m d2 x dt 2 = K( x), d 2 x j dt 2 = K i.

Transkript

1

2 P

3

4

5 m d2 x dt 2 = K( x), m δ ij d 2 x j dt 2 = K i.

6 C W C = K i dx i C δ ij δ ij λδ ij, m m λ d v dt K BA = K AB

7 R 4

8 E 3 R Σ Σ x = R x a, R T R = I, R... E 3 T 1, = 7

9 E 3 = O 3 T 3,... E 3 O 3 T 3 P 1 P 2 ((t 1, x 1 ) ((t 2, x 2 ) t = t 2 t 1 x = x 2 x 1 t x i 2 ψ(t, x) = ψ(t, x) t 2m t = t t t, x i = x k x i x k = Rk i x k i t ψ (t, x ) = 2 2m ψ (t, x ) ψ (t, x ) = ψ (t(t ), x( x ) R E 3 T 3 x = R x a vt t = t b

10 x = x vt t = t (O 3 (T 3 G 3 )) T 1, = 10 G 3... t t x t=0 d2 x dt t d 2 x 2 dt t 2 λt, x ±λ 2 x

11 c c c I I v I I v I I I v I I v

12 I I v v I I v = f ( v ) v v v v = v f ( f ( v ) ) = v g f R + g(0) = 0 g = f ( v ) = ± v, I I L I l < L l < L I I u g g =

13 l = L vt, L = l + vt, L w = T u = l T L < l T l l = L L =: α α = α( v ) l L α(0) = 1 α = L vt l = l + vt L vt = L αl vt = αl l T T = αl l L αl 1 α 1 u = l T, v = αl l T u + v = αl T, u v = l αl l w = L T = L αl l T L αl = αl T αl l α(l αl) = u + v 1 + (1 α 2 ) u, v 1 + (1 α 2 ) u v = 1 + (1 l α2 ) αl l = αl α2 l αl l) α = 1 L = L T = T w = u + v α > 1 w u α < 1 v α 2 1

14 w = ϕ(v, u) w w = ϕ( u, v) w = ϕ(v, u) (1 α 2 ) u v, α = α(v) (1 α 2 (v)) u v = (1 α2 (u)) v u 1 α2 (v) v 2 = 1 α2 (u) u 2 K 1 α 2 = Kv 2, K 1 c2 K = 0 c c α = 1 v2 c 2 α = L L L = 1 v2 c 2 L < L P 1 P 2 I l = 0 T = αt ( ) =

15 w = u + v 1 + (1 α 2 ) u v = u + v 1 + uv c 2 u = c w = c v c v < c α u > c u = c2 v w = v > 0 u > c w = v = c2 u w = w = P 1 P 2 L = l u = w = c2 v L > l > l P 2 P 1 λν = c v c E = c v ν E = c E /λ = c λ (1 v c ) = ν S(1 v c )

16 λ E = (c + v)τ S = c+v ν S ν E = c E /λ E = ν S 1+ v c v v r v δ ν E = ν S 1 v2 c v r c δ ν S v = v r v r = 0 ν E = ν S 1 v c 1 + v c ν E = ν S 1 v2 c 2

17 P 1 0 I I P 1 P 2 L = l + vt α T = L αl v = l + vt α 2 l αv = T + 1 v ( 1 α 2 ) l α α = 1 v2 c 2, t T, x L, t T, x l x = x + vt, t = 1 v2 c 2 γ := 1 α = 1 1 v2 c 2 t + v c 2 x 1 v2 c 2

18 x = γ (x vt) ( t = γ t vx ) c 2 ( ) x = f x + at + b x c t = gt d x a, b, c, d v v y = f(v)y f(v) = f( v) y = f( v)y f 2 = 1 f = 1 y = y z = z c v x = x + x x = x v v 2 v x = x x ( x v = 0)

19 (v, 0, 0) v, (x, 0, 0) x, (0, y, z) x ( t = γ t ) v x c 2 x = x + x = γ ( x vt ) + x = x + (γ 1) x γ vt = x + γ 1 v 2 ( v x) v γ vt v c d x dt = c d x dt = c c 2 ( t) 2 ( x) 2 = c 2 ( t ) 2 ( x ) 2 v = (v, 0, 0) t = γ ( t v ) c 2 x x = γ ( x v t) c 2 ( t 2) ( x ) [ 2 = γ 2 (c 2 v 2) ( ) ] ( t) 2 v 2 + c 2 1 ( x) 2 = c 2 ( t 2) ( x) 2 ( s) 2 := c 2 ( t 2) ( x) 2 ( t) 2 = 1 c ( s)2 x=0

