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30 3 30.1.............. 3 30.2........................... 4 30.3...................... 5 30.4........................ 6 30.5.................................. 8 30.6............................... 9 30.7..................... 10 30.8...................... 11 2
30 ( ) 30.1 : S g = 1 d 4 x R 16πG G R g µν = det g R 2 S g = 1 d 4 x B, B = Γ ρ ρµγ µσ σ Γ λ µνγ µν λ 16πG ( ) Γ λµν = 1 2 ( λg µν + ν g λµ + µ g νλ ) g µν η µν = diag(1, 1, 1, 1) µν h µν : g µν = η µν + h µν. g µν h µν 2 h µν 3
B B = 1 4 λh µν λ h µν 1 4 µh µ h + 1 2 µh ν h µν 1 2 µh λ ν λ h µν h = h µ µ f µν = h µν 1 2 η µνh f µν f = f µ µ = h B h µν = f µν 1 2 η µνf B = 1 4 λf µν λ f µν 1 8 µf µ f 1 2 µf λ ν λ f µν = 1 + ( 1 ) S g = 1 16πG d 4 x ( 1 4 λf µν λ f µν 1 8 µf µ f 1 2 µf λ ν λ f µν 1 S g = d 4 x (2η µρ η νσ η µν η ρσ ) η αβ α f µν β f ρσ 128πG 1 d 4 x η ρσ µ f µρ ν f νσ + S I 32πG S I f µν 3 ) 30.2 µ f µν = 0 S f = 1 d 4 x η ρσ µ f µρ ν f νσ 32πG 4
S c : S g = S g + S f + S c 1 = d 4 x (2η µρ η νσ η µν η ρσ ) η αβ α f µν β f ρσ + S I + S c. 128πG S g S g = i d 4 xd 4 y f µν (x)k µν,ρσ (x, y)f ρσ (y) + S I + S c, 2 K µν,ρσ (x, y) = i 64πG (η µρη νσ + η µσ η νρ η µν η ρσ ) x δ 4 (x y) K µν,ρσ (x, y) ( ) µν,ρσ (x, y) d 4 y K µν,ρσ (x, y) ρσ,αβ (y, z) = δµν αβ δ 4 (x z), δµν ρσ = 1 ( δ ρ 2 µ δν σ + δµδν) σ ρ δµν ρσ : A µν = A νµ δ ρσ µνa ρσ = A µν, δµν ρσ = δµν σρ = δνµ ρσ = δ ρσ µν iɛ µν,ρσ (x, y) = 16πG ( η µρ η νσ +η µσ η νρ η µν η ρσ) d 4 k i e ik (x y) (2π) 4 k 2 + iɛ 30.3 φ(x) m S φ = d 4 x ( ) 1 2 gµν µ φ ν φ m2 2 φ2 5
ɛ µνρσ 4 det g = ɛ µνρσ (η µ0 + h µ0 )(η ν1 + h ν1 )(η ρ2 + h ρ2 )(η σ3 + h σ3 ) = 1 h 00 + h 11 + h 22 + h 33 + ( 2 ) = 1 h + ( 2 ) = det g = 1 + 1 2 h + ( 2 ) g µν = η µν h µν + ( 2 ) ( ) 1 S φ = d 4 x 2 ηµν µ φ ν φ m2 2 φ2 + S gφ, S gφ = d 4 x ( 12 hµν µ φ ν φ + 14 ) ηµν h µ φ ν φ m2 4 hφ2 + (4 ) f µν ( S gφ = d 4 x f µν 1 ) 2 µφ ν φ + m2 4 η µνφ 2 + (4 ) 30.4 m 1, m 2 30.1 30.1: 6
