1 2 1.1............................ 2 1.2............................... 3 1.3.................... 3 1.4 Maxwell.................... 3 1.5.......................... 4 1.6............................ 4 1.7................................... 4 2 5 2.1................... 5 2.2........................... 5 2.3.......................... 5 2.4............................... 6 2.5.................... 6 2.6...................... 7 2.7...................... 7 2.8............................... 8 2.9................................... 8 3 10 3.1............................. 10 3.2.................................. 10 3.3........................... 10 3.4.................................. 11 3.5......................... 11 3.6......................... 12 3.7............................... 12 3.8............................. 13 3.9............................ 13 3.10............................ 15 3.11............................. 15 3.12.............................. 16 3.13................................... 17 3.14................................... 17 3.15................................... 17 1

4 18 4.1.......................... 18 4.2............................... 19 4.3.................... 19 4.4............................ 20 4.5................................... 21 5 21 5.1............................ 21 5.2......................... 22 5.3............................... 22 5.4............................... 23 5.5......................... 24 5.6............................ 24 5.7........................... 24 5.8............................. 25 5.9................... 25 5.10.................... 25 5.11................................... 27 5.12................................... 28 6 28 6.1........................... 28 6.2................................. 28 6.3............................. 29 7 29 1.1 `' ` ' Σ Σ 2

1.2 Σ t Σ t Σ r 2 r 1 Σ Σ r 2 r 1 Σ Σ v x x x = x vt y = y z = z t = t ṙ = r = ṙ v r 1.3 F = m r F A B = F B A ΣΣ Σ : F = m r Σ : F = m r 1.4 Maxwell = (µ 0 ε 0 ) 1/2 + v 3

(Mihelson{Morley) π 0 1054 1.5 ΣΣΣ ` ' 1.6 (Doppler) 1.7 Maxwell Page 290: 1,2,3 4

2.1 (x 1 x 10 ) 2 + (x 2 x 20 ) 2 + (x 3 x 30 ) 2 = onst (x 1 x 10) 2 + (x 2 x 20) 2 + (x 3 x 30) 2 = onst (1) 2.2 x ν x ν = (x ν x ν0 ) x ν = α (2) (1) x ν x α + 1 x α 2 α,β 2 x ν x α x β x α x β + (2) α ν x ν x α + 1 x α 2 α,β 2 x ν x α x β + x α x β 2 = λ ν x 2 ν (3) (3)( x 1, x 2, x 3 )x ν x ν x ν = a ν + α b να x a x ν = α b να x a (4) 2.3 x ν x ν (4) (relativity of diretion)(homogeneity) (isotropy) (4) 5

2.4 OP Σ Σ Σ Σ (0, 0, 0, 0) Σ Σ x 2 + y 2 + z 2 = t 2 x 2 + y 2 + z 2 = t 2 s 2 = t 2 x 2 y 2 z 2 = 0 s 2 s 2 = t 2 x 2 y 2 z 2 = 0 s 2 = t 2 x 2 y 2 z 2 0 ( ) s 2 = 0 (x 1, y 1, z 1, t 1 ) (x 2, y 2, z 2, t 2 ) s 2 = (t 2 t 1 ) 2 (x 2 x 1 ) 2 (y 2 y 1 ) 2 (z 2 z 1 ) 2 s 2 = s 2 (5) A 2 = 1 2.5 ΣΣ x = x(t)x = x (t ) (x, t) (x, t ) x = y = z = t = a 00 x + a 01 y + a 02 z + a 03 t a 10 x + a 11 y + a 12 z + a 13 t a 20 x + a 21 y + a 22 z + a 23 t a 30 x + a 31 y + a 32 z + a 33 t 6

xx yz x = γx + σt y = y z = z t = βx + αt α, β, γ, σv 2.6 Σ Σ vσ Σv dx = dy = dz = 0 dx dt = σ γ = v Σdx = dy = dz = 0 Σ v dx dt = σ α = v α = γ β = ηγ x = γ(x vt) y = y z = z t = γ( ηx + t) (6) γ = 1 η = 0 2.7 (6) (5) γ 2 (t ηx) 2 γ 2 (x vt) 2 = t 2 x 2 7

γ 2 v 2 γ 2 = γ 2 η 2 γ 2 = 1 2η γ 2 2vγ 2 2.8 x = x vt y = y z = z t = t v x v v x = x + vt 2 y = y z = z 2.9 t = t + v x η = v 1 γ = (7) (8) Page 290: 4,5,6,7 8

\ " \ " \ " (I{time) \ "(subjetive time) B C A A AXA \A " (pre{relativity physis) (Eulidean geometry) 9

3.1 O(0, 0, 0, 0)P (x, y, z, t) s 2 = t 2 x 2 y 2 z 2 = t 2 r 2 s 2 s 2 > 0t > r s 2 < 0r > tt s 2 = 0r = t r t t = 0 3.2 P P xy P t O O s 2 = 0 P s 2 > 0 P s 2 < 0 P P 3.3 s 2 = 0 P s 2 > 0 P O P dτ P O P O s 2 < 0 P O P P O 10

