line of the distant source with frequency α : the one appeared displaced with respect to the other [Weyl 1923a, p. 322]. In terms of redshift z, the general definition proposed by Weyl resulted [Weyl 1923b, p. 375]: λ − λe z = 0 = α − 1, (5.52) λe 16We refer to [North 1965, Kerszberg 1986, Bergia-Mazzoni 1999, Goenner 2001 ] for further readings on the role played by Weyl Principle and the reactions to it. Weyl, Lanczos and the redshift-distance law 101 which according to Weyl assumed the form: d z = tan . (5.53) R Therefore, such a relation was linear at small distances. In this relation, R was the constant curvature radius, and d was the “naturally measured distance of the star in the static space at the same moment t at which the observation takes place”[Weyl 1930, p. 306]. In 1923 Weyl did not furnish explicit non-static coordinates, however in a 1930 paper about redshift and relativistic cosmology he used the stationary (expanding) frame which Robertson proposed in 192817: “the cosmology proposed by Robertson - Weyl wrote in 1930 - is identical with the one proposed by me” [Weyl 1930, p. 301]. The coordinates in the stationary frame which Weyl adopted were [Weyl 1930, p. 305]: r = ρ · eτ/R, (5.55) r ³ r ´2 1 − · et/R = eτ/R, (5.56) R and the relation between the observed frequency ν0 and the emitted fre- quency νe was: ³ ´ νe dτ0 ρ = = eτ0/R · e−τe/R = eτ0/R e−τ0/R + . (5.57) ν0 dτe R

At the moment of observation t = τ0 of a star (i.e. of the source) at distance r, from the above equation it followed [Weyl 1930, p. 306]:

ν r d e = 1 + √ = 1 + tan . (5.58) ν0 R2 − r2 R

17It is useful to note that, according to Weyl, a stationary gravitational field was characterized by the metric:

ds2 = −dσ2 + f 2dt2, (5.54) where dσ was the spatial interval, f was the speed of light (c), g0 α=gα 0=0, and space coordinates were not depending on time [Weyl 1921, Engl. tr. p. 241]. 102 The “de Sitter Effect”

Observations revealed that some spiral nebulæ showed relevant displace- ments towards the red, corresponding to a radial velocity of the order of 1000 km/sec. Conclusions about the nature and the distance of these objects were far to be drawn. However, Weyl remarked that it was nec- essary to consider both the possibility of the extragalactic nature of the nebulæ, and the interpretation of redshift which followed from his own analysis [Weyl 1923a, p. 323]. The redshift-distance relation which, on the contrary, was proposed in 1923 by Lanczos derived from a different formulation of de Sitter’s line element suggested by Lanczos in 1922. In a footnote of a 1922 paper about de Sitter’s universe, Lanczos wrote the line element of solution B as [Lanczos 1922, p. 539]: (et + e−t)2 ds2 = − (dφ2 + cos2 φ dψ2 + cos2 φ cos2 ψ dχ2) + dt2. (5.59) 4 Spatial sections were proportional to the factor cosh2 t, revealing the non- static character of such a metric. This solution corresponds to closed spatial sections (k = +1) in an expanding frame [Ellis 1990, p. 100]. It entirely covers the de Sitter hyperboloid and it is non-singular [Earman- Eisenstaedt 1999, p. 203]. In his 1922 paper Lanczos dealt with the question of the aggregation of matter around the equator of de Sitter’s world, i.e. at the so-called mass horizon, which was advocated by Weyl when considering the static form of de Sitter’s universe. In particular, Lanczos criticized Weyl’s attempt to demonstrate such a presence of matter aggregation. Lanczos pursued this point of view in a subsequent paper, which appeared in 1923 and was devoted to the redshift in de Sitter’s universe. In this paper Lanczos pointed out that a massive horizon was not needed to be postulated in solution B, because it was just an apparent singularity, depending on the choice of coordinates. Therefore, also the interpretation of redshift as a distance (gravita- tional) effect was objectionable. In order to deal with the possibility to discriminate whether a redshift depended on the metric structure or was Weyl, Lanczos and the redshift-distance law 103 a pure Doppler effect, according to Lanczos it was necessary to estab- lish invariant geometrical relations. A static coordinate system was not suitable for this purpose18 [Lanczos 1923, p. 169]. Lanczos acknowledged that, through Weyl’s 1923 hypothesis, it was possible to uniquely determine spectral shifts. However, in his own anal- ysis Lanczos argued that the ratio between the frequency of a signal emitted by a source and the observed frequency was equal to the ratio between the proper time of the source and the observer [Lanczos 1923, p. 170]: ν t e = e . (5.60) ν0 t0 Therefore, according to Lanczos, the relation ν ds 0 = 0 (5.61) νe dse was useful to univocally determine redshift. However, any redshift de- pended not only on the relative positions of the source and the observer, but also on the angle between the two intervals. In his analysis Lanczos derived a relation so that the ratio of observed and emitted wavelengths depended only on the two angles that the line element of the source and the line element of the observer formed with the line which connected them [Lanczos 1923, p. 177]: ν cos γ 0 = e . (5.62) νe cos γ0 Such a relation, according to Lanczos, was valid both in special relativ- ity and in general relativity. Since redshift was determined by the same quantities in both theories, and since in special relativity a spectral dis- placement was interpreted as a Doppler effect, Lanczos concluded that

18According to Lanczos, the static form of the universe of de Sitter was not sta- tionary. Lancozs considered that a universe was stationary “if the coefficients of its metric are independent of time in a coordinate system in which the masses are at rest on average. (...) A necessary and sufficient condition for this - Lanczos asserted - is that the time lines of our coordinate systems are geodesics. Therefore the static solution given by de Sitter is not an example of a stationary world” [Lanczos 1924, Engl. tr. p. 363]. 104 The “de Sitter Effect” any redshift had to be interpreted as a Doppler effect, and there was not any shift which depended on the metric, thus in particular there were not gravitational shifts [Lanczos 1923, p. 177].

Figure 5.4: The universe of de Sitter in the version proposed by Lanczos in 1922. Spatial sections at constant time are closed (k =+1). Such coordinates cover the whole hyperboloid without singularities [adapted from Lord 1974, p. 123].

Lanczos then studied the Doppler shift predicted through this version of the metric of de Sitter’s universe. He derived the relation [Lanczos 1923, p. 184]: νe a a 1 + z = = cos − sin sinh τ0, (5.63) ν0 R R where a was the geodesic distance between the source and the observer,

R the curvature radius, and τ0 the time of light reception. Therefore, such a theoretical relation was approximately linear with regard to the Silberstein’s contributions 105 distance, however with a time-dependent additional term19. As seen, in 1923 the question of the interpretation of the prepon- derance of redshift measurements through the line element of de Sitter’s universe was far to be clarified. The de Sitter effect represented a pos- sible solution, however puzzling for the arbitrariness of possible expla- nations of redshift. The general tendency of particles to scatter, which was suggested by Eddington and, mutatis mutandis, by Weyl, was not systematically confirmed by observations. It was this general recession which was strongly criticized by Silberstein, a Polish-American physicist.

5.4 Silberstein’s contributions

Silberstein dealt with shifts in de Sitter’s universe in many papers, the majority of which appeared in 1924. He used the static form of the line element and proposed a relation between velocity and distance. Such a relation was obtained in a different way with respect to Weyl’s 1923 result and was approximately linear for very distant objects. However, Silberstein proposed that such a relation was valid both for receding and for approaching objects, discarding the assumption that there was a general tendency of particles to scatter in de Sitter’s universe. In his own analysis, Silberstein pointed out that globular clusters, and not spiral nebulæ, were useful to confirm such a theoretical relation and to determine the curvature radius of de Sitter’s world20.

19With regard to the summary of redshift contributions proposed by Ellis (see Chapter 2), if an expanding frame is used, as by Weyl in 1930 and Lanczos in 1923, and, as we shall see, also by Lemaˆıtrein 1925 and Robertson in 1928, one obtains expansion redshifts (zC )[Ellis 1990, p. 101; Goenner 2001, p. 123]. 20It is worth noting that in several papers Silberstein referred to Shapley’s data. Following [Flin-Duerbeck 2006, p. 1091], Shapley’s rejection of the existence of ex- ternal galaxies could have played a significant role in Silberstein’s approach to the interpretation of his own theoretical relation and the determination of the value of the curvature radius. However, Shapley was disturbed by the polemical style which Silberstein used in some of his own papers. See [Smith 1979, p. 144] for further 106 The “de Sitter Effect”

Already in 1922, in the first edition of his book “The theory of general relativity and gravitation”, Silberstein mentioned de Sitter’s interpreta- tion of redshift related to the g44 term. Disregarding the contribution of gravitational shift due to the potential of objects themselves, the remain- ing spectral displacement observed in B stars, as seen in section 5.1.1, could be explained by the decrease of g44 term in de Sitter’s metric, lead- ing to a possible determination of the curvature radius. According to Silberstein, “there is, for the present, nothing cogent in the attribution of the said remainder of spectrum shift to the dwindling of g44 with mere distance, and it would certainly be premature either to reject or to ac- cept the results of this attractive piece of reasoning” [Silberstein 1922, pp. 136-137]. Two years later, Silberstein pointed out that, with regard to the ac- tual contributions to this second-order effect due to mere distance, “the distance-effect is insolubly amalgamated with the velocity - or usual Doppler - effect” [Silberstein 1924a, p. 350]. The total shift was pro- r portional to the first power of R , both for receding and for approaching objects. “What concerns us here - Silberstein wrote in 1924 - is the actu- ally interesting case of a star and an observer behaving as free particles, i.e. describing world-geodesics, when they cannot be at rest relatively to each other” [Silberstein 1924a, p. 350]. Silberstein acknowledged that it was Weyl’s merit to have proposed a general treatment of redshift in terms of world-lines. However, the universal tendency of matter to scatter which Weyl had suggested was “an arbitrary hypothesis” [Silber- stein 1924c, p. 909], and the “mythical assumption” [Silberstein 1924a, p. 350] that the world-lines belonged to a pencil of geodesics diverging towards the future was a “sublime guess” [Silberstein 1924c, p. 909]. Also Eddington’s suggestion of a universal scattering which appeared in Eddington’s book “The mathematical theory of relativity” was consid- ered by Silberstein as “a fallacy based upon a hasty analysis” [Silberstein readings on Shapley’s reaction to Silberstein’s papers. Silberstein’s contributions 107

1924a, p. 350], and was “entirely undesirable” [Silberstein 1924c, p. 909]. The result which Weyl proposed in 1923, i.e. the redshift-distance relation21 r z = tan , (5.64) R was contradicted both by measurements of the motion of the Andromeda nebula, which showed a blueshift, i.e. an approaching velocity of the or- der of v ' −316 km/sec, and by the approaching motion of other spirals. Therefore, according to Silberstein, globular clusters represented suitable objects to take into consideration more than spirals, being globular clus- ters “equally interesting and probably as distant celestial objects (...). Among them, negative radial velocities are by no means an exception. Rather the contrary” [Silberstein 1924c, p. 909]. Moreover, referring to data furnished by Shapley, Silberstein acknowledged that estimates about the radial velocities and distances of globular clusters were known with small errors. Silberstein considered the static form of the line element of de Sitter’s empty space-time: r r ds2 = −dr2 − R2 sin2 (dψ2 + sin2 ψdθ2) + cos2 c2dt2. (5.65) R R From this metric, and by considering dψ = 0 and dθ = 0, the geodesic equation which represented the most general radial motion of a free par- ticle was [Silberstein 1924c, p. 910]: s R dσ cos2 σ v = ± cos σ 1 − ≡ ± , (5.66) c dt γ2 c where µ ¶− 1 c dt v2 2 γ = cos2 σ = 1 − 0 . (5.67) ds c2 r In these relations, σ corresponded to R , and v0 was the particle velocity at the instant of emission.

21 δλ In his papers, Silberstein used the symbol D = λ , which corresponds in our notation to z = λo−λe . λe 108 The “de Sitter Effect”

In order to compute the Doppler effect, Silberstein used the general principle, formulated by Weyl, which related spectral displacements to the ratio of the proper time of the observer and of the source: ds z = − 1. (5.68) ds0 For ds0 = c dα, being dα the interval of time between two light-signal emissions from the source (initially at r = 0) to the observer, it followed:

c(dt − dα) = sec σ dr, (5.69) sec σ dr cdt = q . (5.70) cos2 σ 1 − γ2 Therefore Silberstein found for what he called “the complete Doppler effect” the form [Silberstein 1924c, p. 912]: " s # cos2 σ h v i z = γ 1 ± 1 − − 1 = γ 1 ± sec σ − 1. (5.71) γ2 c

The positive sign corresponded to receding objects, while the negative sign to approaching ones. The distance from the observer and the source at the moment of reception was r = Rσ. “The Doppler effect - Silberstein noted - is in general by no means a universal function of distance alone” [Silberstein 1924c, p. 913]. Actually, in the general formula of the Doppler effect there were two terms: a term depending on the individual velocity v0, which was domi- r nant near the observer, and a second term depending upon R , which, ac- cording to Silberstein, was significant for very remote celestial objects22. Silberstein also proposed a method to separate these two effects by ob- serving the displacement of spectral lines at six months intervals [Silber- stein 1924d, Silberstein 1924e, Silberstein 1924f ]. However, Eddington criticized such an attempt. The velocities (V1 and V2) of the observer at six months intervals were considered in the reference frame of the Earth

22 These contributions are equivalent to a Doppler (zD) and a gravitational (zG) effect, respectively, in the present summary of several contributions to redshifts. Silberstein’s contributions 109

with respect to the Sun. They could not be related to the velocity v0, which was the velocity a star had at some epoch in the remote past or future, “so that - Eddington wrote - (v0 − V1) has no obvious relevance to the problem” [Eddington 1924, p. 747; see also Douglas 1924 ].

v0 For near stars, the contribution of c was dominant with respect to cos σ, thus in this approximation the Doppler effect reduced to the special relativistic effect [Silberstein 1924a, p. 350]: r c ± v z = 0 − 1. (5.72) c ∓ v0 On the contrary, for the most distant celestial objects, the approxima- tions cos σ ' 1 and γ ' 1 could be used, and the redshift was [Silberstein 1924a, p. 351; Silberstein 1924b, p. 363]: v r z = ± sin σ = ± sec σ ' ± . (5.73) c R It was such a linear relation which Silberstein used in order to determine the value of the curvature radius of de Sitter’s world by radial velocities, both positive and negative, and distances of seven globular clusters. He obtained a mean value of R = 6 · 1012 AU [Silberstein 1924a, p. 351], which was almost confirmed also by (positive) velocities and distances of the two Magellanic Clouds [Silberstein 1924b, p. 363]. In addition, being the parallax p predicted through de Sitter’s model: a p ' tan p = , (5.74) R sin σ it followed the relation: a z = ± , (5.75) p R which, according to Silberstein, removed “every doubt as to the physical meaning of r”[Silberstein 1924c, p. 914]. The most distant spiral which was known at that time, NGC 584, showed a radial velocity of about v = +1800 km/sec. Therefore, by using R = 6 · 1012 AU, such an object would have been at a distance r = 3.6 · 1010 AU. “Huge as this may seen - Silberstein noted - it will 110 The “de Sitter Effect” be remembered that Shapley’s latest estimate of the semi-diameter of our galaxy is only four times smaller. (...) Whether these estimates will or will not fit into the general scheme of modern galactic and extra- galactic astronomy, is not known to me and must be left to the scrutiny of specialists” [Silberstein 1924c, p. 917]. Weyl, in reply to [Silberstein 1924c], made a list of some objectionable aspects of Silberstein’s analysis. First, the radial distance r used by Silberstein was “very artificial”, because it was the distance of the star, i.e. the source, from the observer at the moment of observation, “but - Weyl noted - in the static space of the star”. On the contrary, in his own analysis, Weyl had considered r “the distance of the star measured in the static space of the observer at the moment of observation” [Weyl 1924, p. 476]. With regard to Silberstein’s disapproval of Weyl’s hypothesis of diverging world-lines, Weyl replied that “going further than de Sitter and Eddington, I strongly emphasized the necessity of adding an assumption regarding the undisturbed state of stars, if anything in the theoretical line regarding the displacement to the red is to be formulated” [Weyl 1924, p. 476]. Finally, according to Weyl, Silberstein used the same assumption introduced by Weyl for both future and past direction of time, a method which appeared “quite abstruse” to Weyl [Weyl 1924, p. 477]. In subsequent papers, Silberstein showed that, from velocity and dis- tance of ten objects (eight clusters and the Magellanic Clouds), a linear relation was actually confirmed by plotting, as suggested by Henry N. Russell (1877-1957), the modulus of the redshift: r = |z| R. Silberstein, however, discarded data belonging to other three globular clusters (NGC 5904, 6626, 7089), which velocities were “suspiciously small’ [Silberstein 1924d, p. 602]. From data of these ten objects, the value of the curvature radius, through the rigorous formula r2 v2 z = + 0 , (5.76) R2 c2 was of the order of R ≥ 9.1· 1012 AU [Silberstein 1924d, p. 602], while the approximate linear formula led to a world radius of de Sitter’s universe Silberstein’s contributions 111 not exceeding R = 8 · 1012 AU [Silberstein 1924e, p. 819].

