Relativity Lite

92 | Relativity Lite right-hand side of equation [6]) is more conventionally written in terms of the cosmological constant, Λ , so we transform equation (6) to read A G R G R G R R = − + ≡ − + 4 3 4 3 4 3 3 0 π ρ π ρ π ρ Λ (7) The required “substance” with negative pressure could simply be the vacuum energy of “empty” space Casimir’s work led us to. This adds a new term to equation (4a), the first Friedmann equation that gives the expan- sion velocity of the universe: S 2 /c 2 = R max /R − k + 1/3 Λ R 2 /c 2 (8a) One sees that early enough in the expansion of the universe, R is small, and its square is smaller still, so that Λ will have no discernible effect. But eventually, R will get big enough and its square will drive the expansion speed S to larger and larger values. This first Fried- mann equation, too, gives a runaway acceleration of the universe that is just starting in the visualization equation (8b) with k = 1 , (8b) There are candidates for what is driving the acceleration of the universe other than the vacuum energy of “empty” space, and there also are some unresolved problems with calcu- lating the vacuum energy density. See, for instance, the work of Robert Caldwell, Rahul Dave, and Paul Steinhardt on a form of dark energy called “quintessence.” * We will leave readers to explore such on their own—after, of course, listening to some more music, say Youssou N’Dour’s song “Liggey” (Live In London, 2003). * R. R. Caldwell, R. Dave, and P. J. Steinhardt, Phys. Rev. Lett. 80 (8), 1582–85 (1998).