20 ( l) 2 = ( s) 2 t=0 ds 2 = c 2 dt 2 (d x) 2 ds ds 2 ds 2 { } }{{} { } }{{} = }{{} x 0 = ct c = c = / c = 1 c = 1 t = γ (t vx) t = vx x = γ (x vt) x = vt v = δ t 2 x 2 = t 2 x 2

21 t = iτ τ 2 + x 2 = τ 2 + x 2 τ, x τ = τ φ x φ i x = τ φ + x φ φ ϕ = iχ t = t χ x χ x = t χ + x χ χ = v φ = iχ

22

23 Q I I P v = 1 γ v > 1 x = iξ, t = iτ ξ τ v 1 v > 1 v > 1 v s = v s = 1 + ϵ c =

24 ω dω k dk > 1 ω(k) = k k 1 P = {Q : ( s) 2 = 0} P P P =

25 P P I x t = 0 x = γ(x vt) x = γ x x = γ 1 x = 1 v 2 x L = x > x = L y = y, z = z c

26 t = l c v = y, 1 v 2 = y y = ( v c ) I x = 0 t = γ t t = t 1 v 2 I x = 0 t = γ t t = t 1 v 2 T = T + T = T 1 v 2 T T v2 2 v 1

27 v T = d v T d 2 v 0f.v 0 d A B A B t A t B = t B T A t B = 1 2 (t A + t A ) ds 2 = dt 2 d x 2 = dt 2 d x 2 = dt 2

28 P ds ds ds = 1 v 2 dt s 1 < s 2 T 2 = T 1 1 v 2 T 2

29 v = 0.994c µ γ = 29.3, b g m/s γ = γ SR (1+αβ 2 +α 2 β 4 ) α , α v v t = γ(t + v x ) x = x + γ 1 v 2 v( v x ) + γ vt

30 x = ut [ x = u + γ 1 ] v 2 ( v u) + γ v w = x t = u γ + γ 1 γv 2 v( v u) + v 1 + v u ( γ 1 γv 2 = γ ) γ + 1, γ2 1 = y 2 v 2 v, u, w u, v, w w R w R w v u v u w = w( v, u) =: v u u v w = u+ v 1+ v u u v w = v + 1 γ u t

31 δ v v δ u v + δ v = v δ u v = (v, 0, 0) I δ u, δ v I v, δ v, v + δ v γ( v + δ v) γ( v) I I (1 γ)δθ I δ u I I I I I δθ T = (γ 1)δΘ I I

32 ω T = dθ T dt = (γ 1) dθ dt I I I I δθ = v ( v + δ v) v v + δ v = v δ v v 2 δ Θ = v δ v v 2 γ 1 (v 0) v2 2 w T (v 0) 0 v v ω T = (γ 1) v 2 w T

33 P x = (x 0, x 1, x 2, x 3 ) = (x i ) = (x 0, x) P, Q x, y y x P Q ( s) 2 = ( x 0 ) 2 ( x) 2 ( x) = η ik x i x k i=0 k=0 x = η ik x i x k +1 0 η = (η ik ) = u, v uv = 1 [ (u + v) 2 u 2 v 2] = η ik u i v k = u 0 v 0 u v = vu 2 uv = 0 v u = 0 x i = L i k xk a i x i = L i k xk

34 u = (u i ) u i = L i k uk u 2 = u 2 u v = uv u, v uv = 0 u u 2 > 0 u 2 = 0 u 2 < 0 u 0 > 0 u u 0 < 0 u t t u 0 = γ ( u 0 v u ) > 0, u 0 > 0, v u v u < u u 0 u ( u = ± ) u 2, 0 t u u = u ( 0, ) u 2, 0, 0 x u = (±1, 1, 0, 0)) (±a, a, 0, 0) u i = λu i, λ = γ(1 v) = 1 v 1 ± v ) v u u v v u u v v = λu

35 P P = {f x i = f i (x), η ik dx i dx k = η ik dx i dx k} η mn f m x i f n x k = η ik x j ( ) f m x i = ±1 f ( 2 f m f n η mn x j x i x k + f m x i ( 2 f m f n η mn x i x k x j + f m x k ( 2 f m f n η mn x k x j x i + f m x j f η mn = η nm m n 2 2 f n ) x j x k = 0 (ijk) ( ) 2 f n ) x i x j = 0 (kij) () 2 f n ) x k x i = 0 (jki) ( ) ( 2 f m f n η mn x j x i x k + f n 2 f m ) x k x i x j = 0 f n 2 f m η mn }{{ x k } x i x j = 0 A mk ( ) f n (A mk ) x k 2 f m x i x j = 0 f i x k Li k x i = f i (x) = L i k xk + c i

