Transcription

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3 i Newton Galilei , (tangent vector) (curvature) (Geodesics)

4 ii (Tetrad) Newton

5 Newton Newton f = m a (1.1.1) 3. Newton ( ) (inertial system) Galilei Galilei 2 Newton 1 Isaac Newton ( ) ( ) 2 Galileo Galilei ( ) ( )

6 2 1 2 S S S S x x. t = 0 t = 0 t = 0 2 S S x v S t ( ) x, y, z S x, y, z ( 1.1) 1.1: x = x vt, y = y, z = z, t = t (1.1.2) S S dx dt d 2 x dt 2 = dx dt v, dy dt = dy dt, dz dt = dz dt, (1.1.3) = d2 x dt, d 2 y 2 dt 2 = d2 y dt, d 2 z 2 dt 2 = d2 z dt 2 (1.1.4) S Newton m d2 x dt 2 = f (1.1.5) f S. S f f = f (1.1.4) (1.1.5) m d2 x dt 2 = f (1.1.6)

7 S (1.1.5) Galilei ( ) ( ) / ( 3 ( 1-2 ) 10 ( 0.5 %) ) 30km/s ( % ) Galilei ( ) Kepler 3 Kepler ( ) Kepler Newton Johannes Kepler

8 S S S S S 1 S x, y, z, t x, y, z, t 1 2. xy x y xz x z z = 0 z = 0 y = 0 y = 0 x, t y = κ(v)y, z = κ(v)z (1.2.1) κ x, y, z, t v y, z. v v κ. κ(v) v S S S x v y = κ( v)y, z = κ( v)z (1.2.2). (1.2.1), (1.2.2) (κ( v )) 2 = 1 κ( v ) = ±1 (1.2.3) v 0 y y, z z κ( v ) = 1 y = y, z = z (1.2.4)

9 t = t = 0 S O t O c S P ( (x, y, z)) t s 2 x 2 + y 2 + z 2 (ct) 2 = 0 (1.2.5) S P S (x, y, z ) P S t s 2 x 2 + y 2 + z 2 (ct ) 2 = 0 (1.2.6) x, y, z, t x, y, z, t 1 ( ) (1.2.5) (1.2.6) ( ) s 2 0 s 2 = α(v)s 2 (1.2.7) α(v) x, y, z, t v s 2 x 2 + y 2 + z 2 (ct) 2 = x 2 + y 2 + z 2 (ct ) 2 = s 2 (1.2.8) 4. (x, t) (x, t ) x 2 (ct) 2 = x 2 (ct ) 2 (1.2.9) x = ax + bt, t = fx + gt (1.2.10) a, b, f, g v (1.2.10) (1.2.9) x, t a 2 c 2 f 2 = 1 ab c 2 fg = 0 (1.2.11) g 2 b 2 /c 2 = 1

10 6 1 a = ± cosh θ, b = c sinh θ = ca tanh θ, ( ) (1.2.12) f = ± 1 c sinh θ, g = cosh θ = cf tanh θ θ tanh θ = b ca ( ) (1.2.13) θ S O S v x O x = y = z = 0 ax + bt = 0, y = 0, z = 0 (1.2.14) x = b a t (1.2.15) b a = v (1.2.16) tanh θ = v c β = v c (1.2.17) cosh θ = (1 tanh 2 θ) 1/2 = 1 1 β 2 sinh θ = β 1 β 2 (1.2.18) x = ± x vt 1 β 2, (1.2.19)

11 t = ± t (v/c2 )x 1 β 2 (1.2.20) v 0 x x, t t x = x vt, 1 β 2 t = t (v/c2 )x 1 β 2 (1.2.21) Lorentz 4 Poincaré 5 Einstein Einstein 2 2 Lorentz (1.2.21) x = x + vt 1 β 2, t = t + (v/c 2 )x 1 β 2 (1.2.22). (1.2.21) v v, (t, x, y, z) (t, x, y, z ). S S (1.2.21) Lorentz Lorentz. Lorentz Lorentz Maxwelll 6 Galilei Galilei Michelson 7 -Morley 8 Newton 4 Hendrik Antoon Lorentz ( ) Jules Henri Poincaré ( ) 6 James Clerk Maxwell ( ) ( ) 7 Albert Abraham Michelson ( ) Edward Morley ( )

12 Galilei Lorentz ( Eötvös (1885,1889, )) (, glücklichste Gedanke meines Lebens ) ( ) ) ( ), ( ) 3

13 , R n R n R n U U a ( ) ɛ [ n ] 1/2 x a (x µ y µ ) 2 < ɛ µ=1 x U U (open set) R n 1. R n 2. U 1, U 2, U k U 1 U 2 U k R n 3. ( ) {U λ } λ Λ λ Λ U λ R n

14 10 2 R n (topology) R n X O 3 O X (topology) X O (X, O) (topological space) 1. X O O 2. U 1, U 2, U k O U 1 U 2 U k O 3. {U λ } λ Λ, U λ O λ Λ U λ O X U O (U O) U X 2 (X, O) (Y, O ) f : X Y Y U f 1 (U) X ( : ɛ δ ) U, V f : U V (1 1)

15 (X, O) (Y, O ) f : X Y f (homeomorphism) 1. f : X Y 2. f : X Y f 1 : Y X Hausdorff X 2 x, y x U y V U V = X Hausdorff R n Hausdorff Hausdorff M M {U α } U α n R n ψ α 1. p M 1 U α M = α U α 2. ψ α : U α R n ψ α (U α ) R n ψ α U α ψ α (U α ) (homeomorphism) 3. U α U β ( : ) ψ β ψ 1 α : ψ α (U α U β ) ψ β (U α U β ) ψ α (U α U β )( R n ) ψ β (U α U β )( R n ) C ( ) {U α, ψ α } M n C U α ψ α (chart,local coordinates) {U α, ψ α } (atlas)

16 {U α, ψ α } {V λ, φ λ } U α V λ φ λ ψ 1 α : ψ α (U α V λ ) φ λ (U α V λ ) ψ α (U α V λ ) φ λ (U α V λ ) C 2 {U α, ψ α } {V λ, φ λ } M 2. M paracompact [ 1] n R n [ 2] 2 S 2 S 2 = {(x 1, x 2, x 3 ) R 3 (x 1 ) 2 + (x 2 ) 2 + (x 3 ) 2 = 1} S 2 O ± i (i = 1, 2, 3) O ± i = {(x 1, x 2, x 3 ) S 2 ± x i > 0} O ± i (i = 1, 2, 3), f + 1 (x 1, x 2, x 3 ) = (x 2, x 3 ) D = {(x, y) R 2 x 2 + y 2 < 1} f ± i (f ± j ) 1 C S 2 2 n M n M M p M M p M (p, p ) M M M, N

17 M ψ α : U α ψ α (U α ) M ψ β : U β ψ β (U β ) M M φ αβ : U αβ ψ αβ (U αβ ) R n+n U αβ = U α U β, ψ αβ (p, p ) = [ψ α (p), ψ β (p )] M M n + n M M M M (product manifold) [ 3] n S 1 n S 1 S 1 n n (n-torus ) m M f p M (U, ψ) f ψ 1 R m ψ(u) M f C M C F(M) M f, g F(M) a R f + g af f + g F(M), af F(M) fg fg F(M) C m M n N f : M N M p (U, ψ) f(p) N (V, φ) φ f ψ 1 C f p C f M C f

18 14 2 M N C f : M N C (1 1 ) C f (diffeomorphism) M N (diffeomorphic) 2? n 4 n ( ) n = 4 n ( ) ( (Exotic Differentiable Structures) S. K. Donaldson 1982, Seiberg-Witten 1994) n n 6 n (Exotic sphere, J. W. Milnor 1956) R( ) C( ) K V V K

19 V 2 a, b a, b V a + b ( ) (a) (b) (a + b) + c = a + (b + c) a + b = b + a (c) a V a + 0 = a V 0 0 (d) a V a + ( a) = 0 V a a 2. V a K λ a λ V λa (a) (b) (c) (d) (λµ)a = λ(µa) 1a = a λ(a + b) = λa + λb (λ + µ)a = λa + µa K V, K

20 V r a 1,, a r r λ 1,, λ r λ 1 a λ r a r = 0 λ 1 = = λ r = 0 a 1,, a r 1 V n 1 (n + 1) 1 n V dim V = n n V n 1 {a 1,, a n } V (basis) V x 1 x = n x i a i i= (tangent vector) S 2 3 R 3 p R n? ( )