3.4 OP Σ(x 1, t 1 )(x 2, t 2 ) t 2 > t 1 Σ (x 1, t 1) (x 2, t 2) t 2 > t 1 t = t v x t 2 t 1 = t 2 t 1 v (x 2 x 1 ) = (t 2 t 1 ) = (t 2 t 1 ) vu 1 v2 ( 1 v ) x 2 x 1 t 2 t 1 u < Σ Σvv < u <, v < > uv t 2 t 1 > 0 3.5 11

3.6 Σ (x 1, t 1 ) (x 2, t 2 ) x 2 x 1 > t 2 t 1 r > t Σ Σ t 2 > t 1 t 2 t 1 = t 2 t 1 v (x 2 x 1 ) = (t 2 t 1 ) ) u (v 2 u v 2 u > 0 v 2 u = 0 t 2 < t 1 t 2 = t 1 3.7 Σ (t 2 = t 1 ) Σ (t 2 t 1 ) Σ U m t m r mn U n U n t n = t m + rmn Σ 12

3.8 Σ τ Σ Σ τ s 2 = τ 2 ΣΣ vσ s 2 = t 2 (x 2 x 1 ) 2 = t 2 v 2 t 2 τ 2 = t 2 v 2 t 2 t = τ > τ dτ 3.9 Σ l C 1 C 2 Σ v C C C 1 C C 1 C 2 0C C 2 Σ l v (1) C C 2 C (2) Σ C 2 C C 2 C Σ τ t = τ τ = l v ΣC l v = < l v τ 13

C C 1 P C C 2 Q Σ P (0, 0) Q(l, l v ) Σ P (0, 0) Q(0, τ) τ = l v v l = l v < l 2 v s 2 = τ 2 = l2 v 2 l2 = l2 v 2 ) ( Σ C 2 l v C τσ Σ C C 1 0ΣC 1 C 2 Σ Σ C C 1 C 2 R Σ R(l, δ) Σ R(l, 0) l = 0 = δ v l 2 δ = l vδ = l v2 l = l v l Σ C 2 R(l, δ) Q(l, l v ) l v δ = l ) ( = τ v < τ 2 Σ C 2 R Q s 2 = τ 2 l 2 = 2 l 2 ) ) ( l (1 2 v2 = 2 l 2 v 2 ( v 2 ) 2 ΣΣ Σ Σ 14

3.10 CΣC Σt C Σv(t )C dt Σ dt dt = (t ) C Σ dt t = dt = > dt = t (t ) t C t C C CC 3.11 xσ P 1 P 2 ΣP 1 P 2 Σ Σ (x 1, t 1 ) (x 2, t 2 ) Σ (x 1, t 1 ) (x 2, t 2 ) x 1 = x 1 vt 1 t 1 = t 1 v x 1, x, t 2 = x 2 vt 2 2 = t 2 v x 2 t 1 = t 2 x 2 x 1 = x 2 x 1 l = l 0 15

( ) 1 x 2 x 1 = (x 2 x 2 ) 2 t 2 = t 1 3.12 Σ u x = dx, u y = dy, u z = dz dt dt dt Σ Σxv Σ u = dx dt x = x vt, t = t v x dx = dx vdt = u x v dt dy = dy, dz = dz dt = dt v dx = u x = dx dt 1 v u x dt = u x v 1 vux u y = dy = 2 u dt y, u 1 vux z = dz = dt 2 Σ u Σ u z 1 vux u x = u x + v u x 1 + vu x = 1 + vu x 2 u y = 2 u 1 + vu y, u z = 2 u x 1 + vu z x (v, u ) u x + v u x u x + v, u y u y, u z u z 16

3.13 u < u < dt dx 3.14 dt 2 (dx 2 + dy 2 + dz 2 ) = dt 2 (dx 2 + dy 2 + dz 2 ) dx dt = u dx ( 2 u 2 )dt 2 = ( u 2 )dt 2 = u dt u < u 2 > 0 u 2 > 0 u < Σ Σ n nσ Σxv u x = u x + v 1 + vu x u x = n + v 1 + v n u x n + ( 1 1 n 2 ) v v u x = n + v 1 v n n + ( 1 1 n 2 ) v 3.15 17

dτ Page 290: 8,9 4.1 x = x 1, y = x 2, z = x 3, it = x 4 x 1 = x 1 vt a = = γx 1 + iβγx 4, x β = v γ 0 0 iβγ 0 1 0 0 0 0 1 0 iβγ 0 0 γ 4 = t v x 1 = iβγx 1 + γx 4 1, γ = = os θ 0 0 sinθ 0 1 0 0 0 0 1 0 sin θ 0 0 os θ (9) sin θ = iβγ, os θ = γ, tan θ = iβ Σ Σ v Σ x 1 = onstσ ( π 2 + θ) dx 4 = tan( π dx 1 2 + θ) dx 1 = 1 dx 1 dx 4 i dt = v i = tan θ a 1 = ã = γ 0 0 iβγ 0 1 0 0 0 0 1 0 iβγ 0 0 γ ãa = I 18 = os θ 0 0 sinθ 0 1 0 0 0 0 1 0 sin θ 0 0 os θ

