Hartle Gravity December 6, 2008 1 20.1 Lorentz boost (4.33) (α β γ γv 0 0 γv γ 0 0 0 ) x α x β 0 0 1 0 0 0 0 1 (20.6a) (5.9) 20.2 (a) xβ x α x α x γ xβ x γ δ β γ (b) (20.6a) xγ x α (a) xγ x α a α δ γ β aβ a γ (20.6b) 20.3 a t a x a y a z 1 0 0 0 0 sin θ cos φ r cos θ cos φ r sin θ sin φ 0 sin θ sin φ r cos θ sin φ r sin θ cos φ 0 cos θ r sin θ 0 a t a r a θ a φ. (7.2) at 1 0 0 0 a r 0 sin θ cos φ sin θ sin φ cos θ a θ 0 r cos θ cos φ r cos θ sin φ r sin θ a φ 0 r sin θ sin φ r sin θ cos φ 0 at a x a y a z at a x a y a z 1 0 0 0 cos θ cos φ 0 sin θ cos φ r cos θ sin φ 0 sin θ sin φ r 0 cos θ sin θ sin φ r sin θ cos φ r sin θ r 0 at a r a θ a φ
Hartle Gravity December 6, 2008 2 20.4 ( f) α f x α (5t/2M 2, r 2 /M 2, 0, 0). 20.5 (a) : eˆ0 eˆ0 (1 2M/r) 1 [ 1 + (2M/r)] 1, eˆ1 eˆ1 (1 2M/r) 1 [ (2M/r) + 1] 1. eˆ0 eˆ1 (1 2M/r) 1 (2M/r) 1/2 (1 2M/r) 1 (2M/r) 1/2 0. (b) (eˆα ) α g αβ (eˆα ) β g αα (eˆα ) α g αβ 0, ifα β: (eˆ0 ) α ( 1, (1 2M/r) 1 (2M/r) 1/2, 0, 0), (eˆ1 ) α ((2M/r) 1/2, (1 2M/r) 1, 0, 0), (eˆ2 ) α (0, 0, r, 0), (eˆ3 ) α (0, 0, 0, r sin θ), (c) {e α } e α e β δ α β e α e β 0, if α β e α e α /g αα (e α ) β (e α ) β /g αα δ β α/g αα
Hartle Gravity December 6, 2008 3 (e 0 ) 0 g 1 00, (e1 ) 1 g 1 11, (e2 ) 2 g 2 22, (e3 ) 3 g 3 33 7 (a) (e α ) β g αβ η ˆα ˆβ eˆα eˆα eˆα /η ˆα ˆα η ˆα ˆβe ˆβ (eˆ0 ) α (eˆ0 )α, (eî) α (eî) α (20.81a) (20.81b-d) (d) a a α e α aˆα eˆα aˆα eˆα a β aˆα (eˆα ) β aˆα (eˆα ) β (eˆα ) β (eˆα ) β aˆα [4 + 3(2M/r) 1/2 (1 2M/r) 1, 4(2M/r) 1/2 +3(1 2M/r) 1, 0, 0], aˆα [ 4 3(2M/r) 1/2 (1 2M/r) 1, 4(2M/r) 1/2 +3(1 2M/r) 1, 0, 0] aˆα η ˆα ˆβa ˆβ (0, 3M, 0, 0) r 3M aˆα (4+3 6, 9+4È2 3, 0, 0) aˆα ( 4 3 6, 9 + 4È2, 0, 0) 3 20.6 (20.7) (20.13) e α (a) e α (a β e β ) a β e α (e β ) a β δ α β a α, a(e α ) a β e β (e α ) a β δ α β a α. 20.7 (a) e α g αβ e β (20.11) (20.23) e α e β g αγ e γ e β g αγ g β γ g αγ g γβ δ α β g γα e α g αβ e β g γαe α g γαg αβ e β
Hartle Gravity December 6, 2008 4 δ β γ e β e γ e α g αβ e β (b) e α g αβ e β (20.23) e α e β g αγ e γ e β g αγ δ β γ g αβ. 20.8 a a α e α a β e β (20.6a) e α x β x α e β 20.9 w α t β αβ xγ x δ x β t δ x α x β x δ γδ xγ w x α γ. 20.10 (20.48) β v α vα x β x γ x β v α x γ x γ x β δš γ xα v x x δ x γ x β x α v δ + x γ x δ x γ x β 2 x α x γ x δ v δ (20.45) ν v µ xβ x ν x µ x α β v α δν γ δ µ δ v δ + δ γ x γ ν x µ x α 2 x α v δ v µ + x µ x γ x δ x ν x α {x α } 2 x α x ν x δ v δ v α v α x α 2 x δ + β x β x δ x β x γ v γ, (1) {x α } (20.50) Γ α βγ x α x δ 2 x δ. (2) x β x γ 2 x δ 2 x δ x β x γ x γ x β Γ α βγ Γ α γβ
Hartle Gravity December 6, 2008 5 Γ α βγ Γ α γβ MTW 20.11 γ t αβ t αβ x γ Γ δ αγt δβ Γ δ βγt αδ. 20.12 Γ θ φφ sin θ cos θ, Γφ θφ Γφ φθ cot θ φ w θ Γ θ θφ wθ cot θ A B w C A ( B w C ) A ( B w C )+Γ C DA ( B wd ) Γ D BA ( D wc ) A φ w φ A ( φ w φ ) + Γ φ φa ( φ wφ ) Γ φ φa ( φ wφ ) A ( φ w φ ) θ φ w φ csc 2 θ w 0 e φ (e φ ) φ φ w φ cot θ 20.13 β u α β u α Γ γ αβ u γ, a α u β β u α u β β u α u β Γ γ αβ u γ. u α [(1 2M/r) 1/2, 0, 0, 0], u α g αβ u β [ (1 2M/r) 1/2, 0, 0, 0]. a α u t t u α u t Γ t αtu t u t Γ t αtu t B Γ t αt α t, Γ t rt (M/r 2 )(1 2M/r) 1 a α [0, (M/r 2 )(1 2M/r) 1, 0, 0] a α g αβ a β [0, M/r 2, 0, 0] (20.61) 20.14.