Figure 5.5: Linear relation between the modulus of spectral displacement (velocity) and distance of ten globular clusters proposed by Silberstein in 1924. Note the three objects which Silberstein discarded for their “suspicious small” velocities [from Silberstein 1924d, p. 602].

Silberstein further developed the theoretical redshift relation for any inertial motion, i.e. not exclusively for radial motions. He derived the general relation [Silberstein 1924g, p. 623]:

cos2 σ γ z = r ³ ´ − 1, (5.77) cos2 σ p2 1 ∓ 1 − γ2 1 + R2 sin2 σ where p and γ were integration constants. In 1929, as we shall see, such a relation was taken into account by Tolman in his own investigation of redshifts in de Sitter’s universe. Silberstein’s contributions marked a crucial passage to the consider- ations on the universe as a whole through general relativity and the in- terpretation of astronomical observations. Indeed several debates, some- times at polemical level, arose about his systematic analysis to confirm a suitable curvature radius through his own linear theoretical relation and observations of astronomical objects. As seen, Eddington and Weyl inde- pendently criticized Silberstein’s proposal for some theoretical features; 112 The “de Sitter Effect” furthermore, as we will see in next chapter, Lundmark and Str¨omberg denied the correctness of Silberstein’s result with regard to observational aspects. Nevertheless, Silberstein’s proposal played an important role in Lemaˆıtre’sanalysis about de Sitter’s universe which Lemaˆıtreproposed in 1925, and also Robertson and Tolman mentioned Silberstein’s result in their own contributions about the de Sitter effect which appeared in 1928 and 1929, respectively. In 1930 Silberstein collected his views concerning relativistic cosmol- ogy in the book “The size of the universe”. In this book Silberstein maintained the proposal of a shift relation, both for red and for blue shifts, which was derived from the static metric of de Sitter’s universe. He did not mention other works published at the end of the 1920’s, only referring to the original papers by Einstein and de Sitter. Silberstein also pursued the objection to the general tendency of particles to scatter suggested by Weyl, and, moreover, criticized Hubble’s measurements of distances in extra-galactic nebulæ23 [Silberstein 1930 ].

5.5 Lemaˆıtre’s1925 notes

A possible relation between spectral displacement and distance in de Sitter’s world was considered by Lemaˆıtrein a short note on the universe of de Sitter which appeared in 1925 [Lemaˆıtre 1925a, Lemaˆıtre 1925b], two years before the fundamental paper which Lemaˆıtrewrote about a non-empty expanding universe. In this 1925 note on de Sitter’s universe, which was presented to the Society of Physics at Washington, Lemaˆıtreshowed that, through appropriate new coordinates, “the field is found homogeneous but not statical. Furthermore the geometry is Euclidean. The singularity at de Sitter’s horizon disappears” [Lemaˆıtre 1925b]. Indeed Lemaˆıtreproposed a transformation in such a way that space in de Sitter’s universe had

23See [Robertson 1932 ] for a strong criticism of Silberstein’s book. Lemaˆıtre’s1925 notes 113 null curvature (k = 0), and spatial sections were proportional to an exponentially time-dependent term. This result was independently found also by Robertson in 1928, as a stationary form of de Sitter’s universe24. In 1924-1925 Lemaˆıtrewas Ph.D. candidate at the Massachusetts In- stitute of Technology; during his stay in United States, he directed his work in two lines: theoretical astrophysics and the theory of relativity. Lemaˆıtrehad previously been (in 1923) a student of Eddington at Cam- bridge Observatory, just in the period when Eddington’s “Mathematical Theory of Relativity” appeared. Therefore, the suggestion by Eddington about the non-static character of de Sitter’s world played an important role in the attention which Lemaˆıtregave to the universe as a whole. However, as noted in [Lambert 2000, pp. 70-81], it was a conference by Silberstein which Lemaˆıtreattended in 1924, and subsequent discussions between Lemaˆıtreand Silberstein about the Doppler effect and the mea- sure of curvature radius in de Sitter’s universe, which most influenced Lemaˆıtre’searly approach on relativistic cosmology25. Following Lemaˆıtre’snotation (with c = 1), the model of the universe

24It is important to note that the result found by Lemaˆıtreand Robertson would have corresponded years later to the exponentially expansion predicted in the steady state model, and the exponential form of de Sitter’s universe is at present used in order to describe the expansion of vacuum energy dominated universes. 25With regard to a dynamical interpretation of de Sitter’s world, in the draft of a 1963 obituary for Roberston stored at LemaˆıtreArchive, Lemaˆıtrewrote: “I was better prepared to accept it following an opinion expressed by Eddington. (...) The errors by Silberstein have been very stimulating. I had myself had a long discussion with him [Silberstein] in 1924 at a British Association Conference in Toronto and my work, as possibly later on the work of Robertson, results as a large part as a reaction against some unsound aspects of Silberstein’s theories” [Lemaˆıtre Archive, Box D32]. It seems very probable that, through Silberstein’s papers, Lemaˆıtrebecame aware of Weyl’s assumption and results about redshifts. In some undated sheets which are stored at LemaˆıtreArchive, Lemaˆıtrerepeated the calculations about the de Sitter effect as in [Silberstein 1924c], referring also to the 1923 works of Weyl mentioned in previous sections [Lemaˆıtre Archive, Box R3]. 114 The “de Sitter Effect” proposed by de Sitter was written with line element:

ds2 = R2[−dχ2 − sin2 χ(dθ2 − sin2 θ dφ2) + cos2 χ dτ 2]. (5.78)

Such a description of de Sitter’s world was objectionable, since there was a preferred point, the “center at χ = 0”: free particles or rays of light did not move along geodesics of the space, except for those passing through such a center. The purpose of the note on de Sitter’s universe - as Lemaˆıtrewrote - was “to look for a separation of space and time which is free from this objection” [Lemaˆıtre 1925a, p. 188]. By new coordinates, the line element could be written as: ¡ ¢ ds2 = R2 − cosh2 τ 0[dχ02 + sin2 χ0(dθ02 + sin2 θ0dφ02)] + dτ 02 . (5.79)

This result was equivalent to the “expanding frame” obtained by Lanczos. In this form the radius of space was constant at any place, but was variable with time, since it was proportional to cosh τ 0. According to Lemaˆıtre,by such a new metric “the central point in space is removed, but now a central time has been introduced” [Lemaˆıtre 1925a, p. 189]. Indeed, at any instant τ 0 6= 0, there were no geodesics of space which were geodesics of the universe, i.e., quoting Lemaˆıtre,“the origin of time becomes a time absolutely distinct from every other” [Lemaˆıtre 1925a, p. 189]. By the transformation: r χ = arcsin , (5.80) t 1 τ = ln(t2 − r2), (5.81) 2 and by changing polar coordinates to Cartesian ones, (r, θ, φ) → (x, y, z), the line element was: −dx2 − dy2 − dz2 + dt2 ds2 = R2 . (5.82) t2 Therefore the geometry of spatial sections was Euclidean, and the cur- vature was null. Time coordinate t was expressed by coordinate T : Z dt T = ± = ± ln t, (5.83) t Lemaˆıtre’s1925 notes 115 and t = 0 represented the infinite past or future. The metric assumed the form: ds2 = R2[−e± 2T (dx2 + dy2 + dz2) + dT 2], (5.84) where the coefficient −e−2T referred to parallel geodesics to the future, while −e2T referred to parallel geodesics in the past direction. This choice of coordinate, according to Lemaˆıtre, could free from the objection of introducing the spurious asymmetry in space and time as in de Sitter’s original metric26, which “is not simply the mathematical appearance of center of an origin of coordinates, but really attributes distinct absolute properties to a center” [Lemaˆıtre 1925a, p. 192]. Lemaˆıtrethen considered the Doppler effect, and calculated the shift predicted by such a metric. He found the same result obtained by Silber- stein, except, as Lemaˆıtrepointed out, for the interpretation of the sign in such an effect. However, Lemaˆıtredid not refer to observational data.

Lemaˆıtreconsidered two light sources, Me and M0, of the same proper interval ds, which described geodesics of constant x, y, z: ds dt dt = e = 0 . (5.85) R te t0

The equation of the light ray from Me to M0 was t0 = te+r. By supposing the light source Me at the origin, as done by Silberstein, Lemaˆıtrefound for the Doppler shift the linear form [Lemaˆıtre 1925a, p. 191]: ∆λ λ − λ dt t r = e 0 = e − 1 = e − 1 = − = − sin χ. (5.86) λ0 λ0 dt0 t0 t0 The observer was supposed to describe geodesics passing neither through the origin, nor at a minimum distance to it. This Doppler effect was ob- tained in the case t = 0 in the past, meaning that lines of the universe converged in the future. The same result, however with the opposite sign,

26With regard to the interpretations of the universe of de Sitter, Lemaˆıtrewrote in 1929 that “Weyl a montr´equ’il pouvait s’interpret´ercomme un espace Euclide´en o`ula confirguration forme´epar les points mat´erielsse dilate en restant semblable `aelle-mˆeme. Lanczos a donn´eune interpretation analogue pour un espace ferm´e” [Lemaˆıtre 1929, p. 31]. 116 The “de Sitter Effect”

Figure 5.6: The ‘exponential form’ of the universe of de Sitter proposed by Lemaˆıtrein 1925 and independently by Robertson in 1928. Spatial sections at constant time have flat geometry (k = 0) [adapted from Lord 1974, p. 124]. was obtained by reversing the sign of t, and supposing geodesics converg- ing in the past [Lemaˆıtre 1925a, p. 192]. Thus, the two solutions found by Silberstein (with both the positive and the negative sign) could not be compounded. Lemaˆıtrewrote about Silberstein’s result that “no way is found of introducing his double sign without spoiling the homogeneity of the field” [Lemaˆıtre 1925b]. “Our treatment - Lemaˆıtreremarked at the end of his note - evidences the non-statical character of de Sitter’s world, which gives a possible interpretation of the mean receding motion of spiral nebulæ” [Lemaˆıtre 1925a, p. 192]. According to Lemaˆıtre,the observed redshifts could be physically interpreted as a feature of de Sitter’s universe. In particular, redshift measurements were interpreted in such a version of de Sitter’s Shifts in de Sitter’s universe according to Robertson and Tolman 117 universe as relative motions through space. However, quoting Lemaˆıtre,“de Sitter’s solution has to be abandoned, not because it is not static, but because it does not give a finite space without introducing an impossible boundary” [Lemaˆıtre 1925a, p. 192]. Lemaˆıtrepointed out in a Ph.D. report of June 1925 that an infinite space represented “a very unsatisfactory feature” in the consideration of the universe as a whole [Lemaˆıtre Archive, Box 8]. In 1925, Lemaˆıtreat- tended a talk by Hubble about Cepheid stars as distance indicators, and in the same year he visited Lowell Observatory, where Slipher was dealing with measurements of relevant spectral displacements in (extragalactic) nebulæ. These important observational aspects and the fundamental re- quirement of a finite space marked the connection between theoretical cosmology and astronomical observations which Lemaˆıtredeveloped in following years [Lambert 2000, p. 91]. As we shall see, Lemaˆıtrecon- sidered a finite spherical universe filled by matter which world radius was depending on time. He investigated the homogeneous and isotropic model of the universe proposed by Einstein, inserting a world radius in- creasing with time. In such a truly expanding model of the universe, which Lemaˆıtrediscovered in 1927, redshifts were directly related to the ratio of time-dependent world-radii, involving the actual cosmological in- terpretation of redshift.

5.6 Shifts in de Sitter’s universe according to Robertson and Tolman

The fact that de Sitter’s universe could be regarded as spatially open was realized, as seen above, by Lemaˆıtrein a note published in 1925. Three years later, also Robertson investigated the empty universe of de Sitter. In a paper which appeared in 1928, Robertson, unaware of Lemaˆıtre’sproposal, proposed a similar result, and, dealing with Doppler shifts, suggested that a linear relation between velocities and distances 118 The “de Sitter Effect” existed in de Sitter’s universe. Furthermore, Robertson appreciated that such a theoretical law could account for available data furnished by Hub- ble and Slipher. By a suitable transformation of coordinates, Robertson proposed a cosmological model which was equivalent to the original model of de Sit- ter, but, as Robertson remarked, differed in its physical interpretations. In such a version of de Sitter’s world, there were neither the paradoxes of the mass-horizon, nor the arrest of time at the horizon. The geometry of space proposed by Robertson had “the advantage of simplicity” because it was Euclidean [Robertson 1928, p. 847]. However, the line element explicitly depended on time: “consequently - Robertson wrote - natural processes are not reversible, but space-time is isotropic in time, in the sense that at any time the line element has the same form as at any other” [Robertson 1928, p. 847]. The model of de Sitter’s universe in the form proposed by Robertson corresponded to a stationary model, i.e., quoting Robertson, “it presents the same view to observers at different times” [Robertson-Noonan 1968, p. 346]. This version of the universe of de Sitter, according to Robertson, was the only non-static stationary model: “the fundamental world lines - Robertson noticed - expand away from each other, but they also present the same appearance at any cosmic time” [Robertson-Noonan 1968, p. 365] The empty model of de Sitter represented a suitable description of the actual world, since the total matter in the world could be assumed to have “little effect on its macroscopic properties” [Robertson 1928, p. 835]. The line element, following Robertson notation, was: dρ2 ds2 = − − ρ2(dθ2 + sin2 dφ2) + (1 − k2ρ2)c2dτ 2, (5.87) 1 − k2ρ2 q λ 1 1 where k = 3 = R . The singularity at ρ = k = R involved the presence of the mass-horizon. Robertson proposed the transformation:

ρ = r ekct, (5.88) 1 τ = t − log(1 − k2r2e2kct). (5.89) 2kc Shifts in de Sitter’s universe according to Robertson and Tolman 119

The line element became:

ds2 = −e2kct(dr2 + r2dθ2 + r2 sin2 dφ2) + c2dt2, (5.90) or equivalently, by Cartesian coordinates:

ds2 = −e2kct(dx2 + dy2 + dz2) + c2dt2. (5.91)

The coordinate r assumed in this frame infinite values, while in the origi- nal form it was 0 < ρ ≤ R; through such a transformation the singularity was removed from the finite region [Robertson 1928, p. 836]. The new form of de Sitter’s line element was thus “dynamical, in that its coeffi- cients depend on time t”[Robertson 1928, p. 837]. Although the mass horizon paradox did not appear in this new frame, and space was there unlimited, Robertson noted that the observable world was not unlimited: de Sitter’s universe presented a kind of cos- mological horizon. “The closed character - Robertson pointed out - is maintained in the sense that the only events of which we can be aware must occur within a sphere of finite radius” [Robertson 1928, p. 837]. Investigating geodesic equations, it followed that R was the maximum distance from which particles or light could enter in causal connection with an observer at the origin: R represented “our observable universe” [Robertson 1928, p. 839]. Following [Earman-Eisentaedt 1999, p. 202], in fact, the Robertson form of de Sitter’s line element is non-singular, how- ever it covers only a portion of de Sitter’s hyperboloid, and space-time is not free of singularities for the choice of logarithmic coordinate. Dealing with the properties of his own form of de Sitter’s line ele- ment, Robertson found a linear relation between velocity and distance. “The measurements of velocities - Robertson noted - can, in principle at least, be reduced to the measurements of distances. (...) The mea- surement of the radial component offers more difficulty; in practice it is accomplished by means of the Doppler effect” [Robertson 1928, p. 843]. Therefore Robertson developed the theory of Doppler effect in de Sitter’s universe. He considered the observer at the origin of coordinates, and he 120 The “de Sitter Effect” supposed that gravitational shifts could be neglected. Robertson studied the relation between time intervals of what he denoted as “equivalent” observers, i.e. all observers at constant (r, θ, φ)[Robertson 1928, p. 837].

