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3 6 6.1 N ( ) 2 x µ, x µ + dx µ (µ = 1, 2,, N) 2 ds ds 2 = g µν (x)dx µ dx ν g µν = g µν (x) : g µν = g νµ g µν / x λ = 0 (flat) g µν = δ ν µ δ ν µ g µν g µν : g µν g νλ = δ λ µ. A µ A µ = g µν A ν 3

4 A µ g λµ g λµ A µ. g λµ g µν A ν = δν λ A ν = A λ A λ = g λµ A µ (x, y) x = x ay, y = y ( a 0 ) (x, y ) ds 2 = dx 2 + dy 2 = (dx + ady ) 2 + dy 2 = dx 2 + (1 + a 2 )dy 2 + 2a dx dy x 1 = x, x 2 = y g 11 = 1, g 22 = 1 + a 2, g 12 = g 21 = a g µν / x λ = 0 3 (x, y, z) x = r sin θ cos φ, y = r sin θ sin φ, z = r cos θ, 0 r <, 0 θ π, 0 φ < 2π 3 (r, θ, φ) ds 2 = dx 2 + dy 2 + dz 2 = (sin θ cos φ dr + r cos θ cos φ dθ r sin θ sin φ dφ) 2 + (sin θ sin φ dr + r cos θ sin φ dθ + r sin θ cos φ dφ) 2 + (cos θ dr r sin θ dθ) 2 = dr 2 + r 2 dθ 2 + r 2 sin 2 θ dφ 2 x 1 = r, x 2 = θ, x 3 = φ g 11 = 1, g 22 = r 2, g 33 = r 2 sin 2 θ ( 0 ) g µν / x λ 0 4

5 (µ ν g µν ) 0 3 r (θ, φ) ds 2 = r 2 dθ 2 + r 2 sin 2 θ dφ 2 ( r ) (θ, φ) 2 ( 6.1) 6.1: dx µ dx µ = x µ x ν dxν µ = / x µ µ = xν x µ ν T µν λ = x µ x ν x ρ x σ 5 xκ T x λ ρσ κ

6 T µν λ 1 dx µ ds A µν, B ν C µ = A µν B ν ν C µ = A µν B ν = x µ x ν x ρ x σ Aρσ xλ x B ν λ = x µ x λ x ρ x σ Aρσ B λ = x µ x ρ δλ σa ρσ B λ = x µ x ρ Aρσ B σ = x µ x ρ Cρ C µ C µ, B ν A µν g µν : g µν = xρ x µ x σ x ν g ρσ. δ ν µ = xρ x µ x ν x σ δσ ρ δµ ν g µν 6.4 φ = φ(x) µ φ : µφ = xν x νφ. µ A µ = A µ (x) : ( ) µa ν = xρ x ν x µ ρ x σ Aσ = xρ x ν x µ x ρa σ + xρ 2 x ν σ x µ x ρ x σ Aσ. 6

7 2 2 : µ A ν = µ A ν + Γ ν λµa λ. µ ( ) Γ ν λµ µ A ν ( µ A ν ) = µa ν + Γ ν λµa λ. x ρ x ν ( ) x µ x ρa σ = xρ x ν σ x µ ρ x σ Aσ + Γ ν x λ λµ x σ Aσ xρ x ν ( ρ A σ + Γ σ λρa λ) = xρ x ν x µ x σ x µ x ρa σ + xρ 2 x ν σ x µ x ρ x σ Aσ + Γ ν λµ x σ Aσ σ, λ x ρ x ν x µ x λ Γλ σρa σ = xρ 2 x ν x µ x ρ x σ Aσ + Γ ν x λ λµ x σ Aσ A σ x λ x λ x σ Γ ν λµ = xρ x ν x µ x λ x σ / x κ Γλ σρ xρ x µ 2 x ν x ρ x σ. Γ ν κµ = x ν x λ x σ x κ x ρ x µ Γλ σρ xσ x κ x ρ x µ 2 x ν x σ x ρ : : µ φ = µ φ. : µ (AB) = ( µ A)B + A µ B. 7

