( )... ds.....

...... 3.1.. 3.1.. 3.1: 1775. g a m I a = m G g, (3.1) m I m G. m G /m I. m I = m G (3.2)............. 1

2............ 4.................. 4 ( )... ds.....

3.2 3 3.2 A B. t x. A B. O. t = t 0 A B t...... 3.2: A B. 3.2 {x α } ds 2 = g µν (x α )dx µ dx ν (3.3). g µν x α 10. g µν η µν g µν η µν. -. 3.3 A B. A B 4 δ B A ds = 0. (3.4) d δ. dδ = δd. λ L = 1 2 3.3: A B. 4. ( ) 2 ds = 1 dλ 2 g µνẋ µ ẋ ν, (3.5)

4 ẋ µ = dx µ /dλ x µ (λ) x µ (λ) + δx µ (λ). (3.4) B A 1 2 δldλ = 0. (3.6) L L < 0 L > 0. λ 4 s. (3.6) L δl. λ B B A A L B ẋ α δẋα dλ = A δl(x α, ẋ α )dλ = B A L ẋ α d(δxα ) = ( L x α δxα + L ) ẋ α δẋα dλ = 0. (3.7) ( ) B L B ẋ α δxα A A ( ) d L dλ ẋ α δx α dλ = 0. 3.3 A B δx α. L - ( ) d L dλ ẋ α L = 0. (3.8) xα. ds 4.. (3.5) - (3.8) d ( gαβ ẋ β) 1 g µν dλ 2 x α ẋµ ẋ ν = 0. (3.9) g µα g αβ = δ µ β ẍµ ẍ µ + Γ µ αβẋα ẋ β = 0, (3.10) Γ µ αβ = 1 2 gµν (g αν,β + g βν,α g αβ,ν ). (3.11) g µν,β = g µν / x β. Γ µ αβ (Christoffel). α β 40.. (3.9) g µν x α g αβ ẋ β. τ λ 4 u α. 2. 2 ds 2 = dr 2 + r 2 dθ 2.. (3.11)...

3.3 5 L = 1 2 (ṙ2 + r 2 θ2 ). - (3.8) (3.10) r r θ 2 = 0, θ + 2 r ṙ θ = 0. (3.12) Γ r θθ = r, Γ θ θr = Γ θ rθ = 1 r, (3.13).. (3.10) ẍ µ. ẍ µ. r r r θ 2. (3.12). (3.13).... (3.11). 3.3.. 19. 43. 19 20.... 20 10 7 43..... 1859 9 12 38.. p.364-365. ( ) 1915 11 18.

6 ds 2 = (1 2M r )c2 dt 2 + (1 2M r ) 1 dr 2 + r 2 (dθ 2 + sin 2 θdφ 2 ). (3.14). M = Gm/c 2 G m. (ct, r, θ, φ). 2M/r. M 1.5. 2M/r 2 10 8. 0.8 10 7. 4 10 6.. L = 1 2c 2 τ ( ) 2 ds = 1 ( 1 2M dτ 2 r ) ṫ 2 + 1 ( 2c 2 1 2M r ) 1 ṙ 2 + 1 2c 2 r2 ( θ2 + sin 2 θ φ 2) (3.15) t φ - (3.8) ( 1 2M ) ṫ = E, (3.16) r r 2 sin 2 θ φ = h. (3.17) E h. θ d dτ (r2 θ) r 2 sin θ cos θ φ 2 = 0. θ π/2. θ π/2 θ π/2. r. (3.16) (3.17) E 2 1 ( c 2 ṙ2 h2 c 2 r 2 1 2M r L = 1/2. ) = 1 2M r. (3.18). r φ. (3.17) τ., u = 1/r. (3.18) φ ṙ = dr du dφ du dφ dτ = h du dφ. d 2 u dφ 2 + u = Mc2 h 2 + 3Mu 2. (3.19) M = Gm/c 2 O(c 2 ) (3.19). (3.19)., u = 1 + e cos φ, p

3.4 7 p = h2 Mc 2, e. O(c 2 ). (3.19) O(c 4 ). O(c 2 e 2 ) d 2 u dφ 2 + u = 1 6Me cos φ + p p 2, 1 = +. p 1 p 3M p 2 p r = [( 1 + e cos 1 3Gm c 2 p ) ]. (3.20) φ φ 2π 6Gmπ/c 2 p.. 43.5.. (3.18) τ (3.17) r r φ 2 = Gm r 2 3Gmh2 c 2 r 4. (3.21).. 3.4 3.2. 3.1... (3.11).. A g αβ (A) = η αβ, g αβ,γ (A) = 0. (3.22) (3.22). ( ). η αβ... (3.22)

8 {x α } {x α }. (3.11). (3.10). Γ µ α β = xµ x µ x α x α. x β Γ µ x β αβ + xµ 2 x µ x µ x α x β = xµ x α x µ x α x β Γ µ x β αβ 2 x µ x a x α x β x α x β x β. (3.23) (3.23)... A (3.22) A. (3.10) A A d 2 x α = 0. (3.24) dλ2 λ 4 s. A x α x α = as + b x β = 0 (β α) a b s. (3.23) A. (3.22) A A 1 3 4 A. A. A. LGS. A A.. {x µ } g αβ A Γ µ αβ (A). A. {x µ } {x µ } Γ µ α β (A). (3.23) x µ = δ µ µ x µ + 1 2 δµ µ Γ µ αβ (A)xα x β. (3.25) 3.5... η αβ dt dτ. g αβ η αβ. dτ = g 00 dt g 00.