S gφ M = 2 2 16πG (η µρ η νσ +η µσ η νρ η µν η ρσ i ) (p 1 p 3 ) 2 ( i 1 ) ( 2 p 1µp 3ν + m2 1 4 η µν i 1 ) 2 p 2ρp 4σ + m2 2 4 η ρσ = i16πg ( p1 p (p 1 p 3 ) 2 2 p 3 p 4 + p 1 p 4 p 2 p 3 p 1 p 3 p 2 p 4 ) + m 2 2 p 1 p 3 + m 2 1 p 2 p 4 2m 2 1m 2 2 2 2 p µ 1 = (E 1, p) µ, p µ 2 = (E 2, p) µ, p µ 3 = (E 1, p ) µ, p µ 4 = (E 2, p ) µ, E i = θ M = p 2 + m 2 i i4πg ( m 2 p 2 sin 2 (θ/2) 1 m 2 2 + 2(m 2 1+m 2 2) p 2 + 4(E 1 E 2 + p 2 ) p 2 cos 2 (θ/2) ) dσ dω = 1 64π 2 (E 1 +E 2 ) 2 M 2 = G2( m 2 1m 2 2 + 2(m 2 1+m 2 2) p 2 + 4(E 1 E 2 + p 2 ) p 2 cos 2 (θ/2) ) 2 4(E 1 +E 2 ) 2 p 4 sin 4 (θ/2) p m i dσ dω = G2 m 2 1m 2 2m 2 4 p 4 sin 4 (θ/2), m = m 1m 2 m 1 +m 2 0 : q 2 1q 2 2m 2 64π 2 p 4 sin 4 (θ/2) q 1 q 2 /(4π) Gm 1 m 2 ( ) 2 7
p m i dσ dω = 4G2 p 2 tan 4 (θ/2) G 2 30.5 X i (i = 0, 1, 2, 3) i b i µ = Xi x µ (vierbein) 4 (vielbein) : dτ 2 = g µν dx µ dx ν = η ij dx i dx j g µν = η ij b i µb j ν b i µ(x ) = X i X j x ν x µ bj ν(x) A i (x) Ã µ = b i µa i 8
η ij, η ij g µν, g µν 30.6 Φ = Φ(x) δφ = Φ Φ = 1 2 ɛ ijs ij Φ ( ɛ ji = ɛ ij, S ji = S ij ) ɛ ij S ij S ij = 0, S ij = γ ij, S ij = M ij. γ ij = 1 4 [ γi, γ j ], (M ij ) kl = η ik δ j l ηjk δ i l, γ i ( ) µ Φ δ µ Φ = µ δφ = 1 2 ɛ ijs ij µ Φ + 1 2 ( µɛ ij )S ij Φ µ ɛ ij : D µ Φ = µ Φ + 1 2 ω ijµs ij Φ, ω jiµ = ω ijµ. ω ijµ D µ Φ Φ : δd µ Φ = 1 2 ɛ ijs ij D µ Φ. : D µ φ = µ φ ( φ ), D µ ψ = µ ψ + 1 2 ω ijµγ ij ψ ( ψ ), D µ A i = µ A i + ω i jµa j D µ A i = µ A i ω j iµa j D µ η ij = D µ η ij = D µ δ i j = 0. ( A i ), ( A i ), 9
: D µ g ρσ = D µ (b i ρb iσ ) = 0 D µ b i ν = 0 D µ b i ν = µ b i ν + ω i jµb j ν Γ λ νµb i λ ω ijλ = b i µ b j ν Γ µνλ b j µ λ b iµ i, j : λ g µν = Γ µνλ + Γ νµλ ( ) SO(3, 1) ( 1956) ω ijµ A a µ (ij) a S ij igt a 30.7 ψ(x) : S ψ = d 4 x ( ) i 2 b i µ ψγ i D µ ψ + c.c. m ψψ. c.c. m D µ = µ + (1/2) ω ijµ γ ij b i µ = δ µ i 1 2 hµ i + (2 ), ω ijµ = 1 2 ( ih jµ + j h iµ ) + (2 ) S ψ = d 4 x ( i ψγ µ µ ψ m ψψ ) + S gψ, S gψ = + ( i d 4 x h 4 ψ λ λ ψ i 4 ψγ λ λ ψ m ) 2 ψψ d 4 x h ( µν i 4 ψγ ν µ ψ + i ) 4 ψγ µ ν ψ + (4 ) 10