8 A ν B ν A ν B ν µ (A ν B ν ) = µ (A ν B ν ). ( µ A ν + Γ ν λµa λ )B ν + A ν µ B ν. ( µ A ν )B ν + A ν µ B ν Γ ν λµa λ B ν + A ν µ B ν A ν µ B ν = 0. 2 ν λ A λ A λ µ B λ = µ B λ Γ ν λµb ν 2 C µν A µ, B ν C µν = A µ B ν : µ T νρ σ = µ T νρ σ + Γ ν λµt λρ σ + Γ ρ λµt νλ σ Γ λ σµt νρ λ λ g µν = 0 Γ λµν = 0 λ g µν Γ λµν ( )

9 6.7 λ g µν = λ g µν Γ ρ µλg ρν Γ ρ νλg µρ = λ g µν Γ νµλ Γ µνλ 0 : λ g µν = 0 λ g µν = Γ µνλ + Γ νµλ. φ 2 ν φ = ν φ µ ν φ = µ ν φ Γ λ νµ λ φ. µ, ν ( µ ν ν µ )φ = (Γ λ µν Γ λ νµ) λ φ. 0 λ φ Γ λµν = Γ λνµ 2 ( ) λ g µν + ν g λµ + µ g νλ = (Γ µνλ + Γ νµλ ) + (Γ λµν + Γ µλν ) + (Γ νλµ + Γ λνµ ) 2 Γ λµν Γ λµν = 1 2 ( λg µν + ν g λµ + µ g νλ ) ( ) 9

10 6.8 A ρ 2 µ, ν µ ν A ρ = µ ν A ρ Γ λ νµ λ A ρ + Γ ρ λµ ν A λ = µ ( ν A ρ + Γ ρ λνa λ ) + Γ ρ λµ( ν A λ + Γ λ σνa σ ) + ( µν ) = µ Γ ρ λνa λ + Γ ρ λµγ λ σνa σ + ( µν ) = ( µ Γ ρ σν + Γ ρ µλγ λ νσ)a σ + ( µν ). µ, ν ( µ ν ν µ )A ρ = R ρ σµνa σ, R ρ σµν = µ Γ ρ σν + Γ ρ µλγ λ νσ ( µν ) = µ Γ ρ σν ν Γ ρ σµ + Γ ρ µλγ λ νσ Γ ρ νλγ λ µσ R ρ σµν R ρσµν = g ρλ R λ σµν = g ρλ µ Γ λ σν + g ρλ Γ λ µτγ τ νσ (µν ) = µ Γ ρσν ( µ g ρλ )Γ λ σν + Γ ρµτ Γ τ νσ (µν ) = 1 2 µ( ρ g σν + ν g ρσ + σ g νρ ) (Γ ρλµ + Γ λρµ )Γ λ σν + Γ ρµλ Γ λ νσ (µν ) = 1 2 µ ρ g σν µ σ g ρν Γ λρµ Γ λ σν (µν ). : R ρσµν = R ρσνµ = R σρµν = R µνρσ, R ρσµν + R ρνσµ + R ρµνσ = 0. N R ρσµν ρ, σ µ, ν 10

11 (ρ, σ) (µ, ν) N(N 1)/2 N(N 1)(N 2 N + 2)/8 N = 2 1, N = 3 6, N = 4 21 N = R µν = R ρ µρν ( ) R µν = g ρσ R ρµσν = g σρ R σνρµ = R νµ. R µν = ρ Γ ρ µν ν Γ µ + Γ ρ Γ ρ µν Γ ρ σµγ σ ρν, Γ µ = Γ ν νµ Γ µ ν Γ µ = µ Γ ν R = g µν R µν = R µ µ (* ) R µν = R ρ µνρ 6.10 λ R ρ σµν = λ µ Γ ρ σν λ ν Γ ρ σµ + ( Γ ) λ R ρ σµν + ν R ρ σλµ + µ R ρ σνλ Γ 0 : λ R ρ σµν + ν R ρ σλµ + µ R ρ σνλ = 0. 11