3.5 9 2.. 1. A B A B. A τ 1 τ 3 B. A B: τ 2 = (τ 1 + τ 3 )/2 A τ 2 B A.. A B B.... 4 4. 1 (ct, x i ). A B t A t B t A = t B........... (ct, x i ). ct = f(ct, x i ), x i = x i. (3.26)........ g αβ t x i (stationary).. (3.14) (3.43). g 0i (static).. (ct, x i ) g αβ. A B 2 (cdt, dx i )... ds 2 = 0 g 00 c 2 dt 2 + 2g 0i cdtdx i + g ij dx i dx j = 0

10 A B cdt = g 0idx i ± (g 0i dx i ) 2 g 00 g ij dx i dx j g 00. (3.27) g 0i = 0 g 00 < 0 cdt ±. A B dt G B A dt R cdt G = g 0idx i hij dx i dx + j, (3.28) g 00 g00 cdt R = g 0idx i hij dx i dx + j,. (3.29) g 00 g00 3.6. h ij = g ij g 0ig 0j g 00. A t t + dt G B t + dt G + dt R A. A t + (dt G + dt R )/2 B t + dt G A t B t + (dt G dt R )/2. (3.28) (3.29) A t B t g 0i dx i /cg 00. g 0i dx i = 0. A t B t. A B. A B B C A. A g0i t t dx i. g 00. g0i g 00 dx i = 0. (3.30) g0i g 00 dx i. (3.30) g 0i... (3.14).. g 0i.....

3.6 11 3.6.. 2.2 1. 1. 3.5 t.. 1 2. ( ) ( )..... 1... 2.3 1 1. 4 (x α, u α ). 1 1 T α. 1. 1 3 4 u α 3 u α. T α T α = T α + T α. (3.31) T α 4 uα T α. 2.3 T α = ( uβ c T β) u α c = ( 1c 2 uα u β ) T β. (3.32) T α 4, T β u β /c. T α = T α T α = ( δ α β + 1 c 2 uα u β ) T β. (3.33)

12. (3.32) (3.33) T α = πα β T β, T α = h α βt β, (3.34) π α β = 1 c 2 uα u β, h α β = δ α β + 1 c 2 uα u β. (3.35). (3.34). πβ α hα β 2 π αβ = 1 c 2 u αu β, h αβ = g αβ + 1 c 2 u αu β. (3.36). π αβ h αβ 1 3.. g αβ T α Kβ = (π αβ + h αβ )T α Kβ = π αβt α Kβ, (3.37) g αβ T K α β = (π αβ + h αβ )T K α β = h αβt K α β, (3.38) π αβ. h αβ. πν α h ν β = 0. (3.39). (3.35) u α u α = c 2. {x α } 4 u 0 = c g00, u i = 0. (3.40) u 0 = c g 00, u i = cg 0i g00. (3.41) h 00 = h 0i = 0, h ij = g ij g 0ig 0j g 00. (3.42). (2.18) ds 2 = (1 ω2 r 2 c 2 )c2 dt 2 + dr 2 + r 2 dθ 2 + dz 2 + 2ωr2 cdtdθ. (3.43) c ω.

3.6 13 π 3. 4. 4.. r = R. dr dx α (r) = (0, dr, 0, 0). h αβ dx α (r) dxβ (r) = h rrdrdr = dr 2, (3.42) (3.43). 2 R 0 dr = 2R. r = R dx α (θ) = (0, 0, dθ, 0) h αβ dx α (θ) dxβ (θ) = h θθdθdθ = 2π π/ 1 ω 2 R 2 /c 2 π. E 0 R 2 1 ω 2 R 2 /c 2 dθ2. R 1 ω2 R 2 /c dθ = 2πR 2 1 ω2 R 2 /c. 2 (3.14) (3.16) E. E. (3.14) (3.41) 4 u α = ( c 1 2M/r, 0, 0, 0). (3.44) 1 4 p α = (cṫ, ṙ, θ, φ), = d/dτ τ. 4 c. (2.17) (3.32) En En = p α u α = u 0 p 0 = c 2 dt/dτ (3.16) E 1 2M r dt dτ. Ec 2 En =. (3.45) 1 2Gm c 2 r r En = Ec 2 Ec 2. (3.45) En Ec 2 r Albert Einstein, The meaning of relativity, fifth edition, 1954, Princeton University Press, p.59-61

14 En. En. r En.. En = hν. r... 3.7 3.1. 3.2 (3.23). 3.3 Robertson-Walker ( ) dr ds 2 = c 2 dt 2 + R 2 2 (t) 1 kr 2 + r2 dθ 2 + r 2 sin 2 θdφ 2, (3.46) k. (1). (2). (3). ( ds 2 = c 2 dt 2 + R 2 (t)γ ıj dx i dx j x 0 = ct Ṙ = dr/dt.) Γ 0 ij = γ ijrṙ/c, Γi 0j = δi jṙ/(cr), Γr rr = kr/(1 kr 2 ), Γ r θθ = r(1 kr2 ), Γ r φφ = r(1 kr2 ) sin 2 θ, Γ θ rθ = Γφ rφ = 1/r, Γθ φφ = sin θ cos θ, Γφ φθ = cos θ/ sin θ, 3.4 (3.17) r 2 sin 2 θ φ = h. ( φ.) 3.5 ds 2 = g αβ dx α dx β x α.