hirameki_09.dvi

2009 July 31 1 2009 1 1 e-mail: mtakahas@auecc.aichi-edu.ac.jp

2 SF 2009 7 31

3 1 5 1.1....................... 5 1.2.................................. 6 1.3..................................... 7 1.4............................... 8 1.5.......................... 10 1.6............................ 11 1.7....................... 14 1.8.................................... 16 2 18 2.1... 18 2.2............................ 19 2.3.............................. 20 2.4.................... 22 2.5................................... 25 3 26 3.1.................................. 26 3.2................................... 27 3.3.............................. 29 3.4................................ 31 3.5..................................... 32 3.6... 34 4 36 4.1...................... 36 4.1.1.................................. 37 4.1.2... 39 4.1.3.............................. 40 4.2... 41 4.3... 43 4.3.1................................... 43 4.3.2................................ 44 4.3.3................................ 44 4.4................................. 45 4.5... 45

4 5 47 5.1.......... 47 5.2................................... 48 5.3........................... 52 6 54 6.1........................... 54 6.2...................... 55 6.3..................................... 55 6.4................................ 56 6.5.......................... 59 6.6..................................... 61 6.6.1............................... 61 6.6.2.............................. 62 6.6.3.............................. 63 7 65 7.1....................... 65 7.2...................... 66 7.3.............................. 67 7.4................................. 68 7.5 )... 68 7.6...................... 69

5 1 1.1 ( ) ( ) ( ) ( ) ( )

1 6 ( ) ( ) ( ) 1.2 ( ) ( ) ( ) ( ) ( ) ( ) 1.5

1 7 ( ) ( ) vˆr c ( ) (r ) r v ˆθ 0 v ˆφ 0 ( ) ZAMO (Zero Angular Momentum Observer) (Bardeen 1973) ZAMO FIDO (Fiducial Obserber) 1.3 ( ) ( ) ( ) ( ) ( ) ( )

1 8 A B ( ) ( M m ) r F g F g Mm/r 2 G ( ) [J(=kgm/t 2 )] [kg 2 /r 2 ] G (1798 ) G =6.67 10 11 Nm 2 /kg 2 (1.1) II 1.4 ( ) ( ) ( ) ( ) ( ) ( )

1 9 1.1: ( ) 2 ( ) z ( ) a/m ( ) ( ) ( ) ( ) Smarr (1973) 1.1( ) (stretched horizon ) (event horizon) r H ( ) (θ) 1 r θ φ ( ) 2 ( ) r H ( ) ( ) ( ) ( ) ( ) 1 r H = M + M 2 a 2 M a ( ) 2

1 10 Bardeen et al. (1972) ( ) 1.5 ( ) ( ) ( ) ( ) ( ) Q

1 11 1.2: ( ) ( ) ( ) Q A 1.6 BH

1 12 1.2 ( ) Q 1.4( ) ( ) ( ) ( ) ( ) 1.3: ( )

1 13 1.4: ( ) ( ) ( ) 1.4 C F ( ) ( ) 1.5 90 ( ) ( ) 1.5 ( ) 1.5

1 14 ( ) Q ( ) 1.7 ( ) ( ) ( )

1 15 ( ) ( ) ( ) 1.5: ( ) ( ) ( ) ( ) ( ) ( ) ( ) 3 km ( ) ( )

1 16 ( ) ( ) SF 1.8 Δg Δg =(2m/r 3 )Δr 3 r 3 (Δr) (r =2m) (Δg) H =(1/4m 2 )Δr (M = M ) g E Δg 10 9 g E (M =10 6 M ) Δg 10 3 g E 3 D 2 z μ dλ 2 = R μ ανβ kα k β z ν (1.2) ( )

1 17 ( ) (Δg) =g E r =(2mΔr/g E) 1/3 (Δr/4m 2 g E) 1/3 4 2 (Δr =2) 10 300 10 5 4 c G 1 m =1.5 10 3 g E =10 16 ( ) 1

= 18 2 ( ) SF 2.1 0 1 ( ) 2 3 z x z t y t = t 2 3 4 x t y t = t 1 2.1: 1 1 1 3 1 3+1=4 4

2 19 2.2 4 4 2.2 ( ) ( ) ( ) 2.2 ( ) ( ) t y x ( ) ( ) ( 4 ) ( ) 2.2: (x-y ) t (manifold

2 20 GPS 4 4 4 3+1 2.3 ( ) ( ) ( ) ( ) ( ) ( ) ()