12 δ ρ µ λ R σν ν R σλ + µ R µ σνλ = 0 g σν λ R 2 ν Rλ ν = 0 ν (Rλ ν 1 ) 2 δν λr = 0. G µν = R µν 1 2 gµν R µ G µν = N dv = d N x = det g. g det g d N x = dx 1 dx 2 dx N. d N x = x det x d N x = det x x dn x dv N x µ dv = d N x. 3 3 x µ = (r, θ, φ) µ = det g = 1 r 2 r2 sin 2 θ = r 2 sin θ dv = d 3 x = r 2 sin θ dr dθ dφ 12

13 6.12 g µν g νσ = δ σ µ g νσ δg µν + g µν δg νσ = 0 g µρ g νσ δg µν + δ ρ ν δg νσ = 0 δg ρσ = g ρµ g σν δg µν. = ( ) det g = det g g µν. g µν = 1 det g = 1 g µν 2 g µν 2 det g g µν = 1 2 g µν. µ = µ g ρσ = 1 g ρσ µ g ρσ = 1 g ρσ (Γ ρσµ + Γ σρµ ) = Γ µ g ρσ 2 2 A µ µ A µ = ( µ A µ + Γ µ νµa ν ) = µ A µ + ( ν )A ν = µ ( A µ ). 2 F µν µ F µν = µ ( F µν ). 2 T µ λ µ T µ λ = µ( T µ λ ) 1 2 λ g µν T µν ( µ Γ ν = 1 µ ) ν = µ ν log µ Γ ν = ν Γ µ 13

14 6.13 dx I = ds = dλ g µ dx ν µν dλ dλ. λ 2 x µ (λ) δx µ (λ) : δi = 0 I dx δ g µ dx ν µν dλ dλ = 1 dx (g ρ dx σ ) 1/2 dx ρσ δ (g µ dx ν ) µν 2 dλ dλ dλ dλ = 1 dλ ( λ g µν δx λ dxµ dx ν 2 ds dλ dλ + 2g dx µ dδx ν ) µν dλ dλ = 1 2 dx µ dx ν dx µ dδx ν λg µν ds dλ δxλ + g µν ds dλ = 1 2 dx µ dx ν λg µν ds dλ δxλ d dx (g µ ) µλ δx λ + d ( ). dλ ds dλ δx λ 1 2 dx µ λg µν ds δi = dx ν dλ d dλ dx (g µ ) µλ = 1 2 dx µ dx ν λg µν ds dλ dx ν dx µ νg µλ dλ ds g d 2 x µ µλ dλds = 1 ( ) dx µ dx ν λ g µν ν g µλ µ g νλ 2 ds dλ g d 2 x µ µλ dλds dx µ dx ν = Γ λµν ds dλ g d 2 x µ µλ dλds. ds d dλ (g 2 x µ λµ dλds + Γ λµν 14 dx µ dλ dx ν ) δx λ ds

15 δx λ (λ) g λµ d 2 x µ dλds + Γ λµν dx µ dλ g ρλ dλ/ds d 2 x ρ ds 2 + Γρ µν dx µ ds dx ν ds = 0. dx ν ds = 0 d 2 x ρ /ds 2 = (θ, φ) ds 2 = r 2 dθ 2 + r 2 sin 2 θdφ 2 r θ = x 1, φ = x 2 g 11 = 1 g 22 = r 2, g 11 = r 2, g 22 = r 2 sin 2 θ, g 12 = g 21 = 0. 1 r 2 sin 2 θ, g 12 = g 21 = g 22 = r2 sin 2 θ θ = 2r 2 sin θ cos θ Γ 122 = 1 2 1g 22 = r 2 sin θ cos θ, Γ 212 = Γ 221 = 1 2 1g 22 = r 2 sin θ cos θ 0 Γ 1 22 = g 11 Γ 122 = sin θ cos θ, Γ 2 12 = Γ 2 21 = g 22 Γ 212 = cos θ sin θ 0 Γ µ = Γ ν νµ Γ 1 = Γ 2 21 = cos θ sin θ, 15 Γ 2 = 0.