Hartle Gravity December 6, 2008 6 20.15 λ x α (λ) u(λ) dxα (λ) dλ uu 0 σ f(λ) x(σ) x(f(λ)), u(σ) u(f(λ)) u α (λ) dxα (λ) dλ dx α (σ) f u α (σ)f f dσ dσ dλ df(λ) dλ 0 ( uu) α (λ) u β (λ)[ β u α (λ)+γ α γβu γ (λ)u β (λ)] (f ) 2 u β (σ) β u α (σ)+f u β (σ)u α (σ) β f (f ) 2 ( u u) α (σ)+f u α (σ) f u β (σ) β f dσ dx β df σ dλ dσ dσ x β 4f f d2 f(λ) dλ 2 σ f(λ) u u Ku K f /(f ) 2 σ aλ + b f(λ) aλ + b, K f /(f ) 2 0 u u 0 20.16 Finkelstein-Eddington (v, r, θ, φ) v t + r + 2M ln r 2M 1 ξ / t / v (1, 0, 0, 0) ξ v 1 ( ξ ξ) α ξ β (ξ α,β + Γ α γβξ γ ) Γ α vv 1 2 gαδ (2g δv,v g vv,δ ) (1 2M ) 1 0 0 r 1 0 0 0 g αβ 0 0 r 2 0 0 0 0 r 2 sin 2 θ 0 1 0 0 g αβ 1 1 2M 0 0 r 1 ( 0 0 0 ξ ξ) α 1 2 gαδ δ (1 r 2 1 0 0 0 r 2 sin 2 θ
Hartle Gravity December 6, 2008 7 2M r ) 1 2 gαr r (1 2M r ) M r 2 g αr ( ξ ξ) α ( ξ ξ) α 8< :M M if α v, r 2 (1 2M ) if α r. (3) r 2 r r 2M ( ξ ξ) r 0, ( ξ ξ) v 1 4M 1 4M ξv ξ ξ 1 ξ K 1 4M 4M ξ ξ g vv (ξ v ) 2 (1 2M r ) r2m 0 null ξ ξ Kξ null ξ v dv dσ 1, ξ r 0 r 2M p. 259 (12.5) 0 (1 2M 2M ) dv dσ + 2 dr dσ > ξr dr dσ 0 Killing ξ / t null null f(x α ) r 2M 0 (v, r, θ, φ) l α f/ x α (0, 1, 0, 0), l α g αβ l β (1, 0, 0, 0) ξ α ξ 19 ξ null 20.17 γ g αβ g αβ,γ + Γ α δγg δβ + Γ β δγ gαδ. (4)
Hartle Gravity December 6, 2008 8 Γ α δγg δβ 1 2 gαɛ g βδ (g ɛδ,γ + g ɛγ,δ g δγ,ɛ ). (5) g βδ g ɛδ g βδ g δɛ δ β ɛ > g βδ g ɛδ,γ g ɛδ g βδ,γ (5) 1 2 gαɛ g βδ g ɛδ,γ 1 2 gαɛ g ɛδ g βδ,γ 1 2 δα δg βδ,γ 1 2 gαβ,γ (4) Γ β δγ gαδ 1 2 gαδ g βɛ (g ɛδ,γ + g ɛγ,δ g δγ,ɛ ). (6) δ ɛ Γ β δγ gαδ 1 2 gαɛ g βδ (g δɛ,γ + g δγ,ɛ g ɛγ,δ ). (7) (5) g δɛ g ɛδ (4) γ g αβ 0 20.18 ξ α g αβ ξ β g α1 α ξ β + β ξ α α ξ β Γ γ βα ξ γ + β ξ α Γ γ αβ ξ γ α g β1 + β g α1 2Γ γ αβ ξ γ 2Γ γ αβ ξ γ g γδ ξ γ ( α g δβ + β g δα δ g αβ ) ξ δ ( α g δβ + β g δα δ g αβ ) α g β1 + β g α1 1 g αβ 1 g αβ 0 x 1 α ξ β + β ξ α 0