2 21 2.3: = 0 = 0 2.4: ( ) ( ). ( ) 2.3 ( ) ( ) ( ) ( )

2 22 ( 2.4 ) ( ) 2.4 90 ( ) 90 2.4 ( ) ( )

2 23 1 ( ) 2.5 r t τ r t C G 1 c = G =1 ( ) (c = G =1 ) Time / M 70 60 50 40 30 20 10 t : Schwarzschild coordinate time τ : proper time 0 0 2 4 6 8 10 r / M 2.5: (r =9M ) r ( t τ) M kg ( ) c =2.998 10 8 m/s G =6.67259 10 11 N m 2 /kg 2 GM/c 2 GM/c 3 c = G =1 M 2M ISCO 6M 2 r =9M τ =30M M km M GM/c 2 1 2 ISCO(innermost stable circular orbot) ( )

2 24 M c G M GM/c 3 M c G ( ) M M =1.989 10 30 kg m km ( ) 30M ( ) 10 6 10 8 M m 1 =1.50 10 11 m 2.5 t ( ) ( τ: ) (r =2M) r ( ) ( ) ( ) 2.5 30M r =0 ( ) ( ) ( ) ( ) ( ) ( )

2 25 ( ) 2.5 ( )

26 3 ( ) 3.1 ( ) 1) ( ) r =0 ( ) 2) 3) 4)

3 27 ( ) 3.2

3 28 z r sinθ dφ dr r dθ dz θ r dy dx x y φ r sinθ dφ 3.1: x y z dx dy dz r θ φ dr rdθ r sin θdφ x, y, z t 1 A B A B (Δs) 2 =(Δx) 2 +(Δy) 2 +(Δz) 2 (3.1) A B Δt (Δs) 2 =(Δx) 2 +(Δy) 2 +(Δz) 2 c 2 (Δt) 2 (3.2) cδt cδt Δt c Δt ( 1 ()

3 29 ) r x θ z φ (Δs) 2 =(Δr) 2 + r 2 (Δθ) 2 + r 2 sin 2 θ(δφ) 2 c 2 (Δt) 2 (3.3) ( ) (Δθ) 2,(Δφ) 2 1 Δθ, Δφ θ, φ 3.1 ( ) 3.3 ( ) ( ) (Δs) 2 =(Δx) 2 +(Δy) 2 +(Δz) 2 c 2 (Δt) 2 (3.4) ( ) Δs Δx Δy Δz ( ) ds 2 = c 2 dt 2 + dx 2 + dy 2 + dz 2 (3.5)

3 30 ( Δx ) ( ) ds 2 = c 2 dt 2 + dr 2 + r 2 dθ 2 + r 2 sin 2 θdφ 2 (3.6) t x y z t r θ φ x x 0 x 1 x 2 x 3 ds 2 = g 00(cdx 0 ) 2 + g 11(dx 1 ) 2 + g 22(dx 2 ) 2 + g 33(dx 3 ) 2 (3.7) g 00 = 1 g 11 =1 g 22 =(x 1 ) 2 g 33 =(x 1 ) 2 sin 2 (x 2 ) g ( ) g ( ) μ( ) ν( ) g μν μ ν ds 2 = dtdφ ( dx 0 dx 3 ) dtdr ( dx 0 dx 1 ) ( ) μ ν 4 4=16 ds 2 = 3 3 ds 2 = g μνdx μ dx ν (3.8) μ=0 ν=0 g μν x μ ( g μν = g μν(x 0,x 1,x 2,x 3 ) ) μ ν ds 2 = g μνdx μ dx ν (3.9) ( ) g μν R μν, R g μν 2 R μν 1 gμνr = κtμν (3.10) 2 N N S km S S km S (x, y, z) S =(2, 1, 0) S x =2 S y = 1 S z =0 S a ( a = x, y, z ) A abc... a, b, c,... 2 R μν ( ), R ( )

3 31 ( ) μ ν 4 4 4=16 10 μ μ = t, x, y, z μ = t, r, θ, φ (principle of general relativity) T μν T μν =0 T μν 0 ds 2 = ( 1 2m r ) c 2 dt 2 + 1 ( 1 2m r ) dr 2 + r 2 dθ 2 + r 2 sin 2 θdφ 2 (3.11) m = GM/c 2 (M ) r =2m g tt =0 g rr = 3.4 τ σ