21 v = (v 1,..., v n ) (directional derivative) µ vµ ( / x µ ), ( ) (Leibnitz) d df(x) (f(x)g(x)) = g(x) + f(x)dg(x) dx dx dx M M R C F(M) p M v v : F(M) R 1. v(f + g) = v(f) + v(g), v(af) = av(f) 2. v(fg) = f(p)v(g) + g(p)v(f) ( f, g F(M), a R) p V p (M) V p (M) v V p (M) (v 1 + v 2 )(f) = v 1 (f) + v 2 (f) (av)(f) = a(v(f)). c v V p (M) v(c) = 0. f F(M) cf v(cf) = cv(f) v(cf) = cv(f) + f(p)v(c)

22 18 2 f(p)v(c) = 0 f v(c) = M n M p(p M) p V p (M) V p (M) dim V p (M) = n. dim V p (M) = n V p (M) n 1 V p (M) p O ψ : O U( ψ(o)) R n f F(M) f ψ 1 : U R C µ = 1,..., n X µ : F(M) R X µ (f) = x (f µ ψ 1 ) (2.2.1) ψ(p) (x 1,, x n ) R n X 1,..., X n 1 X 1,..., X n, V p (M) R n ( 2 ) F : R n R C a = (a 1,..., a n ) R n F (x) = F (a) + n (x µ a µ )H µ (x), ( x µ R n ) (2.2.2) µ=1 C H µ H µ (a) = F x µ (2.2.3) x=a F = f ψ 1 a = ψ(p) f(q) = f(p) + n [x µ ψ(q) x µ ψ(p)]h µ (ψ(q)), ( q O) (2.2.4) µ=1 ( p q O ). v V p (M) v X 1,..., X n

23 f v (2.2.4) v v 0 n v(f) = v[f(p)] + [x µ ψ(q) x µ ψ(p)] v(h µ ψ) q=p + = µ=1 n (H µ ψ(q)) v[x µ ψ x µ ψ(p)] q=p µ=1 n (H µ ψ(p))v(x µ ψ) (2.2.5) µ=1 (2.2.3) H µ ψ(p) X µ (f) f F(M) n v(f) = v µ X µ (f) (2.2.6) µ=1 v µ x µ ψ v v µ = v(x µ ψ) (2.2.7) v X µ n v = v µ X µ (2.2.8) µ= V p (M) {X µ } (coordinate basis) X µ / x µ ψ {X µ} {X µ} X µ (chain rule; ) X µ = n ν=1 x ν x µ X ν (2.2.9) ψ(p) x ν ψ ψ 1 ν (2.2.8),(2.2.9) v v ν v µ v ν = n µ=1 v µ x ν x µ (2.2.10)

24 20 2 (2.2.10) M C ( )R( R ) M C C : R M C p M C T V p (M) f F(M) T (f) f C : R R p (T (f) = d(f C)/dt) ψ X µ M x µ ψ M C R n x µ (t) f F(M) T (f) = d dt (f C) = µ x (f µ ψ 1 ) dxµ dt = µ dx µ dt X µ(f) (2.2.11) T µ T µ = dxµ dt (2.2.12) p M p V p (M) q M V q (M) V q (M) V p (M) q p 3 ( ) p q 0 V p (M) V q (M) M v p M v p V p (M) V p (M) V q (M) v f F(M) C, p M v p (f) v(f) M f v(f) v X µ v v µ

25 (oneparameter group of diffeomorphism) φ t C R M M t R φ t : M M t, s R φ t φ s = φ t+s ( φ t=0 ) φ t v p M φ t (p) : R M t = 0 p φ t (orbit) t = 0 v p M 1 v v M v v (integral curves) (M p M 1 p v p? p ( ) dx µ dt = v µ (x 1,..., x n ) (2.2.13) R n v µ { / x µ } v µ t = 0 v v φ t 1 2 v w, [v, w] [v, w](f) = v(w(f)) w(v(f)) (2.2.14) v w (commutator) 2 X µ X ν 0 0 X 1,, X n

26 ( ) ( ) 3 3 (3 ) 3 ) 1 p n ) p l F F n l ( n, l) F (stress tensor) V ( V = V p (M) ) V R f : V R a, b V λ R f(a + b) = f(a) + f(b) f(λa) = λf(a)

27 V V f, g V (f + g)(a) = f(a) + g(a) (µf)(a) = µf(a) V V V (dual vector space) V (dual vectors) V ω e 1,..., e n V e 1,..., e n V e µ (e ν ) = δ µ ν (2.3.1) µ = ν δ µ ν = 1 0 {e µ } V V {e µ } dim V = dim V e µ e µ V V (isomorphism) {e µ } V V V ( V 2 ) V = (V ) (V ) V v V, V R v V ω V v (ω ) = ω (v) V V 1 dim V = dim V V V V V ( (dual) 2 ) V V V (k, l) (tensor) T

28 24 2 T : V }. {{.. V } } V. {{.. V } R k l χ j, ω 1,, ω k V u i, v 1,, v l V λ, µ R T (ω 1,, ω k, ; v 1,, λu i + µv i,, v l ) =λt (ω 1,, ω k, ; v 1,, u i,, v l ) + µt (ω 1,, ω k, ; v 1,, v i,, v l ) T (ω 1,, λω j + µχ j,, ω k ; v 1,, v l ) =λt (ω 1,, ω j,, ω k ; v 1,, v l ) + µt (ω 1,, χ j,, ω k ; v 1,, v l ) ) (0, 1) (1, 0) V V V (1, 0) (k, l) T (k, l) V {e µ } V {e µ } T (k, l) n k+l ( n = dim V = dim V ) 2 i ( ) j (contraction) C : T (k, l) T (k 1, l 1) T (k, l) n C(T ) = T (..., e σ,... ;..., e σ,...) (2.3.2) σ=1 {e σ } V {e σ } T i j σ C(T ) {e µ } 2 (k, l) T (k, l ) T

29 (k + k, l + l ) T T (k + k ) v 1,..., v k+k (l + l ) w 1,..., w l+l T T (v 1,..., v k+k ; w 1,..., w l+l ) = T (v 1,..., v k ; w 1,..., w l )T (v k+1,..., v k+k ; w l+1,..., w l+l ) T T (outer product) {e µ } V {e ν } n k+l {e µ1 e µk e ν 1 e ν l } T (k, l) (k, l) T T = T µ 1 µ k ν1 ν l e µ1 e µk e ν 1 e ν l (2.3.3) T µ 1 µ k ν1 ν l T {e µ } (components) T (2.3.3) T µ 1 µ k ν1 ν l T C(T ) (C(T )) µ 1 µ k 1 ν1 ν l 1 = n T µ 1 σ µ k 1 ν1 σ ν l 1 (2.3.4) σ=1 T T µ 1 µ k ν 1 ν S = T T l S µ 1 µ k+k ν1 ν l+l = T µ 1 µ k ν1 ν l T µ k+1 µ k+k νl+1 ν l+l (2.3.5) V M p V p (M) V p (M) p (cotangent space)

30 26 2 V p (M) (cotangent vector) V p (M) (contravariant) Vp (M) (covariant) 2 2 V p (M) / x 1,..., / x n Vp (M) dx 1,..., dx n [dx µ dx µ ( / x ν ) = δ µ ν ] v v µ = n µ=1 v µ x µ x µ (2.3.6) ω {dx µ } ω µ (2.3.1) (2.3.6) ω µ = n µ=1 ω µ x µ x µ (2.3.7) (k, l) T T µ 1 µ k ν 1 ν l = n T µ1 µk x µ 1 ν1 ν l x µ 1 µ 1 ν l x µ k x µ k x ν 1 x ν 1 xν l x ν l (2.3.8) (2.3.8) (tensor transformation law) M p V p (M) ( ) v ω (C ) v ω(v) (k, l) T ω 1,..., ω k v 1,..., v l T (ω 1,..., ω k ; v 1,..., v l ) (metirc) 2 2 2

31 g V p (M) V p (M) (0, 2) (symmetric), v 1, v 2 V p (M) g(v 1, v 2 ) = g(v 2, v 1 ) (nondegenerate), v V p (M) g(v, v 1 ) = 0 v 1 = 0 M (metric) g (0, 2) ( ) g g µν g = µ,ν g µν dx µ dx ν (2.3.9) g ds 2 (2.3.9) ds 2 = g µν dx µ dx ν (2.3.10) µ,ν (2.3.10) g p (orthonormal basis) e 1,..., e n µ ν g(e µ, e ν ) = 0 µ = ν g(e µ, e ν ) = ±1 p g(e µ, e µ ) = +1 g(e µ, e µ ) = 1 + (signature) (positive definite) ( ) p M g V p (M) (0,2) V p (M) V p (M) R g v g(, v) V p (M) Vp (M) g 1 1

