Abstract
Allow me, first of all, to thank the International Academy of Philosophy of Science for inviting me to speak to you and with you on this occasion. My subject is “the geometric structure of the universe”. I would never have dared to choose it, but I made no resistance when Evandro Agazzi proposed it to me; so I assume full responsibility for this immodest attempt to cover in one lecture such a broad and intricate subject. Indeed, within the standard framework of contemporary cosmology the geometric structure of the universe is the structure of the universe tout court; at any rate, it is all there is to its large-scale structure. However, other lecturers will elaborate further on its more significant aspects. Thus my talk is mainly intended to provide a necessary, but necessarily schematic conceptual background for the fuller view of present-day physical cosmology you will be given later.
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Notes
A. Einstein, “Kosmologische Betrachtungen zur allgemeinen Relativitätstheorie”, K Preuß. Ak. der Wissenschaften, Sitzungsberichte, 1917, pp. 142–152.
W. de Sitter, “On the relativity of inertia. Remarks concerning Einstein’s latest hypothesis”, K Nederlandse Ak. van Wetenschappen, Proceedings, 19: 1217–1225 (1917); “On the curvature of space”, K Nederlandse Ak. van Wetenschappen, Proceedings, 20: 229–243 (1917); “On Einstein’s theory of gravitation, and its astronomical consequences”, Monthly Notices of the Royal Astronomical Society, 78: 3–28 (1917).
A. Friedmann “Über die Krümmung des Raumes”, Zeitschrift für Physik, 10: 377–386 (1922); “Über die Möglichkeit einer Welt mit konstanter negativer Krümmung”, Zeitschrift für Physik, 21: 326–332 (1924).
G. Lemaitre, “Un univers homogène de masse constant et de rayon croissant, rendant compte de la vitesse radiale des nébuleuses extra-galactiques”, Annales de la Société Scientifique de Bruxelles, A 47: 49–59 (1927).
K. Gödel, “An example of a new type of cosmological solutions of Einstein’s field equations of gravitation”, Reviews of Modern Physics, 21: 447–450 (1949).
B. Riemann, “Über die Hypothesen, welche der Geometrie zugrunde liegen”, Abhandlungen der K. Gesellschaft der Wissenschaften zu Göttingen, 13 (1867), p. 149.
H. Poincaré, “Sur la dynamique de l’électron”, Rendiconti del Circolo Matematico di Palermo, 21: 129–175 (1906). A five-page abstract of this paper was published in the Comptes Rendus of the Académie des Sciences of June 5, 1905 (CR, 140: 1504–8). Einstein’s first paper on Special Relativity was submitted on June 30th, 1905.
A. Einstein, “Autobiographisches”; in P.A. Schilpp, ed., Albert Einstein, Philosopher-Scientist, Evanston, The Library of Living Philosophers, 1949, p. 64.
From a manuscript, in the Pierpont Morgan Library in New York, as quoted in A. Pais, Subtle is the Lord The Science and the Life of Albert Einstein, Oxford, Clarendon Press, 1982, p. 178.
Namely, the gravitational redshift and the bending of light-rays in a gravitational field. See A. Einstein, “Über das Relativitätsprinzip und die aus demselben gezogenen Folgerungen”. Jahrbuch der Radioaktivität und Elektronik. 4: 411–462 (1907).
Minkowki’s lecture of 1907 was not printed until 8 years later: H. Minkowski, “Das Relativitätsprinzip”, Jahresbericht der Deutschen Mathematiker-Vereinigung,24: 372–382 (1915). In 1908 he published “Die Grundgleichungen für die elektromagnetischen Vorgänge in bewegten Körper” (Nachrichten der K. Gesellschaft der Wissenschaften zu Göttingen, Math.-phys. Kl.,1908, pp. 53–111), and delivered his famous lecture “Raum und Zeit” (Phys. Zeitschrift,10: 104–111 (1909)).
Einstein tells of this new insight in the preface to the Czech edition of his “popular exposition” of Relativity. See J. Stachel, “Einstein’s Search for General Covariance, 1912–1915”,Einstein Studies, 1: 63–100 (1989).
A. Einstein, “Die Feldgleichungen der Gravitation”, K Preuß. Ak. der Wissenschaften, Sitzungsberichte,1915, pp. 844–847; cf. Ibid., pp. 778–786, 799–801, and ref. 12.
K. Schwarzschild, “Über das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie”, K. Preuß. Ak. der Wissenschaften, Sitzungsberichte, 1916, pp. 189–196; J. Droste, “The field of a single centre in Einstein’s theory of gravitation, and the motion of a particle in that field”, K Nederlandse Ak. van Wetenschappen, Proceedings, 19: 197–215 (1916)
Einstein derived this result using approximation methods in November 1915. See A. Einstein, “Erklärung der Perihelbewegung des Merkur aus der allgemeinen Relativitätstheorie”, K Preuß. Ak. der Wissenschaften, Sitzungsberichte, 1915, pp. 831–839.
A. Einstein, “Kosmologische Betrachtungen zur allgemeinen Relativitätstheorie”, K. Preuß. Ak. der Wissenschaften, Sitzungsberichte, 1917, pp. 142–152.
V.M. Slipher, “The radial velocity of the Andromeda Nebula”, Lowell Observatory Bulletin, N.58 (1913); “Spectrographic observations of nebulae”, Popular Astronomy, 23: 21–24 (1915); “A spectrographic investigation of spiral nebulae”, Proceedings of the American Philosophical Society, 56: 403–409 ( 1917 ). Einstein’s other piece of astronomical lore was that stars are uniformly distributed in space. Again this would soon be disproved by American astronomers wielding the new long-range telescopes; but for the purpose of Einstein’s argument it would do just as well to invoke the uniform distribution of galaxy clusters, or, at least, superclusters, which until very recently was uncontestedly supported by observational astronomy.
A. Einstein, “Prinzipielles zur allgemeinen Relativitätstheorie”, Annalen der Physik, 55: 241–244 (1918), p. 241.
A. Einstein, ref. 1, p. 146.
W. de Sitter, reference 2.
A. Friedmann, reference 3.
W. de Sitter, “On the expanding universe and the time-scale”, Monthly Notices of the Royal Astronomical Society,93: 628–634 (1933), p. 631.
A systematic exposition of the singularity theorems, with proofs and full references to the original papers, is contained in S.W. Hawking and G.F.R. Ellis, The Large Scale Structure of Space-Time, Cambridge, Cambridge University Press, 1973. See also R.M. Wald, General Relativity, Chicago, The University of Chicago Press, 1984, Chapter 9.
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Torretti, R. (1991). The Geometric Structure of the Universe. In: Agazzi, E., Cordero, A. (eds) Philosophy and the Origin and Evolution of the Universe. Synthese Library, vol 217. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-3598-6_2
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