3 32 (i) (ii) r, θ, φ (FIDO: fiducial observers) ( ) 3.5 τ ( ) ( (Δs) 2 = 1 2m ) (cδt) 2 1 + ( r 1 2m )(Δr) 2 + r 2 (Δθ) 2 + r 2 sin 2 θ(δφ) 2 (3.12) r ( Delta ) Δr =Δθ =Δφ =0 (Δs) 2 = (Δτ) 2

3 33 3.11 (Δs) 2 = (1 2m/r)(Δt) 2 [(Δs) 2 ] 1 Δt = ( ) 1 2m 1/2 Δτ (3.13) r 1/(1 2m/r) 1/2 lapse finction 3 (r =3m) 1 1.7 r =2.9m r =2.8m r =2.7m r =2m r =2m r =2m r =2m (event horizon) : r (Δθ =Δφ =0 ) (Δτ =0) Δl Δr ( Δr = 1 2m ) 1/2 Δl (3.14) r r =3m 1m 0.58m Δt =Δr =0 Δτ =Δl =0 r c c

3 34 c a = 0 Light Rays a = 0 Wave Fronts Black Hole Black Hole 3.2: m 2m ( ) 3.6 ( ) r =0

3 35 2π Δr Δl (t, r, θ, φ) ( ) l 2π ( ) r Schwarzschild radial coordinate ( ) proper radial distance R (proper radial diistance) r (Schwarzschild radial coordinate) R 2M = r 2M 1 2M r dr [ ] R =2M + r(r 2M)+2M ln r/2m 1+ r/2m R 2M 3.3: (a) (b) ( ) (c) ( ) (a) ( ) (b)

36 4 ( ) 4.1 I/II ( ) ( ) ( ) ( )

4 37 ( ) ( ) 4.1.1 4.1: 4 1 1 H 4 2 He + 2e+ + ν (4.1) ( ) ( )

4 38 ( ) ( ) ( ) 10 7 K ( ) 10 10 K K 10 7 K ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

4 39 β β β 1000 10 10 10 39 10 15 m ( ) TNT ( ) ( ) 1000 ( ) 4.1.2 4.2 4.2: ( II ) 50 60 Fe ( ) Fe Fe E =Δm c 2 (Δm c 2 )/A A

4 40 ( ) 10 8 4.08 ev 1932 ( ) 7 3 Li + 1 1 H 2 4 2He (4.2) E = mc 2 0.2 17.4MeV 4.1.3 K ( )

4 41 4.3: 4.2 ( ) X X X Cyg X- 4.4: X ( ) ( ) B X 8.7 X

4 42 ( ) ( ) ( ) ( ) X Cyg X- X X 100keV X ( ) ( 100km ) ( X ) X X Cyg X- ( ) ( ) 1. ( ) 2. ( ) ( ) 3. ( ) 4. X 5. 6.

4 43 7. ( ) 4.3 4.3.1 (QSO) ( ) 1000 ( ) ( ) ( ) ( )

4 44 4.5: M87 5900 4.3.2 1943 ( ) 4.3.3

4 45 4.4 ( ) ( ) 10 15 g ( ) 6 10 27 g cm [ 146.6m 230.4m 230.4m (1/3)] 10 13 g 4.5 : ( )

4 46 : κ GM rc 2 λ GM r 3 c 2 AGN κ 1 λ λ

47 5 II 5.1 5.1: ( ) 545.5 m 10 m 3 /sec 335 10 6 W ( ) (http://www.kurobe-dam.com/whatis/index.html )

5 48 Q 1 ( ) (1) ( ) U = mgh m [kg/s] 1 g =9.8m/s 2 h (2) (3) A (1) 5326.3 10 6 W, (2) 6.3%, (3) 0.59 10 13 %. 1 5.2 II r>r e R e m r ( V (r) = GM ) em x 2 dx = GM em (5.1) r M e F = m GM e R 2 e = mg (5.2) R e = 5371 km M e =5.97 10 24 kg g =9.8 m/sec 2 g R e M e GM e = gre 2 V (r) = mgr2 e r (5.3) E = 1 2 mv2 mgr2 e r (5.4) 1 0.59 10 13 % 10 1 % 10 8 10 10 %

