Abstract
Lemaître’s work on the geometric nature of singularities and his speculations concerning the applications of quantum physics to cosmology are confronted with later achievements in these fields. His works on the global structure of the de Sitter solution and the appearance of “non-regular” points in the Schwarzschild solution led to the conclusion that the “vanishing of the radius of the universe” is a generic property of cosmological models. This conclusion was strengthened when Lemaître proved that, against Einstein’s intuition, space anisotropy (in Bianchi I models) does not remove the singularity. This is why Lemaître regarded the initial singularity as a “geometric support” of his Primeval Atom hypothesis. This hypothesis was not yet a quantum gravity idea (in the present sense of this expression), but it was certainly an application of quantum physics to the early stages of cosmic evolution. The beginning itself is aspatial and atemporal, and both space and time emerge only when the simplicity of the Primeval Atom gives place to physical multiplicity. How do the problems with which Lemaître struggled appear in the light of the present state of cosmological research?
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Hilbert proposed the following criterion: “By [non-regular point] I mean that a line element or a gravitational field g μν is regular at a point if it is possible to introduce by a reversible, one-one transformation a coordinate system, such that in this coordinate system the corresponding functions g' μν are regular at that point, i.e., they are continuous and arbitrarily differentiable at the point and in a neighbourhood of the point, and the determinant g' is different from 0” (quoted from (Earman 1995, p. 6)). This seemingly correct criterion does not distinguish between genuine singularities and coordinate singularities; more on this topic can be found in Earman (1995).
- 2.
In Einstein’s interpretation of Mach’s Principle the metric structure of space-time should be determined by the mass distribution in the universe.
- 3.
In Einstein’s words, this singularity is only an “apparent violation of continuity, as can readily be shown by a suitable change of coordinates”; see Earman and Eisenstaedt (1999).
- 4.
“Thus, he concluded, with the cautionary proviso of ‘until proof to the contrary’, that de Sitter’s solution must be regarded as having a genuine singularity and thus this ‘solution does not accord with the field equations for any choice of coordinates’” (Earman and Eisenstaedt 1999, p. 193).
- 5.
The typescript is preserved in the Lemaître archive in Louvain-la-Neuve.
- 6.
It was defined as T = ρ−3p in units in which c = 1; Lemaître called T invariant density.
- 7.
- 8.
At Lemaître’s archive in Louvain-la-Neuve his calculations of all solutions are preserved (for details see Godart and Heller (1985, p. 57)). At that time Lemaître did not know that these solutions were already published by Alexander Friedmann.
- 9.
Caustics are regions in which families of geodesics do not form smooth space-time submanifolds; they are singularities of families of geodesics rather than singularities of space-time.
- 10.
A Cauchy surface is a spacelike hypersurface that no inextendible history of a particle or photon (i.e., no timelike or null curve) in a given space-time intersects more than once. A Cauchy surface is a global Cauchy surface if every such history in a given space-time intersects it exactly once.
- 11.
World models with non-compact time slices are called open world models.
- 12.
Technically, a trapped surface is a compact two-dimensional submanifold of space-time such that the expansion of both outgoing and ingoing future directed null geodesics, that are orthogonal to this submanifold, is everywhere negative. Physically, this means that a trapped surface is formed provided there is enough matter accumulated in a small region of space-time. Such situations are expected to occur in gravitational collapse.
- 13.
Space-time is globally hyperbolic if it has a global Cauchy surface.
- 14.
The rest of the present section is based on this reference, and all quotations are taken from it.
References
Bojowald, M. (2007). Singularities in quantum gravity, arXiv:gr-qc/0702144.
Clarke, C. S. J. (1998). Generalized hyperbolicity in singular space-times, Class. Quantum Gravity, 15(1998), 975–984.
Craps, B. (2006). Big bang models in string theory, arXiv:hep-th/0605199.
de Sitter, W. (1917). On Einstein’s theory of gravitation and its astronomical consequences. Third paper, Monthly Notices of the Royal Astronomical Society, 78, 3–28.
DeWitt, B. S. (1967). Quantum theory of gravity. I. The canonical theory. Physical Review, 160, 1113–1148.
Earman, J. (1995). Bangs, Crunches, Whimpers, and Shrieks. Singularities and acausalities in relativistic spacetimes. New York/Oxford: Oxford University Press.
