hirameki_09.dvi
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- あきひろ みやくぼ
- 9 years ago
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1 2009 July [email protected]
2 2 SF
3
4 )
5 ( ) ( ) ( ) ( ) ( )
6 1 6 ( ) ( ) ( ) 1.2 ( ) ( ) ( ) ( ) ( ) ( ) 1.5
7 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 ( ) ( ) ( ) ( ) ( ) ( )
8 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 = Nm 2 /kg 2 (1.1) II 1.4 ( ) ( ) ( ) ( ) ( ) ( )
9 : ( ) 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
10 1 10 Bardeen et al. (1972) ( ) 1.5 ( ) ( ) ( ) ( ) ( ) Q
11 : ( ) ( ) ( ) Q A 1.6 BH
12 ( ) Q 1.4( ) ( ) ( ) ( ) ( ) 1.3: ( )
13 : ( ) ( ) ( ) 1.4 C F ( ) ( ) ( ) ( ) 1.5 ( ) 1.5
14 1 14 ( ) Q ( ) 1.7 ( ) ( ) ( )
15 1 15 ( ) ( ) ( ) 1.5: ( ) ( ) ( ) ( ) ( ) ( ) ( ) 3 km ( ) ( )
16 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) ( )
17 1 17 ( ) (Δg) =g E r =(2mΔr/g E) 1/3 (Δr/4m 2 g E) 1/3 4 2 (Δr =2) c G 1 m = g E =10 16 ( ) 1
18 = 18 2 ( ) SF ( ) 2 3 z x z t y t = t x t y t = t 1 2.1: =4 4
19 ( ) ( ) ( ) 2.2 ( ) ( ) t y x ( ) ( ) ( 4 ) ( ) 2.2: (x-y ) t (manifold
20 2 20 GPS ( ) ( ) ( ) ( ) ( ) ( ) ()
21 : = 0 = 0 2.4: ( ) ( ). ( ) 2.3 ( ) ( ) ( ) ( )
22 2 22 ( 2.4 ) ( ) ( ) ( ) ( )
23 ( ) 2.5 r t τ r t C G 1 c = G =1 ( ) (c = G =1 ) Time / M t : Schwarzschild coordinate time τ : proper time r / M 2.5: (r =9M ) r ( t τ) M kg ( ) c = m/s G = 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 ISCO(innermost stable circular orbot) ( )
24 2 24 M c G M GM/c 3 M c G ( ) M M = kg m km ( ) 30M ( ) M m 1 = m 2.5 t ( ) ( τ: ) (r =2M) r ( ) ( ) ( ) M r =0 ( ) ( ) ( ) ( ) ( ) ( )
25 2 25 ( ) 2.5 ( )
26 26 3 ( ) 3.1 ( ) 1) ( ) r =0 ( ) 2) 3) 4)
27 3 27 ( ) 3.2
28 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 ()
29 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)
30 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 ( )
31 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 ( 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
32 3 32 (i) (ii) r, θ, φ (FIDO: fiducial observers) ( ) 3.5 τ ( ) ( (Δs) 2 = 1 2m ) (cδt) ( 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
33 (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) 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
34 3 34 c a = 0 Light Rays a = 0 Wave Fronts Black Hole Black Hole 3.2: m 2m ( ) 3.6 ( ) r =0
35 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 36 4 ( ) 4.1 I/II ( ) ( ) ( ) ( )
37 4 37 ( ) ( ) : H 4 2 He + 2e+ + ν (4.1) ( ) ( )
38 4 38 ( ) ( ) ( ) 10 7 K ( ) K K 10 7 K ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
39 4 39 β β β m ( ) TNT ( ) ( ) 1000 ( ) : ( II ) Fe ( ) Fe Fe E =Δm c 2 (Δm c 2 )/A A
40 4 40 ( ) ev 1932 ( ) 7 3 Li H 2 4 2He (4.2) E = mc MeV K ( )
41 : 4.2 ( ) X X X Cyg X- 4.4: X ( ) ( ) B X 8.7 X
42 4 42 ( ) ( ) ( ) ( ) X Cyg X- X X 100keV X ( ) ( 100km ) ( X ) X X Cyg X- ( ) ( ) 1. ( ) 2. ( ) ( ) 3. ( ) 4. X 5. 6.