36 ( γ L = (L i k ) = b ) a T M η mn L m i L n k = η ik L T ηl = η ( ) ( γ b T 1 0 T ) ( ) ( γ a T 1 0 a M T 0 1 = T ) ( 1 0 T ) b M }{{} b T γ a M T ( ) L 1 γ b T = a M T ( ) ( ) ( γ a LL 1 T γ b T 1 0 = b M a M T = T ) 0 1 γ 2 a 2 = 1, γ b + M a = 0 L 1 x = 0 x 0 = γx 0 x = ax 0 I I v = x x 0 = a γ γ 2 a 2 = 1 v = a = a 1 + a 2 < 1 v γ = ± 1 1 v 2 1 v 2 L = L R L v ( ) ( ) γ γ v T γ a T L v = γ v 1 + γ2 = v v T a 1 + a at 1+γ 1 + γ }{{} γ 1 v 2 L R = LL 1 v = LL v ( ) ( ) γ a T γ a T = b M a 1 + a at 1+γ ( ) γ 2 a 2, γ a T a T a2 1+γ = at γ b + M a, b a T + M + M a at 1+γ ( 1 0 = T ), 0 R

37 a γ = γ 1 M a = γ b ( 1 + γ ) b a T = b a T 1 + γ 1 + γ R = M b a T 1 + γ L R x 0 x 0 ds 2 d x 2 L 1 I I b γ I I R a = M a b γ a2 1 + γ b = γ b γ γ b = b = R v L u = ( γu γ u u T γ u u 1 + γ2 u 1+γ u u u T L u L v L w( v, u) u v ), L v = ( ) γ a T L := L u L v = b M ( γv γ v v T γ v v 1 + γ2 v 1+γ v v v T ) γ = γ v γ u (1 + v u) γv 2 a = γ v γ u v + γ u u + γ u ( u v) v = γ v + 1 γ v u γ v γv 1+γ v ( v u) v } 1 + v u {{ } w( v, u)

38 b = γ w( u, v) M = 1 + ( ) γ2 v v v T + γ2 u u u T γ v γ u + γ v γ u 1 + u v u v T 1 + γ v 1 + γ u (1 + γ v )(1 + γ u ) L u v L = L R L w( v, u) ( = LR w( v, u) L R = L w( u, v) L R ) R = M b a T 1 + γ u, v R R = v T +... u T v u v u = v γ γ u w, a = γ u u +... v = γ w( v, u) v w u 1 u γ u γ2 v 1 + γ v v v T + γ v u v T b = a( v u) γv ( u + v) ( b a T γv u v 2 T + γ v v γ v v T + u T + = γ 2 v u v T + γ v v u T + γ 2 v R = M 1 ( 1 + γ b a T 1 + γ v ( γ γ v (1 + v u) }{{} u 1+γ u 1+γv ) γ2 v ( u v) v T 1 + γ v ) v v T ( 1 + γ v u v 1 + γ v γ2 v 1 + γ v ) u v T γ v v u T 1 + γ v ( ) ) 1 u v 1 + γ v (1 + v u) γ2 v 1 + γ v = γ2 v 1 + γ v 1 + γ v

39 = 1 + γ v 1 + γ v ( u v T v u T ) R x = x a n x x + γ v 1 + γ v [ u( v x) v( u x)] }{{} ( v u) x γ v α = α n v u γ2 v w v 1 + γ v 1 + γ v w v γ2 v v I a = v v ( w = v + v) 1 + γ v t 0 : ω T = γ2 v v d v 1 + γ v dt γ < 0 γ 1 R = 1 T = P = L L = ±1 L + L = +1 ( ) ( ) L L 0 0 > 0 L0 0 1 (L0 0 )2 L α 0 Lα 0 = 1 L + L + := L + L T P

40 x(t) v = d x dt v < 1 : s x i (s) u i := dxi ds dxi ds = = u = ( dt ds, d x ) = dt ( 1, d x ) = γ(1, v) v α = uα ds ds dt u 0 s = t u = (1, 0) v 1, γ 1 u (1, v) u 2 = η ik dx i ds dx k ds = ds2 ds 2 = 1 u u dt ds > 0 a i := dui ds = d2 x i ds 2 0 = d ds (η iku i u k ) = 2η ik u i a k ua = 0 a a = (0, b), b = d v dt b a 2 = b 2 = b 2 b = (b, 0, 0) u(s) = ( f(s), f(s), 0, 0) a 2 = f 2 (s) = b 2 f = ±b(s s 0 ) = bs

41 x 0 (0) = x 1 (0) = 0 s ( 1 x(s) = u(s )ds = b (bs), 1 ) ((bs) 1) b = = m p = mu (m, m v) p (p 1, p 2, p 3 ) = γm v ym p 0 p 0 = m 1 v 2 v 1 m + mv2 2 p 0 E = m 1 v 2 =

