7.5. THE UNIVERSE 459
whereM(r) = 4 πr^3 ρ/3, andρis the mass density. Thus, (7.5.10) can be rewritten as follows
(7.5.11) r′′=−
4
3
πGrρ.Make the nondimensional
r=R(t)r 0 ,
whereR(t)is the scalar factor, which is the same as in the FLRW metric (Ma and Wang,
2014e). Letρ 0 be the density atR=1. Then we have
(7.5.12) ρ=ρ 0 /R^3.
Thus, equation (7.5.11) is expressed as
(7.5.13) R′′=−
4 πG
3ρ 0
R^2,
which is the dynamic equation of Newtonian cosmology.
Multiplying both sides of (7.5.13) byR′we have
d
dt
(R ̇^2 −
8 πG
3ρ 0
r) = 0.
Hence, (7.5.13) is equivalent to the equation
(7.5.14) R ̇^2 =
8 πG
3ρ 0
R−κ,whereκis a constant, and we shall see thatκ=kc^2 , andk=− 1 , 0 ,or 1.
8.The Friedmann cosmology.The nonzero components of the Friedmann metric areg 00 =− 1 , g 11 =R^2
1 −kr^2
, g 22 =R^2 r^2 , g 33 =R^2 r^2 sin^2 θ.Again by the Cosmological Principle (Roos, 2003 ), the energy-momentum tensor of the Uni-
verse is in the form
Tμ ν=
pc^2000
0 g 11 p 0 0
0 0 g 22 p 0
0 0 0 g 33 p
.
By the Einstein gravitational field equations
Rμ ν=−
8 πG
c^4(Tμ ν−1
2
gμ νT),DμTμ ν= 0 ,we derive three independent equations
R ̈=−^4 πG
3(
ρ+3 p
c^2)
(7.5.15) R,
RR ̈+ 2 R ̇^2 + 2 kc^2 = 4 πG(
ρ−p
c^2)
(7.5.16) R^2 ,
ρ ̇=− 3(
R ̇
R
)(
ρ+p
c^2