hirameki_09.dvi
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- あきひろ みやくぼ
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1 2009 July mtakahas@auecc.aichi-edu.ac.jp
2 2 SF
3
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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
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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
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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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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 =
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34 2 2 h = h/2π 3 V (x) E 4 2 1 ψ = sin kxk = 2π/λ λ = h/p p = h/λ = kh/2π = k h 5 2 ψ = e ax2 ガウス 型 関 数 1.2 1 関 数 値 0.8 0.6 0.4 0.2 0 15 10 5 0 5 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
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基礎から学ぶトラヒック理論 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. http://www.morikita.co.jp/books/mid/085221 このサンプルページの内容は, 初版 1 刷発行時のものです. i +α 3 1 2 4 5 1 2 ii 3 4 5 6 7 8 9 9.3 2014 6 iii 1 1 2 5 2.1 5 2.2 7
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( ) 200133 3 3 3 3, 7 347 57 10 i ii iii -1- -2- -3- -4- 90011001700mm -5- 4.2 1991 73.5 44.4 7.4 10.5 10.5 7.4 W 3 H 2.25 H 2.25 7.4 51.8 140.6 88.8 268.8m 5,037.9m 2 2mm 16cm916cm 10.5 W 3 H 2.25 62.8
atomic line spectrum emission line absorption line atom proton neutron nuclei electron Z atomic number A mass number neutral atom ion energy
1 22 22.1 atomic line spectrum emission line absorption line atom proton neutronnuclei electron Z atomic number A mass number neutral atom ion energy level ground stateexcited state ionized state 22.2
3-2 PET ( : CYRIC ) ( 0 ) (3-1 ) PET PET [min] 11 C 13 N 15 O 18 F 68 Ga [MeV] [mm] [MeV]
![3-2 PET ( : CYRIC ) ( 0 ) (3-1 ) PET PET [min] 11 C 13 N 15 O 18 F 68 Ga [MeV] [mm] [MeV]](/thumbs/91/106519999.jpg "3-2 PET ( : CYRIC ) ( 0 ) (3-1 ) PET PET [min] 11 C 13 N 15 O 18 F 68 Ga [MeV] [mm] [MeV]")
3 PET 3-1 PET 3-1-1 PET PET 1-1 X CT MRI(Magnetic Resonance Imaging) X CT MRI PET 3-1 PET [1] H1 D2 11 C-doxepin 11 C-raclopride PET H1 D2 3-2 PET 0 0 H1 D2 3-1 PET 3-2 PET ( : CYRIC ) ( 0 ) 3-1-2 (3-1
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
![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](/thumbs/85/92098187.jpg "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")
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)
1 3 3 3 10 16 24 26 26 41 43 43 43 48 53 61 62 70 () 73 75
001Y219E 1 3 3 3 10 16 24 26 26 41 43 43 43 48 53 61 62 70 () 73 75 90 4 1 3 2 3 3 3 1 2 3 3 4 4 1 2 1 2 90 1983 83 82.6ha 1,800 3 1,036 83 86 1 1-1 1986 2 (1993 ) 3(1997 ) 4(2001 ) 93 97 01 93 97 01 1