Mathematical Principles of Theoretical Physics

7.6. THEORY OF DARK MATTER AND DARK ENERGY 477

we obtain from (7.6.22) that

(7.6.24) 3 α 2 = 24 kπ α 1 +k+ 1.

By the relation (7.6.24) from (7.6.22), we can also derive, in the same fashion as above,
the dark energy formula (7.6.23).

7.6.3 PID gravitational interaction formula

Consider a central gravitational field generated by a ballBr 0 with radiusr 0 and massM. It is
known that the metric of the central field atr>r 0 can be written in the form

(7.6.25) ds^2 =−euc^2 dt^2 +evdr^2 +r^2 (dθ^2 +sin^2 θdφ^2 ),

andu=u(r),v=v(r).
In the exterior ofBr 0 , the energy-momentum is zero, i.e.

Tμ ν= 0 , forr>r 0.

Hence, the PID gravitational field equation for the metric (7.6.25) is given by

(7.6.26) Rμ ν−

1

2

gμ νR=−∇μ νφ, r>r 0.

whereφ=φ(r)is a scalar function ofr.
By (7.1.25) and (7.1.26), we have

R 00 −

1

2

g 00 R=−

1

r

eu−v

[

v′+

1

r

(ev− 1 )

]

,

R 11 −

1

2

g 11 R=−

1

r

[

u′−

1

r

(ev− 1 )

]

,

R 22 −

1

2

g 22 R=−

r^2

2

e−v

[

u′′+

(

1

2

u′+

1

r

)

(u′−v′)

]

,

∇ 00 φ=−

1

2

eu−vu′φ′,

∇ 11 φ=φ′′−

1

2

v′φ′,

∇ 22 φ=−re−vφ′.

Thus, the fields equations (7.6.26) are as follows

(7.6.27)

v′+

1

r

(ev− 1 ) =−

r

2

u′φ′,

u′−

1

r

(ev− 1 ) =r(φ′′−

1

2

v′φ′),

u′′+

(

1

2

u′+

1

r

)

(u′−v′) =−

2

r

φ′.