kcte Being le = r e the distance between a light source and the observer 27 at the origin , light emitted from the source in the interval te, te + dte arrived at the origin in the interval t0, t0 + dt0, and the relation between time intervals was [Robertson 1928, p. 843]:

dte dt0 = . (5.92) 1 − kle Since the proper-time of atom vibrations was the same at the origin and at the source, such a relation led to a shift ∆λ given by: ∆λ λ − λ dt kl ≡ 0 e = 0 − 1 = e . (5.93) λe λe dte 1 − kle Such a shift “ would be attributed in practice” [Robertson 1928, p. 843] to the Doppler effect due to a velocity of recession v: s v ∆λ 1 + c zD = = v − 1. (5.94) λe 1 − c Therefore it followed the relation:

v = ckle(1 + O[k]). (5.95)

Thus, through the application of Doppler effect for receding sources, the observer at the origin would have measured a velocity v ' ckle of the source [Robertson 1928, p. 843]. Robertson further developed such a relation to the general case of a source which had a non-vanishing coordinate velocity δe ≡ (δr, δθ, δφ), i.e. he considered the case when “the proper time of vibration of the moving atom is not the same as the time interval measured by a stationary

27 Note that in his 1928 paper Robertson used the notation l0, dt0, dt for, respec- tively, the measured distance of the source from the origin, the time interval measured by the source, and the time interval measured by the observer at the origin [Robertson

1928, p. 843]. In our notation these quantities become le, dte, dt0, respectively. Shifts in de Sitter’s universe according to Robertson and Tolman 121 observer” [Robertson 1928, p. 843]. The radial component of the velocity resulted:

kcte vr ' e δr + ckle. (5.96)

“If we assume - Robertson wrote - that there is no systematic correlation of coordinate velocity with distance from the origin, we should expect that the Doppler effect would indicate a residual positive radial velocity of distant objects because of the term ckle”[Robertson 1928, p. 844]. The stationary form of the line element of de Sitter’s universe pre- dicted the relation: v l z = ' . (5.97) c R Essentially, the metric of de Sitter’s universe predicted a residual mo- tion of recession through Doppler effect, which, by averaging proper motions, was the cause of the “excess of recessional velocity of spiral nebulæ” [Robertson 1928, p. 847]. Such a linear relation, as Robertson acknowledged, was found also by Weyl in 1923. Comparing his own result (equation 5.97) with Silberstein’s one, Robertson remarked that in his own velocity-distance relation it was not possible to introduce a negative sign, unless changing the sign of k [Robertson 1928, p. 844]. Robertson pointed out that such a theoretical linear relation was nearly verified from distances of extragalactic nebulæ obtained by Hub- ble in [Hubble 1926b], and from velocities proposed by Slipher which appeared in [Eddington 1923 ], giving a radius of de Sitter’s universe of about R = 2 · 1027 cm [Robertson 1928, p. 845]. Therefore, in his 1928 paper Robertson proposed that, neglecting gravitational shift and eliminating proper motion, the effect of recession predicted by de Sitter’s cosmology was linear, and was approximately confirmed by observational data of spirals. In a subsequent paper which appeared in 1929 Robertson acknowl- edged that already in 1925 Lemaˆıtrehad discovered the same coordinate system Robertson independently proposed in 1928. In this paper Robert- son stated that the only stationary cosmological models (in the sense seen 122 The “de Sitter Effect” above) were those of Einstein and de Sitter. These models, according to Robertson, arose “from particular cases of a class of solutions whose gen- eral member defines a non-stationary cosmology” [Robertson 1929, p. 822]. The general form of the line element was [Robertson 1929, p. 826]: Ã ! dr2 ds2 = −e2f(t) + r2 dθ2 + r2 sin2 θ dφ2 + c2dt2. (5.98) r2 1 − R2 In the retrospect, this form actually corresponds to the general line ele- ment of an expanding FLRW frame, where the spatial curvature is arbi- trary [Ellis 1989, p. 379]. With regard to the question of the Doppler effect, in the general case the resulting Doppler shift was attributed to a velocity of recession:

v = c tanh [f(t0) − f(te)] . (5.99)

“Our choice of coordinates in the actual universe - Robertson concluded - has been such that the above considerations apply to the residual Doppler effect, after having averaging to eliminate the effect due to accidental ‘proper’ motions. Of the two stationary cosmologies, only that of de Sitter will show such a residual effect, as in Einstein’s f is constant” [Robertson 1929, p. 827]. Interestingly, Robertson mentioned in a note of this paper the solu- tions proposed by Friedmann in 1922 and 1924 which described an ex- panding universe. The cosmological consequences of Friedmann’s models were fully acknowledged only in 1930. In 1929 Robertson found these so- lutions objectionable, because they were non-stationary. These solutions, quoting Robertson, introduced “untenable assumptions on the matter- energy tensor, and require that Einstein’s field equations be satisfied instead of making full use of the intrinsic uniformity of such a space” [Robertson 1929, p. 828]. Just before the rise of the theory of the expanding universe, also Tol- man took into account the redshift-distance relation predicted through de Sitter’s static model. In 1929 a detailed paper by Tolman appeared Shifts in de Sitter’s universe according to Robertson and Tolman 123 about the astronomical implications of the line element of de Sitter’s universe. In this paper, Tolman studied the metric of de Sitter’s universe in the form which Eddington already proposed in 1923: µ ¶ 1 r˜2 ds2 = −¡ ¢dr˜2 − r˜2dθ2 − r˜2 sin2 θ dφ2 + 1 − dt2. (5.100) r˜2 R2 1 − R2

According to Tolman, this equation was “the most suitable for our pur- poses, since the variables occurring in it will later appear to be the most natural ones to correlate with the results of astronomical measurements” [Tolman 1929a, p. 248]. With regard to the geodesic equations, Tolman found the relation: r dr˜ r˜2 h2 h2 = ± k2 − 1 + − + , (5.101) dt R2 r˜2 R2 where h and k were constants of integration: h could assume positive or negative values according to the direction of motion of the particle, and k was a positive parameter, “if - Tolman wrote - we are going to interpret t as the time and exclude the possibility that the proper time s and coordinate time t could ever run backward” [Tolman 1929a, p. 249]. In his analysis of Doppler effect, Tolman considered both cases of the observer and the emitting source at the origin of coordinates. In the first case, with the observer at the origin, the source was at a distancer ˜ dr˜ and moved with a velocity dt at the time of light-signal emission. The interval ds was proportional to the unshifted wavelength. The period dt0 of the light arriving at the origin was related to the period of emission dt², and the time dte which was taken by light to traverse the radial distance dr˜ dr˜ = dt dt²:

dt0 = dt² ± dte, (5.102) where k dr˜ dte = ¡ ¢ ds, (5.103) r˜2 2 dt 1 − R2 124 The “de Sitter Effect” and k dt = ds. (5.104) ² r˜2 1 − R2

Therefore, since dt0 was proportional to the wavelength observed at the origin, the Doppler effect was [Tolman 1929a, p. 259]: q 2 r˜2 h2 h2 k ± k − 1 + R2 − r˜2 + R2 z = − 1. (5.105) r˜2 1 − R2 In the second case, with the source at the origin and the observer at distancer ˜ from the source, spectral displacement was:

1 − r˜2 z = q R2 − 1. (5.106) 2 r˜2 h2 h2 k ∓ k − 1 + R2 − r˜2 + R2 Tolman remarked that the complete Doppler formula which Silberstein had proposed in 1924 was correct only for the special case h = 0 and k = 1 [Tolman 1929a, p. 261]. Furthermore, Tolman proposed a formulation of the Doppler effect dr˜ in the case dt = 0, i.e. when the particle was at the perihelion of his hyperbolic orbit. In this case, beingr ˜m the minimum distance from the origin to the perihelion, the Doppler effect resulted:

2 2 2 r˜m h h 1 − 2 + 2 − 2 R r˜m R z = 2 − 1. (5.107) r˜m 1 − R2 According to this formula, the Doppler effect at the perihelion was pos- itive and could be interpreted as due to a motion of recession, although the source had no radial velocity at the time of emission. “The produc- tion of a positive Doppler effect by a source which is at rest - Tolman wrote - was the original reason which led investigators to hope that the de Sitter universe might furnish an explanation for the preponderating shift toward the red in the light of extragalactic nebulæ” [Tolman 1929a, p. 262]. With regard to the astronomical implications, Tolman admitted that there was an approximate uniform distribution of nebulæ in regions of Shifts in de Sitter’s universe according to Robertson and Tolman 125 space accessible to telescopes. The average Doppler effect, moreover, ap- peared to be approximately proportional to the first power of the distance [Tolman 1929a, p. 266]: r˜ z ∝ . (5.108) av R The observed preponderance of shifts towards the red could be recon- ciliated with theoretical Doppler effect only by some special assumptions for the values of k and R. However, as Tolman pointed out, “we cannot greatly increase k or decrease R without getting larger Doppler effects than those actually observed” [Tolman 1929a, p. 268]. Alternatively, a reconciliation was obtained through another special assumption, i.e. by assuming that the number of nebulæ “which make perihelion at a given radius increases very rapidly with the radius” [Tolman 1929a, p. 269]. Actually, the great preponderance of positive velocities involved strong ad hoc requirements for the values of the parameters k and h. With k = 1 and h = 0, the theoretical Doppler shift reduced to r˜ z = , (5.109) R only if the negative sign in the complete Doppler formula was neglected. Moreover, the value of the theoretical average Doppler effect at any radius r˜ was found by Tolman of the order of [Tolman 1929a, p. 270]

1 r˜2 z ≥ , (5.110) av 2 R2 which “would not appear to account for the very rare occurrence of neg- ative Doppler effects and for linear increase in average Doppler effect with distance, without the imposition of restrictions on the parameters determining the orbits of the nebulæ” [Tolman 1929a, p. 274]. Therefore, according to Tolman, “the de Sitter line element (...) does not appear to afford a simple and unmistakably evident explanation of our present knowledge of the distribution, distances, and Doppler effects for the extra-galactic nebulæ” [Tolman 1929a, p. 273]. “Further obser- vational material on the nebulæ - Tolman concluded - would be of great 126 The “de Sitter Effect” importance. In particular it is desirable to be certain as to the form of the relation between Doppler effect and distance, and also as to the rela- tive frequency of negative and positive Doppler effects” [Tolman 1929a, p. 274]. Tolman investigated in another paper the possible line elements of the universe. The natural requirement was, according to Tolman, the condition of homogeneity and isotropy, together with the possibility to write the line element in a form which was static respect to time. “The requirement of spherical symmetry - Tolman wrote - is an obvious one to impose, since otherwise the universe regarded on large scale would have different properties in different directions. The requirement of symmetry with respect to past and future time means that the large scale behaviour of the universe is reversible, and the static form of the line element means that by and large the universe is in a steady state” [Tolman 1929b, p. 298]. A further requirement was that both the density of matter and the pressure were constant throughout the universe, and the spatial velocity had to be zero. Therefore, the solutions proposed by Einstein and de Sitter (and the model of the special theory of relativity) were the only models which satisfied the mentioned requirements. However, at the end of his paper Tolman suggested that other models could be investigated, by imposing different requirements. “In particu- lar - Tolman concluded - it should be noted that our assumption of a static line element takes no explicit recognition of any universal evolu- tionary process which maybe going on. The investigation of non-static line elements would be very interesting” [Tolman 1929b, p. 304]. As seen, at the end of the 1920’s the question of the explanation of red- shift measured in galaxies through theoretical relativistic models of the universe, which was foreshadowed in the discussions about the de Sitter effect, was actually close to a solution. Investigations of non-static and non-stationary models of the universe, in the light of the observational evidence of a cosmic recession, inaugurated in 1930 a second renewal of cosmology. Shifts in de Sitter’s universe according to Robertson and Tolman 127 1 − ´ 2 2 ˜ r R − 1 ³ / ¶ 2 2 h R + 2 2 r R ˜ r h 1 − − 0 ' ± 2 2 τ ˜ r R σ 2 ¢ r R sec sinh 1 + ¡ c v a R 1 2 − ± 2 + k sin = r R q − 2 σ ¢ d R a R ± ˜ r χ R sin ' ± ¡ k l c 2 R 1 v ± µ = ' = tan = cos = = sin ' = redshift (velocity)-distance relation z z z z z z z z = 0 = +1 = 0 = 0 k k k k frame static static stationary, stationary, static stationary, stationary, static de Sitter, 1917 Eddington, 1923 Weyl, 1923 + 1930 Lanczos, 1923 Silberstein, 1924 Lemaˆıtre,1925 Robertson, 1928 Tolman, 1929

Table 5.1: Summary of different interpretations of the model of de Sitter and different formulations of the redshift-distance law related to the de Sitter effect. Part of this summary is based on [Ellis 1990, p. 100].

Chapter 6

Observational investigations of redshift relations

In this chapter the first observations on large scale during the 1920’s, i.e. the beginning of observational cosmology, are presented. In par- ticular, a critical reconstruction is here proposed of attempts to find a suitable relation between velocities and observable quantities, such as the apparent diameter and the distance of nebulæ, and therefore to possibly confirm the de Sitter effect.

6.1 The nature of the nebulæ around 1920

In the book “Stellar movements and the structure of the universe”, which appeared in 1914, Eddington devoted a chapter on the Milky Way, star clusters and nebulæ. The general view which Eddington supported was that there was firstly an inner and flattened star system, whose density diminished from the center outwards, and secondly a certain number of star clouds around the inner system, which “make up the Milky Way. It is to the inner system - Eddington suggested - that our knowledge of stellar motions and luminosity relates. Whether the other clouds are continuous with the inner system or whether they are isolated,

129 130 Observational investigations of redshift relations is a question at present without answer” [Eddington 1914, p. 233]. With regard to spiral nebulæ, Eddington remarked that these systems were not correlated to the planetary of irregular ones, which on the contrary intimately belonged to our stellar system. Despite “direct evidence is entirely lacking as to whether these bodies are within or without the stellar system, (...) the island universe theory is much to be preferred as a working hypothesis; and its consequences are so helpful as to suggest a distinct probability of its truth ” [Eddington 1914, pp. 242-243]. The possibility to obtain a reliable distance of the nebulæ would have permitted to discriminate between the galactic or extragalactic position of these objects, and thus to understand their nature1. This challenge was solved in 1925, thanks to the most powerful telescope which oper- ated during the 1920’s, the Hooker 100-inch reflector at Mt. Wilson, and to the man who directed such a telescope towards the deepest space ever observed until then, namely Hubble, who inaugurated the cosmo- logical observations, even better measurements, of objects at very large distances2. Hubble, as we shall see, determined the distance of some nebulæ by

1See [Fernie 1970 ] for the historical quest for the nature of the nebulæ. 2It is important to note that in 1922 Ernst Opik¨ (1893-1985) estimated the distance of the Andromeda nebula at about 450’000 pc. He did not base his own calculations on some empirical distance indicators. Indeed Opik¨ determined the distance of M31 through a method based on the observed rotational movement. “Assuming that the centripetal acceleration at a distance r from the center is equal to the gravitational acceleration due to the mass inside the sphere of radius r - Opik¨ wrote - an expression is derived for the absolute distance”[Opik¨ 1922, p. 406]:

E sin ρ ³v ´2 D = 0 , (6.1) i ω where v0 was the velocity of motion along a circular orbit, ρ the angular distance from the center, i the apparent luminosity, ω the orbital velocity of the Earth, and E was the energy radiated per unit mass, which Opik¨ assumed the same as for our Galaxy. However, this result did not attract much interest, nor the method stimulated further developments. The nature of the nebulæ around 1920 131 using the period-luminosity relation typical of Cepheid variable stars3. As mentioned in Chapter 3, such a fundamental relation was suggested by Henrietta Leavitt in 1912. Leavitt used the catalogue of 1777 variable stars in the two Magellanic Clouds which was given by studying plates obtained at the Harvard Observatory at Arequipa (Peru). She took into account 25 variables in the Small Magellanic Cloud, for which both peri- ods and brightness at maxima and minima were available. Leavitt then noted that “a straight line can readily be drawn among each of the two series of points corresponding to maxima and minima, thus showing that there is a simple relation between the brightness of the variables and their periods” [Leavitt-Pickering 1912, p. 2]. It was the Danish astronomer Ejnar Hertzsprung (1873-1967) who considered the result proposed by Leavitt, and first calibrated such a relation in 1914, allowing to deter- mine the absolute magnitude at the maximum of brightness [Hertzsprung 1914 ]. By using statistical parallaxes of 13 Cepheids, Hertzsprung ob- tained the zero-point of such a calibration curve, and proposed a relation between the absolute visual magnitude (MV ) and the period (P ). In modern notation, the relation found by Hertzsprung was [Fernie 1969, p. 708]:

< MV >= −0.6 − 2.1 log P, (6.2) with P measured in days. Therefore, the determination of apparent magnitude could be used to obtain the distance of such objects through the distance modulus equation. Shapley used such a relation in 1918 to determine the extent of the Milky Way, by assuming that globular clusters belonged to it and formed a galactic halo. In his paper “On globular clusters and the structure of the galactic system”, Shapley pointed out that, in order to determine the distance of globulars, “the Cepheid variables (...) are of so much greater weight, because of more definite knowledge of the dispersion of absolute

3We refer to [Fernie 1969 ] for a detailed reconstruction of the history of the period- luminosity relation. 132 Observational investigations of redshift relations brightness, that the other types [B stars and red giants] can best be used as checks or as secondary standards” [Shapley 1918, p. 43].