43 ( ) (QSO) ( ) 1000 ( ) ( ) ( ) ( )
44 : M ( ) 4.3.3
45 ( ) ( ) g ( ) g cm [ 146.6m 230.4m 230.4m (1/3)] g 4.5 : ( )
46 4 46 : κ GM rc 2 λ GM r 3 c 2 AGN κ 1 λ λ
47 47 5 II : ( ) m 10 m 3 /sec W ( ) ( )
48 5 48 Q 1 ( ) (1) ( ) U = mgh m [kg/s] 1 g =9.8m/s 2 h (2) (3) A (1) W, (2) 6.3%, (3) % 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 = 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) % 10 1 % %
49 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 = M ) E <0 L = r p p = mv (r, φ) L = L = mr(r φ) =constant v 2 =(ṙ) 2 +(r φ) 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)
50 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
51 5 51 A Black Hole 0.9 V/m Black Hole 0.9 V/m B x/m y/m x/m 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)
52 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 ( ) ( ) ( ) (
53 5 53 ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) Ω F Ω F > ω H BH ω H Ω F < ω H BH 5.4: ( ) ( ) ( ) ( ) ( )
54 X ( ) S
55 : ( ) 6.2 X 6.3
56 : X (Cygnus X-1) η ( ) 6000 X ( 26km) 20% ( ) ( B ) 500 ( X-1 ) (?) 6.4 ( )
57 : 6.4: ( ) ( ) ( ) ( ) ( ) ( )
58 6 58 ( ) ( ) ( ) 90 ( ) 6.4 ( ) ( ) 6.3 ( ) ( BH y [M] x [M] 6.5: ( )
59 6 59 ) 6.4 ( ) 6.5
60 X 6.6: ( radio/spectrum/ radiow/window.htm ) 6.7: (Sgr A ) ( ) ( ) Q (VLA, VLBI)
61 : NGC ? II ( ) ( ) ( ) ( ) ( ) ( )
62 : ( ) 1 ( ) ( ) ( ) 1 II
63 : ( II ) 6.10 ( ) ( ) ( ) ( ) ( ) ( ) ( ) c 6.11:
64 6 64 ( ) ( ) G.Walker, Nature, 378, 332 (1995) ( p.50 ) 6.12: (MCG ) ( ) BH 6.13: Cygnus X-1
65 65 7 ( ) SF ( ) 7.1 ( ) ( ) ( )
66 7 66 ( ) (M/M ) (M/M ) M ( ) ( ) ( ) ( 1 1
67 7 67 ( ) Q ( ) 7.1: 2 Schwarzschild black hole Kruskal coordinate system ( )
68 ) ( ) SF ( ) ( ) TV SF ( ) SF ( ) ( ) (
69 7 69 ) ( ) ( ) ( ) ( ) 7.6 ( ) 7.2: ( ) ( :Worm3.jpg )
70 (?) ( ) ( ) ( ) ( ) 70
71 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
72
24.15章.微分方程式
m d y dt = F m d y = mg dt V y = dy dt d y dt = d dy dt dt = dv y dt dv y dt = g dv y dt = g dt dt dv y = g dt V y ( t) = gt + C V y ( ) = V y ( ) = C = V y t ( ) = gt V y ( t) = dy dt = gt dy = g t dt
ma22-9 u ( v w) = u v w sin θê = v w sin θ u cos φ = = 2.3 ( a b) ( c d) = ( a c)( b d) ( a d)( b c) ( a b) ( c d) = (a 2 b 3 a 3 b 2 )(c 2 d 3 c 3 d
A 2. x F (t) =f sin ωt x(0) = ẋ(0) = 0 ω θ sin θ θ 3! θ3 v = f mω cos ωt x = f mω (t sin ωt) ω t 0 = f ( cos ωt) mω x ma2-2 t ω x f (t mω ω (ωt ) 6 (ωt)3 = f 6m ωt3 2.2 u ( v w) = v ( w u) = w ( u v) ma22-9
4 2 4.1: =, >, < π dθ = dφ = 0 3 4 K = 1/R 2 rdr + udu = 0 dr 2 + du 2 = dr 2 + r2 1 R 2 r 2 dr2 = 1 r 2 /R 2 = 1 1 Kr 2 (4.3) u iu,r ir K = 1/R 2 r R
1 4 4.1 1922 1929 1947 1965 2.726 K WMAP 2003 1. > 100Mpc 2. 10 5 3. 1. : v = ȧ(t) = Ha [ ] dr 2. : ds 2 = c 2 dt 2 a(t) 2 2 1 kr 2 + r2 (dθ 2 + sin 2 θdφ 2 ) a(t) H k = +1 k *1) k = 0 k = 1 dl 2 = dx
1 180m g 10m/s 2 2 6 1 3 v 0 (t=0) z max t max t z = z max 1 2 g(t t max) 2 (6) 1.3 2 3 3 r = (x, y, z) e x, e y, e z r = xe x + ye y + ze z. (7) v =
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5 36 5................................................... 36 5................................................... 36 5.3..............................
