Mathematical Principles of Theoretical Physics

4.2. PHYSICAL SUPPORTS TO PID 195

Consider the influence of cosmic microwave background (CMB)radiation, the energy-
momentum tensor can be approximatively written as

(4.2.13) Tμ ν=

(

−g 00 ρ 0

0 0

)

,

whereρis the energy density, a constant.
For the metric (4.2.12), the nonzero components of the Ricci tensor are

(4.2.14)

R 00 =−eμ−ν

[

u′′

2

+

u′

r

+

u′

4

(u′−v′)

]

,

R 11 =

u′′

2

−

v′

r

+

u′

4

(u′−v′),

R 22 =e−v

[

1 −ev+

r

2

(u′−v′)

]

,

R 33 =sin^2 θR 22.

On the other hand, equations (4.2.10) can be equivalently written as

(4.2.15)

Rμ ν=−

8 πG

c^4

(Tμ ν−

1

2

gμ νT),

∇μTμ ν= 0 ,

and by (4.2.13),
T=gμ νTμ ν=g^00 T 00 =−ρ.

Thus, the Einstein field equations for the spherically symmetric gravitation fields (4.2.15) are
in the form

(4.2.16)

R 00 =

4 πG

c^4

g 00 ρ,

R 11 =−

4 πG

c^4

g 11 ρ,

R 22 =−

4 πG

c^4

g 22 ρ,

∇μTμ ν= 0.

Now, we deduce that the equations (4.2.16) have no solutions. ByDμTμ ν=0, we have

Γ^010 T 00 =

1

2

u′ρ= 0 ,

which implies thatu′=0. Hence, by (4.2.14) we have

R 00 = 0 ,

which is a contradiction to the first equation of (4.2.16). Therefore the equations (4.2.16)
have no solutions.
However, if we consider this example by using the field equations derived from PID, then
the problem must have a solutions; see the theory of dark matter and dark energy in Chapter
7.