{γ µ, γ νλ } + {γ ν, γ µλ } = 0 f µν S gψ = d 4 x f ( µν i 4 ψγ ν µ ψ + i 4 ψγ µ ν ψ i 8 η ψγ µν λ λ ψ + i 8 η µν λ ψγ λ ψ + m 2 η ψψ ) µν + (4 ) 30.8 m 1, m 2 30.2 30.2: M = 16πG (η µρ η νσ +η µσ η νρ η µν η ρσ i ) (p 1 p 3 ) 2 ( i 1 4 p 1µū 3 γ ν u 1 1 4 p 3µū 3 γ ν u 1 + m ) 1 4 η µνū 3 u 1 ( i 1 4 p 2ρū 4 γ σ u 2 1 4 p 4ρū 4 γ σ u 2 + m ) 2 4 η ρσū 4 u 2 = iπg (p 1 p 3 ) 2 ( (p 1 +p 3 ) (p 2 +p 4 ) ū 3 γ µ u 1 ū 4 γ µ u 2 + ū 3 (/p 2 +/p 4 )u 1 ū 4 (/p 1 +/p 3 )u 2 4m 1 m 2 ū 3 u 1 ū 4 u 2 ) 11
u i = u si (p i ), /p i = γ µ p iµ (/p m)u s (p) = 0 (p 1 + p 3 ) µ = (2m 1, 0) µ, (p 2 + p 4 ) µ = (2m 2, 0) µ, ū 3 γ µ u 1 = (2m 1, 0) µ δ s 3 s 1, ū 4 γ µ u 2 = (2m 2, 0) µ δ s 4 s 2, ū 3 u 1 = 2m 1 δ s 3 s 1, ū 4 u 2 = 2m 2 δ s 4 s 2 p, p, θ M = i4πgm2 1m 2 2 p 2 sin 2 (θ/2) δs 3 s 1 δ s 4 s 2 dσ dω = M 2 64π 2 (m 1 +m 2 ) 2 = G2 m 2 1m 2 2m 2 4 p 4 sin 4 (θ/2) δs 3 s 1 δ s 4 s 2 p µ 1 = p (1, 0, 0, 1) µ, p µ 2 = p (1, 0, 0, 1) µ, p µ 3 = p (1, sin θ, 0, cos θ) µ, p µ 4 = p (1, sin θ, 0, cos θ) µ u 1 = { 2 p (1, 0, 0, 0) T 2 p (0, 0, 0, i) T, u 2 = { 2 p (0, i, 0, 0) T 2 p (0, 0, 1, 0) T, u 3 = { 2 p (c, s, 0, 0) T 2 p (0, 0, is, ic) T, u 4 = { 2 p ( is, ic, 0, 0) T 2 p (0, 0, c, s) T, c = cos(θ/2), s = sin(θ/2). u s ( p) = γ 0 u s (p) 2 p (c, s, is, c) µ (s 1 = s 3 = +1) ū 3 γ µ u 1 = 2 p (c, s, is, c) µ (s 1 = s 3 = 1) 0 (s 1 s 3 ), 2 p (c, s, is, c) µ (s 2 = s 4 = +1) ū 4 γ µ u 2 = 2 p (c, s, is, c) µ (s 2 = s 4 = 1) 0 (s 2 s 4 ) 12
(p 1 +p 3 ) µ = 2 p (1, sc, 0, c 2 ) µ, (p 2 +p 4 ) µ = 2 p (1, sc, 0, c 2 ) µ i8πg p 2 1 + 3 cos2 (θ/2) sin 2 (s 1 = s 3 = s 2 = s 4 ) (θ/2) M = i8πg p 2 3 + cos2 (θ/2) tan 2 (s 1 = s 3 s 2 = s 4 ) (θ/2) 0 (s 1 s 3 s 2 s 4 ) G 2 p 2 ( 1 + 3 cos 2 ) 2 (θ/2) 4 dσ dω = M sin 2 (s 1 = s 3 = s 2 = s 4 ) (θ/2) 2 64π 2 (2 p ) = G 2 p 2 ( 3 + cos 2 ) 2 (θ/2) 2 4 tan 2 (s 1 = s 3 s 2 = s 4 ) (θ/2) 0 (s 1 s 3 s 2 s 4 ) 13
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