Mathematical Principles of Theoretical Physics

410 CHAPTER 7. ASTROPHYSICS AND COSMOLOGY

The Universe and all stars are in the momentum-flow state, i.e. they are fluid spheres.

To investigate the interiors of the Universe, galaxies and stars, we need to develop dynamic

models for fluid spheres.

The precise equations of fluid sphere should be defined in the Riemannian metric space

as follows:

(7.1.75) ds^2 =g 00 (x,t)c^2 dt^2 +gij(x,t)dxidxj forx∈M^3 ,

whereM^3 is the spherical space. The gravitational field equations are expressed as

Rμ ν−

1

2

gμ νR=−

8 πG

c^4

(7.1.76) Tμ ν−DμΦμ, (gj 0 =g 0 j= 0 , 1 ≤j≤ 3 ),

where

Tμ ν=gμ αgν β

[

εαεβ+pgα β

]

,

andεμis the 4D energy-momentum vector.

For the fluid component of the system, it is necessary to simplify the model by making

some physically sound approximations.

Approximation Hypothesis 7.9.The metric (7.1.75) and the stationary solutions of the fluid

dynamical equations are radially symmetric.

Under Hypothesis7.9, the metric (7.1.75) is as in (7.1.12), or is written in the following

form

(7.1.77) ds^2 =−ψ(r)c^2 dt^2 +α(r)dr^2 +r^2 dθ^2 +r^2 sin^2 θdφ^2 ,

and the fluid dynamic equations are rewritten as

(7.1.78)

∂P

∂ τ

+

1

ρ

(P·∇)P=ν∆P+μ∇(divP)−∇p−

c^2 ρ

2 α

dψ

dr

( 1 −β(T−T 0 ))~k,

∂T

∂ τ

+

1

ρ

(P·∇)T=κ ̃∆T+Q(r),

∂ ρ

∂ τ

+divP= 0 ,

whereP= (Pr,Pθ,Pφ),∇P, ̃∆T,(P·∇)P,divPare as in (7.1.57)-(7.1.60),∇pis as in (7.1.56),
~k= ( 1 , 0 , 0 ), and

(P·∇)T=Pr

∂T

∂r

+

Pθ

r

∂T

∂ θ

+

Pφ

rsinθ

∂T

∂ φ

.

The gravitational field equation (7.1.76) for the metric (7.1.77) is radially symmetric,

therefore

Φν=Dνφ, φ=φ(r).

Thus we have

D 0 D^0 φ=

1

2 α ψ

ψ′φ′, D 1 D^1 φ=

1

α

φ′′−

1

2 α^2

α′φ′,

D 2 D^2 φ=D 3 D^3 φ=

1

rα

φ′, DμDνφ= 0 forμ 6 =ν.