42 E 0 = mc 2 = p 0 v 1 p 2 = (p 0 ) 2 p 2 = m 2 p 0 = m 2 + p 2, v = p p 0 = p m 2 + p 2 = p E dp ds = ma = K Ku = 0 K = d p ds = γ d p dt K 0 = dp0 ds = γ de dt = γ

43 N A=1 p A = N p 1 + p 2 = p 3 + p 4 ( ) p 0 1 A = m A + T A, T A = m A 1 1 v 2 m 1 + T 1 + m 2 + T 2 = m 3 + T 3 + m 4 + T 4 p 1 + p 2 = p 3 + p 4, p A = m A v A 1 va 2

44 m 3 = m 1, m 4 = m 2 T 1 + T 2 = T 3 + T 4, T 4 = 1 2 m A v A 2 p A = m A v A p = q 1 + q 2 p 2 = q q q 1 q 2 q 0 1 q0 2 q 1 q 2 > 0 q A = m 2 = m (q 0 1q 0 2 q 1 q 2 ) (q 0 A )2 m 2 A < q0 A A m2 A < m2

45 m = 0 p i = m dxi ds = 0 m 0 s 0 v 1 u x p p 2 = 0 p = ( p, p) 0 p = k, k = (ω, k) ω = k = 2π λ = 2π λν = c k I (v,0,0) I ϑ 1 k = ω ϑ ϑ 0

46 1 γ k = ω ϑ ϑ = ω γv 0 γv γ 1 1 ϑ ϑ 1 0 ω = γω(1 v ϑ) ω ϑ = γω( v + ϑ) ω ϑ = ω ϑ) ( ) () ( ) / : ϑ = ϑ v 1 v ϑ / : ϑ = ( ϑ 2 = ϑ v γ(1 v ϑ) ) 1 + v 1 v ϑ 2 2 ϑ + 2 ϑ = 1 v ϑ ϑ ϑ = π 2 ϑ = ( π 2 ϑ ) α = v α 20 α = k ϕ k 0, 3 α = v c v 1 ϑ = 180 v, k

47 ω ω α > α e γ p + q = p + q m 2 = q 2 = (p p + q) 2 = p 2 + p 2 2pp + +2q(p p ) + q 2 q(p p ) = pp e q = (m, 0), p = (ω, k)

48 m (ω ω ) = 2 ωω (1 ϑ) ( ϑ = ( p, p ) ) 2π ωω ω = 2π λ λ λ = h (1 ϑ) m h mc mc Q B p L e n p + p p + p + π 0 e + + e 2γ π 0 2γ, n p + + e + ν e = (E pot + E kin p + e + 2γ E B = m p + m e m 1 2 m eα 2 = 13, 56 (α = e2 c 1 ) 137 M M = E B m p + m e E B E B m p = 1 2 α2 m e m p 10 8

49 M M 10 2 M M 0, 4 X? m A x A X P ( ) d x a d x a p A = m A m A ds A dt X(t) p 0 A x A (t) p 0 A C X = p p 0 C v, v C p 0 2 > p0 1 I CM : A p A = 0 I CM l ik = x i p k x k p i x i l α = ϵ αβγ x β p γ l i = ϵ ijmn u i l mn ϵ ijkl ϵ u A lik A

50 S[l] L l = P2 P 1 ds l T2 S[x(τ)] = m ẋ2 dτ T 1 Ldτ ( d ) dτ τ S τ f(τ) x x f ẋ2 ẋ 2 f S f τ 2 > τ 1 τ 2 < τ 1 x 0 τ :, x 0 τ : τ = t L = mc 2 1 v2 c 2 mc2 + m 2 ( ) d x 2 dt S[x(λ)] = 1 2 λ2 λ 1 ( ) dx dλ dλ λ aλ + b

51 d2 x dλ 2 = 0 m > 0 λ s m S = S λ π i = L ẋ i = mη ikẋ k dx k ẋ2 }{{} = mη ik ẋdτ=ds ds ( πα = p α, π 0 = p 0) V = n 1 = 1 f : V R, v ±ϵ f(αv + βw) = αf(v) + βf(w) ( 0, ϵ 0) f(v) = f i v i V := { }

52 (αf + βg)(v) = αf(v) + βg(v) V = V = v uv u (v) = uv = η ik u k v i = u i v i V V u i := η ik u k ϵ u u u i..., u i... u uv = u i v i = u i v i η 2 = η il η lk = δ i k (u i ) = (η ik u k ) = (u 0, u) u i = η ik u k, η ik (η 1 ) ik = η ik u i = η ik u k = η ik L k l ul = η ik L k l ηlm u m = ( (L T ) 1) m u i m =: Li m u m, L T ηl = η (L T ) 1, η ηlη = (L T ) 1 x i x i

