Spacetime

Cosmological Term

Our student Hari Onoda continues to work on Einstein's cosmology (first published in 1917 as "Cosmological Considerations on the General Theory of Relativity", Preussische Akademie, Sitzungsberichte) and its sequel. Recall that Einstein introduced a new term Λ (or sometimes λ) into his field equations (see Einstein Equation), and this seemed to be conducive to satisfying his relativity principle and Mach's principle, at the same time dispensing with dependence on "boundary conditions" for solving the field equations. The result was a static spherical unverse. Onoda traced also Einstein's later discussion with de Sitter and others (her graduation thesis and other presentations).

Einstein's 1917 paper "Cosmological Considerations"

Recently, John Earman published a paper on the "cosmological term" Λ, a review essay on the history of this term. It is quite instructive, since it covers both the prehistory of Λ and its sequel up to the present, relating to such wide topics as the "expanding universe" (big-bang model), astronomical observations of "red shift" of quasi-stellar objects, the "vacuum energy density of quantum fields", the "inflationary cosmology" ("inflation" means, briefly, an accelerated expansion), and the "missing mass" in the universe. The reader is referred to the following:

Earman, "Lambda: The constant that refuses to die", Arch. Hist. Exact Science 55 (2001), 189-220.

As a supplement, let me add a quotation and a figure for illustrating the notions of "vacuum" and the "cosmological constant" in recent astronomy.

Imagine taking a region of space and removing from it all of the matter, radiation, and other substances we could conceivably remove. The reslting state is referred to as the 'vacuum' ... The vacuum has the lowest energy of any state, but there is no reason in principle for that energy to be zero. ... In the General Theory of Relativity, however, any form of energy affects the gravitational field, so the vacuum energy becomes a potentially crucial ingredient. To a good approximation ..., we believe that the vacuum is the same everywhere in the universe, so the vacuum energy density is a universal number which we call the cosmological constant. More precisely, the conventionally defined cosmological constant Λ is proportional to the vacuum energy density ρΛ ; ("Cosmological Constant" by Sean M. Carroll, Encyclopedia of Astronomy and Astrophysics, Nature Publishing Group, 2001 [on the web: http://www.ency-astro.com/])

Now, the average energy density in the universe ρ may be expressed in terms of the density parameter Ω (which contains the gravitational constant G, Hubble constant H, and the velocity of light c),

Ω　＝　(8πG/3HHcc)ρ.

And this may be decomposed into a sum of contributions from different sources, (1) density parameter for matter ΩM and (2) desnity parameter for the cosmological constant (for vacuum) ΩΛ. Notice that the density parameter Ω = ΩM +ΩΛ is directly related with the curvature of spacetime. The curvature is negative, zero, and positive, according as Ω<1, Ω=1, and Ω>1. Further, the present conjecture about the actual universe may be represented in the following figure, which is adapted from Carroll's Figure 1 in the quoted article.

Last modified Sept. 17, 2005. (c) Soshichi Uchii

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