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1 .1, θ, ϕ d = A, t dt + B, t dtd + C, t d + D, t dθ +in θdϕ.1.1 t { = f1,t t = f,t { D, t = B, t =.1. t A, tdt e φ,t dt, C, td e λ,t d.1.3,t, t d = e φ,t dt + e λ,t d + dθ +in θdϕ.1.4 { = f1,t t = f,t { D, t =,t B, t =.1.5, t.1.4 d = e φ,t dt + e λ,t d +, t dθ +in θdϕ.1.6, t m = m 8

2 . 9 Γ t tt = φ Γ t = λe λ φ Γ t t =Γ t t = φ Γ t θθ = e φ Γ = λ Γ tt = φ e φ λ Γ t =Γ t = λ Γ θθ = e λ Γ ϕϕ = e λ in θ Γ θ tθ =Γ θ θt = Γ ϕ tϕ =Γ ϕ ϕt = Γ t ϕϕ = e φ in θ Γ θ θ =Γ θ θ = Γ ϕ ϕ =Γ ϕ ϕ = : Γ θ ϕϕ = in θ co θ Γ ϕ θϕ =Γ ϕ ϕθ =cotθ..1.7 R αβ μ Γ μ αβ β Γ μ αμ +Γ μ γμγ γ αβ Γ μ γβγ γ αμ.1.8 : 8πGT t t = G t t = λ + e φ + G α β = R α β 1 δα βr μ μ.1.9 G α β =8πGT α β λ e λ πGT t = G t = e φ φ + λ.1.1 8πGT = G φ = + e λ + φ e φ πGT θ θ = G θ θ = λ + λ φ λ φ + λ + e φ + φ + φ λ φ + φ λ + e λ.1.14 ϕ, ϕ θ, θ. 1 1 pefect fluid ideal fluid

3 . 3 T μν =ρ + pu μ u ν + pg μν T μ ν =ρ + pu μ u ν + pδ μ ν..1 u μ 4 ρ p T μ ν u μ, θ, ϕ 4 u μ = dx μ /dτ u t = dt dτ = e φ u t = g tt u t = e φ, u = u θ = u ϕ = : T tt = ρe φ, T = pe λ, T θθ = p, T ϕϕ = p in θ..3 T t t = ρ, T = T θ θ = T ϕ ϕ = p..4 T μ ν;μ = : T μ ν;μ = 1 gt μ ν 1 g μα g x μ x T μα..5 ν g g μν.1.6 g = e φ+λ in θ t- T μ t;μ = 1 gt μ t 1 g μα g x μ t T μα = 1 gt t t 1 ġtt T tt +ġ T +ġ θθ T θθ +ġ ϕϕ T ϕϕ g t = 1 ρe φ+λ 1 e φ+λ t = ρ ρ φρ λρ + φρ φt t t + λt + T θ + θ T ϕ ϕ λ + p =...7 =

4 T μ ;μ = θ- ϕ- 1 ρe φ+λ 1 φ ρ +λ p +4 e φ+λ t p = p + φ ρ + φ p =...8 T μ θ;μ = 1 p in θ 1 coθ in θ θ in θ p = p =, θ..9 T μ ϕ;μ = p =, ϕ..1 ρ + λ + ρ + p =,..11 φ + 1 ρ + p p = : 8πGρ = λ + e φ + + λ e λ = φ + λ.3. 8πGp = φ + e λ + φe φ πGp = λ + λ φ λ φ φ + + λe +φ + φ λ φ + φ λ + e λ.3.4 φ, λ,, ρ, p 5 4 p ρ p = pρ

5 tt, t 3 dτ e φ dt : u τ = 1 d e φ dt = e φ.3.5 u =e φ = e φ φ }{{} = e φ λ e φ u λ =u u = u..3.7 Γ e λ.3.8 Γ =Γ +Γ Γ = e λ + λ πGρ =1+u + u + λ e λ }{{} = Γ / = [ 1 + u Γ ]..3.1 Γ= 1+u G 4πρ d.3.11 = 4 e λ d d λ Γ=e.3.11 : m, t,t 1 =,=.3.1 4πρ d

6 Γ e λ =1+u Gm, Gm, t u = Γ u = e φ φe φ πGp = φ + e λ + φe φ 1 = φ +1 Γ u ue φ πGp = Γ p ρ + p ue φ Gm φ u e t = Gm/ p 1+u ρ + p Gm 4πGp u ρ + eφ + u eφ ρ + p = t 6 u m p ρ φ= log g [ u 1 t = eφ 1+u Gm p ρ + p + Gm ] +4πGp.3. t = eφ u.3.1 ρ u t = eφ +u ρ + p.3. φ = 1 p.3.3 ρ + p m =4πρ.3.4