53 v(t, x) u(x) ϵ(t, x) π(t, x) ϵ π P 0 (t) = d 3 x ϵ(t, x) P (t) = d 3 x π(t, x) P = (P 0, P ) φ φ (x ) = φ(x) φ = φ g 1 g P v i (x ) = L i k vk (x) v i (x ) = L k i v k (x) φ x i iφ φ,i (= (dφ) i ) φ φ x k = x i x k x i = L i k φ x k dx i = L i k dxk dx k = ( L 1) k i dx i

54 xk x i = ( L 1) k i = L i k d 3 x Σ {f(x) = } df 0 f,i n i (x) : n i (x)v i = 0 x n 2 = ±1 n n n 2 = 0 Σ n Σ Σ Σ v A V (v 0, v 1, v 2, v 3 ) V (v 0, v 1, v 2, v 3 ) V (e 0, e 1, e 2, e 3 ) = 1 {e i } e 0 = (1, 0), e 1 = (0, 1, 0, 0), e 2 = (0, 0, 1, 0), e 3 = (0, 0, 0, 1) {e i } v A = v j A e j

55 v v 0 3 V (v 0,..., v 3 ) = = ϵ ijkl v i v v3 3 0 v j 1 v 2 k v3 l +1 (ijkl) (1234) ϵ ijkl = 1 (ijkl) (1234) 0 ϵ dv = V ( dx 0 e 0, dx 1 e 1, dx 2 e 2, dx 3 e 3 ) = dx 0 dx 1 dx 2 dx 3 d 4 x dv = dx 0 dx 1 dx 2 dx 3 σ(v 1, v 2, v 3 ) := V (n, v 1, v 2, v 3 ) n v 1, v 2, v 3 n n v 1, v 2, v 3 σ = 0 n i ϵ ijkl v j 1 v 2 kv 3 l ( Σ ) x i (ξ 1, ξ 2, ξ 3 ) ξ H H 3 n, x, x, x ξ 1 ξ 2 ξ 3 x Σ dσ = σ(dv 1, dv 2, dv 3 ) dv i α = xi ξ α dξ α dσ = n 0 n 1 n 2 n 3 x0 x 0 x 0 ξ 1 ξ 2 ξ 3 x1... ξ 1 x3 ξ 3 } {{ } χ( ξ) dξ 1 dξ 2 dξ 3 }{{} d 3 ξ d 3 ξ= (ξ) (ξ d 3 ξ ) n i dσ i x xi ξ α ξ α x i ( ξ).

56 dσ i = ϵ ijkl x j ξ 1 x k ξ 2 x l ξ 3 d 3 ξ L = +1 χ( ξ) n x j x k x l ϵ ijkl ξ 1 ξ 2 ξ 3 =,... x i,... x 3 ) ± (x0 (ξ 1, ξ 2, ξ 3 ) (dσ i ) 0 ( v, x ) x x ξ 1 ξ 2 ξ 3 = 0 v v i dσ i = 0 dσ i n i dσ i = ±n i dσ Σ df = n d F d F = ndf E 3 dv dσ t Σ = I A t x α, x β dσ 0 = 0, dσ α = dtdf α df α = ϵ αβγ dx β dx γ dσ = n i dσ i = dt n d F = dtdf, dσ α = n α dtdf... d 4 x,... d 3 ξ G f Σ v Σ G f dσ = f dv = H f G f(x) d 4 x R 4 ( ( )) ( ) x ξ χ ξ d 3 ξ R 3 dσ i n dσ = 1 ϵ 2 αβγdx β dx γ dx β dx γ (xβ,x γ ) (ξ 1,ξ 2 ) dξ1 dξ 2

57 Σ v i dσ i = H ( ( )) v i x ξ x j x k x l ϵ ijkl ξ 1 ξ 2 ξ 3 d3 ξ R 3 H A 3 Σ S 2 i v i dv = G G v i dσ i G n i G n i n i t i > 0 t n i G = I A v i dσ i = v n dt df n G G G t2 0 v 0 dv = d 3 ( x) x 0 v 0 dt G H 0 t 1 ( x) = d 3 x [ v 0 (t 2 ( x), x) v 0 (t 1 ( x), x) ] H 0 = v 0 dσ 0 G