32 28 2 g 1 1 paracompact (1 )Riemann paracompact 1 (1 ) Lorentz T µ 1 µ k ν1 ν l (1),(2)

33 (abstract index notation) (k, l), k l T a 1 a k b1 b l T abc de (3, 2) (2.3.4) ) T abc de 2 1 T abc be (2, 1) 2 T abc de S a b T abc des f g (4, 3) T µνλ σρ T abc de g (0, 2) g ab v a g ab v b v a (= g ab v b ) g ab V p (M) Vp (M) g ab g ab (2, 0) (g 1 ) ab g ab g ab g bc = δ a c δ a c (V p (M) V p (M) ) ω a g ab ω b ω a (3, 2) T abc de T a b cde g bf g dh g ej T afc hj (0, 2) T ab (v a, w a ) T ab v a w b T ab

34 30 2 (v a, w a ) T ab v b w a T ba T ab = T ba T ab (0, 2) T ab T (ab) = 1 2 (T ab + T ba ) (2.4.1) T [ab] = 1 2 (T ab T ba ) (2.4.2) (0, l) T a1 a l T (a1 a l ) = 1 T aπ(1) a l! π(l) (2.4.3) π T [a1 a l ] = 1 δ π T aπ(1) a l! π(l) (2.4.4) π 1,..., l π δ π 1, 1 T (ab)c [de] = 1 4 [T abc de + T bac de T abc ed T bac ed] (2.4.5) (0, l) T a1 a l 1 (differential 1-form) T a1 a l = T [a1 a l ] (2.4.6)

35 g ab M (intrinsic) ) 3.2 ( 5 ) 2 p q V p (M) V q (M) p q

36 ? 3 1 (M, g ab ) ) 3.1 M (derivative operator) ( (covariant derivative) ), (k, l) (k, l + 1) 5 T a 1 a k b1 b l T (k, l) T c T a 1 a k b1 b l a ( a 5 1. : A, B T (k, l) α, β R c (αa a 1 a k b1 b l +βb a 1 a k b1 b l ) = α c A a 1 a k b1 b l +β c B a 1 a k b1 b l 2. Leibnitz :

37 A T (k, l), B T (k, l ) e [A a 1 a k b1 b l B c 1 c k d1 d l ] = [ e A a 1 a k b1 b l ]B c 1 c k d1 d l + A a 1 a k b1 b l [ e B c 1 c k d1 d l ] 3. : A T (k, l) d (A a 1 c a k b1 c b l ) = d A a 1 c a k b1 c b l 4. : f F(M) t a V p (M) t(f) = t a a f 5. (Torsion free): f F(M) a b f = b a f 5 5 ) v a, w b a f [v, w](f) = v{w(f)} w{v(f)} = v a a (w b b f) w a a (v b b f) = {v a a w b w a a v b } b f (3.1.1) [v, w] b = v a a w b w a a v b (3.1.2) ( ) ψ { / x µ } {dx µ } (ordinary derivative) a

38 34 3 T a 1 a k b1 b l T µ 1 µ k ν1 ν l c T a 1 a k b1 b l (T µ 1 µ k ν1 ν l )/ x σ 5 ( ) ψ a ψ a ct a 1 a k b1 b l T a 1 a k b1 b l? (4) 2 a a 2 ω b f a (fω b ) a (fω b ) Leibnitz a (fω b ) a (fω b ) = ( a f)ω b + f a ω b ( a f)ω b f a ω b = f( a ω b a ω b ) (3.1.3) ( (4) ) p a ω b a ω b ω b p (3.1.3) a ω b a ω b p ω b. a ω b a ω b p ω b ω b p ω b ω b ω b. ω b ω b p 0 (2 2 ) p 0 f (α) µ (α) b ω b ω b = n α=1 f (α) µ (α) b (3.1.4) (3.1.3) p a (ω b ω b ) a (ω b ω b ) = α { a (f (α) µ (α) b ) a (f (α) µ (α) b )} = α f (α) ( a µ (α) b a µ (α) b ) = 0 (3.1.5)

39 p f (α) = 0 a ω b a ω b = a ω b a ω b (3.1.6) a a p p (0, 2) (1) ( a a ) p (1, 2) C c ab 2 a a a ω b = a ω b C c abω c (3.1.7) C c ab (5) C c ab ω b = b f = b f a b f = a b f C c ab c f (3.1.8) a b f a b f a b C c ab = C c ba (3.1.9) (3.1.9) a a (3.1.7) (4) t a ω a (4) ( a a )(ω b t b ) = 0 (3.1.10) ( a a )(ω b t b ) = (C c abω c )t b + ω b ( a a )t b (3.1.11) ω b ω b [( a a )t b + C b act c ] = 0 (3.1.12) a t b = a t b + C b act c (3.1.13)

40 36 3 a a C c ab T T (k, l) a T b 1 b k c1 c l = a T b 1 b k c1 c l + i C b i ad T b 1 d b k c1 c l j C d ac j T b 1 b k c1 d c l (3.1.14) 2 a a C c ab a C c ab (3.1.14) a n C c ab n 2 (n + 1)/2 (3.1.14) a a C c ab Γ c ab (Christoffel symbol) a t b = a t b + Γ b act c (3.1.15) (3.1.15) Γ b ac a a t a C v a t a a v b = 0 (3.1.16) (parallelly transported) t a a T b 1 b k c1 c l = 0 (3.1.17) (3.1.15) (3.1.16) t a a v b + t a Γ b acv c = 0 (3.1.18)

41 t dv ν dt + t µ Γ ν µλv λ = 0 (3.1.19) µ,λ v a v a v a (3.1.19) p p q V p (M) V q (M) (connection) g ab 2 v a w a g ab v a w b (3.1.16) v b w c t a a (g bc v b w c ) = 0 (3.1.20) t a v b w c a g bc = 0 (3.1.21) (3.1.21) a g bc = 0 (3.1.22) a a g ab a g bc = 0 a. a a C c ab

42 38 3 C c ab C c ab (3.1.14) C c ab 0 = a g bc = a g bc C d abg dc C d acg bd (3.1.23) C cab + C bac = a g bc (3.1.24) C cba + C abc = b g ac (3.1.25) C bca + C acb = c g ab (3.1.26) (3.1.24) (3.1.25) (3.1.26) C c ab (3.1.9) 2C cab = a g bc + b g ac c g ab (3.1.27) C c ab = 1 2 gcd { a g bd + b g ad d g ab } (3.1.28) C c ab (3.1.22) g ab a, ((3.1.22) ) (3.1.14) (3.1.28) a a Γ c ab = 1 2 gcd { a g bd + b g ad d g ab } (3.1.29) Γ ρ µν = 1 { g ρσ gνσ 2 x + g µσ µ x g } µν ν x σ σ (3.1.30)

43 3.2. (curvature) (curvature) C p q V p (M) V q (M) Riemann Riemann a ω a f fω a 2 a b (fω c ) = a (ω c b f + f b ω c ) = ( a b f)ω c + b f a ω c + a f b ω c + f a b ω c (3.2.1) b a (fω c ) (3.2.1) 3 b a (fω c ) ( a b b a )(fω c ) = f( a b b a )ω c (3.2.2) ( (3.1.3) ) p ( a b b a )ω c p ω c ( a b b a ) p p (0, 3) ω c a b ω c b a ω c = R abc d ω d (3.2.3) R abc d R abc d Riemann R abc d p M p 2 S t s 3.3 s = 0

44 40 3 t t = t s t s v a p S ) v a v a ω a v a ω a t v a ω a δ 1 δ 1 = t t (va ω a ) ( t/2,0) (3.2.4) t 2 δ 1 δ 1 = tt b b (v a ω a ) ( t/2,0) = tv a T b b ω a ( t/2,0) (3.2.5) T b s (3.1.16) T b b v a = 0 δ 2, δ 3, δ 4 2 t δ 1 δ 3 δ 1 + δ 3 = t{v a T b b ω a ( t/2,0) v a T b b ω a ( t/2, s) } (3.2.6) δ 2 δ 4 s 0 t s 1 v a ω a v a ) t s 1 v a ω a 2 (3.2.6) 1 t = t/2 ( t/2, 0) ( t/2, s) v a T b b ω a s 1 ( t/2, s) v a ( t/2, 0) v a 1 ) 1 ( t/2, s) T b b ω a, ( t/2, 0) (T b b ω a ) ss c c (T b b ω a ) (S c t ) t, s 2 δ 1 + δ 3 = t s v a S c c (T b b ω a ) (3.2.7)