Figure 6.1: Magnitude-period curve of Cepheid stars which Shapley took into account in order to study the structure of the Milky Way [from Shapley 1918, p. 44].

Shapley obtained < M >= −0.4 and < M >= −4.0 for Cepheids in the Small Magellanic Clouds and in the local system with, respectively, periods less than a day and periods longer than a day. Furthermore, he fixed the absolute photographic magnitude for the 25 brightest stars in 69 globular clusters at M = −1.5 [Shapley 1918, p. 44]. Having in this way determined their distances, Shapley concluded that “the globular clusters outline the extent and arrangement of the total galactic orga- nization. Adopting this view of the stellar system, all known sidereal objects become part of a single enormous unit, in which the globular clus- ters and Magellanic Clouds (...) are clearly subordinate factors” [Shapley 1918, p. 50]. At the end of this paper, Shapley listed the observational evidences unfavorable to the “island universe” hypothesis. He took up these objections in a subsequent 1919 paper “On the existence of external galaxies”. This paper contained the main issues that Shapley would have The nature of the nebulæ around 1920 133 faced in 1920 in the famous discussion “The scale of the universe”, i.e. in the so-called “Great Debate” with Curtis. According to Shapley, the arguments against the stellar interpretation of spiral nebulæ were: the relevant size of our Galaxy, the internal motions in spirals measured by Adrian van Maanen (1884-1947) (which, however, Shapley acknowledged to be still uncertain), and the absolute magnitude of Novæ, which “would far transcend any luminosity with which we are acquainted” [Shapley 1919, p. 266]. Also the first relevant radial velocities measured in spirals by Slipher (which will be described in next section) seemed to oppose the extragalactic nature of nebulæ: “high speed - Shapley noted - is not a condition impossible of production by the forces in our galactic sys- tem” [Shapley 1919, p. 265]. Shapley concluded that “the evidence now supporting the island universe interpretation appears unconvincing. (...) We have, however, no evidence that somewhere in space there are not other galaxies; we can only conclude that the most distant sidereal or- ganizations now recognized (globular clusters, Magellanic Clouds, spiral nebulæ) cannot successfully maintain their claims to galactic structure and dimensions” [Shapley 1919, p. 268]. On the contrary, during the “Great Debate” Curtis advocated the theory that spirals were island universes. According to him, the main favorable points were “the tremendous space-velocities of spirals” [Curtis 1920, p. 326], which could not be related in any way with the thirty- fold smaller velocities measured in stars, together with the evidences of a spiral structure also in our system. Other favorable points were the spectrum of the great part of spirals, which “is practically identical with that given by a star cluster” [Curtis 1920, p. 326], and eventually the presence of Novæ in those systems. Curtis illustrated in a scheme that spirals were distributed in greatest numbers around the poles of our Galaxy: nearly 400’000 and 300’000 spi- ral nebulæ were placed, respectively, around the north and south galactic pole4. “Our stellar system - Curtis wrote - is shaped like a thin lens, and 4It is worth noting that Curtis proposed such a scheme already in 1917 [Curtis 134 Observational investigations of redshift relations is perhaps 3’000 by 30’000 light years in extent. In this space occur nearly all the stars, nearly all the new stars, nearly all the variable stars, most of the diffuse and planetary nebulæ, but no spiral nebulæ”[Curtis 1920, p. 320].

Figure 6.2: Position of nebulæ with respect to the Milky Way according to Curtis [from Curtis 1920, p. 320].

6.2 Slipher and the radial velocities of spi- rals

Around 1920, questions about spiral nebulæ have as the main top- ics their positions in space, i.e. their still unknown distances, and the astonishing evidence that some of these objects showed large spectral displacements, which were interpreted as large velocities for the habit to directly relate spectral shift z to the corresponding velocity. 1917, p. 100]. However, in the 1917 scheme the number of spirals which, according to Curtis, were placed around the poles was about 100’000, both on the north side and on the south side of the Milky Way. The comparison between the 1917 and the 1920 schemes is therefore useful in order to have a snapshot of the new estimates of the amount of spirals by observations accumulated during those years. Slipher and the radial velocities of spirals 135

The first radial velocity of a nebula was determined in 1912 by Slipher. From that year for over a decade Slipher supplied these fundamental ob- servations. As seen in the previous chapter, radial shifts, and in partic- ular redshifts, became during the 1920’s the most important empirical evidence supporting the model of the universe proposed by de Sitter, be- cause, unlike Einstein’s model, this solution of field equations predicted spectral displacements for objects in it. At Lowell Observatory, Slipher used a 24-inch refractor with a high dispersion prism and a very short-focus (fast) camera. Indeed, “no choice of the telescope - Slipher noted - as regards aperture, or focal-length, or ratio of aperture to focus, will increase the brightness of the spectrum of an extended source” [Slipher 1915, p. 22]. On the contrary, the spectrograph and the camera were the most important factors to take into account for the brightness. Slipher acknowledged that “I have given more attention to the ve- locity since the study of the spectra had been undertaken with marked success by Fath at Lick and Mt. Wilson, and by Wolf at Heidelberg” [Slipher 1915, p. 22]. The first object which Slipher observed was the Andromeda nebula: “the early attempts - he noted - recorded well the continuous spectrum crossed by a few Fraunhofer groups, and were par- ticularly encouraging as regards the exposure time required” [Slipher 1913, p. 56]. By taking several spectrograms from September, 1912 to December, 1912, the observations revealed a mean radial velocity of v = −300 km/sec. According to Slipher, “the magnitude of this velocity, which is the greatest hitherto observed, raises the question whether the velocity-like displacement might not be due to some other cause, but I believe we have at the present no other interpretations” [Slipher 1913, p. 56]. Therefore, the conclusion was that the Andromeda nebula was approaching the solar system with this unexpected velocity. By 1915 Slipher obtained spectral displacements of 15 nebulæ, the most part of which showed redshift, i.e. positive velocity. Among them, the objects NGC 1069 and NGC 4594 had about v = +1100 km/sec. “As 136 Observational investigations of redshift relations well as may be inferred - Slipher remarked - the average velocity of the spirals is about 25 times the average stellar velocity” [Slipher 1915, p. 23]. In a subsequent paper, which appeared in 1917, Slipher proposed a list of the radial velocity of 25 spiral nebulæ, and also compared his own results with velocities measured by Pease, Wright and Moore [Slipher 1917 ]. It is important to recall the fact that in 1917 de Sitter related mea- surements of radial velocity to the cosmological consequences of his own world-model, however by referring only to three nebulæ. By quoting what Hubble wrote in 1936, “Slipher’s list of 13 velocities, although published in 1914, had not reached de Sitter, probably as a result of the disruption of communications during the war” [Hubble 1936, p. 109].

6.3 The K term and the solar motion

The possibility to relate the relevant radial velocities of spiral nebulæ to the stellar system was soon realized by some astronomers around 1916, in the framework of the determination of the solar motion towards the so-called Sun apex. The components of the motion of the Sun, indeed, corresponded to (−X, −Y, −Z) through the general formula:

v = X cos α cos δ + Y sin α cos δ + Z sin δ, (6.3) where v was the observed radial velocity of a star (or of a group of stars), and (α, δ) were, respectively, the right ascension and declination of such a star. Already at the end of 1915, O. H. Truman, at the Iowa State Ob- servatory, suggested that “it seems likely that a distinct motion of our own spiral nebula with respect to them [spiral nebulæ] would manifest itself, and conversely, if we can find such a motion of our system with respect to the nebula, it will be quite strongly urged upon us that they are other sidereal system” [Truman 1916, p. 111]. Referring to Slipher’s The K term and the solar motion 137

1915 measurements, Truman concluded that “our nebula is moving with a velocity of +670 km/sec in the direction of R.A. 20 hrs, Dec −20◦. (...) I think the determination of radial velocities of spiral nebulæ should be rigorously pursued, in order to quickly find out whether the above results are indeed real” [Truman 1916, p. 112]. In 1916, Reynold Young (1886-1977) and William Harper (1878-1940) (Dominion Observatory, Canada), in front of the results proposed by Slipher, noted that “it seems quite possible that our solar system and the whole universe or spiral to which it belongs may be rushing through space with a speed as yet undetermined, but of the same order of magnitude as other spiral nebulæ” [Young-Harper 1916, p. 134]. Indeed observations of such radial motions in spirals could be used to determine the direction and speed of our system. By using data of 15 nebulæ, Young and Harper found the velocity of v = −598 ± 234 km/sec for what they denoted as the “velocity of the universe” [Young-Harper 1916, p. 135]. The relation of the stellar system to radial motions of spiral neb- ulæ was faced in 1916 also by George Paddock (1879-1955), astronomer at Lick Observatory, who further developed the equation of motion by taking into account the so-called K term. The method which Paddock suggested referred to what Campbell had already proposed in 1911 in order to determine the solar motion from stellar velocities. In 1911, indeed, Campbell found that a constant K term, which Campbell interpreted as a “systematic error”, had to be added in the equation of the solar motion [Campbell 1911, p. 93]:

v = v0 cos d + K, (6.4) where

cos d = cos δ0 cos δ cos(α0 − α) + sin δ0 sin δ. (6.5)

Campbell proposed the above equation by taking into account 35 groups of B stars; v was the mean observed radial velocity of each of the 35 groups, v0 was the velocity of the Sun with respect to the system of B 138 Observational investigations of redshift relations stars, d was the average angular distance of each group of B stars, and

(α0, δ0) were the right ascension and the declination of the Sun apex. “An error of obscure source - Campbell concluded in 1911 - causes the radial velocities of Class B stars to be observed too great by a quan- tity, K, amounting to several kilometers” [Campbell 1911, p. 105]. In 1913, in his famous book “Stellar motions”, Campbell highlighted that “the universe of Classes B to B9 stars is expanding, with reference to the instantaneous position of the solar system as a center, at the rate of 4.93 km/sec. (...) A personal equation in the measurements of the spectrograms, systematically positive, amounting to 5 km/sec, cannot be regarded as possible” [Campbell 1913, p. 203]. As seen in previous chapter, in 1917 de Sitter took into account this puzzling systematic red- shift of B stars which Campbell had pointed out. De Sitter proposed that the great part of this shift could be interpreted as a spurious ve- locity through the g44 potential in his own cosmological solution of field equations. In 1916, Paddock related such a K term to the velocities of spiral nebulæ. He inaugurated, as we will see, a method which would have been used by several astronomers “for the purpose of finding a possible space motion of our system of stars, including the Sun, relative to a possible system of spirals of which our stellar system may be a unit, and the spirals each perhaps a system of stars” [Paddock 1916, p. 109]. Interestingly, Paddock, in front of the relevant average positive radial velocities of groups of spirals at the north and south galactic poles, noted that “accordingly a solution for the motion of the observer through space should doubtless contain a constant term to represent the expanding or systematic component whether there be actual expansion or a term in the spectroscopic line displacements not due to velocities” [Paddock 1916, p. 113]. Such a required constant term was represented by the K term. Paddock used the same form of the equation of motion with the K term already found by Campbell. He analyzed the results proposed by Young and Harper about nebulæ, and obtained a K term ranging The K term and the solar motion 139 from + 248 to + 295 km/sec [Paddock 1916, p. 114]. In order to explain such a large value of the K term, Paddock argued that “the algebraic average numbers of the velocities of spirals will probably diminish with increasing numbers of observed velocities. Probably, likewise, the value of the apparent systematic term will diminish, so that it may therefore be concluded that its appearance here is the result of insufficient data” [Paddock 1916, p. 115].

6.3.1 Wirtz and de Sitter’s cosmology

Accounting for a large K term, a possible explanation of radial ve- locities of nebulæ through the cosmology of de Sitter was proposed in 1924 by Wirtz, a German astronomer sometimes called “the European Hubble without a telescope” [Sandage 2005, p. 500]. Wirtz started his astronomical activity at Strasbourg Observatory in 1902, where, by using a large refractor with small focal ratio, the main researches were directed to the study of the nebulæ [Duerbeck-Seitter 2005, p. 168]. In 1918, now in Kiel, Wirtz considered the relation of the K term to the motion of spirals, as done by Paddock in 1916. Wirtz obtained a value of K = +656 km/sec: “if one gives this value a literal interpretation - Wirtz wrote - the system of spiral nebulæ disperses with the velocity 656 Km/sec relative to the momentary position of the solar system as center” [Wirtz 1918, p. 115. Engl. tr. in Seitter-Duerbeck 1999, p. 238]. In 1922 Wirtz, in order to calculate the solar motion, took into ac- count measurements related to 29 spirals. He found a notable value of K = +840 ± 141 Km/sec [Wirtz 1922, p. 351]. Moreover, by investigat- ing radial motions and observational properties of such a group of spirals, Wirtz suggested that a linear relation existed between velocity and abso- lute magnitude, in the sense that the nearest nebulæ showed a tendency to approach, whether the distant ones receded from our galactic system [Wirtz 1922, p. 352]. 140 Observational investigations of redshift relations

It was in 1924 that Wirtz clearly related his own statistical work on radial velocity measurements to the model of the universe proposed by de Sitter. In his 1924 paper “De Sitters Kosmologie und die Radialbe- wegungen der Spiralnebel” (The de Sitter cosmology and radial motions of spiral nebulæ), Wirtz appreciated that both Einstein’s and de Sit- ter’s worlds corresponded to limiting cases, the actual world being an intermediate state between them [Wirtz 1924, p. 21]. Referring to the question of the mass horizon in the universe of de Sitter with which de Sitter, Eddington and Weyl previously dealt, the slowing down of natural clocks at increasing distances from the origin actually was, according to Wirtz, a phenomenon accessible to astronomical observations. Spectral displacements towards the red corresponded to such an effect, together with the redshift effect suggested by Eddington due to the acceleration of particles which increased with distances in de Sitter’s world. The fact that, by inserting test particles, the empty world of de Sitter became non-static was a remarkable feature rather than an objectionable one. Through the theory of de Sitter, indeed, it could be predicted an increas- ing redshift for increasing distances. However, as Wirtz pointed out, even though Doppler radial velocities of many nebulæ were known, distance measurements were not available. Nevertheless, the apparent diameter of nebulæ (Dm), i.e. the apparent measure of major axes, could be taken into account, by supposing that the linear diameter was nearly the same for all spirals. Therefore radial motions, i.e. radial velocities, should have increased with decreasing apparent diameters measured in spiral nebulæ [Wirtz 1924, p. 23]. Wirtz used the list of 41 radial velocities ob- tained by Slipher which Eddington had published in his 1923 book “The mathematical theory of relativity”, together with the value v = +700 Km/sec for NGC 2681. With regard to the apparent diameter of these 42 objects, Wirtz referred to 1918 works of Curtis at Lick Observatory, and 1917-1920 data by Pease at Mt. Wilson Observatory. The analysis of the nebulæ which approximately had the same logarithm of apparent diameter revealed that Doppler radial velocities decreased for groups of The K term and the solar motion 141 spirals with increasing apparent diameters. Vice versa, by considering groups of spirals with the same radial velocities, the largest velocities corresponded to the smallest logarithms of apparent diameter. In the first case Wirtz determined the average relation [Wirtz 1924, p. 24]:

v = 914 − 479 · log (Dm). (6.6)

In the second case he determined the average relation:

log (Dm) = 0.96 − 0.000432 · v. (6.7)

The average velocity was v = +574 km/sec. Thus, according to Wirtz, it was clear that the radial motion of spiral nebulæ remarkably increased with increasing distance. Actually, a diagram with values of velocities and logarithmic diameters showed a V -shaped or triangular form. Wirtz deduced from such a relation that apparently small nebulæ had either very small or large velocities, while apparently large nebulæ showed only small velocities: “the dispersion of the linear dimensions of the nebulæ - Wirtz noted - fills the triangular plane in such a way that among the near nebulæ absolutely small and large objects are visible, while in the depth of space only the absolutely largest are subject to observing their radial motions” [Wirtz 1924, p. 24. Engl. tr. in Seitter-Duerbeck 1990, p. 376]. For the largest nebulae, Wirtz determined the relation between velocity and logarithmic diameter in the form:

v = 2200 − 1200 · log (Dm). (6.8)

