Mathematical Principles of Theoretical Physics

474 CHAPTER 7. ASTROPHYSICS AND COSMOLOGY

and byTμ ν=diag(c^2 ρ,g 11 p,g 22 p,g 33 p), we have

T 00 −

1

2

g 00 T=

c^2

2

(

ρ+

3 p

c^2

)

,

Tkk−

1

2

gkkT=

c^2

2

gkk

(

ρ−

p

c^2

)

for 1≤k≤ 3 ,

φ 00 −

1

2

g 00 Φ=

1

2 c^2

(

φtt−

3 Rt

R

φt

)

,

φkk−

1

2

gkkΦ=

1

2 c^2

gkk

(

φtt+

Rt

R

φt

)

for 1≤k≤ 3.

Thus, we derive from (7.6.4) two independent field equations as

R′′=−

4 πG

3

(

ρ+

3 p

c^2

)

R−

1

6

φ′′R+

1

2

(7.6.5) R′φ′,

R′′

R

+ 2

(

R′

R

) 2

+

2 c^2

R^2

= 4 πG

(

ρ−

p

c^2

)

+

1

2

φ′′+

1

2

R′

R

(7.6.6) φ′.

We infer from (7.6.5) and (7.6.6) that

(7.6.7) (R′)^2 =

8 πG

3

R^2 ρ+

1

3

R^2 φ′′−c^2.

By the Bianchi identity:

∇μ(∇μ νφ+

8 πG

c^4

Tμ ν) = 0 ,

we deduce that

(7.6.8) φ′′′+

3 R′

R

φ′′=− 8 πG

(

ρ′+

3 R′

R

ρ+

3 R′

R

p

c^2

)

.

It is known that the energy densityρand the cosmic radiusR(also called the scale factor)
satisfy the relation:

(7.6.9) ρ=

ρ 0

R^3

, ρ 0 the density atR= 1.

Hence, it follows from (7.6.9) that

ρ′=− 3 ρR′/R.

Thus, (7.6.8) is rewritten as

(7.6.10) φ′′′+

3 R′

R

φ′′=−

24 πG

c^2

R′

R

p.

In addition, making the transformation

(7.6.11) φ′′=

ψ

R^3

,