58 α v α }{{} G xα dv = dtdf α 2 (t,x β,x γ ) α v α dx α H α x α 1 (t,xβ,x γ ) = dtdf [v ( α α t, x α 2 (t, x β, x γ ), x β, x γ) ( v α t, x α 1 (t, x β, x γ ), x β, x γ)] H α = dtv α df }{{} α G n αdf = G v α dσ α G G α v α dv = Σ i v G α dσ α = i G v α dσ α Q Q(t) = d 3 x ρ(t, x) }{{} = {x 0 =} dσ 0 ρ ( = ) dσ i j i, ρ j 0, dσ α = 0

59 Σ Q Σ = Σ dσ i j i Q Σ Σ dσ i dσ i j i j ρ = j 0 j Σ = I A A dσ α = dt df α = dt n α df n A R 3 t2 Q Σ = dx 0 j df t 1 A = t 1 t 2 A n j α = e α α I A dq dt = j df df j i Q Q Q Σ j j Q i j i = 0 j S j = 0 S Σ 1 Σ 2 0 = G i j i dv }{{} = j i dσ i = G j i dσ i Σ 2 j i dσ i Q Σ2 = Q Σ1 Σ 1 Σ 2 Σ 1 Σ Σ = Q Σ2 dq Q Σ1 Q dt = Σ ρd3 x = Σ jd 3 x = Σ j df = 0

60 Σ 1 Σ 2 0 = Q Σ2 Q Σ1 = G d 4 x i j i i j i = 0 x G x Q {t=0} = Q {t =0} I, I

61 Σ j ( 0 = G 1 G 2 ) d 4 x i j i = Q Σ Q Σ j Q j i,i = 0 j i = j i + f ik,k f ik = f ki j i,i = 0 Q = Q f ik Σ dσ i f ik,k }{{} = dσ ik f Σ }{{} ik = 0, P i PΣ i = Σ T ik (x)dσ k P i Σ Σ Σ T ik n k n T ik T ik u k = Li mt mn u n = L i mt mn (L T ) k n u k = Li ml k nt mn u k u T ik = L i ml k nt mn T ik (x ) = L i ml k nt mn (x) : T ik (x) j T 00 = ϵ T 0α = S α, S

62 T α0 = π α, π T αβ = α ( e β ) = α ( e β ) = I A T ik A dp α dt = T αβ df β T αβ T αβ = T β α n n T αβ = pδ αβ, p > 0 K = p ndf p T αβ = T δ αβ, T > 0 T K n T αβ T αβ v 0 > 0 d F v K α = 2T αβ df β K > 0 m v T αβ v K α = T αβ df β 2 < 0 T αβ T ik T ik

63 S u(x) v = ϵ S v ϵ = i j i = 0 Q Σ Σ Σ T ik,k = 0 P Σ i Σ Σ T ik,k = 0 T ik L ik Σ = t=const d 3 x(x i π k x k π i ), π i = T i0 kovariant L ik Σ = Σ (xi T kl x k T il )dσ l Σ T ik,k = 0 T ik = T ki, L ik (x i T kl x k T il ),l = 0 T ki + x i T kl,l T ik x k T il,l = 0 T ik = T ki T αβ = T βα T ik T ki S ik = s ikl dσ l s ikl = s kil J ik = L ik + S ik J ik T ik,k = 0 T ki T ik + s ikl,l = 0 u i (x) T αβ T αβ = p δ αβ, S =0, n =0 ϵ 0 T ik = p p 0 p

64 ϵ p T ik = (ϵ + p)u i u k pη ik ϵ, p = 5 T ik,k = 0 p = p(ϵ) 5 T ik,k = 0 ṗu i + (ϵ + p) u i p,i = 0 : u k k p = 0 T ik = ϵu i u k u i = 0 T ik,k = 0 n i (x) = n(x)u i (x), n(x) = N = Σ ni dσ i, Σ n i,i = 0 n(x) u i n(0, x) n Σ Σ s(x) de = T ds pdv v 1 n e = ϵ n x ϵ =

65 S s i E 1 T = S E, E = p0 1 T kβ i S p i = T = 1 v 2 T ds = δq T β i δq i rev δq E f M (v) v 2 e mv2 2kT f J (v) γ 5 e mγ kt ( 0 f J (v)dv = 4π 0 dp p 2 e mγ kt = 1, p = γmv) β i p i (p, q) T : V V }{{} V V R }{{} p q w = w i e i, v = v i e i, e i (e j ) = δ i j T (w (1),..., w (p) ; v (1),..., v (q) ) = T i 1...i p j 1...j q w (1) i 1... w (p) i p v (j 1) 1... v (j p) q }{{} p + q L GL(V ) T i i...i p j 1...j p = L i 1 m1 L ip m p L n 1 j 1 L nq j q T m 1...m p n1...n q L = (L i k ), L i k = ((L T ) 1 ) i k

