The cosmological constant and the expansion of space
Shall we put a little motion into Einstein’s universe or a little matter into de Sitter’s?
Sir Arthur Eddington, address to the Royal Astronomical Society, January 1930, cited in John Farrell, The Day Without Yesterday, Thunder’s Mouth Press, New York, 2005. |
We contemplated earlier a simplified version of the Einstein Field Equations: /converting-the-thought-experiment-into-mathematical-equations.html. This now merits further consideration[1].
Einstein’s pure equations suggested a universe that would not be stable. It would either collapse under its own weight or it would expand, but it never occurred to Einstein that instead of collapsing, the universe his equations were describing might actually be expanding. In common with everyone else at the time, Einstein just assumed that the universe as a whole was static and unchanging, and he considered that his equations should support this picture. So, two years after his seminal paper on general relativity, in 1917 Einstein wrote a paper entitled “Cosmological Considerations of the General Theory of Relativity” in which he introduced the cosmological constant lambda (Λ) as a kind of buttress which counterbalanced the universal effects of gravity at a cosmic level in order to construct a model of the universe as a closed system, spherical, and in perfect equilibrium and not changing over time.
Λ’s value was unknown but it represented “a cosmic repellent to the force of gravity, a sort of antigravity that stabilised the world on the large scale of the universe as a whole” and maintained the universe in a state of equilibrium. At smaller scales, that of the solar system, its effects would be negligible.
With the introduction of the term, Einstein’s equations came out looking something like this:
Gμν - Λgμν = 8πTμν
Shortly after the publication of Einstein’s paper, the Dutch astronomer, William de Sitter, argued that Einstein’s equations could in fact be used to posit a static universe that was entirely devoid of matter, a suggestion which was anathema to Einstein, since his whole theory was premised upon spacetime curving in the presence of matter.
Then, in two separate papers published in 1925 and 1927 and building upon both the Einstein and de Sitter models, the French cleric, physicist and astronomer Georges Lemaître postulated a new dynamic model for the universe which was evolving with red-shifted nebulae illustrating space-time expansion, which was expanding according to a law under which nebulae were receding at radial velocities directly proportional to their distance – a direct precursor to Hubble’s law not formulated unto two years later. In other words, Lemaître was attributing the recession to the actual expansion of space-time itself.
Lemaître’s 1927 paper suggested an expansion of the universe beginning from an initial Einstein static state, perhaps for an infinite period in the past, which could then have evolved, expanding ultimately into the flat de Sitter state of virtually empty space – in other words not one which emanated from a so-called big bang. That came later in the form of Lemaitre’s primaeval atom or cosmic egg, a theoretical concept which made him famous as the father of the Big Bang. But in the meantime, “Lemaître had at a single stroke created the first consistent, evolving model of the universe that beautifully incorporated Hubble’s findings on the apparent velocity distance relation (even before Hubble had published them). Hubble’s law might just as easily been called Lemaître law”[2].
It was the whole idea of a dynamic universe that Lemaître introduced at the time, though but not its origin, a dynamic that Einstein objected to, but close examination of then static Einstein model revealed that there had to be some kind of beginning of all physical processes in order to work as a cosmological model. It wasn’t until 1931 that Lemaître published a paper which laid the groundwork for what ultimately became the big bang model of the universe, suggesting that at some level space and time must themselves be quantized and thus directly tying the notion of the universe to quantum processes.
Meanwhile, in 1929 Hubble published his empirical findings corroborating the universe’s expansion from the redshift of distant galaxies which were found to be receding at velocities directly proportional to their distance, following which Einstein began to question the necessity for Λ, and in 1931, he removed it from his theory for good, calling it his “greatest blunder”. But Lemaître was not so sure that it should be removed. He believed the cosmological constant was more than just a mathematical balancing factor, and that it represented an actual physical force – the vacuum energy of space, an energy that was essentially undetectable at the local level of the solar system and stars, but one that was very real on the scale of the galaxies and the universe as a whole. In his own work, he included it on the right side of the field equations which govern energy density, whereas Einstein had placed it on the left, signifying its purely geometric state, to prop up the static model[3]. History has proven Lemaître right following the discovery of vacuum energy as a positive force influencing the universe’s expansion[4]. Seventy years later, in 1998, in order to explain the movements of supernovae and the implications flowing from the discovery of “dark energy”, astronomers restored it. So even when Einstein was wrong, he was right, only the value he had ascribed to it was understated, being artificially contrived to produce a static universe, not an expanding one[5]!
The signpost to this expanding universe had been there all along in Einstein’s own equations in 1917. “Had he followed where his equations were prompting him to go, he could have specifically predicted the expansion of the universe more than a decade before evidence for it was actually discovered. This would have been the greatest single prediction in the history of science”[6].
[1] This account is heavily reliant on John Farrell’s The Day Without Yesterday – Lemaître, Einstein and the Birth of Modern Cosmology, Thunder’s Mouth Press, New York, 2005, and in particular pages 8-9, 160-1, 80, 65, 110, 104, 106-7, 116-7. See also https://worldhistoryproject.org/1917/albert-einstein-publishes-first-paper-on-cosmology.For a different aspect of the cosmological constant, see String theory, spatial expansion and the cosmological constant
[2] Ibid, 98
[3] Cf the same technique (transferring Λ to the right hand side) employed by Lawrence Krauss in the link at the top of the page: /converting-the-thought-experiment-into-mathematical-equations.html
[4] Ibid.
[5] Robert P Crease, The Great Equations, Norton, New York, 2010, 200, and footnote 34 on pp 293-4.
[6] Farrell, op cit, 13.