According to Wirtz, such an empirical law, together with the velocity- magnitude relation which he had proposed in 1922, could be interpreted just through the properties of de Sitter’s world model. 142 Observational investigations of redshift relations 6.4 Astronomers at work: Lundmark and Str¨omberg

As seen, in 1924 the connection between the theoretical relativistic description of the universe as a whole and observational cosmology was analyzed and complicated through some important contributions. In that very year, indeed, Silberstein took into account the static form of de Sitter’s universe and observations of globular clusters. By consid- ering the modulus of spectral shift, he proposed that a linear relation existed between shift and distance. However, according to Silberstein, the recession of astronomical objects at large distances was not system- atic. Furthermore, as just seen, in 1924 Wirtz followed Eddington’s and Weyl’s suggestions about a general tendency to scatter of test particles in de Sitter’s world. Wirtz argued that the evidence of a relevant value of the K term could be explained just through solution B (the de Sit- ter’s model), and proposed that a linear relation existed between velocity (redshift) and logarithm of apparent diameter. Therefore, beside the still unsolved question of the nature of the neb- ulæ, some puzzling questions marked the first steps in the rise of scien- tific cosmology: did observations reveal a general systematic recession of nebulæ? If so, which was the form of such a redshift-relation and the in- terpretation of redshift? How to reconcile the de Sitter effect, predicted by the metric of the truly empty de Sitter’s universe, with the actual non-empty universe? Eventually, which were the distances of nebulæ and which was the radius of the universe? The relation of the spirals to the stellar system and the determination of the curvature of space-time in de Sitter’s universe were faced by the Swedish astronomer Knut Lundmark in 1924 and 1925. At the time of the 1920 discussion between Shapley and Curtis about the nature of the nebulæ, Lundmark agreed with Curtis on the large distances at which spirals were placed, and seemed to prefer the extragalactic position of Astronomers at work: Lundmark and Str¨omberg 143 them. However, by using Hubble’s words, Lundmark “was noncommit- tal” [Hubble 1936, p. 89]. With regard to the motion of spirals, Lund- mark had pointed out in 1920 that “it is obviously by no means out of the question, that the great radial velocities found for the spirals are not real, but that the measured displacements of spectral lines represent other phenomenon than a Doppler-effect” [Lundmark 1920, p. 47]. In 1924, Lundmark discussed the results obtained by Silberstein, and carefully examined the question of the nature of measured redshifts. By using data of the Andromeda nebula obtained by Wright at Lick Ob- servatory, Lundmark showed that the spectral displacement was nearly constant for 16 lines ranging from 3970 A˚ to 4860 A˚, with the average value [Lundmark 1924, p. 748]: ∆λ z = = 0.00116 ± 0.00008. (6.9) av λ Therefore, according to Lundmark, “the shift is evidently a Doppler one. The same applies to the velocities of the globular clusters. Another question is, whether such a large Doppler shift represents motion in the line of sight alone or is caused in other ways?” [Lundmark 1924, p. 748]. Lundmark was skeptical about Silberstein’s proposal that the motions of spirals and globular clusters showed any effect of the curvature of space- time. In order to compute the motion of the Sun, Lundmark used data of h ◦ 18 globular clusters, and found the values A = 20 .4, D = +60 , v0 = 305 km/sec, K = +31 km/sec, while with data of 43 spirals it resulted A = h ◦ ◦ 20 .3, D = +75 ± 30 , v0 = 651 ± 135 km/sec, K = +793 ± 88 km/sec [Lundmark 1924, p. 748]. “The explanation may simply be - Lundmark pointed out - that our local system as a whole has the motion found above relatively to the huge systems of globulars and spirals” [Lundmark 1924, p. 749]. However the question remained open until 1929, when more information was accumulated. Furthermore Lundmark criticized in his 1924 analysis the determina- tion of the curvature radius of the universe proposed by Silberstein, who, quoting Lundmark, “has not given, and will probably not be able to give, 144 Observational investigations of redshift relations any justification for the use of the velocities of the globular clusters for a determination of R”[Lundmark 1924, p. 750]. Since for these objects the K term was small, Lundmark suggested that this result indicated that globulars were both nearer than spirals, and “little if at all affected by the slowing down of atomic vibrations in distant objects in de Sitter’s world which might be erroneously interpreted as a motion of recession” [Lundmark 1924, p. 750]. Moreover, Silberstein’s result was objection- able because Silberstein had used in his own calculations selected values of available radial velocities, excluding just objects which did not give a constant value of the curvature radius R. By using data of 18 globular clusters, Lundmark concluded that “there is little or no correlation be- tween v and r, in contrary to what is to expected from the theory of Dr. Silberstein” [Lundmark 1924, p. 752]. From such data, by assuming the possibility to set a value for R, it followed a mean value R ' 19.7 · 1012 km, three times larger than the radius adopted by Silberstein.

Figure 6.3: Diagram with radial velocities and distances of globular clusters proposed by Lundmark in 1924 [from Lundmark 1924, p. 753]. Astronomers at work: Lundmark and Str¨omberg 145

Since the value of R was uncertain, Lundmark suggested that other classes of distant objects could be used in order to calculate the curvature radius. He took into account radial velocities and distances of, respec- tively, 30 Cepheid stars, 8 Novæ, 27 O stars, 29 R stars, 25 N stars and 31 Eclipsing variables5. For these classes of objects any progression seemed to be absent when the radial velocities were plotted according to the corresponding distances. The average values of the curvature radius were, respectively, 7.5, 41, 4.0, 6.7, 2.3, 2.7 · 1012 km [Lundmark 1924, pp. 756-763]. At the end of this 1924 paper, Lundmark investigated the relation between velocity and distance of 44 spiral nebulæ. He estimated the distance of the Andromeda nebula at 200’000 pc by means of the Novæ maximum brightness method, and he used such a value as the unit of dis- tance scale. Lundmark hypothesized “that the apparent angular dimen- sions and the total magnitudes of the spiral nebulæ are only dependent on the distance” [Lundmark 1924, p. 767]. Lundmark then concluded that “there may be a relation between the two quantities, although not a very definite one” [Lundmark 1924, p. 768]. This issue was faced also in a subsequent paper published in 19256, where the Swedish astronomer remarked that “a rather definite correlation is shown between apparent dimensions and radial velocity, in the sense that the smaller and presum- ably distant spirals have the higher space-velocity” [Lundmark 1925, p.

5With regard to Cepheid distances, Lundmark followed the derivation proposed by Shapley, who used data of proper motion together with the period-luminosity law. Distances of Novæ were determined by assuming that the mean absolute maximum magnitude had almost a constant value. 6It is interesting to note that at the end of this 1925 paper about motions and distances of spirals, Lundmark discussed the question of the extension of the universe, recalling Charlier hierarchical model which has been mentioned in Chapter 3 of present thesis. “Our present knowledge - Lundmark wrote - as to the space-distribution of the stars and the spirals can be summed up in the statement: our stellar system and the system of spiral nebulæ are constructed according to the conceptions expressed in the Lambert-Charlier cosmogony”[Lundmark 1925, p. 893]. 146 Observational investigations of redshift relations

867]. In this paper Lundmark returned on the K term, and suggested that its value was not constant in space. He proposed that the solar motion could be expressed through7 [Lundmark 1925, p. 867]:

v = X cos α cos δ + Y sin α cos δ + Z sin δ + k + lr + mr2. (6.10)

Here k, l, m were constants, and r was the relative distance of a spiral derived through the apparent diameter and the total absolute magnitude, through the assumption of the same absolute dimension of spirals. By using data of 44 spirals, and by taking the distance of the Andromeda nebula as the distance unit, it followed:

K = +513 + 10.365r − 0.047r2 km/sec. (6.11)

“According to the above expression - Lundmark noted - the shift reaches its maximum value, 2250 km/sec, at some 110 Andromeda units, which (...) corresponds to a distance of 108 light years. (...) One would scarcely expect to find any radial velocity larger than 3000 km/sec among spirals” [Lundmark 1925, p. 867]. In 1925 another important analysis on the subject appeared by Str¨om- berg, at Mt. Wilson Observatory. In this paper Str¨omberg summarized all 63 measured radial velocities of globular clusters and “non-galactic nebulæ” [Str¨omberg 1925, pp. 354-355]. According to Str¨omberg, be- side the interesting fact that the large solar velocity which was obtained from these data “indicated that a fundamental reference system could be defined by them8”, such an analysis was useful to ascertain “whether the velocities give any evidence of a curvature of space-time” [Str¨omberg 1925, p. 353]. With regard to the Sun’s apex, Str¨omberg obtained the values α = 315◦, δ = +62◦. The velocity was about 350 and 300 km/sec, which

7Note that Lundmark wrote sin δ instead of cos δ in the first term on the right-hand side. 8Str¨omberg had previously dealt with the existence of a universal world frame in [Str¨omberg 1924 ]. Astronomers at work: Lundmark and Str¨omberg 147 was determined, respectively, by data of spirals and globulars [Str¨omberg 1925, p. 356]. The value of the K term resulted +616 km/sec, however, as suggested also by Lundmark in the same year, Str¨omberg noted that “the assumption of a constant K term for the nebulæ is probably only approximately correct” [Str¨omberg 1925, p. 358]. Dealing with the curvature of space, Str¨omberg recalled the conse- quence that from the static form of de Sitter’s universe a redshift in spectral lines was expected, due to the apparent slowing down of atomic vibrations and producing “a fictitious positive radial velocity” [Str¨omberg 1925, p. 359]. Therefore, radial velocities could be used in order to verify their relation to distances. Str¨omberg pointed out in his analysis that Sil- berstein’s conclusion about both red and blue shift “cannot be regarded as conclusive. One thing is obvious, however. If the nebulæ studied are at about the same distance (...) of globular clusters, we cannot regard the large positive K term as a de Sitter effect, as the K term for the clusters must be very small” [Str¨omberg 1925, p. 359]. Since distances of spirals were not known, it was reasonable to have an indication of the distances by assuming the same total brightness. However, no reliable correlation between velocities and distances resulted from data. “De Sitter’s effect - Str¨omberg pointed out - can be regarded as disproved by the clusters if their distances are of the same order as those of the nebulæ. Silberstein’s effect seems possible, but cannot be established by the data” [Str¨omberg 1925, p. 361]. Therefore Str¨omberg concluded that it was not possible to confirm any dependence of radial motion on distance from the Sun. As seen, through the authoritative contributions of Lundmark and Str¨omberg, the explanation of relevant redshifts came to a standstill. In particular, there was not a clear interpretation of the nature of redshift which was shared by scientific community. In 1925, dealing with the question of the finiteness of the universe, Archibald Henderson (1877-1963) wrote that “if, as now appears prob- able, the spirals are isolated systems, this recession must be explained, it appears, either as a wholesale error or else as a relativistic effect. (...) 148 Observational investigations of redshift relations

Figure 6.4: Summary of velocities measured in globular clusters and nebulæ up to 1925 [from Str¨omberg 1925, pp. 354-355]. Astronomers at work: Lundmark and Str¨omberg 149

Much additional data will be required and many further researches made before it will be possible categorically to decide between the infinite, limiteless, Euclidean universe of Newton and the finite, unbounded, non- Euclidean universe of Einstein or of de Sitter” [Henderson 1925, p. 223].

On the contrary, Campbell remarked in 1926 that “the motions of the spirals seem to free them from the charge that they are retainers of our stellar system. (...) Although other conditions than the radial velocity of the light source as a whole are known to displace spectral lines from their normal positions, there seems now to be no inclination to doubt that the large displacements observed by Slipher are chiefly and perhaps wholly Doppler-Fizeau effect” [Campbell 1926, p. 80].

In 1927 large radial velocities in (extragalactic) nebulæ were discussed in the book “Astronomy” by Russell, Raymond Dugan (1878-1940) and John Stewart (1894-1972). Such velocities indicated a solar motion of about 400 km/sec, and the mean velocity of spirals was about v = +700 km/sec. Referring to de Sitter’s suggestion of spurious velocities, in front of the observed recession these authors pointed out that “whether this represents a real scattering of the nebulæ away from this region where the Sun happens to be is very doubtful. It may arise from some other cause” [Russell-Dugan-Stewart 1927, p. 850.]

The question of the nature of the nebulæ, as already mentioned, was eventually solved by Hubble, who showed in 1925 that spirals were true extragalactic systems. Again, as we shall see, it was just Hubble who clarified the issue about the form of redshift relation. Hubble empirically established in 1929 that such a relation between redshift (velocity) and distance among distant nebulæ existed, was actually linear (at least from available data), and could represent a possible confirmation of the de Sitter effect. 150 Observational investigations of redshift relations 6.5 Hubble and the universe of galaxies

The contributions given by Hubble represent a milestone in the his- tory of astronomy and cosmology9. Quoting Osterbrock and Sandage, Hubble “was a demon of energy, observing and making new discoveries in quick succession” [Osterbrock 1990, p. 273], and “in only 12 years, from 1924 to 1936, brought to an almost modern maturity the four foun- dations of observational cosmology, even as its principles are practiced today” [Sandage 1998, p. 2]. Indeed, venturing into the realm of the nebulæ, Hubble clarified that spirals were really extragalactic systems, i.e. galaxies (1925), then proposed a galaxy classification system shaped like a tuning-fork (1926), established that the relation between velocities and distances of nebulæ was linear (1929), and inaugurated a program of galaxy counts as a function of magnitudes, N(m), attempting to measure the curvature of space-time (1934). As mentioned in Chapter 3, Hubble proposed a very large distance of the Andromeda nebula (M31) by using 60-inch and 100-inch reflectors at Mt. Wilson Observatory. Indeed, observing outer regions of M31 and M33, “a survey of the plates made with the blink comparator - Hubble noted - has revealed many variables among the stars, a large proportion of which show the characteristic light-curve of the Cepheids” [Hubble 1925a, p. 252]. Among variables and Novæ discovered in such nebulæ, Hubble determined period and magnitude of 22 Cepheids in M33 and 12 in M31, by taking, respectively, 65 plates and 130 plates. Photographic magnitudes were obtained from 12 comparisons of selected areas with the 100-inch telescope, with exposures from 30 to 40 minutes [Hubble

9A vast literature exists about the life and works of Hubble. For further readings we refer to [Hetherington 1996, Christianson 1995, Christianson 2004, Sandage 1989, Osterbrock-Brashear-Gwinn 1990 ]. Comprehensive studies about the developments of redshift observations in nebulæ and the question of an empirical velocity-distance relation can be found in [Smith 1979, Smith 1982 ]. Moreover, a great description of measurements of distances and radial velocities in extragalactic nebulæ was proposed by Hubble himself in his own famous book “The realm of the nebulæ” [Hubble 1936 ]. Hubble and the universe of galaxies 151