9 8 3............................................. 3.......................................... 4.3............................................ 4 5 3 6 3..................................................
168 13 Maxwell ( H ds = C S rot H = j + D j + D ) ds (13.5) (13.6) Maxwell Ampère-Maxwell (3) Gauss S B 0 B ds = 0 (13.7) S div B = 0 (13.8) (4) Farad
13 Maxwell Maxwell Ampère Maxwell 13.1 Maxwell Maxwell E D H B ε 0 µ 0 (1) Gauss D = ε 0 E (13.1) B = µ 0 H. (13.2) S D = εe S S D ds = ρ(r)dr (13.3) S V div D = ρ (13.4) ρ S V Coulomb (2) Ampère C H =
7 9 7..................................... 9 7................................ 3 7.3...................................... 3 A A. ω ν = ω/π E = hω. E
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2012 10 13 1,,,.,,.,.,,. 2?.,,. 1,, 1. (θ, φ), θ, φ (0, π),, (0, 2π). 1 0.,,., m Euclid m m. 2.., M., M R 2 ψ. ψ,, R 2 M.,, (x 1 (),, x m ()) R m. 2 M, R f. M (x 1,, x m ), f (x 1,, x m ) f(x 1,, x m ).
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I 1 1 1.1 1. 3 m = 3 1 7 µm. cm = 1 4 km 3. 1 m = 1 1 5 cm 4. 5 cm 3 = 5 1 15 km 3 5. 1 = 36 6. 1 = 8.64 1 4 7. 1 = 3.15 1 7 1 =3 1 7 1 3 π 1. 1. 1 m + 1 cm = 1.1 m. 1 hr + 64 sec = 1 4 sec 3. 3. 1 5 kg
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1.1 ft t 2 ft = t 2 ft+ t = t+ t 2 1.1 d t 2 t + t 2 t 2 = lim t 0 t = lim t 0 = lim t 0 t 2 + 2t t + t 2 t 2 t + t 2 t 2t t + t 2 t 2t + t = lim t 0
A c 2008 by Kuniaki Nakamitsu 1 1.1 t 2 sin t, cos t t ft t t vt t xt t + t xt + t xt + t xt t vt = xt + t xt t t t vt xt + t xt vt = lim t 0 t lim t 0 t 0 vt = dxt ft dft dft ft + t ft = lim t 0 t 1.1
V(x) m e V 0 cos x π x π V(x) = x < π, x > π V 0 (i) x = 0 (V(x) V 0 (1 x 2 /2)) n n d 2 f dξ 2ξ d f 2 dξ + 2n f = 0 H n (ξ) (ii) H
199 1 1 199 1 1. Vx) m e V cos x π x π Vx) = x < π, x > π V i) x = Vx) V 1 x /)) n n d f dξ ξ d f dξ + n f = H n ξ) ii) H n ξ) = 1) n expξ ) dn dξ n exp ξ )) H n ξ)h m ξ) exp ξ )dξ = π n n!δ n,m x = Vx)
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A A 506J A15-6 A15-8 A15-8 A15-11 A15-11 A15-14 A15-19 A15-20 A15-24 A15-24 A15-26 A15-27 A15-28 A15-30 A15-32 A15-35 A15-35 A15-38 A15-38 A15-39 A15-40 A15-43 A15-43 A15-47 A15-47 A15-47 A15-47 A15-49
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γ 5 J, M α J, M α = c JM JM J, M c JM e ipr p / M p = 0 M J(J + 1) / Λ p / M J(J + 1) / Λ ~ 1 / m π m π ~ 138 MeV J P,I = 0,1 π 1, π, π 3 ( ) ( π +, π 0, π ) ( ), π 0 = π 3 π ± = m 1 π1 ± iπ ( ) π ±,
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64 3 g=9.85 m/s 2 g=9.791 m/s 2 36, km ( ) 1 () 2 () m/s : : a) b) kg/m kg/m k
63 3 Section 3.1 g 3.1 3.1: : 64 3 g=9.85 m/s 2 g=9.791 m/s 2 36, km ( ) 1 () 2 () 3 9.8 m/s 2 3.2 3.2: : a) b) 5 15 4 1 1. 1 3 14. 1 3 kg/m 3 2 3.3 1 3 5.8 1 3 kg/m 3 3 2.65 1 3 kg/m 3 4 6 m 3.1. 65 5