7 p = φ e φ dt dt φ Γ t = t e = λ λ t + t e λ = u + u e λ =..4.1 p = Γ d = e φ dt, t + d +, t dθ +in θdϕ.4. Γ p = φ = Γ Γ t =u u t G m t + Gm t = G m t.4.3 Γ m.3.14 : t = Γ 1+ Gm t = d.4.4 Gm +Γ 1 Gm +Γ 1 Gm Γ 1 Γ 1 Γ inh 1 Γ 1 > 1 3/ Gm t t = 9Gm 3/ Γ 1= Gm 1 Γ Gm 1 Γ 1 Γ 1 Γ in 1 Γ 1 < 3/ Gm.4.5 t = t = t > = t <t t t.4.4 Γ 1 Γ 1/3 9Gm t t 9Gm 3/, t t t /3.4.6, t m 1/3 G t t /3 + t 1/3 4Gm t t 1/

8 t / d dl = 1 Γ.3.4, t d 1 t Γ 1/3 4Gm t t 1/3 d, ρ, t = m 4π 1 t t.4.9 t t, t dv p =4π d 1πGm t t td.4.1 Γ Γ t t t / =.4.7 t t dl t t /3, dv p t t, ρ t t , t.1 d = e φ dt + e λ d + dθ + in θdϕ Gm ρ + p p p + Gm +4πGp = = Gρ + pm +4πp3,.5. 1 Gm/ TOV p ρ c c, 4πp 3 m 3p ρ 3c c, Gm m M.5.3

9 . 36 Gm.5. TOV ρ, p, m 3 p = Gρm.5.4 m = 4πx ρdx.5.5 p = pρ.5.5 = m = TOV i : ρ ii :.5. p = = R m mr iii :.3.14 e λ = 1 1 Gm/.5.6 iv tt : φ = 1 p ρ + p = Gm +4πp3 Gm..5.7 e φ =1 [ Gm +4πpx 3 ] φ = x Gmx dx Gm +4πpx 3 eφ =exp x Gmx dx.5.8 > p =.5.8 Gm φ = xx Gm dx = 1 1 =ln x Gm x Gm x x =ln 1 Gm e φ =1 Gm >.5.9

10 . 37 <.5.8 [ e φ Gm ] =exp xx Gm dx Gm +4πpx 3 x Gmx dx = 1 Gm [ ] Gm +4πpx 3 exp x Gmx dx = 1 Gm [ p ] dp exp ρ + p TOV : dp d = Gρ + pm +4πp3 1 Gm/.6.1 ρ =ρ.6. m.6.1 dp d = 4πGρ + pρ 3 /3+p 3 1 8πGρ /3 1 dp 4πG m = 4π 3 ρ p + ρ p + ρ /3 d = 1 8πGρ /3 = 1πG 1 8πGρ 1/κ dp p + ρ p + ρ /3 ρ d = ρ 1/κ +1/κ 1 1/κ..6.4 p + ρ p + ρ /3 κ 8πGρ /3.6.1 = R p = <R p.6.4 p = 3 p + ρ p + ρ /3 =1+ρ 1 3 p + ρ /3 =3 1 κ R <R 1 κ..6.5

11 . 38 p M 4πρ R 3 /3 1 p + ρ /3= ρ 1 κ 3 R 3 1 κ 1 = ρ 1 κ κ R 1 κ 1 κ p =ρ 1 κ R 3 1 κ R 1 κ = 3M 4πR e λ = 1 1 Gm/ 1 GM /R 3 1 GM /R 3 1 GM /R 1 GM /R 3 [ e φ =exp ] Gm +4πpx 3 x Gmx dx >R p =,ρ = m = m 1 1 R e λ = φ = = 1 1 Gm/ = 1 1 8πGρ /3 = GM /R 3 R 1 x 1 x GM Gm +4πpx 3 x Gmx dx GM x GM x dx + dx = 1 ln R x x GM 4πGx 3 p + ρ /3 x κ x4 dx..6.9 = 1 R ln 1 GM R a 1 κ R, ux 1 κ x..6.1 = R R = ln 4πGx ρ 1 κ x 3 κ x 1 κ x 1 κ x 3 1 κ R 1 κ 1 κ x 3 1 κ R 1 κ x dx x dx = a u du 3a u 3a u a 3a u φ =lna + ln[3a u] lna =ln e φ = GM 1 GM R R 3