45 3.2. (curvature) 41 p δ 2 δ 4 v a ω a δ(v a ω a ) = t s v a {T c c (S b b ω a ) S c c (T b b ω a )} = t s v a T c S b ( c b b c )ω b = t s v a T c S b R cba d ω d (3.2.8) 2 T a S a (2 2 (3.1.2) ) Riemann (3.2.3) (3.2.8) ω a v a ( t s 2 ) δv a = t s v d T c S b R cbd a (3.2.9) Riemann (3.2.3) (3.1.14) Riemann t a ω a 5 (3.2.3) 0 = ( a b b a )(t c ω c ) = a (ω c b t c + t c b ω c ) b (ω c a t c + t c a ω c ) = ω c ( a b b a )t c + t c ( a b b a )ω c = ω c ( a b b a )t c + t c ω d R abc d (3.2.10) ( a b b a )t c = R abd c t d (3.2.11) T c 1 c k d1 d l k ( a b b a )T c 1 c k c d1 d l = R i abe T c 1 e c k d1 d l + i=1 l R e abdj T c 1 c k d1 e d l (3.2.12) Riemann 4 j=1

46 R abc d = R bac d (3.2.13) 2. R [abc] d = 0 (3.2.14) 3. a ( a g bc = 0 ) R abcd = R abdc (3.2.15) 4. Bianchi [a R bc]d e = 0 (3.2.16) (1) R abc d (3.2.3) (2) ω a a [a b ω c] = 0 (3.2.17) (3.1.14) a a C c ab = Γ c ab (3.1.9) ω d 0 = 2 [a b ω c] = [a b ω c] [b a ω c] = R [abc] d ω d (3.2.18) (2) (3) (3.2.12) g ab 0 = ( a b b a )g cd = R abc e g ed + R abd e g ce = R abcd + R abdc (3.2.19) 1,2,3 Riemann R abcd = R cdab (3.2.20) (4) Bianchi (3.2.12) ( a b b a ) c ω d = R abc e e ω d + R abd f c ω f (3.2.21) a ( b c ω d c b ω d ) = a (R bcd e ω e ) = ω e a R bcd e + R bcd e a ω e (3.2.22)

47 3.2. (curvature) 43 (3.2.21) (3.2.22) a, b, c R [abc] e e ω d + R [ab d f c] ω f = ω e [a R bc]d e + R [bc d e a] ω e (3.2.23) d 1 (3.2.14) 2 ω e ω e [a R bc]d e = 0 (3.2.24) (4) (Bianchi ) Riemann trace trace 0 (1) (3) Riemann (1 3 ) Ricci R ac R ac = R abc b (3.2.25) (3.2.20) R ac R ac = R ca (3.2.26) R Ricci R = R a a (3.2.27) trace 0 Weyl C abcd n 3 R abcd = C abcd + 2 n 2 (g 2 a[cr d]b g b[c R d]a ) (n 1)(n 2) Rg a[cg d]b (3.2.28) Weyl Riemann (1),(2),(3) trace 0 (conformal tensor)

48 44 3 Bianchi (3.2.16) R ab a R bcd a + b R cd c R bd = 0 (3.2.29) d b d a R c a + b R c b c R = 0 (3.2.30) a G ab = 0 (3.2.31) G ab = R ab 1 2 Rg ab (3.2.32) G ab Einstein Einstein Bianchi 2 (3.2.31) Einstein 3.3 (Geodesics) a (geodesic) ( ) T a T a a T b = 0 (3.3.1) T a a T b = αt b (3.3.2) α (3.3.2) (3.3.1) (3.3.1)

49 3.3. (Geodesics) 45 (3.3.1) affine parametrization ψ R n x µ (t) (3.1.19) T a T µ dt µ + Γ µ σνt σ T ν = 0 (3.3.3) dt σ,ν (2.2.12) T µ T µ = dxµ dt (3.3.4) d 2 x µ dt 2 + σ,ν Γ µ dx σ dx ν σν dt dt = 0 (3.3.5) (3.3.5) n x µ (t) n 2 x µ dx µ /dt (3.3.5) p M T a V p (M) p T a (exponential map) V p (M) M T a V p (M) (M p T a ) p affine M T a affine t = V p (M) V p (M) n R n p Riemann p R n (3.3.5) Christoffel

50 46 3 Γ µ σν p Riemann a g ab Gaussian 2 S (n M (n 1) ) p S S V p (S) M V p (M) (n 1) V p (S) ( g ab ) n a V p (M) n a S Riemann n a V p (S) n a g ab n a n b = 0 n a V p (S) S p S g ab n a n b = ±1 n a Gaussian p S p n a S ( 1 ) (x 1,, x n 1 ) S ( ) t p S x 1,, x n 1 S p S ( ) q (x 1,, x n 1, t) Gaussian (t ) S t S 0 = S S t n a S t ) X a 1,..., X a n 1 X a n b b (n a X a ) = n a n b b X a = n a X b b n a = 1 2 Xb b (n a n a ) = 0 (3.3.6) 2 n a X b M ) 3 S n a n a = ±1

51 3.3. (Geodesics) 47 n a n a M S n a X a = 0 (3.3.6) S ( ) Riemann g ab M C C l l = (g ab T a T b ) 1/2 dt (3.3.7) T a C t Lorentz ( + +) g ab T a T b < 0 g ab T a T b = 0 g ab T a T b > 0 (3.3.7) 0 τ τ = ( g ab T a T b ) 1/2 dt (3.3.8) Lorentz ) s = s(t) S a = (dt/ds)t a l = (g ab S a S b ) 1/2 ds = (g ab T a T b 1/2 dt ) ds = l (3.3.9) ds 1 R n (3.3.7) l = b a [ µ,ν g µν dx µ dt ] 1/2 dx ν dt (3.3.10) dt

52 48 3 C(a) = p C(b) = q l Lagrangian l δl = b a [ µ,ν g µν dx µ dt dx ν dt ] 1/2 { α,β g αβ dx α dt d(δx β ) dt σ g αβ dxα δxσ xσ dt (3.3.11) ( ) g ab T a T b = 1 = µ,ν g µν dx µ dt { } b dx α d(δx β ) 0 = g αβ + 1 g αβ dx α dx β a dt dt 2 x σ dt dt δxσ dt α,β σ { b = d ) } dx (g α αβ + 1 g αλ dx α dx λ δx β dt (3.3.12) dt dt 2 x β dt dt a α,β δx β δx β (3.3.12) α g αβ d 2 x α dt 2 α,λ λ dx ν dt g αβ dx λ dx α x λ dt dt + 1 g αλ dx α dx λ 2 x β dt dt = 0 (3.3.13) α,λ Γ σ αλ (3.1.30) (3.3.13) (3.3.5) 2 affine ) Lagrangian } dx β dt dt L = µ,ν g µν dx µ dt dx ν dt (3.3.14) Γ µ σν Lagrangian

53 3.3. (Geodesics) 49 (3.3.14) Euler-Lagrange (3.3.5) Γ µ σν Riemann Lorentz 2 2 Einstein (4 3 γ s (t) 1 s R γ s affine t (t, s) γ s (t) 1 ( ) γ s (t) 2 Σ Σ s t T a = ( / t) a T a a T b = 0 (3.3.15) X a = ( / s) a (deviation vector) X a γ s (t) affine t t = b(s)t + c(s) X a T a dγ s (t ) ds = γ s(t ) s + γ s(t) dt t ds?? g ab X a T a s t g ab T a T b t ) s X a T a

54 50 3 T b b X a = X b b T a (3.3.16) (3.3.6) X a T a γ s (t) s t C(s) t = 0 ( ) γ s (t) affine t = 0 X a T a = 0 X a T a = 0 v a = T b b X a v a a a = T c c v a = T c c (T b b X a ) (3.3.17) a a Riemann a a = T c c (T b b X a ) = T c c (X b b T a ) = (T c c X b )( b T a ) + X b T c c b T a = (X c c T b )( b T a ) + X b T c b c T a R cbd a X b T c T d = X c c (T b b T a ) R cbd a X b T c T d = R cbd a X b T c T d (3.3.18) (3.3.18) (geodesic deviation equation) a a = 0 R abc d = 0 ( - v a = T b b X a = 0 - ) R abcd a a