5 49 (5.4) ( ) 2 v(r) = E + mgr2 e (5.5) m r E (r ) E 0 E <0 v g moon (1/6)g earth R moon (1/3.6)R earth c (Schwarzschild radius) (M gal =2 10 11 M ) E <0 L = r p p = mv (r, φ) L = L = mr(r φ) =constant v 2 =(ṙ) 2 +(r φ) 2 1 2 m(ṙ2 + r 2 φ2 ) GMm = E = constant (5.6) r 1 2 mṙ2 + L2 2mr 2 GMm = E (5.7) r φ V eff (r) L2 2mr 2 GMm r (5.8)

5 50 v r ṙ (5.7) E = 1 2 v2 r + V eff (5.9) (5.8) r r 0 r r V 0 r 5.2: r E>0

5 51 A Black Hole 0.9 V/m Black Hole 0.9 V/m B 0.8 0.7 8 6 2 x/m 2 6 8 8 6 2 2 6 y/m 8 0.8 0.7 8 6 2 x/m 2 6 8 8 6 2 2 6 y/m 8 5.3: ( ) ( :L =0) ( :L 0) ( ) (Black Hole) (Event Horizon) (Schwarzschild) φ r [( d 1 2M ) ] ṫ =0, (5.10) dλ r d [ r 2 sin 2 θ dλ φ ] =0. (5.11)

5 52 ṫ = dt/dλ φ = dφ/dλ λ τ/m [ ] E/m ( 1 2M ) ṫ = constant r (5.12) L/m r 2 sin 2 θ φ = constant (5.13) r m 2 ṙ 2 = E 2 V (r) (5.14) ( V (r) 1 2M r ) )(m 2 + L2 r 2 (5.15) 5.3 ( ) ( ) ( ) (

5 53 ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) Ω F Ω F > ω H BH ω H Ω F < ω H BH 5.4: ( ) ( ) ( ) ( ) ( )

54 6 6.1 X ( ) S

6 55 6.1: ( ) 6.2 X 6.3

6 56 6.2: X (Cygnus X-1) η ( ) 6000 X 1967 8.7 ( 26km) 20% ( ) ( B ) 500 (http://en.wikipedia.org/wiki/cygnus X-1 ) (?) 6.4 ( )

6 57 6.3: 6.4: ( ) ( ) ( ) ( ) ( ) ( )

6 58 ( ) ( ) 6.3 6.4 6.3 ( ) 90 ( ) 6.4 ( ) ( ) 6.3 ( ) ( 6 4 2 0 BH y [M] -2-4 -6-8 -10-8 -6-4 -2 0 2 4 6 8 x [M] 6.5: ( )

6 59 ) 6.4 ( ) 6.5

6 60 6.6 X 6.6: (http://www.shokabo.co.jp/sp radio/spectrum/ radiow/window.htm ) 6.7: (Sgr A ) ( ) ( ) Q (VLA, VLBI)

6 61 6.8: NGC4258 6.6? II 6.6.1 ( ) ( ) ( ) ( ) ( ) ( )

6 62 6.9: ( ) 1 ( ) ( ) 6.6.2 ( ) 1 II

6 63 6.10: ( II ) 6.10 ( ) ( ) ( ) 6.6.3 ( ) ( ) ( ) ( ) c 6.11:

6 64 ( ) ( ) G.Walker, Nature, 378, 332 (1995) ( p.50 ) 6.12: (MCG-6-30-15) ( ) BH 6.13: Cygnus X-1

65 7 ( ) SF ( ) 7.1 ( ) ( ) ( )

7 66 ( ) 6.57 10 6 (M/M ) 16.5 10 6 (M/M ) 10 14 M 40 7.2 ( ) ( ) ( ) ( 1 1

7 67 ( ) Q 7.3 2 ( ) 7.1: 2 Schwarzschild black hole Kruskal coordinate system ( )

7 68 7.4 7.5 ) ( ) SF ( ) ( ) TV SF ( ) SF ( ) ( ) (

7 69 ) ( ) ( ) ( ) ( ) 7.6 ( ) 7.2: ( ) (http://ja.wikipedia.org/wiki/ :Worm3.jpg )

(?) ( ) ( ) ( ) ( ) 70

S II II F Gravitation C.W.Misner, K.S.Thorne, & J.A.Wheeler, W.H.Freeman and Company: New York (1970) Black Holes, White Dwarfs, and Neutron Stars S.L.Shapiro & S.A.Teukolsky, JohnWiley &Sons: New York (1983) Black Holes: The Membrane Paradigm K.S.Thorne, R.H.Price, & D.A.Macdonald, Yale Univ. Press: New Haven and London (1986) ( ) (2007) 71