12 . 39 <R d = GM 1 GM dt 4 R R GM 1 d + dθ +in θdϕ.6.13 R 3.6. >R p =,ρ =,m = M e λ = 1 1 GM /.6.14 Gm +4πpx 3 φ = x Gmx dx = = 1 ln 1 GM >R GM x GM x dx d = 1 GM dt + 1 GM 1 d + dθ +in θdϕ.6.16 = R Sch : R Sch GM = GM.6.17 c.6.3 <R GM > 1 GM >,.6.18 R 3 R 3 1 GM 1 GM > 3 1 GM 1 >.6.19 R R 3 R

13 R >R Sch = GM c,.6. GM R < 8 9 R > 9 8 R Sch.6.1 R < 9 8 R Sch = 9 8πG 8 3 ρ R 3 R > c 3πGρ.6. M =4πρ 3 /3 d dt = GM = 4πG 3 ρ.6.3 τ dyn 3 τ dyn.6.4 4πGρ.6. R R.6.4 : d = 1 R Sch dt + d 1 R Sch / + dθ +in θdϕ.6.5 4π π +Δ Δl Δ.6.5 Δl = Δ 1 / > Δ..6.6

14 l 1.1: > 1 1 l 1 x / l 1 = 1 d /1 1 / = / dx x 1 x..6.7 dx 1 x x 1 x = x x ln 1+ 1 x x l 1 = [ x + 1 ] /1 1 1 x ln 1+ 1 x / = 1 / 1 1 / 1 + ln = 1 / 1 1 / 1 + ln > / / 1 / / l ln l 1 > 1 1 = l 1 l 1 = 1 / + ln / + /

15 m = 4πx ρxdx in θddθdϕ d 3 p = g g θθ g ϕϕ ddθdϕ = e λ in θddθdϕ dv p = d π dϕ π dθ g g θθ g ϕϕ =4π e λ d.7. d 3 p dv p M p = ρdv p = 4πx ρxe λ 4πx ρx dx = dx Gm/x ρ ρ m ρ I : m = ρ = ρ m + ρ I.7.4 ρ m + ρ I 1 Gm x dv p..7.5 Gm/x 1, ρ I ρ m m = ρ m + ρ I 1 Gm x dv p ρ m + ρ I 1 Gm dv p x Gmρ m ρ m dv p + ρ I dv p dv p..7.6 x }{{}}{{}}{{} =M m =U I =U G M p M m + U I M p >M m >m M p M m M m m M p m 5 M p >m>m m U I

16 d = e φ,t dt + e λ,t d + dθ +in θdϕ , t =.8.1 = = =, =1.1.1 e φ λ = λ = λ e λ 1 =.8.3 φ + 1 e λ 1 =.8.4 φ + φ λ φ + φ λ e λ = λ + φ = λ + φ = C 1 =.8.6 C 1 = λ e λ = d d e λ =1 e λ =1+ C.8.7 C.6. M C = GM.8.7 λ +λ λ e λ = φ = λ

17 : d = 1 GM dt + 1 GM 1 d + dθ +in θdϕ.8.9 Bikhoff.9 :.9.1 d = g μν dx μ dx ν d = g tt t, dt + g t, d + g θθ t, dθ +in θdϕ.9.1 d = dt + at [ W xdx + x dθ +in θdϕ ].9. g tt = 1 d = dt + γ ij dx i dx j : d = e φx,t dt + e λx,t dx + x, t dθ +in θdϕ.9.4 R R = λ + λ φ λ φ + + φ φ + λ φ + λ + λ + e φ φ e λ

18 φx, t, λx, t =lnat+ 1 ln W x x, t =atx ȧ R =6ä a [ W a a xw + x 1 1 ] W.9.7 W xw K K.9.8 x W W x W x = 1 1 Kx.9.9 Robeton Walke [ ] d = dt + at dx 1 Kx + x dθ +in θdϕ, T μν T μν =ρ + pu μ u ν + pg μν.9.11 φx, t, λx, t =lnat 1 ln1 Kx x, t =atx [6.6] 8πGρa x +Λa x = 3ȧ x 3Kx.9.13 =ȧ a a ȧ πGpa x +Λa x = Kx aäx ȧ x πGp +Λa x = aäx ȧ x Kx.9.16

19 . 46 : G α β +Λδ α β =8πG T α β Λ : ȧ = 8πG a 3 ρ K a + Λ 3,.9.18 ä a = 4πG 3 ρ +3p+Λ ä = G 4π ρ +3p Λ a a 3 4πG }{{} a ρ Λ = Λ 8πG, p Λ = Λ 8πG.9.1 ρ ρ + ρ Λ p p + p Λ Λ.9.18 ä a = 4πG ρ ȧ 3 a +ρ + Λ ρ = 3ȧ ρ + p..9.3 a i matte: p ρ ii adiation: p = ρ/3 iii comological contant: p = ρ iv dak enegy: p = wρ

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