55 Christoffel Γ c ab ω a (3.2.3) b ω c = b ω c Γ d bcω d (3.4.1) a b ω c = a ( b ω c Γ d bcω d ) Γ e ab( e ω c Γ d ecω d ) Γ e ac( b ω e Γ d beω d ) (3.4.2) R abc d ω d = [ 2 [a Γ d b]c + 2Γ e c[aγ d b]e]ω d (3.4.3) Γ c ab (3.1.9) (3.4.3) ω d ω d R abc d R σ µνρ = x ν Γσ µρ x µ Γσ νρ + (Γ α µργ σ αν Γ α νργ σ αµ) (3.4.4) α g ab R abc d g µν (3.1.30) Γ σ µν (3.4.4) R µνρ σ Ricci (3.4.4) R µρ = ν = ν R µνρ ν ( ) x ν Γν µρ Γ ν x µ νρ + (Γ α µργ ν αν Γ α νργ ν αµ) ν α,ν (3.4.5) g µν g ab g µν (g µν ) (g µν ) g g = det(g µν ) (3.4.6)

56 52 3 Christoffel Γ a ab (3.1.30) Γ a aµ = Γ ν νµ = 1 g να g να (3.4.7) 2 x µ ν ν,α ν,α (3.4.9) g να g να x µ Γ a aµ = 1 1 g 2 g x = µ = 1 g a T a = a T a + Γ a abt b = µ g x µ (3.4.8) x µ ln g (3.4.9) 1 g x ( g T µ ) (3.4.10) µ (Tetrad) a R abc d { / x µ } (nonholonomic) (e µ ) a (e µ ) a (e ν ) a = η µν (3.4.11) η µν = diag( 1,, 1, 1, 1) 4 {(e µ ) a } tetrad (3.4.11) η µν (e µ ) a (e ν ) b = δ a b (3.4.12) µ,ν

57 ω aµν = (e µ ) b a (e ν ) b (3.4.13) ω λµν = (e λ ) a (e µ ) b a (e ν ) b (3.4.14) ω aµν = (e µ ) b a (e ν ) b = (e ν ) b a (e µ ) b = ω aνµ (3.4.15) ω aµν = ω aνµ (3.4.16)

58

59 R 3 R 3 (x 1, x 2, x 3 ) (Cartesian coordinates) R (x 1, x 2, x 3 ) 2 x x D D 2 = (x 1 x 1 ) 2 + (x 2 x 2 ) 2 + (x 3 x 3 ) 2 (4.1.1) 2 h ab (4.1.1) 2 (δd) 2 = (δx 1 ) 2 + (δx 2 ) 2 + (δx 3 ) 2 (4.1.2) ((2.3.10) ) ds 2 = (dx 1 ) 2 + (dx 2 ) 2 + (dx 3 ) 2 (4.1.3) h ab = h µν (dx µ ) a (dx ν ) b (4.1.4) µ,ν (h µν = diag(1, 1, 1)) h ab h ab

60 56 4 (4.1.4) a h bc = 0 (4.1.5) h ab Γ a bc (3.2.3), h ab (3.3.5) (3.3.7) (4.1.1) (4.1.4) ( ) Riemann R 3 Riemann R 3 0 (4.1.1) Riemann R Taylor ) )

61 R 4 x 1, x 2, x 3 t R 4 (global inertial coordinate system) 10 Poincaré 1 1 t = x 0, x 1, x 2, x 3 2 x x I(c = 1 ) I = (x 0 x 0 ) 2 + (x 1 x 1 ) 2 + (x 2 x 2 ) 2 + (x 3 x 3 ) 2 (4.2.1) (4.2.1) (metric of spacetime) η ab 3 η ab = η µν (dx µ ) a (dx ν ) b (4.2.2) µ,ν=0 η µν = diag( 1, 1, 1, 1) {x µ } η µν a a η bc = 0 (4.2.3) η ab 0 η ab η ab, Lorentz R 4

62 58 4 η ab 2 (proper) Poincaré ) η ab T a T b 0 ( ) (proper time) τ τ = ( η ab T a T b ) 1/2 dt (4.2.4) t (t ) T a τ ) τ u a 4 (4-velocity) τ 4 u a u a = 1 (4.2.5)

63 u a a u b = 0 (4.2.6) a η ab (4.2.6) 0 ( (4.2.26) ) m m 4 p a p a = mu a (4.2.7) )4 v a E = p a v a (4.2.8) 4 p a v a = u a (4.2.8) E = mc 2 c = 1 η ab 4 (stressenergy-momentum) T ab 4 v a T ab v a v b T ab v a v b 0 (4.2.9) x a v a T ab v a x b x a y a v a T ab x a y b x a y a

64 60 4 T ab = ρu a u b + P (η ab + u a u b ) (4.2.10) u a 4 T ab ρ P a T ab = 0 (4.2.11) (4.2.11) ρ, P, u a u b u a a ρ + (ρ + P ) a u a = 0 (4.2.12) (P + ρ)u a a u b + (η ab + u a u b ) a P = 0 (4.2.13) P ρ, u µ = (1, v), vdp/dt P ρ t + (ρ v) = 0 (4.2.14) { } v ρ + ( v ) v = P t (4.2.15) (4.2.11) (4.2.14) Euler (4.2.15) (4.2.11) 4 v a 1 b v a = 0 T ab J a = T ab v b (4.2.16) 4 (4.2.11) a J a = 0 (4.2.17) (4.2.17), 4 V 3 S J a n a ds = 0 (4.2.18) S

65 n a 4.1 (4.2.18) (4.2.11) (4.2.11) 2 Klein- Gordon a a φ m 2 φ = 0 (4.2.19) φ T ab = a φ b φ 1 2 η ab( c φ c φ + m 2 φ 2 ) (4.2.20) T ab (4.2.9) (4.2.19) (4.2.11) E B F ab F ab = F ba F ab 6 4 v a E a = F ab v b (4.2.21) B a = 1 2 ɛ ab cd F cd v b (4.2.22) ɛ abcd ɛ abcd ɛ abcd = 24 ɛ 0123 = 1 F ab Maxwell a F ab = 4πj b (4.2.23) [a F bc] = 0 (4.2.24)

66 62 4 j a 4 F ab 0 = b a F ab = 4π b j b (4.2.25) Maxwell b j b = 0 J a q F ab u a a u b = q m F b cu c (4.2.26) Lorentz F ab T ab = 1 4π {F ac F b c 14 η abf de F de } (4.2.27) T ab (4.2.9) j a = 0 Maxwell a T ab = 0 j a 0 T ab Poincaré (4.2.24) A a ( ) F ab = a A b b A a (4.2.28) A a Maxwell a ( a A b b A a ) = 4πj b (4.2.29) χ a χ A a (4.2.28) F ab χ Lorentz a a χ = b A b (4.2.30) a A a = 0 (4.2.31)

67 (4.2.29) a a A b = 4πj b (4.2.32) Maxwell ( ) A a = C a exp(is) (4.2.33) C a S (phase) j a = 0 ((4.2.32) ) a a S = 0 (4.2.34) a S a S = 0 (4.2.35) ((4.2.31) ) C a a S = 0 (4.2.36) f a f f t a t a a f = 0 (4.2.35) S k a = a S k a k a = 0 (4.2.35) 0 = b ( a S a S) = 2( a S)( b a S) = 2( a S)( a b S) = 2k a a k b (4.2.37) k a (4.2.35) (4.2.37) a a ) Lorentz 4 v a ) ω = v a a S = v a k a (4.2.38)

68 64 4 (4.2.33) S = 3 k µ x µ (4.2.39) µ=0 {x µ } k µ k µ Fourier Maxwell 0 (4.2.37) Maxwell (support) p p Maxwell 4.3 Maxwell Maxwell Coulomb Newton Einstein ( ) (4.2.6) (4.2.26) ( ) F ab

69 ( ) Newton ( ) 2 ( ) 2 (3.3.18) 980cms 2 Newton 980cms 2 (

70 66 4 ( ) η ab R 4 ( R 4 ) ( ) g ab R 4 (a priori) Einstein g ab M 2 (1) g ab (2) g ab R 4 4 u a ρ P

71 F ab 2 η ab g ab η ab a g ab a 4 u a (g ab ) u a a u b = 0 (4.3.1) a g ab a b = u a a u b 0 f b = ma b m ) ( ) m q F ab Lorentz u a a u b = q m F b cu c (4.3.2) g ab F b c = g bd F dc 4 p a = mu a (4.3.3) E = p a v a (4.3.4) v a 4 T ab T ab = ρu a u b + P (g ab + u a u b ) (4.3.5) a T ab = 0 (4.3.6) u a a ρ + (ρ + P ) a u a = 0 (4.3.7) (P + ρ)u a a u b + (η ab + u a u b ) a P = 0 (4.3.8)

