tomocci 18 7 5...,. :,,,, Lie,,,, Einstein, Newton. 1 M n C. s, M p. M f, p d ds f = dxµ p ds µ f p, X p = X µ µ p = dxµ ds µ p. µ, X µ.,. p,. T M p. M F (M), X(F (M)).. T M p e i = e µ i µ. a a = a i e i. p 1
2 T M p ( ) T M p., ( ) ( ). T M p θ i, θ i e j = δ i j, e i θ i = e i θ i = 1. dx µ. dx µ ν = δ µ ν, µ dx µ = µ dx µ = 1 3 g T M p T M p g(e i, e j ) = η ij = diag(η, ζ,, ζ). (η, ζ), (+,+), (+, ), (,+). θ i g = η ij θ i θ j g 1 = η ij e i e j (η ij = η ij ). g( µ, ν ) = η ij θµθ i ν j = g µν, g 1 (dx µ, dx ν ) = η ij e µ i eν j = g µν g µν g, ( ).. η ik η kj = δ i j, g µρ g ρν = δ µ ν. g = det(g µν ) = ηζ n 1 (det(θ i µ)) 2 = ηζ n 1 (det(e µ i )) 2 = det(g µν ) 1. A, B g(a, B) = η ij A i B j = g µν A µ B ν. α α = g 1 (α, ) α. α A α, A : α A = g(α, A) 2
4 x µ f(x), λ y µ = x µ + λx µ (x) ( λ 1) f(y) f(y) = f(x) + λxf(x). X = X µ µ. f(y) y=x+λx f(x) lim λ 0 λ = Xf(x),.,. y x. y e i (y) x e i (x)., e i (x) = e i (x) + λ X e i (x). X., e i x, e i (x), X e i = e j ω j i X ω i j.,. A = A i (y)e i (y) A (x) = A i (y)e i (x). ω i jµ = ω i j µ A (x) = A i (y)e i (x) = (A i + X µ µ A i )(e i + e j ω j iµx µ ) = A(x) + X µ ( µ A i + ω i jµa j )e i = A(x) + X A(x) Leibnitz X (A i e i ) = ( X A i )e i + A i X e i, f X f = Xf., fx+y = f X + Y. 3
µ = µ, X = X µ µ, X f = X µ µ f = X µ µ f. A µ A = ( µ A i + ω i jµa j )e j., ( µ A) i µ A i. θ i, θ i e j = δ i j µ θ i = ω i jµθ j, µ ν = Γ ρ µν ρ µ A = ( µ A ν + Γ ν µρa ρ ) ν. ( µ A) ν µ A ν. ω i jµ, Γ ρ µν e i = µ (e µ i µ). ω i jµ = θ i ρ( µ e ρ j + Γρ µνe ν j ) Γ ρ µν = e ρ i ( µθ i ν + ω i jµθ j ν) 4
5 Lie, 2 x µ, y µ = x µ + λx µ (x), ( λ 1) A(x) A(y),. y A(y) x, f(y) Ã(x)f(y(x)) = A(y)f(y) Ã(x).., 2 Lie Ã(x) = A(y) y=x+λx = xµ (y) A ν (y) y ν 1 L X A = lim λ 0 λ (. x µ (y) y ν y=x+λx x µ ) A ν (y) A µ (x) y=x+λx L X A = 1 lim λ 0 λ ((δµ ν λ ν X µ )(A ν + λx ρ ρ A ν ) A µ ) µ = (X ν ν A µ A ν ν X µ ) µ L A X = L X A, L A+B = L A + L B, L X (fa) = (Xf)A + fl X A L X f = Xf L fx Y = fl X Y (Y f)x., f X, Y Y µ µ (X ν ν f) = Y µ ( µ X ν ) ν f + Y µ X ν µ ν f, 2 2.., [X, Y ]f = X(Y f) Y (Xf) = (X ν ν Y µ Y ν ν X µ ) µ f. Lie : L X Y = [X, Y ] L X µ = ( µ X ν ) ν, L X dx µ = ( ν X µ )dx ν. Lie. L X α = (X ν ν α µ + α ν µ X ν )dx µ 5 µ
6 Lie Lie., 2 Lie. (x 1, x 2 ) θ (y 1, y 2 ),, Ã 1 (x) = A 1 (y(x)) cos θ + A 2 (y(x)) sin θ Ã 2 (x) = A 1 (y(x)) sin θ + A 2 (y(x)) cos θ (L X A) 1 = x 2 1 A 1 + x 1 2 A 1 + A 2 (L X A) 2 = x 2 1 A 2 + x 1 2 A 2 + A 1 ( X µ = ( x 2, x 1 ))., Ãi (x) = A i (y(x)),. Lie,., X = d/dt fx X Lie L X (fx) = (df/dt)x.,. ω i j,. 6