1925b, p. 140]. “The now familiar period-luminosity relation - Hubble wrote - is conspicuously present” [Hubble 1925a, p. 254]. By assuming that the variable stars were actually related to the spirals, the distance modulus which resulted was m − M = 21.8 for M31, and m − M = 21.9 for M33. Having corrected such values by half the average ranges of the Cepheids (because the original method by Shapley was based on median magnitudes), the final value was about m − M = 22.3 for both objects. This value corresponded to a distance of about 285’000 pc (930’000 light years) [Hubble 1925b, p. 142]. Thus Hubble gave the fundamental confirmation that the “island uni- verse” theory was correct. Hubble furnished through a reliable empirical method, based on distance indicators, an astonishing distance for the An- dromeda nebula, clearly outside the boundary of globular clusters which Shapley had previously proposed for the Milky Way. In the same year, Hubble proposed the distance of another object, the irregular nebula NGC 6822. Through the application of the period- luminosity law to 11 Cepheid variables in such a system, he found a distance modulus of m − M = 21.65, corresponding to a distance of about 214’000 pc. (700’000 light years) Therefore such an object “was definitely assigned to a region outside the galactic system” [Hubble 1925c, p. 409]. Moreover, Hubble remarked that “of especial importance is the conclusion that the Cepheid criterion functions normally at this great distance. (...) This criterion seems to offer the means of exploring extra- galactic space” [Hubble 1925c, p. 432]. With regard to M33, by using data of 35 Cepheids, Hubble set the distance modulus of such a nebula at m − M = 22.1, corresponding to a distance of 263’000 pc (850’000 light years) [Hubble 1926a]. In 1926, in a paper devoted to a general classification of extragalac- tic nebulæ, Hubble explained his own “working hypothesis” on which he would have later based the effort to calculate the distances of other galaxies. In such a paper, where Hubble furnished the distinction among elliptical, normal spiral, barred spiral and irregular nebulæ, Hubble sug- 152 Observational investigations of redshift relations gested that “the various types are homogeneously distributed over the sky, their spectra are similar, and the radial velocities are of the same general order. These facts, together with the equality of the mean mag- nitudes and the uniform frequency distribution of magnitudes, are con- sistent with the hypothesis that the distance and absolute luminosity as well are of the same order for the different types” [Hubble 1926b, p. 332]. The possibility to obtain the absolute magnitude was restricted to a very few number of galaxies which distances were known. Hubble adopted the mean value MT = −15.2 from data referred to 8 nebulæ: our Galaxy, M31, M32, M33, M101, the Magellanic Clouds and NGC 6822. Furthermore, the mean absolute magnitude which Hubble derived from the brightest stars of such systems was MS = −6.3. [Hubble 1926b, p. 356]. Therefore, through this generalization, apparent magnitudes could be used to determine distances. Hubble confined at the end of this paper some considerations on the theoretical cosmological consequences of general relativity and on the possibility to calculate the radius and the mass of Einstein’s universe by estimates of the density of matter [Hubble 1926b, p. 369]. However, fol- lowing [Smith 1982, p. 181], these considerations were slightly confused, revealing some misunderstandings in the difference between solution A (Einstein’s model) and solution B (de Sitter’s model). It was in 1928 that Hubble focused his own attention on the astro- nomical consequences of de Sitter’s solution, in particular to the redshift relation. In that year Hubble met de Sitter and Eddington in Leiden, during the 1928 International Astronomical Union Meeting. Following [Christianson 1995, p. 198], the idea to increase the set of radial velocities obtained by Slipher was suggested to Hubble by de Sitter himself. With regard to 1928, i.e. to such an important date for Hubble’s “cosmological” plans, it is interesting to note that, in his reconstruction of the history of Mt. Wilson Observatory, Sandage remarks that “Robertson, who was my professor of mathematical physics at CalTech in 1951, told me in 1961 (...) that in 1928 he had discussed with Hubble his prediction and Hubble and the universe of galaxies 153 partial verification of an expanding universe based on Friedmann’s 1922 solution. Robertson had used Slipher’s velocities and his own distance estimates based on Hubble 1926 calibration of mean galaxy luminosity to obtain observational results supporting the theory” [Sandage 2005, p. 501]. As seen in previous chapter, in 1928 Robertson derived a linear relation between distance and velocity from the stationary (expanding) metric of de Sitter’s universe, suggesting that such a relation was roughly confirmed by comparing radial velocities measured by Slipher (1923) and distances obtained by Hubble (1926). In 1929, Milton Humason (1891-1972), the man who would have ex- tended the radial velocity scale, wrote that “about a year ago Mr. Hub- ble suggested that a selected list of fainter and more distant extragalactic nebulæ (...) be observed to determine, if possible, whether the absorp- tion lines in these objects show large displacements towards longer wave- lengths, as might be expected on de Sitter’s theory of curved space-time” [Humason 1929, p. 167]. This statement is therefore useful to understand Hubble’s intention with regard to distant nebulæ, i.e. to obtain further measurements of distances and velocities in order to establish the general form of the relation between these observable quantities.

6.5.1 The contributions by Humason

While Hubble attempted to measure distances, Humason dealt with radial velocities of galaxies, giving a sort of continuity in the monumental work of Slipher about spectrographic measurements of relevant redshifts. The first important result in such an exploration of deep space was proposed by Humason in 192710. By using a two-prism spectrograph (with a camera of 3-inch of focal length) at the Cassegrain focus of the 100-inch reflector at Mt. Wilson, Humason measured the radial velocities of M101 and NGC 6822. For the former, Humason observed N1, N2, Hβ

10According to [Hetherington 1996, p. 121], such a 1927 paper by Humason was probably written by Hubble himself. 154 Observational investigations of redshift relations

and Hγ emission lines after exposures of 4 and 5 hours. The weighted mean value from the two plates was about v = +216 km/sec [Humason 1927, p. 317]. For the latter, the mean velocity from the same lines just mentioned, weighted from plates of exposures of 5 and 6 hours, was v = +133 km/sec. “The radial velocity of both NGC 6822 and M101 - Humason concluded - are unusually low for non-galactic objects. This is consistent with the marked tendency already observed for the smaller velocities to be associated with the larger (and hence probably closer) nebulæ and those which are highly resolved” [Humason 1927, p. 318]. Therefore, an empirical progression of velocities with respect to distances was suggested by Humason in this 1927 work. Such a suggestion was clearly related by Humason to the de Sitter effect in another paper, which appeared in 1929 and which was commu- nicated at the same time of Hubble’s famous 1929 article. In this work Humason reported the result about the large velocity of NGC 7619, which was obtained with exposure times of 33 and 45 hours. Humason obtained a mean value of v = +3779 km/sec, twice larger than the available largest velocity of NGC 584, for which Slipher had obtained v = +1800 km/sec. According to Humason, this result, together with what Hubble would have shortly after communicated, suggested “an influence of distance upon the observed line shift-such as would be produced, for example, on de Sitter’s theory, both by the apparent slowing down of atomic vibra- tions with distance and by a real tendency of material bodies to scatter in space” [Humason 1929, p. 167].

6.5.2 Hubble’s 1929 relation

The 1929 paper about the velocities and distances among extragalac- tic nebulæ, which marked a turning point in the rise of scientific cosmol- ogy, was introduced by Hubble as “a re-examination of the question of the K term” [Hubble 1929, p. 168]. Hubble based the determination of distance upon the criteria of abso- Hubble and the universe of galaxies 155 lute magnitude and luminosity which he had already suggested in 1926. He took into account 24 nebulæ which velocities were known, four of them obtained by Humason. For seven of these stellar systems the dis- tance was obtained by direct investigation of many stars in them. Then Hubble calculated the distance of other 13 nebulæ by using the apparent magnitude of their brightest stars and the absolute magnitude criteria

(MS = −6.3), i.e. “the criterion of a uniform upper limit of stellar lu- minosity” [Hubble 1929, p. 170]. For other 4 objects, which were in the Virgo Cluster, Hubble assigned the distance of 2 · 106 pc by using the distribution of nebular luminosity. Since “the data (...) indicate a lin- ear correlation between distances and velocities” [Hubble 1929, p. 170], Hubble proposed a new expression of the equation of the solar motion, in- troducing a direct proportionality between velocity and distance through the K term, which became later known as H, the “Hubble constant”11:

v = X cos α cos δ + Y sin α cos δ + Z sin δ + Kr. (6.12)

By using the 24 nebulæ individually the result was:

K = +465 ± 50 km/sec, (6.13) while the combination of these 24 objects in 9 groups according to their distances implied: K = +513 ± 60 km/sec. (6.14) There were other 22 nebulæ for which the velocities were known, which distances were, however, not available. In order to estimate the K cor- relation term also for these nebulæ, Hubble derived their mean distance from the mean absolute magnitude, and compared this value with the mean radial velocity. It followed a value of K = +530 km/sec at a dis- tance of 1.4 · 106 pc, which agreed with the relation obtained from the first 24 objects. “The results - Hubble pointed out - establish a roughly linear relation between velocities and distances among nebulæ for which velocities have

11See [Trimble 1996 ] for a 1925-1975 history of such a “constant”. 156 Observational investigations of redshift relations been previously published, and the relation appears to dominate the distribution of velocities” [Hubble 1929, p. 173].

Figure 6.5: The linear velocity-distance relation among extra-galactic nebulæ which Hubble proposed in 1929 [from Hubble 1929, p. 172].

Concluding his own investigation, Hubble referred to the possibility that such a relation could actually represent the de Sitter effect, “and hence that numerical data may be introduced into discussions of the gen- eral curvature of space” [Hubble 1929, p. 173]. Both the interpretations of the de Sitter effect, namely the slowing down of atomic vibrations and the scattering of particles, had to be taken into account. According to Hubble, “the relative importance of these two effects should determine the form of the relation (...), and in this connection it may be empha- sized that the linear relation found in the present discussion is a first approximation representing a restricted range in distance” [Hubble 1929, p. 173]. Following [Hetherington 1996 pp. 124-128], Hubble introduced the linearity of the velocity-distance relation by emphasizing the empirical as- pect of his own work, based on the widely accepted method of the K term. The refutation of Silberstein’s results by Lundmark and Str¨omberg, in- deed, was not yet forgotten. Therefore, Hubble cautiously mentioned the theoretical aspects of his work, i.e. the connection to the de Sitter effect, just at the end of the paper, claiming that “new data to be expected in Hubble and the universe of galaxies 157 the near future may modify the significance of the present investigation” [Hubble 1929, p. 173]. The relation which Hubble found between velocity and redshift be- came known as the “Hubble law”:

v = zc = Kr. (6.15)

Such a relation, as already mentioned, marked the turning point in the connection between observations at large distance and theoretical rela- tivistic cosmology. Indeed, despite the fact that, according to Hubble, the linearity of the relation had to be confirmed by future observations12, it was now evident that extragalactic nebulæ, few counterexamples apart13, showed a systematic redshift. However, the cause of such a redshift was still unknown. A possible explanation of redshift was proposed by Zwicky in 1929. Beside the theoretical possibilities which the model of de Sitter offered, Zwicky added and advocated the idea that such spectral shifts could be attributed to a “gravitational drag of light” analogue to the Compton effect [Zwicky 1929, p. 775]. This interpretation became later known as the “tired-light hypothesis”. “According to the relativity theory - Zwicky wrote - a light quantum hν has an inertial and a gravitational hν mass c2 . It should be expected, therefore, that a quantum hν passing a mass M will not only be deflected but it will also transfer momentum and energy to the mass M and make it recoil. During this process, the light quantum will change its energy, and therefore its frequency” [Zwicky 1929, p. 775]. According to Zwicky, spectral displacements towards the red were expected by considering light traveling at a distance L, which

12In addition, Shapley in 1929 criticized the actual linearity of Hubble’s relation [Shapley 1929 ]. However, Shapley accepted such a result in 1930. See [Smith 1982, pp. 183-185] for further readings on Shapley’s reaction to Hubble’s proposal. 13See for example the objects with negative velocity, i.e. with an approaching motion, in the summary proposed by Str¨omberg in 1925. 158 Observational investigations of redshift relations would have lost the momentum: µ ¶ hν 1.4 π Gρ DL hν ∆ = · , (6.16) c c c2 and therefore ∆ν 1.4 π Gρ DL = . (6.17) ν c2 Here ρ was the density of matter in the universe, and D the distance at which the perturbing effect faded out [Zwivky 1929, p. 778]. It was in 1930 that eventually the cosmological explanation of red- shift, i.e. the explanation of spectral shifts as due to the expansion of the universe, entered modern cosmology by revaluing the interpretation which Lemaˆıtrehad pointed out already in 1927.

velocity relation Wirtz, 1924 v = a − b · log (Dm) Lundmark, 1925 v = k + lr + mr2 Str¨omberg, 1925 no definite relation Hubble, 1929 v = Kr

Table 6.1: Summary of different empirical relations proposed during the 1920’s. The observable quantities are the velocity (v), the angular diam- eter (Dm) and the distance (r) of nebulæ (galaxies). Chapter 7

The “third way” between Einstein’s and de Sitter’s solutions

In this chapter the second renewal of relativistic cosmology, i.e. the general acceptance of the model of the expanding universe, is analyzed. The rise of the expanding universe involved the decline of the interest in the de Sitter effect. Indeed, in 1930 truly non-static and non-empty mod- els of the universe, already proposed in 1922 by Friedmann and indepen- dently in 1927 by Lemaˆıtre,were rediscovered by scientific community. As from 1930, such solutions were taken into account in order to explain the observational evidence of the systematic recession of extragalactic stellar systems revealed by Hubble’s observations. The puzzling ques- tions of the origin of extragalactic redshifts and of the redshift-distance relation could be finally rightly interpreted by considering the metric of a universe which world-radius increased in time, as in particular Lemaˆıtre had suggested in his 1927 work.

159 160 The “third way” between Einstein’s and de Sitter’s solutions 7.1 1930: Eddington, de Sitter and the ex- panding universe

During the 1930 January Meeting of the Royal Astronomical Society, Eddington and de Sitter, in front of the sensational result by Hubble that galaxies showed a systematic redshift, agreed on the conclusion that non-static intermediary solutions between Einstein’s and de Sitter’s uni- verses represented a suitable description of the actual universe. “The question now arises - de Sitter pointed out during such a meeting - how can we account for the linear connection between the velocities and the distances?”. Eddington then remarked that ”one puzzling question is why there should be only two solutions. I suppose the trouble is that people look for static solutions. Solution A is such a static solution. So- lution B is, on the contrary, non-static and expanding, but as there isn’t any matter in it that does not matter” [R.A.S. Meeting 1930, pp. 38-39]. Eventually, it was the time to rediscover the expanding model which Lemaˆıtreinvestigated in his 1927 paper “Un univers homog`enede masse constante et de rayon croissant, rendant compte de la vitesse radiale des n´ebuleusesextra-galactiques” (A homogeneous universe of constant mass and increasing radius, accounting for the radial velocity of extra-galactic nebulæ1)[Lemaˆıtre 1927 ]. The original letters and manuscripts which are stored at the de Sitter and LemaˆıtreArchives permit to reconstruct such an important discovery. When the report of the mentioned R.A.S. meeting appeared in “The Observatory” in February 1930, Lemaˆıtrerecalled the attention of Ed- dington on his own 1927 work. Together with copies of his 1927 paper,

1It is important to note that in the proofs of this 1927 paper, Lemaˆıtrehad initially written “variable radius” (“rayon variable”). Before the publication in the Annales de la Soci´et´eScientifique de Bruxelles, he had then changed this word with “increasing radius” (“rayon croissant”) [Lemaˆıtre Archive, Box R10]. In the following pages of present thesis, quotations from the original 1927 paper refer to the 1972 reprinted French version (see bibliography for details). 1930: Eddington, de Sitter and the expanding universe 161

Lemaˆıtresent Eddington a letter in which he noted that “I just read the February number of the Observatory and your suggestion of investigat- ing non-statical intermediary solutions between those of Einstein and de Sitter. I made this investigation two tears ago. I consider a universe of curvature constant in space but increasing in time. And I emphasize the existence of a solution in which the motion of the nebulæ is always a receding one from time minus infinity to plus infinity” [Lemaˆıtre Archive, Box D17].

Figure 7.1: Draft of the 1930 letter which Lemaˆıtresent Eddington. In this letter Lemaˆıtrerecalled the attention of Eddington on the expanding model of the universe which Lemaˆıtrehad already discovered in 1927 [from Lemaˆıtre Archive, Box D17].

Eddington immediately realized the importance of such a solution. On March 19, 1930, Eddington sent de Sitter a copy of the French paper by Lemaˆıtre,adding that “this seems a complete answer to the problem we were discussing”. In a postcard that Eddington attached to the paper, Eddington remarked that “by the way it was the report of your remarks and mine at the R.A.S. which caused Lemaˆıtre to write to me about it. (...) A research student McVittie and I had been worrying at the 162 The “third way” between Einstein’s and de Sitter’s solutions problem and made considerable progress; so it was a blow to us to find it done much more completely by Lemaˆıtre(a blow softened, as far as I am concerned, by the fact that Lemaˆıtrewas a student of mine)” [de Sitter Archive, Relativity Box, A2]. Indeed around 1930 Eddington was working in conjunction with George McVittie (1904-1988) on the stability of Einstein’s spherical universe. With regard to the “brilliant solution” by Lemaˆıtre,“although not expressly stated - Eddington remarked - it is at once apparent from his formulæ that the Einstein world is unstable, an important fact which, I think, has not hitherto been appreciated in cosmological discussions” [Eddington 1930a, p. 668].