72 68 4 (4.3.6) v a a v b = 0 a (T ab v b ) = 0 Gauss 4 J a = T ab v b (v b ) (4.2.18) v a v a = 1 (a v b) = 0 v a (4.3.6) b v a 0 (4.3.6) (4.3.6) (4.3.6) Klein-Gordon, η ab g ab, a a a a φ m 2 φ = 0 (4.3.9) T ab = a φ b φ 1 2 g ab( c φ c φ + m 2 φ 2 ) (4.3.10) a T ab = 0 (4.2.19) α a a φ m 2 φ αrφ = 0 (4.3.11) (4.3.11) α = 1/6 Maxwell a F ab = 4πj b (4.3.12) [a F bc] = 0 (4.3.13)

73 (4.2.27) η ab g ab T ab = 1 {F ac F c b 14 } 4π g abf de F de (4.3.14) (4.3.13) A a ( ) Lorentz A a Maxwell (4.2.32) a a A b R d ba d = 4πj b (4.3.15) Maxwell (4.2.32) (4.2.15) Ricci R d ba d (4.3.15) (4.3.15) a j a = 0 Maxwell A a = C a exp(is) (4.3.16) C a (4.3.16) (4.3.15) j b = 0 b b C a Ricci a S a S = 0 (4.3.17) k a = a S? Newton φ 2 (tidal acceleration) ( x ) φ x 2 (3.3.18) R cbd a v c x b v d v a 4 x a R cbd a v c v d b a φ (4.3.18)

74 70 4 Poisson 2 φ = 4πρ (4.3.19) ρ ( ) G = c = 1 T ab T ab v c v d ρ (4.3.20) v a 4 (4.3.18) (4.3.20) (4.3.19) R cad a v c v d = 4πT cd v c v d R cd = 4πT cd c T cd = 0 Bianchi (3.2.31) c (R cd 1 2 g cdr) = 0 R cd 4πT cd d R = 0 R T = T a a G ab = R ab 1 2 Rg ab = 8πT ab (4.3.21) Bianchi Bianchi (4.3.21) (4.3.21) R = 8πT (4.3.22) R ab = 8π(T ab 1 2 g abt ) (4.3.23) (c = 1 ) T ρ = T ab v a v b (4.3.23) R ab v a v b 4πT ab v a v b (4.3.21) Einstein 1915 Einstein

75 Lorentz g ab M g ab Einstein (4.3.21) Einstein 3 1 R µν g µν 3 4 R µν g µν g µν 2 Einstein Einstein (4.3.21) Maxwell (4.2.32) j a T ab j a Maxwell A a T ab Einstein g ab g ab T ab T ab Einstein Einstein a T ab = 0 a T ab = 0 T ab Einstein P = 0 ( ) a T ab = 0 (4.3.8) a T ab = 0 (Fock 1939; Geroch and Yang 1975) Einstein (geodesic hypothesis) Einstein

76 72 4 a T ab = 0 (Papapetrou 1951; Dixon 1974). 4.4 η ab g ab = η ab + γ ab (4.4.1) γ ab ( η ab γ ab γ µν 1 (4.4.1) Einstein g ab γ ab η ab a γ ab g ab g ab η ab η ab 1 g ab η ac η bd g cd g ab = η ab γ ab (4.4.2) (4.4.1) (4.4.2) γ ab 2 Einsein γ ab 1 Christoffel Γ c ab = 1 2 ηcd ( a γ bd + b γ ad d γ ad ) (4.4.3)

77 γ ab 1 Ricci (3.4.5) R (1) ab = cγ c ab a Γ c cb = c (b γ a)c 1 2 c c γ ab 1 2 a b γ, (4.4.4) γ = γ c c 1 Einstein G (1) ab = R(1) ab 1 2 η abr (1) = c (b γ a)c 1 2 c c γ ab 1 2 a b γ 1 2 η ab( c d γ cd c c γ) (4.4.5) γ ab = γ ab 1 2 η abγ (4.4.6) γ ab Einstein G (1) ab = 1 2 c c γ ab + c (b γ a)c 1 2 η ab c d γ cd = 8πT ab (4.4.7) C φ : M M g ab φ g ab φ φ. 2 γ ab γ ab η ab 2 2 ξ a C Lie γ ab γ ab + L ξ η ab C L ξ η ab a L ξ η ab = a ξ b + b ξ a (4.4.8) γ ab γ ab + a ξ b + b ξ a (4.4.9)

78 74 4 A µ A µ + a χ γ ab C (2.3.8) (2.3.8) γ ab γ ab + a ξ b + b ξ a 1 Einstein b b ξ a = b γ ab (4.4.10) ξ a (4.4.9) b γ ab = 0 (4.4.11) Lorentz Einstein c c γ ab = 16πT ab (4.4.12) Maxwell (4.2.32) (T ab = 0) (4.4.11) (4.4.12) Fiertz Pauli (1939) η ab Newton η ab T ab ρt a t b (4.4.13)

79 t a = ( / x 0 ) a (4.4.13) T ab ( ) ( ) γ ab (4.4.12) (4.4.12) µ = ν = 0 µ, ν 2 γ µν = 0 (4.4.14) 2 γ 00 = 16πρ (4.4.15) 2 Laplace (4.4.14) γ µν = 0 γ µν = constant 0 ). Newton γ ab γ ab = γ ab 1 2 η ab γ = (4t a t b + 2η ab )φ (4.4.16) φ 1 4 γ 00 Poisson 2 φ = 4πρ (4.4.17) d 2 x µ dτ + ( ) ( ) dx Γ µ ρ dx σ 2 ρσ = 0 (4.4.18) dτ dτ ρ,σ x µ (τ) 2 dx a /dτ (1, 0, 0, 0) τ t d 2 x µ dt 2 = Γ µ 00 (4.4.19)

80 76 4 (4.4.16) µ = 1, 2, 3 Γ µ 00 = 1 γ 00 2 x µ = φ x µ (4.4.20) φ a = φ (4.4.21) a = d 2 x/dt 2 η ab (4.4.17) (4.4.21), Newton Newton Newton Newton ( ) Newton η ab 1 T ab = 2t (a J b) ρt a t b (4.4.22) J b = T ab t a 4 Einstein γ ab a a γ 0µ = 16πJ µ (4.4.23) A a = 1 4 γ abt b Lorentz Maxwell J a γ ab γ ab a = E 4 v B (4.4.24)

81 E B A a 4 Lorentz (q = m ) (Lense-Thirring 4 4,Gravity Probe B (2004)) Maxwell Einstein ( (4.4.11) (4.4.12) ) a γ ab = 0 (4.4.25) c c γ ab = 0 (4.4.26) (4.4.25) b b ξ a = 0 (4.4.27) γ ab γ ab + a ξ b + b ξ a (4.4.25) Lorentz A a b b χ = 0 (4.4.28) A a A a + a χ (j a = 0) A 0 Coulomb radiation t = t 0 2 χ = A (4.4.29)

82 78 4 (4.4.28) t = t 0 (4.4.29) χ/ t = A 0 χ f (4.2.32) (4.4.28) f = A 0 + χ/ t (4.4.30) a a f = 4πj 0 (4.4.31) t = t 0 f = 0 (4.4.32) f t = A χ t t = A + 2 χ = 0 (4.4.33) 2 (4.4.32) (4.4.33) (4.4.31) f = 0 A a A a + a χ Lorentz A 0 = 0 (T ab = 0) radiation γ = 0, γ 0µ = 0 (µ = 1, 2, 3) (4.4.27) γ 00 = 0 radiation t = t 0 ( 2 ξ ) 0 t + ξ = γ (4.4.34a) 2 [ 2 ξ 0 + ( ξ t )] = γ t (4.4.34b) ξ µ t + ξ 0 x = γ µ 0µ (µ = 1, 2, 3) (4.4.34c) 2 ξ µ + ( ) ξ0 = γ 0µ (µ = 1, 2, 3) (4.4.34d) x µ t t ξ 0, ξ 1, ξ 2, ξ 3 (4.4.27) xi a ξ a (4.4.25) γ = 0 γ 0µ = 0 (µ = 1, 2, 3)

83 γ 00 = 0 γ = 0 γ ab = γ ab µ = 1, 2, 3 γ 0µ = 0 (4.4.25) γ 00 t = 0 (4.4.35) Einstein (4.4.12) 2 γ 00 = 16πT 00 (4.4.36) T 00 = 0 (4.4.36) γ 00 γ 00 = 0 radiation Einstein ( ) 3 γ ab = H ab exp i k µ x µ (4.4.37) H ab k µ k µ = η µν k µ k ν = 0 (4.4.38) µ,nu µ (4.4.26) radiation (ν = 0, 1, 2, 3 ) µ=0 3 k µ H µν = 0 µ=0 H 0ν = 0 3 H µ µ = 0 µ=0 (4.4.39a) (4.4.39b) (4.4.39c) (4.4.39a) (4.4.39b) ν H 0νk ν = H µν 10 H ab Einstein