7,. X, Y [ X, Y ] = X µ Y ν [ µ, ν ] + [X, Y ] µ µ, [ µ, ν ] [ µ, ν ] X Y Y X [X,Y ]., f Z,. T (X, Y )f = ( X Y Y X [X, Y ])f R(X, Y )Z = ( X Y Y X [X,Y ] )Z. T (torsion tensor field), R (curvature tensor field) Riemann, Riemann. T (X, Y ) = X i Y j T (e i, e j ).. X, Y T ( µ, ν )f = (Γ ρ µν Γ ρ νµ) ρ f = T ρ µν ρ f = T i µνe i f T i µν = θ i ρt ρ µν ω i jµ Γ ρ µν T i µν = 2( [µ θ i ν] + ω i j[µθ j ν] ). [, ],, (, ).. R(X, Y )(fz) = fr(x, Y )Z, Z µ, e i. R( µ, ν ) σ = ρ 2( [µ Γ ρ ν]σ + Γ ρ [µ λγ λ ν]σ) = ρ R ρ σµν R( µ, ν )e j = e i 2( [µ ω i j ν] + ω i k[µ ω k j ν]) = e i R i jµν R ρ σµν R i jµν R i jµν = θ i ρe σ j R ρ σµν. 7
8, η ij, η ij i, j, k,, g µν, g µν µ, ν, ρ,.. g = g µ g = 0, (compatible). g = g µν dx µ dx ν µ Γ µ,ρν + Γ ν,ρν = ρ g µν., Γ ρ,µν = S ρ,µν + 1 2 T ρ,µν µ, ν S ρ,µν = Γ ρ,(µν) 1 2 T ρ,µν = Γ ρ,[µν], S µ,ρν + S ν,ρν = ρ g µν T (µ,ρ ν), S, Γ ρ,µν = Γ (0) ρ,µν T (µ,ν)ρ + 1 2 T ρ,µν. Γ (0) ρ,µν = 1 2 ( µg νρ + ν g µρ ρ g µν ) Γ (0)ρ µν Christoffel., g = η ij θ i θ j e µ i eν j ω ijµ + ω jiµ = 0 e µ i eν j T kµν = 2e µ [i eν j]( µ θ kν + ω klµ θ l ν) i, j, k. ω ijµ = e ρ [i eσ j]θ i µ ρ θ k σ + e ν [i µ θ j]ν e ν [i ν θ j]µ + e ν [it j]νµ + 1 2 θk µe ρ i eσ j T k,ρσ 8
9 2 P, Q C L L = Q P dt X 2 1/2. C t, X = X µ µ = (dx µ /dt) µ., X 2 = g(x, X) = g µν X µ X ν. L, C. L δx µ,. δl = X µδx µ X 2 1/2 Q P Q P dt sgn(x2 ) X 2 1/2 δx µ ( ) ( δ ν µ Xµ X ν dx ν X 2 dt + Γ (0)ν ρσx ρ X σ ) δx µ = g µν δx ν, X µ = g µν X ν sgn(x 2 ) = X 2 / X 2. ( ) ( ) δ ν µ Xµ X ν dx ν + Γ (0)ν ρσx ρ X σ = 0 X 2 dt. t X = (ds/dt)t, g(t, T ) = const. t s = s(t) : dt µ ds + Γ(0)µ ρσt ρ T σ = 0 s (, affine parameter)., X. X = X + λ X X, λ 1., X X = fx. f. X = (1 + λf)x, X. dx µ + Γ (0)µ ρσx ρ X σ g µτ T ρ,στ X ρ X σ = fx µ dt. X µ f, ( ) ( ) δ ν µ Xµ X ν dx ν + Γ (0)ν ρσx ρ X σ g µτ T X 2 ρ,στ X ρ X σ = 0 dt.,,. 9
10 Einstein. Minkowski, ζ = η. Einstein-Hilbert, S g = η d n x ηg g µν R µν 2κ S m = m dτ. R µν = R ρ µρν, dτ 2 = ηg µν dx µ dx ν > 0. S g δg = gg µν δg µν, δg µν = g µα g νβ δg αβ, δr µν = 2 [ρ δγ ρ µ]ν δs g = η κ d n x µ ( ηg g σ[µ δγ ν] νσ) η 2κ d n x ( ηg R µν 1 ) 2 g µνr δg µν R = g µν R µν. S m δdτ = 1ηv 2 µv ν dτδg µν, v µ = g µν dx ν /dτ δs m = ηm d n x dτ v µ (τ)v ν (τ)δ n (x z(τ))δg µν 2 z(τ). T µν = 2 δs m 1 = ηm dτ v ηg δg µν µ (τ)v ν (τ)δ n (x z(τ)) ηg(x). Einstein. R µν 1 2 g µνr = ηκt µν 10
11 Newton, ζ = η. ( ) ηg ab δ ab g 0a 0 g µν ηg 00 1 2 0,, Newton. η = g µν v µ v ν g 00 (v 0 ) 2 v 0 (ηg 00 ) 1/2. dv a dx 0 = 1 v 0 dv a dτ = 1 v 0 Γa µνv µ v ν Γ a 00v 0 Γ a 00 = F a F grad( 1 2 ηg 00).. ηg 00 = 1 + 2φ φ Newton. Einstein trace R µν = κ(t µν 1 n 2 g µνt ). R 00 = R µ 0µ0 a Γ a 00 divf T = g µν T µν g 00 T 00, T 00 1 n 2 g 00T. Gauss T 00 ηmδ n 1 (x) ( 1 1 ) n 2 g 00g 00 T 00 n 3 n 2 T 00 ηm n 3 n 2 δn 1 (x) divf = κm n 3 n 2 δn 1 (x). n 1, F r = n 3 κm Ω n 2 (n 2) r n 2 Ω n 2 n 2 { Ω n 2 = 2 n/2 π (n 2)/2 /(n 3)!! (n even) 2π (n 1)/2 /( n 1 )! 2 (n odd) n = 4 κ = 8πG (G Newton ). 11