Mathematical Principles of Theoretical Physics

7.2. STARS 427

To derive an explicit expression of (7.2.46), we need to compute the covariant derivatives of
the dual gravitational fieldφ:

Dμ νφ=

∂^2 φ

∂xμ∂xν

−Γλμ ν

∂ φ

∂xλ

.

Letφ=φ(r,t). Then we have

D 00 φ=

1

c^2

φtt−

1

2 c^2

utφt−

1

2

eu−vurφr,

D 11 φ=φrr−

1

2 c^2

ev−u(kt+vt)φt−

1

2

vrφr,

D 22 φ=−

r^2

2 c^2

e−uktφt+re−vφr,

D 33 φ=D 22 φsin^2 θ,

D 10 φ=φrt−

1

2 c

(urφt+φrkt+φrvt).

Thus, the field equations (7.2.46) are written as

R 10 =−D 1 D 0 φ,

Rkk=−

8 πG

c^4

(Tkk−

1

2

gkkT)−(Dkkφ−

1

2

gkkΦ) fork= 0 , 1 , 2 ,

which are expressed as

(

1 +

rur

2

)

kt+vt=

8 πGr

c^2

eu+k+vPr+crφrt−

r

2

(7.2.47) (urφt+φrkz+φrvt),

3 ktt+

3

2

kt^2 +vtt+

1

2

(7.2.48) vt^2 +ktvt−ut(kt+vt)

−c^2 eu−k−v

[

urr+

1

2

u^2 r−

1

2

urvr+

2

r

ur

]

=−

8 πG

c^2

(ρ+ 3 p)−c^2

(

D 00 φ+eu−k−vD 11 φ+

2 eu−k

r^2

D 22 φ

)

,

ktt+

3

2

kt^2 +vtt+v^2 t+ 3 ktvt−

1

2

(7.2.49) ut(kt+vt)

−c^2 eu−k−v

[

urr+

1

2

u^2 r−

1

2

urvr−

2

r

vr

]

=

8 πG

c^2

eu(ρ−p)+c^2 (eu−k−vD 11 φ−D 00 φ−

2 eu−k

r^2

D 22 φ),

ktt+

3

2

kt^2 +

1

2

kt(vt−ut)+

2 c^2 eu−k−v

r^2

[

ev+

r

2

(kr+vr−ur)− 1

]

(7.2.50)

=

8 πG

c^2

eu(ρ−p)+c^2 (D 00 φ−eu−k−vD 11 φ),

The equations (7.2.47)-(7.2.50) have seven unknown functionsu,v,k,φ,Pr,ρ,p, in which
Pr,ρ,psatisfy the fluid dynamic equations and the equation of stateintroduced hereafter.