16 R µν = ρ Γ ρ µν ν Γ µ + Γ ρ Γ ρ µν Γ ρ σµγ σ ρν R 11 = 1 Γ 1 (Γ 2 21) 2 = 1, R 22 = 1 Γ Γ 1 Γ Γ 1 22Γ 2 12 = sin 2 θ ( ) R = g µν R µν = g 11 R 11 + g 22 R 22 = 2 r 2. d 2 x ρ : ds + dx µ dx ν 2 Γρ µν ds ds = 0 θ = ( ) 2 dφ d 2 φ sin θ cos θ = 0, ds ds = 0 2 (θ = π/2) ( ) 2 ( ) G µν = R µν (1/2) g µν R [ ] 2 2 ɛ µν R ρµσν = R 1212 ɛ ρµ ɛ σν R µν = g ρσ R ρµσν = g ρσ R 1212 ɛ ρµ ɛ σν = R 1212 ( g 1 ) µν. 16

17 g 1 g 1 g 1 = det(g 1 )(g 1 ) 1 = 1 det g g. R µν = R 1212 det g g µν, R = g µν R µν = R 1212 det g gµν g µν = 2R 1212 det g ( ) G µν = 0 [ ] 0 3 (* ) g µν g µν = δ µ µ (x, y, z), a z = axy a (0, 0, 0) ( saddle point) 6.2: ( ) 17

18 z = axy ds 2 = dx 2 + dy 2 + dz 2 = dx 2 + dy 2 + (aydx + axdy) 2 = (1 + a 2 y 2 )dx 2 + (1 + a 2 x 2 )dy 2 + 2a 2 xy dxdy x = x 1, y = x 2 g 11 = 1 + a 2 y 2, g 22 = 1 + a 2 x 2, g 12 = g 21 = a 2 xy. g 11 = 1 + a2 x a 2 r 2, g 22 = 1 + a2 y a 2 r 2, g 12 = g 21 = a2 xy 1 + a 2 r 2. r = x 2 + y 2. Γ 1 12 = Γ 1 21 = a2 y 1 + a 2 r 2, Γ 2 12 = Γ 2 21 = a2 x 1 + a 2 r 2 0 (1, 1) R 11 = 1 Γ 2 21 (Γ 2 21) 2 = a2 (1 + a 2 y 2 ) (1 + a 2 r 2 ) 2 2 R µν g µν R µν = a 2 (1 + a 2 r 2 ) 2 g µν. R = g µν R µν = 2a 2 (1 + a 2 r 2 ) 2 a m, X i (i = 1, 2, 3), U = U(X) L = m 2 Ẋ 2 U. t ( ) x i ẋ i = dx i /dt L = m 2 g ijẋ i ẋ j U 18

19 g ij = g ij (x) L ẋ k = mg kjẋ j, d dt L x k = m 2 kg ij ẋ i ẋ j k U, L ẋ k = m ( ẋ i i g kj ẋ j + g kj ẍ j) d L : dt ẋ = L k x k m ( g kj ẍ j + Γ kij ẋ i ẋ j) = k U 3 x i = (r, θ, φ) i g 11 = 1, g 22 = r 2, g 33 = r 2 sin 2 θ ( 0 ). Γ 122 = Γ 212 = Γ 221 = r, Γ 133 = Γ 313 = Γ 331 = r sin 2 θ, Γ 233 = Γ 323 = Γ 332 = r 2 sin θ cos θ ( 0 ) k = 1, 2, 3 m ( r r θ 2 r sin 2 θ φ ) 2 = U r, ( m r 2 θ r 2 sin θ cos θ φ 2 + 2rṙ θ ) = U θ, ( m r 2 sin 2 θ φ + 2r sin 2 θ ṙ φ ) + 2r 2 sin θ cos θ θ φ = U φ 19

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