416 CHAPTER 7. ASTROPHYSICS AND COSMOLOGY
Hereα,ψ,Tsatisfy equations (7.1.79)-(7.1.80) with the boundary conditions (7.1.81)-(7.1.84).
The detailed derivation of (7.2.9)-(7.2.10) will be given hereafter.
Theσ-factor (7.2.10) can be expressed in the following form, to be deduced later:
σ(r) =c^2 r 03 ( 1 −δ)β
2 κ^2 r^2eζ(r)
eζ(^1 )·( 1 −δr^2 −η)·(
1
r^2δr^2 +η
1 −δr^2 −η+rξ)
(7.2.11)
·
(
A−
1
κ∫ 1rr^2 Q
1 −δr^2 −ηdr)
for 0≤r≤ 1 ,where
(7.2.12)
η=1
2 r∫r0r^2 ψ′φ′
α ψdr,ζ=∫r0(α
r+rξ)
dr,ξ=8 πG
c^2
αp+φ′′−α′φ′
2 α
for 0≤r≤ 1 ,δis called theδ-factor given by
(7.2.13) δ=
2 mG
c^2 r 0
,
andm,r 0 are the mass and radius of the star.
2.Formula for the relativistic effect. The termFGPin (7.2.7) can be expressed in the
following form:
(7.2.14) FGP=
−ν(
δ+η′
2 r)
∂
∂r(rPθ)
−ν(δ+η′
2 r)∂
∂r(rPφ)
ν
2(
( 2 δr+η′)^2
1 −δr^2 −η+^2 δ+η′′)
Pr+ν 2 ( 2 δr+η′)∂∂rPr
,
whereη,δare as in (7.2.12) and (7.2.13).
The forceFGPis of special importance in studying supernovae, black holes, and the
galaxy cores. In fact, by the boundary conditions (7.1.81)-(7.1.83), we can reduce that the
radial component of the force (7.2.14) on the stellar shell is as
fr=(
2 ν δ^2
1 −δ+δ+φ′′( 1 ))
Pr+δ ν∂Pr
∂r,
which has
(7.2.15) fr∼
2 ν δ^2
1 −δPr→∞ as δ→ 1 (forPr> 0 ).The property (7.2.15) will lead to a huge supernovae explosions as they collapse to the radii
r 0 → 2 mG/c^2. It is the explosive force (7.2.15) that prevents the formation of black holes;
see Sections7.2.6and7.3.3.