Hartle Gravity December 6, 2008 9 20.19 ( l l) α l β (l α,β Γ γ αβ l γ) l β [l α,β 1 2 gγδ (g δα,β + g δβ,α g αβ,δ )l γ ] l β [l α,β 1 2 lδ (g δα,β + g δβ,α g αβ,δ )] l β [l α,β 1 2 lδ g δβ,α ] g δα,β l δ l β g βα,δ l β l δ Lagrange 1 2 g βγ,αu β u γ du α dτ u β u α,β null l l g δβ l δ l β 0,α 0 g δβ,α l δ l β + g δβ l δ,αl β + g δβ l δ l β,α g δβ,α l δ l β + 2g δβ l δ l β,α g δβ g βδ > g δβ l δ,αl β g δβ l δ l β,α 1 2 lβ l δ g δβ,α g δβ l δ l β,α l β l β,α ( l l) α l β l α,β + l β l β,α l α f/ x α 2 f/ x α x β 2 f/ x β x α l α,β l β,α ( l l) α l β l β,α +l β l β,α (l β l β ),α 0 null l l l β l β 0 20.20 (a) Killing α ξ β + β ξ α 0 (x, y, z) Killing / x (1, 0, 0) / y (0, 1, 0) Killing η ( y, x, 0) α, β z x y y x 0 α β x α β y
Hartle Gravity December 6, 2008 10 α x, β y x (x) + y ( y) 1 1 0 (b) / φ ( / x )( x / φ )+( / y )( y / φ ) r sin θ sin φ( / x )+r sin θ cos φ( / y ) y( / x ) + x( / y ) 20.1 φ e φ cos φe y sin φe x e φ e φ g φφ r 2 sin 2 θ e φ ye x + xe y η φ (c) p. 34 (3.3) (x, y) (d, e) x x d, y y e z Killing η y ( / x ) + x ( / y ) (y e)( / x ) + (x d)( / y ) e( / x ) d( / y ) + η 20.21 8.2 Γ α αγ 1 2g αα g αα x γ x γh ln È gαα i α α Box 20.1 (g) Weinberg 4.7 20.22 Weinberg 4.4 20.23 (a) φ 0 > x µν, y 0, z (µ 2 ν 2 )/2 2.7(a)
Hartle Gravity December 6, 2008 11 (b) dx cos φd(µν) µν sin φdφ, dy sin φd(µν) + µν cos φdφ, dz µdµ νdν, ds 2 dx 2 +dy 2 +dz 2 [d(µν)] 2 +(µν) 2 dφ 2 + (µdµ νdν) 2 (µ 2 + ν 2 )(dµ 2 + dν 2 ) + µ 2 ν 2 dφ 2 2.7(b) (c) (b) (µ, ν, φ) Box 20.1 (e) (i) (j) (n) g
Hartle Gravity December 6, 2008 12 det(g ij ) [µν(µ 2 + ν 2 )] 2 f (µ 2 + ν 2 ) 1/2 ( µ f e ˆµ + ν f e ˆν ) + (µν) 1 φ f e ˆφ, V [µν(µ 2 + ν 2 )] 1 µ µν(µ 2 + ν 2 ) 1/2 V ˆµ + ν µν(µ 2 + ν 2 ) 1/2 V ˆν φh ˆφio + (µ 2 + ν 2 1 µ )V f, ν f 2 f [µν(µ 2 + ν 2 )] µν(µ 2 + ν 2 ) 2 µ + µν(µ 2 + ν 2 ) 2 ν, φ f 1/2n + νh (µν) 3 (µ 2 + ν 2 ) φ V (µν) 1 (µ 2 + ν 2 ) µνv ˆφi φ ˆν o 1/2n φ (µ 2 + ν 2 ) 1/2 ˆµ µh V ˆφio e ˆµ +(µν) 1 (µ 2 + ν 2 ) (µ 2 + ν 2 ) 1/2 V µνv e ˆν 1 µ ˆν ν ˆµ +(µ 2 + ν 2 ) (µ 2 + ν 2 ) 1/2 V (µ 2 + ν 2 ) 1/2 V e ˆφ. 20.24 s u (20.63) Leibniz d(s u)/dτ u α (s u)/ x α u (s u) u s u + s u u 0 Ds/dτ u s d(s u)/dτ D(s u)/dτ Ds/dτ u + s Du/dτ 0 20.25 u eˆ0 t, r ( ueˆα ) α u β [(eˆα ) α,β + Γ α γβ(eˆα ) γ ] (1 2M/r) 1 Γ α γt(eˆα ) γ (2M/r) 1/2 [(eˆα ) α,r +Γα γr(eˆα ) γ ] [(1 2M/r) 1 Γ α γt (2M/r) 1/2 Γ α γr](eˆα ) γ (2M/r) 1/2 (eˆα ) α,r.