Earman, J., & Eisenstaedt, J. (1999). Einstein and singularities. Studies in History and Philosophy of Science, 30, 185–235.
Eddington, A. S. (1923). The mathematical theory of relativity. Cambridge: At the University Press (reprinted in 1965).
Einstein, A. (1915). Erklärung der Perihelbewebung des Merkur aus der allgemeinen Relativitätstheorie, Sitzungsber. Preuss Akad Wiss 47(2), 831–839.
Einstein, A. (1931). Zum kosmologischen Problem der allgemeinen Relatvitätstheorie, Sitzungsber. Preuss Akad Wiss phys-math Kl 235–237.
Godart, O., & Heller, M. (1985). Cosmology of Lemaître. Tucson: Pachart.
Hawking, S. W. (1965). Occurrence of singularities in open universes. Physics Letters, 15, 689–690.
Hawking, S. W., & Ellis, G. F. R. (1973). The large scale structure of space-time. Cambridge: Cambridge University Press.
Hilbert, D. (1917). Die Grundlagen der Physik: Zweiter Mitteilung, Königliche Gesellschaft der Wissenschaften zu Göttingen. Mathematische-Physikalische Klasse. Nachrichten, 55–76.
Kragh, H. (1996). Cosmology and controversy. The historical development of two theories of the universe. Princeton: Princeton University Press.
Lambert, D. (2000). Un atome d'univers. La vie et l’oeuvre de Georges Lemaître. Bruxelles: Lessius-Racine.
Lemaître, G. (1925). Note on de Sitter’s universe. Journal of Mathematics and Physics, 4, 37–41.
Lemaître, G. (1927). Un univers homogène de masse constante et de rayon croissant, rendant compte de la vitesse radiale des nèbuleuses extra-galactiques, Annales de la Societé Scientifiques de Bruxelles, ser A, 47, 49–59.
Lemaître, G. (1931a). A homogeneous universe of constant mass and increasing radius accounting for the radial velocity of extra-galactic nebulae. Monthly Notices of the Royal Astronomical Society, 91, 483–489.
Lemaître, G. (1931b). The beginning of the world from the point of view of quantum theory. Nature, 127(3210), 706.
Lemaître, G. (1933). L’Univers en expansion. Annals de la Societe Scientifique de Bruxelles Series A, 53, 51–85.
Lemaître, G. (1958a). Rencontres avec Einstein, Revue des Questions Scientifiques, janvier 1958, 129–132.
Lemaître, G. (1958b). The primaeval atom hypothesis and the problem of the clusters of galaxies. In R. Stoops (Ed.), La Structure et l’Évolution de l’Univers (pp. 1–25). Bruxelles: Institut International de Physique Solvay.
Lemaître, G. (1960). L’étrangeté de l’Univers, La Revue Generale Belge, juin 1960, 1–14.
Lemaître, G. (1978). Un travail inconnu de Georges Lemaître: L’Univers, problème accessible à la science humaine, published by O. Godart and M. Heller, Revue d’Histoire des Sciences, 31, 345–359.
Lemaître, G. (1996). The expanding universe. In M. Heller (Ed.), Lemaître, big bang and the quantum universe. Tucson: Pachart.
Penrose, R. (1965). Gravitational collapse and space-time singularities. Physical Review Letters, 14, 57–59.
Rovelli, C. (2004). Quantum gravity. Cambridge: Cambridge University Press.
Schwarzschild, K. (1916a). Über das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie, Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin. Phys-Math Klasse, 189–196.
Schwarzschild, K. (1916b). Über das Gravitationsfeld einer Kugel aus incompressibler Flüssigkeit, Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin. Phys-Math Klasse, 424–434.
Tipler, F. J., Clarke, C. J. S., & Ellis, G. F. R. (1980). Singularities and horizons—A review article. In A. Held (Ed.), General relativity and gravitation (Vol. 2, pp. 97–206). New York/London: Plenum Press.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Heller, M. (2012). Lemaître, the Big Bang and the Quantum Universe. In: Holder, R., Mitton, S. (eds) Georges Lemaître: Life, Science and Legacy. Astrophysics and Space Science Library, vol 395. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32254-9_5
Download citation
DOI: https://doi.org/10.1007/978-3-642-32254-9_5
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-32253-2
Online ISBN: 978-3-642-32254-9
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)