Universe Einstein's Newtonian gravity gave the energy of the universe as R R
2
8 - G0 3
R0 R
3
=
2E R2
(2.01)
In the relativisitic case, the equation is very similar R R
2
8 - G0 3
R0 R
3
kc2 =- 2 + R 3
(2.02)
Here, k is constant which defines the curvature of the universe. Note that k only appears when divided by R2 (which is not measureable). Thus, to simplify things, we can choose the units of R such that k is either 0, 1, or -1. The value k = 1 represents a universe with positive curvature, like a sphere; k = -1 reflects negative curvature, like a saddle. In the critical case, where k = 0, the universe is flat. The other new variable is a Cosmological Constant. You can think of it as a pressure term which supplies a new repulsive (or attractive) force that is directly proportional to distance, R = R (2.03)
If one wants the universe to be static, then one needs to include this additional term to counter-act the attractive force of gravity. (There are other reasons to add in a Cosmological Constant, but this was the origin...
lec2n