84 80 4? 2 2 (3.3.18) η ab 2 d 2 X µ dt 2 = ν R µ ν00x ν (4.4.40) (X a ) radiation (γ 00 = 0 ) (3.4.4) Riemann R ν00µ = 1 2 γ µν (radiationguage) (4.4.41) 2 t 2 2? Einstein γ ab (4.4.12) γ µν (x) = 4 Λ T µν (x ) x x ds(x ) (4.4.42) Λ x ds = r 2 drdω γ ab (4.4.11) a T ab (4.4.42) ( radiation )

85 Fourier η ab t ˆ γ µν (ω, x) = 1 2π γ µν (t, x) exp(iωt)dt (4.4.43) T µν Fourier (4.4.42) ˆTµν (ω, x ˆ γ ) µν (ω, x) = 4 exp(iω x x )d 3 x (4.4.44) x x exp(iω x x ) (4.4.42) ˆ γ µν ˆ γ 0ν (4.4.11) 3 ˆ γ νµ iωˆ γ 0µ = (4.4.45) x ν (far zone) R 1/ω R exp(iω x x ) exp(iω x x )/ x x exp(iωr)/r ˆT ab { 3 } ˆT µν d 3 x = x ( ˆT αν x µ αν ˆT ) a x a xµ σ=1 = iω ˆT 0ν x µ = iω ( 2 ˆT 0ν x µ + ˆT 0µ x ν ) { = iω 3 } 2 x ( ˆT 0β x µ x ν 0β ˆT ) β x β xµ x ν β=1 = ω2 ˆT 00 x µ x ν (4.4.46) 2 2 Gauss T ab ˆ γ µν (ω, x) = 2ω2 3 µ=1 exp(iωr) ˆq µν (ω) (µ, ν = 1, 2, 3) (4.4.47) R

86 82 4 ˆq µν 4 q µν = 3 T 00 x µ x ν d 3 x (4.4.48) (4.4.47) ˆ γ µν (t, x) = 2 d 2 q µν 3R dt 2 (µ, ν = 1, 2, 3) (4.4.49) t = t R G (1) ab [γ cd] = 0 (4.4.50) R (2) ab = 1 2 γcd a b γ cd γ cd c [a γ b]d ( aγ cd ) b γ cd + ( d γ c b) [d γ c]a d(γ dc c γ ab ) 1 4 ( c γ) c γ ab ( d γ cd 1 2 c γ) [a γ b]c (4.4.51) G (1) ab [γ(2) cd ] + G(2) ab [γ cd] = 0 (4.4.52) G (1) ab [γ(2) cd ] = 8πt ab (4.4.53) t ab = 1 8π G(2) ab [γ cd] (4.4.54) E = t 00 d 3 x (4.4.55) Σ E = t 00 ds a (4.4.56) Σ E = P dt (4.4.57) P = (4.4.58) Q µν = q µν 1 3 δ µνq (4.4.59) M L Ω P rod = 2G 45c 5 M 2 L 4 Ω 6, (4.4.60)

87 (input) (Kirshner et al. 1981) X γ 3K 5 4 3K

88 84 5 (instant of time) (space) : ( ) (homogeneous) 5.1 ( ) Σ t 1 t p, q Σ t g ab (isometry) p q ( C ) 1 (isotropic) ( ) u a ( 5.2) p 2 s a 1, s a 2 V p p u a ) p u a s a 1 sa 2 g ab u a Σ t u a Σ t u a Σ t g ab p Σ t g ab Σ t Σ t h ab (t) Σ t (i) p Σ t q Σ t h ab (ii) Σ t 2 Σ t

89 h ab (3) R abc d 3 h ab p (3) R ab cd p 2 W L : W W (3.2.20) L (h ab W ) W L p 2 p L L L = KI (5.1.1) (3) R ab cd = Kδ c [aδ d b] (5.1.2) (3) R abcd = Kh c[a h b]d (5.1.3) (i) K Σ t (ii) K (5.1.3) Bianchi (3.2.16) 0 = D [e (3) R ab]cd = (D [e K)h c a h b]d (5.1.4) D a Σ t h ab a D a 4 g ab 3 (5.1.4) D e K = 0 0 K Σ t (5.1.3) (K = constant ) (space of constant curvature) K 2 ( ) (isometric) (Eisenhart 1949) Σ t K

90 86 5 K 3 4 R 4 x 2 + y 2 + z 2 + w 2 = R 2 (5.1.5) 3 ds 2 = dψ 2 + sin 2 ψ(dθ 2 + sin 2 θdφ 2 ) (5.1.6) K = 0 3 ds 2 = dx 2 + dy 2 + dz 2 (5.1.7) K 3 4 (Minkowski ) t 2 x 2 y 2 z 2 = R 2 (5.1.8) ds 2 = dψ 2 + sinh 2 ψ(dθ 2 + sin 2 θdφ 2 ) (5.1.9) K = (closed) (open) 4 g ab g ab = u a u b + h ab (t) (5.1.10) t h ab (t) Σ t (a) (b) (c) 4

91 (a) (b) (c) τ 2 τ dψ 2 + sin 2 ψ(dθ 2 + sin 2 θdφ 2 ) ds 2 = dτ 2 + a 2 (τ) dx 2 + dy 2 + dz 2 (5.1.11) dψ 2 + sinh 2 ψ(dθ 2 + sin 2 θdφ 2 ) 3 3 [ dψ 2 + ψ 2 (dθ 2 + sin 2 θdφ 2 ) ] (5.1.11) Robertson-Walker 3 a(τ) a(τ) Einstein 5.2 (5.1.11) Einstein (4.3.21) 1 T ab (Einstein ) (grain of dust) (pressure)

92 88 5 T ab = ρu a u b (5.2.1) ρ ( ) 3K 0 P = ρ/3 5 4 T ab Einstein T ab T ab = ρu a u b + P (g ab + u a u b ) (5.2.2) T ab T ab (5.1.11) G ab (5.2.2) 8πT ab G ab u b T ab u b ) Einstein 0 G ab (5.1.1) Einstein Einstein G ττ = 8πT ττ = 8πρ (5.2.3) G = 8πT = 8πP (5.2.4) G ττ = G ab u a u b G = G ab s a s b s a

93 G ττ G a(τ) ds 2 = dτ 2 + a 2 (τ)(dx 2 + dy 2 + dz 2 ) (5.2.5) (3.1.30) 0 Γ τ xx = Γ τ yy = Γ τ zz = aȧ, (5.2.6) Γ x xτ = Γ x τx = Γ y yτ = Γ y τy = Γ z zτ = Γ z τz = ȧ/a, (5.2.7) ȧ = da/dτ (3.4.5) Ricci R ττ = 3ä/a, (5.2.8) R = a 2 R xx = ä a + 2ȧ2 a 2 (5.2.9) (ä ) R = R ττ + 3R = 6 a + ȧ2 a 2 (5.2.10) G ττ = R ττ R = 3ȧ2 /a 2 = 8πρ, (5.2.11) G = R 1 2 R = 2ä a ȧ2 = 8πP. (5.2.12) a2 2 3ä/a = 4π(ρ + 3P ) (5.2.13) 3ȧ 2 /a 2 = 8πρ 3k/a 2, (5.2.14) 3ä/a = 4π(ρ + 3P ) (5.2.15) 3 k = +1, k = 0, k = 1 (P = 0) P = ρ/3

94 90 5 Dust Radiation Spatial Geometry P = 0 P = 1ρ 3 3-sphere,k = +1 a = 1C(1 cos η) a = C 2 [1 (1 τ/ C ) 2 ] 1/2 τ = 1 C(η sin η) 2 Flat, k = 0 a = (9C/4) 1/3 τ 2/3 a = (4C ) 1/4 τ 1/2 Hyperboloid,k = 1 a = 1C(cosh η 1 ) a = C 2 [(1 + τ/ C ) 2 1] 1/2 τ = 1 C(sinh η η) 2 5.1: Robertson-Walker 5-1 ρ > 0 P 0 (5.2.15) ä < 0 ȧ > 0, ȧ < 0 ) ( ) τ 2 R R v dr dτ = R da a dτ = HR (5.2.16) H(τ) = ȧ/a (Hubble) ( H ) (5.2.16) R v 2 (5.2.16)

95 G ab + Λg ab = 8πT ab (5.2.17) Λ (cosmological constant) G ab g ab (Lovelock 1972). (5.2.17) Einstein Λ 0 Newton Λ Newton ) Λ Λ = 0 ȧ > 0 (5.2.15) ä < 0 T = a/ȧ = H 1 a = 0 a H 1 (5.2.14) a 2