Hartle Gravity December 6, 2008 13 B p.546 eˆ0 eˆ1 t, r tt, tr Γ t tr (M/r 2 )(1 2M/r) 1, Γ r tt (M/r 2 )(1 2M/r) ( u eˆ0 )θ ( u eˆ0 )φ ( u eˆ1 )θ ( u eˆ1 )φ 0 ( u eˆ0 )t (1 2M/r) 1 Γ t rt(eˆ0 )r (2M/r) 1/2 Γ t tr(eˆ0 )t (2M/r) 1/2 (eˆ0 )t,r (2M/r) 1/2 [2Γ t rt(1 2M/r) 1 + (eˆ0 )t,r ] 0, ( u eˆ1 )t (1 2M/r) 1 Γ t rt(eˆ1 )r (2M/r) 1/2 Γ t tr(eˆ1 )t (2M/r) 1/2 (eˆ1 )t,r Γ t rt[(1 2M/r) 1 + (2M/r)(1 2M/r) 1 ] (M/r 2 )(1 2M/r) 2 (1 + 2M/r) 0. ( u eˆ0 )r (1 2M/r) 1 Γ r tt(eˆ0 )t (2M/r) 1/2 Γ r rr(eˆ0 )r (2M/r) 1/2 (eˆ0 )r,r (M/r 2 )(1 2M/r) 1 (2M/r)(M/r 2 )(1 2M/r) 1 + M/r 2 0, ( u eˆ1 )r (1 2M/r) 1 Γ r tt(eˆ1 )t (2M/r) 1/2 Γ r rr(eˆ1 )r (2M/r) 1/2 (M/r 2 )(1 2M/r) 1 + (2M/r) 1/2 (M/r 2 )(1 2M/r) 1 0. eˆ2 θ θr, θt
Hartle Gravity December 6, 2008 14 Γ θ θr 1/r ( u eˆ2 )t ( u eˆ2 )r ( u eˆ2 )φ 0 ( u eˆ2 )θ (2M/r) 1/2 Γ θ θr(eˆ2 )θ (2M/r) 1/2 (eˆ2 )θ,r (2M/r) 1/2 [(1/r)(1/r) 1/r 2 ] 0. eˆ3 φ φr, φt Γ φ φr 1/r ( ueˆ2 )t ( u eˆ2 )r ( u eˆ2 )θ 0 ( u eˆ2 )φ (2M/r) 1/2 Γ φ φr (eˆ3 )φ (2M/r) 1/2 (eˆ3 )φ,r (2M/r) 1/2 [(1/r)[1/(r sin θ)] 1/(r 2 sin θ)] 0. u eˆα 0 20.26 θ φ θ φ Lorentz boost t (9.16)e (s)ˆt [(1 2M/r) 1/2, 0, 0, 0] r (9.72a)e (s)ˆr [0, (1 2M/r) 1/2, 0, 0] 7.23 (20.81) (e (s)ˆα ) β e γ α(eˆγ ) β e γ α Lorentz 20.5(d) Lorentz
Hartle Gravity December 6, 2008 15 α γ 1CA e γ α 0B@ (1 2M/r) 1/2 0 0 0 0 (1 2M/r) 1/2 0 0 0 0 1/r 0 0 0 0 1/(r sin θ) 0B@ 0B@ (1 2M/r) 1 (2M/r) 1/2 0 0 (1 2M/r) 1 (2M/r) 1/2 1 0 0 0 0 1/r 0 0 0 0 1/(r sin θ) (1 2M/r) 1/2 (1 2M/r) 1/2 (2M/r) 1/2 0 0 (1 2M/r) 1/2 (2M/r) 1/2 (1 2M/r) 1/2 0 0 0 0 1 0 0 0 0 1 1CA 1 1CA γ γv 0 0 γv γ 0 0 Lorentz boost 5.9 0 0 1 0 0 0 0 1 γ (1 2M/r) 1/2, v (2M/r) 1/2 r r γ u u obs v È 1 (1 2M/r)/e2 e (1 2M/r)dt/dτ γ (1 v 2 ) 1/2 e/è 1 2M/r 9.3 e 1 v È 2M/r)
e 16