66 p = 0, q = 0 p = 1, q = 0 V = V : v(w ) = w (v) p = 0, q = 1 V p = 1, q = 1 A i j T i...j... i j = ±T j...i... αa βb = C C = αa + βb T (p,q) = V V V V }{{} V V }{{} p V q p q v (1) v (p) w (1) w (q) v (1) w (q) (x (1),..., x (p), y (1),..., y (q) ) := v (1) (x (1) ) v (p) (x p )w (1) (y (1) ) w (q) (y (q) ) T (p,q) T = T i 1...i p j 1...j q e i1 e ip e j 1 e j q T (p,q) = n p+q n = V (e i1 e ip e j 1 e j q ) (e ip +1 e ir e j q+1 e j s ) = e i1 e ir e j 1 e j s : A B = C, A T (p,q), B T (p,q ) C T (p+p,q+q ) ) C i...j... k...l... = A i... k... Bj... l... V = V T = p,q=0 T(p,q) = R V V V V V V V V... R = V 0 id T = T T...i......i... } {{ } (p,q) = Q }{{} (p 1,q 1)

67 T i i A i...j... k... B m... l...j... } {{ } (p,q) } {{ } p,q i...ĵ...m = C k...l...ĵ... }{{} (p+p 1,q+q 1) u i v i = u (v) = v(u ) u i = η ik v k C i...k j...l = Ai...km... j...ln... Bn... m... B (p, q) (p, q ) (p + q, q + p) A ik = T ik, p = 1, q = 1 v i u i v u (1, 1) x i = T i k xk ) : T i k = δi k T ik = δ ik! δ i k V = V = V T ik = η im η kn T mn, T i j = η jkt ik, T ik = η im η kn T mn η ij = η ik η jl η kl η 1 = η 1 η(η 1 ) T η ab η ab = L a i Lb kη ik = η ab η = (L 1 ) T ηl 1 = η L 1 L T ηl = η L + L L\L + ϵ abcd ϵ abcd = L i a L j b L k c L l d ϵijkl = ( L)ϵ abcd ϵ abcd = ϵ abcd

68 ϵ abcd = ( η)ϵ abcd = ϵ abcd ϵ abcd ϵ ijkl δ T i... k... (x ) = L i m Lk n T m... n...(x) j T i... k... (x) = Di... jk... (p, q + 1) G d4 x i T i... k... = G dσ it i... k...

69 E, B [q] = M 1 2 L 3 2 T 1 [ E] = [ B] ( d p dt = q E + 1 ) c v B E = 4πρ B = 0 E + 1 B c t = 0 B 1 E c t = 4π c j B d F = 0 K 12 = q 1q 2 r 2 ( B = 0 ) ( ( E + A) = 0 ) B = A E = ϕ A }

70 ϕ t + A = 0 E = ϕ A = 4πρ B = A A = ϕ A + 4π j ϕ = 4πρ A = 4π j = 2 t 2 A A = A + Λ ϕ ϕ = ϕ Λ t } E, B 0 = ϕ t + A Λ = ϕ t + A Λ Λ = 0 = η ik i k = 2 A i = (ϕ, A) A i A i i Λ i A i = 0 A j = (ρ, j) A j A i = 4πj i e iq A i dx i A i,k k Ak i,k = 4πji E α = α A 0 0 A α = α A A α B α = β A γ γ A β = β A γ + γ A β α, β, γ F ik := i A k k A i

71 0 E 1 E 2 E 3 (F ik ) = 0 B 3 B 2 0 B 1 0 F ik = η im η kn F mn 0 E 1 E 2 E 3 (F ik ) = 0 B 3 B 2 0 B 1 0 F ik F ik (F ik ) E 2 B ( ) 2 2 E B k F ik ( = k i A k k A i) = A i i A i = 0 k F ik = 4πj i i j i = 0 F ik,l + F kl,i + F li,k = 0 df = 0, d F = 4π j ( ) q E + v B K i = qf ik u k ( ) = qγ E v, E + v B K i u i = 0 dp i ds = Ki d p dt = 1 γ K p

72 F = (F ik ) F = LF L T E 1 = E 1 E 2 = γ(e 2 vb 3 ) E 3 = γ(e 3 + vb 2 ) B 1 = B 1 B 2 = γ(b 2 + ve 3 ) B 3 = γ(b 3 ve 2 ) v E (x ) = E (x) ( E (x ) = γ E (x) + v B ) (x) B (x ) = B (x) ( B (x ) = γ B (x) v E ) (x) E i = F ik u k, B i = 1 2 ϵ ijklf jk u l F ik = 1 2 ϵ ikmnf mn 0 B 1 B 2 B 3 Fik = 0 E 3 E 2 0 E 1 0 F = F : E B, B E F ik,k = 0 F ij,k + F ki,j + F jk,i = 4πϵ ijkl j l F ik,k = 4πji, F ik,k = 4π j i, B = 4π ρ, F ij,k + F ki,j + F jk,i = 4πϵ ijkl j l Fij,k + F ki,j + F jk,i = 4πϵ ijkl j l E + B t = 4π j, i j i = 0 j j j j j