96 92 5 τ (5.2.15) ä (P = 0) ρ + 3(ρ + P )ȧ/a = 0 (5.2.18) ρa 3 = constant (5.2.19) (P = ρ/3) ρa 4 = constant (5.2.20) a a 3 a 3 (5 3 a 1 (5.2.19) (5.2.20) (a 0) k = 0 1 (5.2.14) ȧ 0 P 0 a ρ a 3 ( ) a ρa 2 0 k = 0 ȧ τ 0 k = 1 τ ȧ 1 k = 1 (5.2.14) 1 a 2 a a c a c τ a a c (5.2.15) ä k = +1 a c 3

97 (5.2.14) (5.2.15) (5.2.19) (5.2.20) ρ (5.2.14) ȧ 2 C/a + k = 0 (5.2.21) (C = 8πρa 3 /3 ) ȧ 2 C /a 2 + k = 0 (5.2.22) (C = 8πρa 4 /3 ) Robertson-Walker (5.1.11) P 1 τ 1 ω 1 P 2 τ 2 2 ω 2 2 (1), (4 3 ) (2) 4 u a k a ω = k a u a (5.3.1) ( (4.2.38) ) k a (5.3.1) C-3 ξ a Killing ( C ) (isometries) 1 t a t a ξ a

98 94 5 (5.3.6) 3 Killing ξ a k a P 1 Σ 1 k a P 2 Σ 2 k a P 1 Σ 1 ( / x) a k a ( / y) a = k a ( / z) a = 0 ( / y) a ( / z) a Killing P 2 0 k a P 2 Σ 2 ( / x) a ξ a = ( / x) a Killing ξ a P 1 ξ a P 2 ξ a Σ 1 Σ 2 a (ξ a ξ a ) 1/2 P1 = a(τ 1) (5.3.2) (ξ a ξ a ) 1/2 P2 a(τ 2 ) k a u a (k a ) Σ P 1 k a u a 1 = k a [ξ a /(ξ b ξ b ) 1/2 ] P1 (5.3.3) ω 1 = [(k a ξ a )/(ξ b ξ b ) 1/2 ] P1 (5.3.4) ω 2 = [(k a ξ a )/(ξ b ξ b ) 1/2 ] P2 (5.3.5) Killing (k a ξ a ) P1 = (k a ξ a ) P2 ω 2 = (ξb ξ b ) 1/2 P1 = a(τ 1) ω 1 (ξ b ξ b ) 1/2 P2 a(τ 2 ) (5.3.6) (5.3.2)

99 z z = λ 2 λ 1 λ 1 = ω 1 ω 2 1 = a(τ 2) a(τ 1 ) 1 (5.3.7) R τ 2 τ 1 a(τ 2 ) a(τ 1 ) + (τ 2 τ 1 )ȧ (5.3.8) z ȧ R = HR (5.3.9) a Hubble a(τ) τ P? Robertson- Walker P ( ) P P (particle horizon) Robertson-Walker Robertson-Walker ( 5.1 ) dt 2 = dτ 2 + a 2 (τ)(dx 2 + dy 2 + dz 2 ) (5.3.10) t = dτ a(τ) (5.3.11)

100 96 5 τ t (5.3.10) ds 2 = a 2 (t)( dt 2 + dx 2 + dy 2 + dz 2 ) (5.3.12) Minkowski (?) (conformally flat) (5.3.12) ds 2 = dt 2 + dx 2 + dy 2 + dz 2 (5.3.13) (5.3.12) 2 (5.3.13) P τ 0 t (5.3.11) τ 0 a(τ) ατ (α ) Robertson-Walker Minkowski Robertson-Walker t = constant Minkowski k = 0 a(τ) τ 2/3 P > 0 a(τ) Einstein Robertson-Walker (5.3.11) τ 0 τ 0 a(τ) (5.2.14) k

101 Robertson-Walker (a) (b) Robertson-Walker Robertson-Walker 2 Misner 1969 Robertson-Walker (inflationary phase) Robertson-Walker

102 Robertson-Walker a ( ) 1000 (5.2.19) (5.2.20) a 1000 a (k = 0, ±1) a ρ τ τ k = 0 a(τ) = (4C ) 1/4 τ 1/2 (5.4.1) 3 ρ = 32πGτ 2 (5.4.2) G c ρ ρ = n i=1 α i g i π c 5 (kt )4 (5.4.3)

103 n g i α i 1, 7/8 kt 0 (5.4.3) (5.4.1), (5.4.2), (5.4.3) T ρ 1/4 a 1 (5.3.6) ) ) t E a, (5.4.1) t E a/ȧ = 2τ (5.4.4) t I 1 nσc a3 σ τ 3/2 /σ(t ) (5.4.5) n a 3 σ σ (5.4.4) (5.4.5) σ t I t E, t I > t E (T ), ( ) t I t E a 1/1000 Robertson-Walker (G /c 3 ) 1/ cm

104 100 5 (τ = s ρ gcm 3 ) 2 T ab λg ab (λ ) (phase) de Sitter (1) (2) C CP ( ) (3) t E

105 τ = 1 ρ gcm 3 T K (ω ω/a) (T T/a) T 2K τ 1.5 1/6 ( 1MeV τ = 4 ρ = gcm 3 T K 0.5MeV 1.4 τ K 4 He 4 He ( K) ( 2 H) 2 H

106 K 4 He He 25% 4 He 2 H, 3 He, 7 Li 4 He 2 H 2 H He 2 H 25% % 4000K τ K 10 4 (recombination) T 2.7K ( ) Penzias Wilson (1965).

107 M ( M ) 10 5 M τ 10 3 τ 10 7 ( ) ? k = 0, 1 k = 1? 5 2 (5.2.14),(5.2.15) ȧ/a q P 0 q = äa/a 2 (5.4.6) H 2 = 8πGρ/3 k c 2 /a 2 (5.4.7) q = 4πGρ 3H 2 (5.4.8) Ω = 8πGρ/3H 2 (5.4.9) Ω q = Ω/2 (k = 1) Ω > 1 ρ > ρ c 3H 2 /8πG

108

109 (static) (spherically symmetric) Einstein Ricci 0 4 Lorentz (stationary) 1 φ t (time translation symmetry) Killing ξ a ( ) Σ (static) Frobenius Killing ξ a ξ [a b ξ c] = 0 (6.1.1) Σ ξ a 0 Σ Σ ξ a ξ a 0 Σ {x µ } ds 2 = V 2 (x 1, x 2, x 3 )dt µ,ν=1 h µν (x 1, x 2, x 3 )dx µ dx ν (6.1.2) V 2 = ξ a ξ a dtdx µ ξ a Σ

110 106 6 (6.1.2) t t t t + const. (6.1.1) ξ a (twist) (6.1.1) (spherically symmetric ) (isometry group) SO(3) 2 SO(3) r = (A/4π) 1/2 (6.1.3) (θ, φ) 2 ds 2 = r 2 (dθ 2 + sin 2 θdφ 2 ) (6.1.4) 3 r ( R 3 R S 2 r r (radial coordinate) ds 2 = f(r)dt 2 + h(r)dr 2 + r 2 (dθ 2 + sin 2 θdφ 2 ) (6.1.5) (6.1.5) (e 0 ) a = f 1/2 (dt) a (6.1.6a) (e 1 ) a = h 1/2 (dr) a (6.1.6b) (e 2 ) a = r(dθ) a (6.1.6c) (e 3 ) a = r sin θ(dφ) a (6.1.6d)

111 a [a (e 0 ) b] = 1 2 f 1/2 f (dr) [a (dt) b], (6.1.7) [a (e 1 ) b] = 0, (6.1.8) [a (e 2 ) b] = (dr) [a (dθ) b], (6.1.9) [a (e 3 ) b] = sin θ(dr) [a (dφ) b] + r cos θ(dθ) [a (dφ) b], (6.1.10) f = df/dr 1 2 f 1/2 f (dr) [a (dt) b] = h 1/2 (dr) [a ω b]01 + r(dθ) [a ω b]02 + r sin θ(dφ) [a ω b]03, (6.1.11) 0 = f 1/2 (dt) [a ω b]01 + r(dθ) [a ω b]12 + r sin θ(dφ) [a ω b]13, (6.1.12) (dr) [a (dθ) b] = f 1/2 (dt) [a ω b]20 + h 1/2 (dr) [a ω b]21 + r sin θ(dφ) [a ω b]23, (6.1.13) sin θ(dr) [a (dφ) b] + r cos θ(dθ) [a (dφ) b] = f 1/2 (dt) [a ω b]30 + h 1/2 (dr) [a ω b]31 + r(dθ) [a ω b]32, (6.1.14)