Relativity Lite

Cosmology | 75 since one then would have S 2 /c 2 = 1 − 1 = 0 . (Here R max is given * in terms of the current radius R 0 and current density ρ 0 of the universe as 8 π G/3 ρ 0 R 0 3 /c 2 , in which we must have ρ 0 > ρ c for k to be 1.) But the mass is still exerting a force on the universe when it has stopped expanding, so we have a big crunch coming. For those of you not familiar with looking for patterns in equations with your mind’s eye, let us cast equation (4a) into a form for more conventional vision. If we arbitrarily set R max = 4 , then as R increases from 0 to 10, the ratio R max /R drops from a high value to 0.4 at R = 10 . This is the second picture in equation (4b). Next we have k = 1 for all R from 0 to 10, the third picture in equation (4b). If we subtract off the third picture from the second picture, we get the first picture, S 2 /c 2 , with the dropping curve shifted downward by 1 unit at every point and becoming 0 at R = R max = 4 : (4b) When we look at the Hubble expansion in a k = 0 universe, the only value of R that will make S 2 /c 2 = 0 in equation (4a) is R = ∞ , as one can see by trying a series of larger values of R for R max arbitrarily set to 4 units: 4/100 = 0.04, 4/1000 = 0.004, 4/10000 = 0.0004, and so forth. This is also seen in the k = 0 universe pictured in equation (4c). We have extended R 10 times farther than in equation (4b), and even at R = 100 , the curve seems already at 0. In this case, the third picture has k = 0 for all values of R , so the subtraction of 0 does nothing, and the first picture for S 2 /c 2 just duplicates the second one, going to 0 as R → ∞ : (4c) When we look at the Hubble expansion in a k = −1 universe, subtracting a negative is equivalent to adding a positive ( −(−1) = +1 ) and there is no value of R that will make S 2 /c2 = 0 , since we will always have the 1 left over, however small R max /R becomes. In equa- tion (4d), the first picture has the R max /R of the second picture shifted upward to approach 1. This universe will go on expanding forever. * Tai L. Chow, Gravity, Black Holes, and the Very Early Universe an Introduction to General Relativity and Cosmology (Springer, New York, 2008), p. 142, eq. (8.42).