73 A Ã F ik = i Ã k k Ã i B = ϕ K i = qf ik u k qf ik u k d p ( ) ( ) dt = q E + v B + q B v E q q q q = n 2 c j G = I V I dp i = = dt V ( = dt T ik,k d4 x G ( t πi + T iα,α ) d 3 x π i ) d 3 x + V T iα df α V t }{{} V T ik,k = κ i! = = T ik,k κ i = F ik j k j = A ρ A u A ρ A κ = ρ E + j B = q( E + v B)δ (3) ( x z(t)) κ 0 = j E = v κ j = q vδ (3) ( x z(t)) j i = qn i

74 F ik (x)q dτż k δ (4) (x z(τ)) τ=z0 = qf ik v k δ (3) ( x z(t)), v k = (1, v)) F ik j k = + 1 4π F ik F l k,l = 1 4π (F ik F l k ),l 1 4π 1 F i k,l }{{} (F i 2 k,l F i l,k }{{} (F ik,l +F li,k = F kl,i ) = F,i kl = 1 4π (F ik F l k ηil F km F km ),l T ik = 1 4π (F ij Fj k 1 4 ηik F mn F nm ) f ikl,l f ikl = f ilk T ik T ik T ik,k + T ik,k = 0 L ik + L ik ) F kl A i = 4πj i G G(x) = δ (4) (x) A i = 4π d 4 x G(x x )j i (x ) G(x) = 1 (2π) 4 k 0 = ± k d 4 k e ikx k 2 G (x) = δ(r t) 4πr = Θ(t) δ(x2 ) 2π

75 A(x) = 4π d 4 x G (x x )j(x ) +A (x) = } {{ } A (x) d 3 x j (t x x ) x x + A (x) P ( ) i P ( ) i = d 4 x F ik j k, R 4 F = F + F d 4 xf,ik j k = d 4 xj k (x) d 4 x [ ] i G (x x )j k (x ) k G (x x )j i (x ) }{{} x x 1 2 ( ig (x x ) + i G (x x) ) = 1 }{{}}{{} 2 x i G(x x ), i G(x) = i (2π) 3 G (x x ) d 4 kδ(k 2 ) [ Θ(k 0 ) Θ( k 0 ) ] e ikx F := F F = F }{{} F =F F ( P ( ) i P ( ) i = F,ik + 1 ) R 4 2 F,ik }{{} G G G j k

76 z(s) j i (x) = q ds δ (4) (x z(s)) ż i (s) ( ) A, i (x) = 4π = 4πq = 2q = q d 4 x G (x x )j i (x ) s1 dsż i (s)g (x z(s)) dsż i (s)δ((x z(s) ) 2 ) }{{} y ż i (s 0 ) (x z(s 0 ))ż(s 0 ) s 0 s 1 s 1 z 0 (s 1 ) = x 0 s 0 (x z(s 0 )) 2 = 0 x 0 > z 0 (s 0 ) G (x) = Θ(x0 ) 2π δ(x2 ), δ(y 2 (s)) = δ(s s 0) s<s1 2yẏ s=s0 A F T P i = T, ik dσ k K

77 s = (s 2 s 1 ) F T K de dt = 2 ( d 2 ) 2 z 3 q2 dt 2 dp i ds = 2 3 q2 ż i z 2 i = 0 m z i? = q2 ż i z 2 z ż ( : m z = 2 ) 3 q2 z( z) 2 0 = (ż z) = z 2 + ż z ) ż z 2 + z ż m z i = 2 3 q2 (ż i z 2 + z i ) z

78 E = q r [ ] n r z + ( z n) n, n x r dωn i = 0, dωn i n j = 4π 3 δ ij E ( = ) q z r 0 3 r z (t) = z (t r) = z (t) r z (t) m 0 z = q E q E q 2 r z = 2 3 r 0 }{{} δm z, q2 z (m 0 + δm) z = 2 }{{} 3 q2 z. m m 0 δm 1 2 ( E 2 + B 2 )d 3 x m z i = 2 3 q2 z i K ż m z i = 2 3 q2 ( z i + ż i z 2 ) z r 0 q 0 q2 r δm m 0 q m m 0 δm z F

79 m z i = qf, ik ż k q2 ( ż i z 2 + z i).