On March 25, 1930, de Sitter sent Lemaˆıtrean enthusiastic letter, in which he showed his own satisfaction because the ds2 proposed by Lemaˆıtreactually represented a suitable, simple and elegant solution to the cosmological problem: “M. Eddington - de Sitter wrote to Lemaˆıtre - m’a envoy´eil y a quelque jours un exemplaire de votre petit, mais important, m´emoiredu 1927 (...). J’avais moi mˆeme(...) tach´ede trouver une formule pour ds2 que comprendrait les deux solutions que j’ai appel´e A et B (...). Votre solution, simple et ´el´egante, me paraˆıtenti´erement satisfaisante” [Lemaˆıtre Archive, D15]. Replying to this letter on April 5, 1930, Lemaˆıtreappreciated the interest which de Sitter gave to such a work2: “je vous remercie beaucoup de l’int´eretque vous voulez bien t´emoignerpour ma note sur l’univers de rayon variable (...). J’attends avec grand int´eretles pr´ecisionsque vous avez obtenues sur la relation entre v and r et les diagrammes que vous avez montr´eaux membres de la R.A.S.” [Lemaˆıtre Archive, D15].

Eddington and de Sitter further managed for an English translation of Lemaˆıtre’s 1927 paper, which appeared in the Monthly Notices on March, 1931 [Lemaˆıtre 1931a].

2In this letter, Lemaˆıtrealso drew the attention of de Sitter on the 1922 work by Friedmann and on Einstein’s reaction to it. The importance of Lemaˆıtre’s1927 proposal 163

Figure 7.2: Copy of the 1930 letter which de Sitter wrote to Lemaˆıtre,where de Sitter acknowledged that the 1927 proposal by Lemaˆıtrewas a solution to the cosmological problem [from Lemaˆıtre Archive, Box D15].

7.2 The importance of Lemaˆıtre’s1927 pro- posal

As seen in Chapter 5, in 1925 Lemaˆıtrehad faced some features of de Sitter’s universe. By introducing new coordinates, Lemaˆıtrehad shown that the singularity of time-coordinate in the static form of de Sitter universe was removed, however involving a non-static picture of the uni- verse. According to Lemaˆıtre,such a representation of the universe as a whole had to be discarded, because corresponded to an infinite universe (the curvature of spatial sections was k = 0). 164 The “third way” between Einstein’s and de Sitter’s solutions

However, Lemaˆıtre realized that this change of coordinates repre- sented an interesting suggestion for a world-radius changing with time. As he wrote in a note of his 1927 paper (which was not translated in the 1931 English version), by using two coordinates, one spatial and the other temporal, the de Sitter universe in the static form could be represented as the surface of a sphere, where spatial lines corresponded to meridians, whereas time lines corresponded to parallels on the sphere. The largest parallel (the equator) of such a sphere corresponded to a geodesic, and the “pole” of the sphere represented the singularity of time-coordinate. On the contrary, by changing the coordinates, Lemaˆıtreshowed that the homogeneity was finally respected. Now the time lines corresponded to meridians, and spatial lines to parallels: therefore, as Lemaˆıtrepointed out, the world-radius changed with time3. The empty universe of de Sitter, nevertheless, “is of extreme interest as explaining quite naturally the observed receding velocities of extra- galactic nebulæ” [Lemaˆıtre 1931a, p. 483]. On the contrary, as Lemaˆıtre pointed out, Einstein’s truly static and finite universe was in agreement “with the existence of matter, giving a satisfactory relation between the radius and the mass of the universe” [Lemaˆıtre 1931a, p. 483]. A third way between such solutions seemed desirable. Therefore, Lemaˆıtrecon- cluded that “in order to find a solution combining the advantages of those of Einstein and de Sitter, we are led to consider an Einstein uni- verse where the radius of space or of the universe is allowed to vary in an arbitrary way” [Lemaˆıtre 1931a, p. 484]. Lemaˆıtreapproximated the content of the universe to a rarefied gas which was uniformly and homogeneously distributed through space, whose molecules were the extragalactic nebulæ. As he soon recognized, “when

3“Les coordonn´eesrespectant l’homog´en´eit´ereviennent `aprendre pour lignes tem- porelles un syst`emede m´eridienset pour lignes spatiales les parall`elescorrespondants, alors le rayon de l’espace varie avec le temps” [Lemaˆıtre 1927, p. 90]. The author of present thesis would like to thank Prof. Dominique Lambert for having pointed out to him such an important passage in Lemaˆıtre’sanalysis. The importance of Lemaˆıtre’s1927 proposal 165 the radius of the universe varies in an arbitrary way, the density, uniform in space, varies with time” [Lemaˆıtre 1931a, p. 484]. Notably, Lemaˆıtre introduced in cosmology the concept of a time-dependent hypothetical average density of matter in the universe, which Lemaˆıtredenoted with δ. Moreover, he took into account also the contribution by the pressure. Even though the pressure by matter could be considered negligible, the radiation pressure, which Lemaˆıtredenoted with p, could not to be dis- carded. The total energy density was thus:

ρ = δ + 3p. (7.1)

Lemaˆıtrethen considered the line element of the universe (with c = 1) with a time-dependent world radius, R ≡ R(t):

ds2 = −R2dσ2 + dt2, (7.2) being dσ the spatial line element. Field equations reduced to two equa- tions, which described the variation of the world radius with respect to the world content, i.e. to radiation and matter [Lemaˆıtre 1927, p. 91]:

R02 3 3 + = λ + κρ; (7.3) R2 R2

R02 RR00 1 + 2 + = λ − κ p. (7.4) R2 R2 R2 0 dR 00 In these relations λ was the cosmological constant, R ≡ dt , and R ≡ dR0 dt . As mentioned in Chapter 2, these equations, together with the simi- lar solutions (however without the contribution by pressure) which Fried- mann proposed in 1922, became known in cosmology as the Friedmann- Lemaˆıtre(FL) equations. Lemaˆıtre,in his 1927 analysis, noted that the second of these equa- tions could be replaced by introducing the condition of the adiabatic expansion of the universe:

d(V ρ) + p dV = 0, (7.5) 166 The “third way” between Einstein’s and de Sitter’s solutions being V = π2R3 the total volume. This condition, indeed, could be written as: dρ 3R0 + (ρ + p) = 0, (7.6) dt R which expressed the energy conservation and was equivalent to the second (FL) field equation. “The variation of total energy - Lemaˆıtrehighlighted - plus the work done by radiation-pressure in the dilatation of the universe is equal to zero”[Lemaˆıtre 1931a, p. 485]. He then looked for the solution of a universe of constant mass: M = V δ = constant. He introduced α and β, where α was a constant related to the density of matter: α κδ = , (7.7) R3 and β was a constant of integration proportional to the pressure: β κ p = . (7.8) R3 In this way the first (FL) field equation reduced to [Lemaˆıtre 1927, p. 93]: Z dR t = q . (7.9) λ R2 α β [ 3 − 1 + 3R + R2 ] The case of α = β = 0 corresponded to the de Sitter’s solution, and the world-radius could be written in the form already found by Lanczos in 1922: r r 3 λ R = cosh (t − t ). (7.10) λ 3 0 On the contrary, Einstein’s solution was obtained by making β = 0 and by considering a constant radius [Lemaˆıtre 1927, p. 93]: 2 α = κ δ R3 = √ . (7.11) λ

According to Lemaˆıtre, the cosmological constant was related to R0, which was the asymptotic value at t = −∞ from which the radius of the universe R increased without limit. The relation between λ and R0 was expressed in the form: 1 λ = 2 . (7.12) R0 The importance of Lemaˆıtre’s1927 proposal 167

The radius of the actual world then resulted [Lemaˆıtre 1927, p. 94]:

3 2 R = RER0. (7.13)

RE was the radius of the finite universe of Einstein, which depended on the density of matter and increased in time: 2 κδ = . (7.14) RE

The value of R0 could be deduced from the radial velocities of nebulæ, as Lemaˆıtreshowed in a section of his 1927 paper devoted to the “Doppler effect due to the variation of the radius of the universe”: already in the title of this section, Lemaˆıtrepointed out which was the cause of the redshift. From the form of the line element it followed for a ray of light traveling from σ1 to σ2: Z t2 dt σ1 − σ2 = . (7.15) t1 R “A ray of light emitted slightly later - Lemaˆıtreexplained - starts from

σ1 at time t1 + δ t1 and reaches σ2 at time t2 + δ t2”[Lemaˆıtre 1931a, p. 487]. Therefore it followed:

δ t δ t 2 − 1 = 0, (7.16) R2 R1 which involved δ t R v 2 − 1 = 2 − 1 = . (7.17) δ t1 R1 c

R1 and R2 were the radius of the universe at the time of emission and reception, respectively. It was thus Lemaˆıtrewho offered in 1927 the actual cosmological interpretation of redshift, since the ratio of these radii minus 1 corresponded to “the apparent Doppler effect due to the variation of the radius of the universe” [Lemaˆıtre 1931a, p. 487]. Redshifts which were observed in nebulæ were not due to a relative motion between the observer and the observed object, nor to the slowing down of atomic vibrations. “The recession velocities of extragalactic nebulæ - Lemaˆıtre 168 The “third way” between Einstein’s and de Sitter’s solutions pointed out - are a cosmical effect of the expansion of the universe” [Lemaˆıtre 1931a, p. 489]. For sources which were near enough, it was useful to use the approx- imate formula [Lemaˆıtre 1927, p. 96]: v R − R dR R0 R0 = 2 1 = = dt = r, (7.18) c R1 R R R which gave: R0 v = . (7.19) R c r It is interesting to note that Lemaˆıtreactually found a linear relation R0 a˙ between velocities and distances. In present notation we have R ≡ a ≡ v H(t), and the previous relation can be written as c = H r. In 1927 Lemaˆıtrealso derived a value of the proportionality constant (575 km/sec Mpc−1) close to the value which Hubble proposed in his 1929 paper, which became later known as “Hubble constant”. However, such passages in the 1927 French paper were not translated in the 1931 English version. With regard to the radial velocity of nebulæ, Lemaˆıtrebased his calculations on data by [Str¨omberg 1925 ]. For distances, he referred to [Hubble 1926 ], and to the Hubble’s suggestion to use an average absolute magnitude (M = −15.2) for nebulæ. The distance was obtained from the relation [Lemaˆıtre 1927, p. 96]:

log r = 0.2 m + 4.04, (7.20) being m the apparent magnitude. For 42 nebulæ Lemaˆıtre found an average distance of 0.95 Mpc, corresponding to an average radial velocity of 600 km/sec. “On trouverait - Lemaˆıtreadded in a note - 670 km/sec `a1.16 · 106 pc, 575 km/sec `a106 pc. Certains auteurs ont cherch´e`a mettre en ´evidencela relation entre v et r et n’ont obtenu qu’une tr`es faible corr´elationentre ces deux grandeurs. (...) Il semble donc que ces r´esultatsn´egatifsne sont ni pour ni contre l’interpr´etationrelativistique de l’effet Doppler” [Lemaˆıtre 1927, p. 97]. From (7.19) he obtained: R0 v 625 · 105 = = = 0.68 · 10−27 cm−1. (7.21) R c r 106 · 3.08 · 108 · 1010 The importance of Lemaˆıtre’s1927 proposal 169

By using the approximate formula c r R0 = √ , (7.22) v 3

8 Lemaˆıtrefound for the world radius at t = −∞ the value R0 ' 2.7 · 10 pc [Lemaˆıtre 1931a, p. 487]. In order to understand the cause of the expansion of the universe, Lemaˆıtre suggested at the end of his paper that “we have seen that the pressure of radiation does work during the expansion. This seems to suggest that the expansion has been set up by the radiation itself” [Lemaˆıtre 1931a, p. 489].

7.2.1 Rediscovering the models of Friedmann

In 1930, also the cosmological equations proposed by Friedmann in 1922 and 1924 were eventually widely accepted by the scientific commu- nity. Already in 1922, dealing with the question of the curvature of space, Friedmann explained that his purpose was to show that Einstein’s and de Sitter’s solutions “are special cases of more general assumptions, and secondly to demonstrate the possibility of a world in which the curvature of space in independent of the three spatial coordinates but does depend on time” [Friedmann 1922, Engl. tr. p. 49]. In this paper Friedmann took into account a positive spatial curvature of a universe which radius depended on time. Subsequently, in 1924 he considered both a static and a non-static case, now admitting a negative spatial curvature [Friedmann 1924 ]. In 1922, Friedmann proposed a model of an expanding universe, in which he discarded the contribution by the pressure, and where stellar velocities were small with respect to the speed of light. In the expression of the metric, he considered R ≡ R(x4): “R is proportional to the radius of curvature, that will be proportional to time also” [Friedmann 1922, Engl. tr. p. 50]. Furthermore, by an appropriate choice of coordinates, “space can be made orthogonal to time. We cannot offer - Friedmann 170 The “third way” between Einstein’s and de Sitter’s solutions noted - any philosophical or physical justification for these assumption; they simplify the calculations” [Friedmann 1922, Engl. tr. p. 51]. The metric was written as:

2 2 2 2 2 2 2 2 2 2 ds = R (dx1 + sin x1dx2 + sin x1 sin x2 dx3) + M dx4. (7.23)

R2 M was a function of all four coordinates. By putting R = − c2 , Ein- stein’s metric arose if M = 1, whereas de Sitter’s metric corresponded to

M = cos x4. The general case corresponded to M as a function of time coordinate x4. In the latter case, field equations for µ = ν = 4 yielded to [Friedmann 1922, Engl. tr. p. 53]: R02 c2 3 + 3 − λ c2 = 8π Gρ. (7.24) R2 R2 On the contrary, for spatial indexes (µ = ν = 1, 2, 3) it followed: R02 RR00 c2 + 2 + − λ c2 = 0. (7.25) R2 R2 R2

0 dR 00 d2R In these equations it was R = , and R = 2 . By replacing x4 with dx4 dx4 t, the second (FL) equation reduced to: µ ¶ µ ¶ R dR 2 λ = A − R + R3, (7.26) c2 dt 3 and thus: v Z u" # 1 R u x t = t dx + B, (7.27) c λ 3 a A − x + ( 3 )x where a, B, A were arbitrary constants [Friedmann 1922, pp. 53-54]. The density of matter resulted: 3A ρ = , (7.28) c2κ R3 and A was related to M¯ , the total mass in the universe: κ c2M¯ A = . (7.29) 6π2 The quantity λ was not determined. Different world-models therefore 4c2 depended on the value which λ assumed with respect to λE = 9A2 , i.e. The importance of Lemaˆıtre’s1927 proposal 171 to the value of the cosmological constant in Einstein’s static universe.

Notably, the case λ > λE involved a “creation of the world (...); the time since the creation of the world - Friedmann suggested - might be infinite” [Friedmann 1922, Engl. tr. p. 56]. However, in his 1922 and 1924 works Friedmann did not mention any possible relation to astronomical consequences, in particular to redshift, and at the beginning of the 1920’s such papers basically appeared as a mathematical speculation [North 1965, p. 117]. Both the 1922 pa- per by Friedmann and the 1927 paper by Lemaˆıtrewere published in lesser-known journals, which were not strictly focused on astronomical topics. Therefore until 1930 they remained nearly unknown to most of the scientists, nowadays called cosmologists, who were involved in the first debates about relativistic cosmology. As seen, Robertson referred to Friedmann’s paper in his own 1929 analysis. On the contrary, at the time of writing his own 1927 paper, Lemaˆıtrewas not aware of Friedmann’s equations. It was Einstein who pointed out to Lemaˆıtrethe existence of Friedmann’s paper in 1927, as Lemaˆıtreacknowledged at the end of a 1929 conference [Lemaˆıtre 1929, p. 32]. Indeed in 1927 Lemaˆıtrehad the chance to meet Einstein at the fifth Solvay Conference in Bruxelles and became acquainted with Friedmann’s result. Lemaˆıtrecould also briefly talk with Einstein about his own analysis of the expanding uni- verse. With regard to Einstein’s reaction on the contents of Lemaˆıtre’s 1927 paper, Lemaˆıtrereferred some years later that “du point du vue physique, cela lui parait tout `afait abominable” [Lemaˆıtre 1958, p. 129]. According to Einstein, such an expanding solution did not correspond to a physical possibility. This was the same remark that Einstein had already made in 1918 to the de Sitter’s universe, and in 1922 with respect to the result found by Friedmann. In 1922, indeed, Einstein objected in a brief note that Fridemann’s result concerning a non-stationary world “seems suspect” to Einstein himself [Einstein 1922b, Engl. tr. p. 66]. According to Einstein, “from the field equations it follows necessarily that the divergence of the matter 172 The “third way” between Einstein’s and de Sitter’s solutions tensor vanishes. This (..) leads to the condition ∂ρ = 0, which (...) im- ∂ x4 plies that the world-radius R is constant in time” [Einstein 1922b, Engl. tr. p. 66]. However, thanks to Aleksander Krutkoff, Einstein could read a letter where Friedmann explained the details of his own calculations. Therefore in a subsequent paper Einstein accepted that Friedmann’s re- sult was “both correct and clarifying” [Einstein 1923, Engl. tr. p. 67]. Nevertheless, despite the correctness of the mathematical method, Ein- stein still believed that the physical interpretation of Friedmann’s result was not conceivable. In 1931, in front of the empirical evidence that galaxies were receding one to another, Einstein abandoned the cosmo- logical constant which he had introduced in 1917 in order to express in general relativity the static nature of the universe. Now this supposition was contradicted by the observed recession of nebulæ which claimed to the expanding universe4 [Einstein 1931 ].