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15 8 1 15 7 26 1. 2. 15 7 27 15 7 28 1 2 7:13 16:56 0:13 3km 45 346 108 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3. 3.1 26 7 10 1 20cm 2 1 2 45 1/15 3 4 5,6 3 4 3 5 6 ( ) 7,8 8 7 8 2 55 9 10 9 10
2 g g = GM R 2 = 980 cm s ;1 M m potential energy E r E = ; GMm r (1.4) potential = E m = ;GM r (1.5) r F E F = ; de dr (1.6) g g = ; d dr (1.7) g g g
1 1 (gravitation) 1.1 m F a ma = F (1.1) F a m F 1.1 m F a (1.1) m a F m F a m a F F a m 0 0 1.2 (universal gravitation) (potential) M m gravitational force F r F = ; GMm r 2 (1.2) G = 6:67 10 ;8 dyn cm
m d2 x = kx αẋ α > 0 (3.5 dt2 ( de dt = d dt ( 1 2 mẋ kx2 = mẍẋ + kxẋ = (mẍ + kxẋ = αẋẋ = αẋ 2 < 0 (3.6 Joule Joule 1843 Joule ( A B (> A ( 3-2
3 3.1 ( 1 m d2 x(t dt 2 = kx(t k = (3.1 d 2 x dt 2 = ω2 x, ω = x(t = 0, ẋ(0 = v 0 k m (3.2 x = v 0 ω sin ωt (ẋ = v 0 cos ωt (3.3 E = 1 2 mẋ2 + 1 2 kx2 = 1 2 mv2 0 cos 2 ωt + 1 2 k v2 0 ω 2 sin2 ωt = 1
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10 log 10 W W 10 L W = 10 log 10 W 10 12 10 log 10 I I 0 I 0 =10 12 I = P2 ρc = ρcv2 L p = 10 log 10 p 2 p 0 2 = 20 log 10 p p = 20 log p 10 0 2 10 5 L 3 = 10 log 10 10 L 1 /10 +10 L 2 ( /10 ) L 1 =10
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13 13.1 O r F R = m d 2 r dt 2 m r m = F = m r M M d2 R dt 2 = m d 2 r dt 2 = F = F (13.1) F O L = r p = m r ṙ dl dt = m ṙ ṙ + m r r = r (m r ) = r F N. (13.2) N N = R F 13.2 O ˆn ω L O r u u = ω r 1 1:
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128 3 II S 1, S 2 Φ 1, Φ 2 Φ 1 = { B( r) n( r)}ds S 1 Φ 2 = { B( r) n( r)}ds (3.3) S 2 S S 1 +S 2 { B( r) n( r)}ds = 0 (3.4) S 1, S 2 { B( r) n( r)}ds
127 3 II 3.1 3.1.1 Φ(t) ϕ em = dφ dt (3.1) B( r) Φ = { B( r) n( r)}ds (3.2) S S n( r) Φ 128 3 II S 1, S 2 Φ 1, Φ 2 Φ 1 = { B( r) n( r)}ds S 1 Φ 2 = { B( r) n( r)}ds (3.3) S 2 S S 1 +S 2 { B( r) n( r)}ds
. ev=,604k m 3 Debye ɛ 0 kt e λ D = n e n e Ze 4 ln Λ ν ei = 5.6π / ɛ 0 m/ e kt e /3 ν ei v e H + +e H ev Saha x x = 3/ πme kt g i g e n
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確率論と統計学の資料
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4 4. A p X A 1 X X A 1 A 4.3 X p X p X S(X) = E ((X p) ) X = X E(X) = E(X) p p 4.3p < p < 1 X X p f(i) = P (X = i) = p(1 p) i 1, i = 1,,... 1 + r + r
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d (K + U) = v [ma F(r)] = (2.4.4) t = t r(t ) = r t 1 r(t 1 ) = r 1 U(r 1 ) U(r ) = t1 t du t1 = t F(r(t)) dr(t) r1 = F dr (2.4.5) r F 2 F ( F) r A r
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2.4 ( ) U(r) ( ) ( ) U F(r) = x, U y, U = U(r) (2.4.1) z 2 1 K = mv 2 /2 dk = d ( ) 1 2 mv2 = mv dv = v (ma) (2.4.2) ( ) U(r(t)) r(t) r(t) + dr(t) du du = U(r(t) + dr(t)) U(r(t)) = U x = U(r(t)) dr(t)
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