7.3 The decline of the interest in the de Sitter effect

The cosmology of Lemaˆıtre finally solved the problems related to the models of Einstein and de Sitter, which were unsatisfying from the point of view of a suitable comparison with astronomical observations. The expanding universe gave a solution of the puzzling question of the origin of redshift. Therefore, as from 1930, the de Sitter effect, i.e. the interpretation of spectral displacements which the model of de Sitter offered, faded away just because, quoting de Sitter himself, solution B “must be rejected, (...) and the true solution represented in nature must be a dynamical solution” [de Sitter 1930a, p. 482].

4It is interesting to note that in 1923, when both Eddington and Weyl proposed their own interpretations of a cosmical recession in de Sitter’s world, Einstein wrote to Weyl that “if there is no quasi-static world, then away with the cosmological term” [quoted in Pais 1982, p. 288]. The decline of the interest in the de Sitter effect 173

On April 17, 1930, de Sitter wrote to Shapley that “I have been very busy lately on spiral nebulæ and on the relativistic explanation of the big velocities. I had come to the conclusion that my solution B could not be accepted as an adequate explanation, as it supposes the universe to be empty (...). Only very lately I have found the true solution, or at least a possible solution, which must be somewhere near the truth, in a paper published already in 1927 by Lemaˆıtreof Louvain, which had escaped my notice at the time” [de Sitter Archive, Box 17.4 C]. De Sitter acknowledged that non-static and non-empty world-models eventually represented the actual universe. “The important point - the Dutch as- tronomer wrote to Tolman on May 7, 1930 - is that we must look for a dynamical solution of the field equations as the true interpretation of nature” [de Sitter Archive, Relativity Box, A3]. As de Sitter noted, the dynamical solution which Lemaˆıtreproposed “requires all observed ra- dial velocities to have the same sign, while in the solution B the sign was indeterminate for each individual body” [de Sitter 1930a, p. 487]. De Sitter took into account observations of apparent magnitude and angular diameter, in order to test by a method which differed from Hubble’s one the linearity of the observed relation among galaxies. He then confirmed through available data the existence of a linear redshift-distance relation, adding proof to Hubble’s result [de Sitter 1930b]. With regard to the comparison between the de Sitter effect and the explanation of redshift proposed by Lemaˆıtre, Eddington pointed out in 1930 that de Sitter “introduced the slowing down of time at great distances from the origin, which does not occur in the new formulæ. The present description involves fewer paradoxes and is undoubtedly easier to apprehend (...). It has, moreover, the advantage that we now approach de Sitter’s world as the limit of a series of worlds of gradually diminishing density; whereas formerly we had to start with a completely empty world, and very cautiously put a few material bodies in it. (...) Of course it is possible that the recession of the spirals is not the expansion theoretically predicted; it might be some local peculiarity masking a much smaller 174 The “third way” between Einstein’s and de Sitter’s solutions genuine expansion. But the temptation to identify the observed and the predicted expansions is very strong” [Eddington 1930a, pp. 676-677]. In his 1933 report on the astronomical aspect of general relativity, de Sitter remarked that “all that observations tell us is that light coming from great distances, and which therefore has been a long time on the way, is redder when it arrives than when it left its source. Light is red- dened by age: traveling through space, it loses its energy and gets older. Or, expressed mathematically: the wave-length of light is proportional to a certain quantity R, which increases with the passing of time5”[de Sitter 1933, p. 155]. A different interpretation of spectral shift was proposed in 1932 by Edward Arthur Milne (1896-1950). Milne proposed a special relativistic model of the universe. In this kinematical model, by using the cosmolog- ical principle, the universe was bounded, and expanded into an infinite flat space. Following [Harrison 2000, p. 374], such a universe can be (mathematically) transformed into an infinite and unbounded universe within the Robertson-Walker expanding frame. In such a version, Milne’s universe is homogeneous and isotropic, its spatial curvature is negative, and the radius of the universe is R = t, i.e. the universe expands at constant rate. As Milne noted in 1934, in his own special relativistic model “the Doppler shift s remains constant in time for any one patch; it increases at any one epoch of observation as b [the surface brightness] decreases” [Milne 1934, p. 26]. With regard to the cosmological (expansion) shift, also Hubble doubted about its general relativistic interpretation. In 1937, in “The observa-

5It is worth noting that in the draft of such a 1933 report de Sitter used different words. On p. 15 of this draft, de Sitter wrote: “the interpretation of Lemaˆıtre’s non-static solution as an expanding universe, R being the ‘Radius’ of the universe, is not imperative. Apart from any interpretation of the meaning of the quantity R or of the observed redshift as a velocity, we can enounce the net result as follows: the ratio between the observed and emitted wave-length is the same as the ratio of the values of R at the times of observation and emission, respectively. Since R increases with the time, light is reddened by age” [de Sitter Archive, Relativity Box, A8]. The decline of the interest in the de Sitter effect 175 tional approach to cosmology”, Hubble acknowledged that the interpre- tation of redshift as radial motion was the only permissible explanation that was known “until evidence to the contrary is forthcoming” [Hubble 1937, p. 26]. Beside such an interpretation of redshift as velocity-shift, Hubble mentioned also the possibility of redshift as loss of energy in transit. “Redshifts - he wrote - are produced either in the nebulæ, where the light originates, or in the intervening space through which the light travels. If the source is in the nebulæ, then the redshifts are probably velocity-shifts and the nebulæ are receding. If the source lies in the inter- vening space, the explanation of redshifts is unknown but the nebulæ are sensibly stationary” [Hubble 1937, p. 31]. Within this alternative, which corresponded to a universe which was not rapidly expanding, Hubble pointed out that an exact linear relation between redshift and distance had to be expected. On the contrary, redshifts as velocity-shifts involved a departure from the linearity [Hubble 1937, p. 41]: ∆λ = kr + lr2 + mr3 + ... (7.30) λ “Theories may be revised - Hubble concluded - and new information may alter the complexion of things, but meanwhile we face a rather serious dilemma” [Hubble 1937, p. 44]. Alternative proposals apart, “it is now clear - Robertson wrote in his 1933 review on relativistic cosmology - that the existence of the so-called velocity-distance relation formed no essential part of the deduction [of Friedmann-Lemaˆıtrecosmological models], which was based entirely on the evidence for the uniform distribution and state of motion of matter in the large, and on the acceptance of the general theory of relativity (...). If we consider the observed redshift as arising from the nature of space-time, we find in it additional evidence for the theory” [Robertson 1933, p. 83]. New issues characterized the cosmological debates during the first years of the 1930’s. Scientists involved in those discussions dealt, among others, with the concept of the evolution of the universe and its time- 176 The “third way” between Einstein’s and de Sitter’s solutions scale, the cause of the expansion of the universe, the value of the Hubble constant. In 1931 the concept of the beginning of the world entered the relativistic investigation of the universe. In that year Lemaˆıtresuggested the hypothesis of the “primeval atom”, which corresponded to the ini- tial state of the universe with minimum entropy [Lemaˆıtre 1931b]. In 1932 Einstein and de Sitter, whose 1917 rival models were the objects of the cosmological question before the diffusion of the expanding universe, proposed a metric which corresponded to an expanding model, where the contribution by pressure was neglected and the curvature was zero (flat spatial sections). In this model, Einstein and de Sitter did not take into account the cosmological term in field equations. Indeed, through the dynamical solutions rediscovered in 1930, the existence of a finite mean density of matter could be now theoretically achieved without the introduction of λ [Einstein-de Sitter 1932 ]. Whereas on the one hand general relativity offered the theoretical base for a suitable description of the universe as a whole beyond New- tonian cosmology, on the other hand observations on large scale now revealed that the consequences and predictions of such a theory could be now empirically verified or confuted. As seen, such a connection entered modern cosmology just during the debates about the de Sitter effect, which foreshadowed a non-static picture of the universe as a whole, and, moreover, led to the development of cosmology from theoretical specula- tion into an empirical science. Conclusion

In present thesis we critically reconstructed the debates which took place during the 1920’s about the de Sitter effect, i.e. the history of the different predictions and possible confirmations of the redshift-distance relation which was obtained through the metric of de Sitter’s universe. From this analysis it emerges the fundamental role played by the de Sitter effect in the rise of cosmology as an empirical science. Such a redshift-distance relation was the leading thread in the early phases of the modern scientific view of the universe, during the tortuous process from the 1917 beginning of theoretical relativistic cosmology towards the 1930 diffusion of the model of the expanding universe which was claimed by astronomical observations. As shown in present thesis, the 1920’s debates were characterized by a richness of ideas, attempts, controversies and failures related to the cosmological question in the light of relativity revolution. Dealing with the curvature of space-time in astronomical context, scientists involved in cosmological debates from 1917 to 1930 approached and thoroughly analyzed fundamental issues, some of them still present in modern cos- mology. They addressed themselves to the question of which was the most suitable model which represented the actual universe, and began to verify relativistic models through observations. In this framework, it was the interest in the de Sitter effect which led to link together theoret- ical relativistic cosmology with astronomical observations on large scale, inaugurating the modern approach of cosmologists in the comprehension of global properties of the universe.

177 178 Conclusion

In 1917 Einstein showed that a new cosmology was allowed by his new theory of gravitation, overcoming Newtonian difficulties at infinity. In his picture of a static and finite universe, Einstein achieved the re- quirement that, through a suitable metric and by introducing the cosmo- logical constant, space-time was globally influenced by gravitation, and inertia was entirely produced by all masses in the world. In his spherical world, Einstein made the working hypothesis of a uniform and homoge- neous density of matter in order to deal with properties of gravitation and inertia on large scale, foreshadowing the fundamental assumption which became known as the Cosmological Principle. In the same year de Sitter demonstrated that, from a mathematical point of view, an empty universe actually corresponded to another suitable solution of relativistic field equations with the cosmological term. The constant radii of both these finite universes depended on the value of a new universal constant, λ. In Einstein’s intention such a constant acted a a sort of anti-gravity in order to counterbalance gravitational effects on large scale. Nevertheless, as de Sitter soon pointed out, a redshift-distance relation could be derived from the metric of de Sitter’s universe: a mass test in his own empty universe showed a displacement of spectral lines, which de Sitter interpreted as due both to a gravitational shift caused from the form of the metric, and to a Doppler effect from the geodesic equations. Therefore de Sitter related this feature, the so-called de Sitter effect, to observations of shifts in spectral lines from stars and nebulæ, in order to estimate the value of the world-radius of his own model. General relativity offered the possibility to coherently extrapolate laws of physics and properties of light and matter to the universe as a whole, and de Sitter inaugurated the verification of the first two theoretical models with respect to their astronomical consequences. De Sitter’s pioneering attempts did not pass unnoticed. Despite its lack of matter, the empty universe of de Sitter drew the attention of several scientists looking for the best approximation of the actual world. Conclusion 179

In this framework, the de Sitter effect was taken into account to rightly interpret redshift measurements in nebulæ. In his own 1923 analysis, Eddington pointed out that a general cosmic recession was expected in de Sitter’s universe just because of the pres- ence of the cosmological constant. Such a tendency of particles to scatter could roughly account for the astonishing radial velocities measured by Slipher in spiral nebulæ, the most part of which revealed receding mo- tions from the observer. Remarkably, Eddington realized that a suitable description of the actual world corresponded to an intermediate solution between Einstein’s model, which had matter but not motion, and de Sitter’s model, which had motion but not matter. The geometry of de Sitter’s world-model was not uniquely deter- mined, and non-static interpretations of the universe emerged by ap- propriate coordinate changes, as done in the 1920’s by Weyl, Lanczos, Lemaˆıtreand Robertson. Their contributions, which referred to a sta- tionary frame of de Sitter’s universe, showed that theoretical cosmology properly allowed to deal with non-static line elements, i.e. with a non- static picture of the universe. Beside the controversies around the interpretation of relativistic cos- mological models, during the 1920’s first astronomical observations on large scale inaugurated the beginning of the observational approach to cosmology. In 1924 Wirtz realized that the universe of de Sitter represented a suitable model to account for redshift and apparent diameter of nebulæ. In the same year, on the contrary, Silberstein criticized the possibility of a general cosmic recession, and considered the distances of globular clusters in order to verify the de Sitter effect. The correctness of the method and of the result proposed by Silberstein was shortly after denied by Lundmark and Str¨omberg. There was neither a general agreement on the meaning of redshift, nor a clear and widely accepted understanding of the properties of de Sitter’s universe. A suitable test of redshift relations was possible only with a reliable determination of the distances of spiral 180 Conclusion nebulæ. In this controversial picture, the contributions of Hubble marked a turning point in the comprehension of the structure of the universe. Thanks to the revolutionary observations which Hubble furnished dur- ing the 1920’s, spiral nebulæ were accepted as ‘island universes’, i.e. as true extra-galactic stellar systems. Moreover, Hubble finally confirmed in 1929 that such systems receded relatively to one another, and that their radial velocities linearly increased with distances. The puzzling question of the ambiguous interpretation of the de Sitter effect and the meaning of redshift was solved in 1930, when the cosmol- ogy of Lemaˆıtrewas reconsidered in order to explain the cosmic recession of galaxies revealed by Hubble. The model of a non-empty and expand- ing universe which Lemaˆıtrehad already proposed in 1927 provided the proper cosmological interpretation of redshift: the displacement of spec- tral lines was due to the expansion of the universe. As from 1930, the de Sitter effect, which during the 1920’s represented the first hint in the intersection between the new theory of gravitation and observed facts, was seen as an effect of minor importance. The expand- ing universe inaugurated another chapter of modern cosmology. Static and stationary universes were eventually considered as limiting cases of the general dynamical solutions of Friedmann-Lemaˆıtreequations, which brought new questions and new challenges in the investigation of the properties of the universe. From 1917, when Einstein proposed his cosmological considerations in general relativity, the knowledge of the universe as a whole greatly devel- oped towards the new 1930 paradigm of the expanding universe, which resulted the natural and more comprehensive and coherent theoretical interpretation of the observational evidence of a cosmic recession. The advancement of scientific cosmology was, and is nowadays, char- acterized by new ideas, discoveries, changes. The history of the early developments of modern cosmology during the 1920’s reveals how the is- sues faced in that fruitful period represent a remarkable passage towards Conclusion 181 the comprehension of the universe through the laws of physics. “The theory of today - de Sitter wrote in 1932 - is not the theory of tomorrow. (...) Science is developing so very rapidly nowadays, that it would be preposterous to think that we had reached a final state in any subject. The whole of physical science, including astronomy, is in a state of transition and rapid evolution. Theories are continually being improved and adapted to new observed facts. It would certainly not be right to suppose at the present time that we had reached any state of finality. We are, however, certainly on the right track” [de Sitter 1932a, pp. 103-104].

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Acknowledgements

I want to express my deep gratitude to my thesis supervisor, Prof. Giulio Peruzzi, for his patience, encouragement, and thoughtful guidance. A very special thanks to my co-supervisor, Prof. Luigi Secco, for his constant support and interesting suggestions. I am sincerely grateful to my advisor, Prof. J¨urgenRenn, for his kind attention and helpful comments. I am deeply indebted to Dr. Jan Guichelaar, for his help and very useful translations, and to Prof. Dominique Lambert, for interesting discussions. I want to sincerely thank Prof. Frans van Lunteren and Dr. David Baneke for their kind attention and collaboration, and for the permission to reproduce original manuscripts from the de Sitter Archive. My special thanks to Mark Hurn, Adam Green, Tatiana Turato, Claudia Toniolo and Ruth Kessentini, for their kindness and patience. I gratefully acknowledge Liliane Moens, for her great help and the permission to reproduce original manuscripts from the LemaˆıtreArchive. I am grateful to Prof. Donald Lynden-Bell, Prof. Malcolm Longair and Dr. Matthias Schemmel, for their useful suggestions. I also wish to thank Prof. Cesare Chiosi, for his support, and Prof. Bepi Tormen, for helpful comments. Thanks also to Luca and Monica, for their immediate help. Finally, my sincere thanks to my grandmother, Laura, for her great help with translations. This work has been supported in part by the Italian Ministry of the University and Research under the Research Projects of National Interest (PRIN).

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