4.2 dτ = ds T = T dx µ U µ = dxµ dτ V µ = a µνv ν U µ = dx µ dτ = dx µ dt dt dτ = γ dx µ dt = γ(u 1, u 2, u 3, i) v T µν = a µλa ντ T λτ 4.3 φ = k x ωt = k x ω t = φ = onst k µ x µ = k µx µ = onst k µ = k 1 = γ ( k 1 v ω ) k 2 = k 2 k 3 = k 3 ω = γ (ω vk 1 ) k x θ k xθ ( k, i ω ) k 1 = ω os θ, k 1 = ω os θ ( ω k 1 = γ os θ v ) ω = γ ω ( os θ v ) 2 ω = ωγ (1 v ) os θ 19

1 tan θ os2 θ = ω = 2 ω 2 k 1 2 1 = os θ k 1 2 k 1 ω 2 2 k 2 1 ω = 2 ω 2 γ ( γ ( 2 1 v os θ v ) = os θ) 2 ( γ 2 os θ v γ ( os θ v ) sin θ = γ ( os θ v ) Σ ω = ω 0 ) 2 ω = ω 0 γ ( (10) 1 v os θ) v γ 1 ω ω 0 ( 1 v os θ) ω = ω 0 4.4 ω = ω 0 F µ = G µ F µ G µ a µν F ν = a µν G ν F µ = G µ 20

4.5 Page 291: 10,11 ρ J E B 5.1 Q Q = ρ dv = onst ρ 0 dv 0 u dv = 1 u2 dv 2 0 ρ 0 dv 0 = ρ dv 21

ρ = u ρ 0 1 u2 = γ u ρ 0 (11) J = ρu = γ u ρ 0 u (12) U µ = γ(u 1, u 2, u 3, i) (11)(12) J µ = ρ 0 U µ = ρ 0 γ u (u, i) = (γ u ρ 0 u, iγ u ρ 0 ) = (J, iρ) 5.2 J µ = (J, iρ) J + ρ t = 0 J µ = J 1 + J 2 + J 3 + (iρ) = 0 (13) x µ x 1 x 2 x 3 (it) (13) ρ J ρ J ρ J 5.3 2 A 1 2 A t 2 = µ 0J, 2 ϕ 1 2 ϕ t 2 = ρ ε 0 A + 1 ϕ t = 0 (14) 2 1 2 t 2 = x µ x µ (14) A = µ 0 J, ϕ = ρ ε 0 = µ 0 ρ = iµ 0 (iρ) A ϕ A µ = (A, i ) ϕ 22

A µ = µ 0 J µ A + 1 ϕ t = A + ( i ϕ) (it) = A µ x µ = 0 A µ = a µνa ν A x = γ(a x v ϕ) A y = A y A z = A z ϕ = γ(ϕ va x ) 5.4 B = A, E = ϕ A t B 1 = A 3 x 2 A 2 x 3, B 2 = A 1 x 3 A 3 x 1, B 3 = A 2 x 1 A 1 x 2 (15) E 1 = ϕ A 1 x 1 t E 2 = i ( A4 x 2 A 2 x 4 = i ( i ϕ) x 1 i A 1 ) ( (it) = i A4 A ) 1 x 1 x 4 ) ( A4, E 3 = i A 3 x 3 x 4 F µν = A ν x µ A µ x ν (15) (16) (17) F µν = 0 B 3 B 2 i E 1 B 3 0 B 1 i E 2 B 2 B 1 0 i E 3 i E 1 i E 2 i E 3 0 (16) (17) 23

5.5 E = ρ ε 0, B = µ 0 ε 0 E t + µ 0J F µν x ν = µ 0 J µ (18) µ = 4 µ = 1, 2, 3 B = 0, E = B t F µν x λ + F νλ x µ + F λµ x ν = 0 (19) τ = 1, 2, 3, 4 5.6 F µν = a µλa ντ F λτ E 1 = E 1 E 2 = γ(e 2 vb 3 ) B 1 = B 1 B 2 = γ(b 2 + v E 3 ) E 3 = γ(e 3 + vb 2 ) B 3 = γ(b 3 v E 2 ) (20) E = E B = B E = γ (E + v B) B = γ ( B v E ) (21) v v (21) E = E + v B, B = B v B 5.7 F µν J ν f µ f µ = F µν J ν = (f, f 4 ) f = ρe + J B, f 4 = i J E = i w 24

ff 4 w ω S g T [ T ] i T µν = S ig ω 5.8 F µν 1 1 F : F = 2 2 F µνf µν = B 2 1 E2 { ε µνλτ det a = ε αβγδ a 1α a 2β a 3γ a 4δ = 1 ε µνλτ ε µνλτ = a µαa νβ a λγ a τδ ε αβγδ = (det a)ε µνλτ = ε µνλτ i 8 ε µνλτ F µν F λτ = 1 B E 5.9 B = E B 2 1 E2 = 0 B = 1 e k E 1 B E = 0 E B B = E E B 5.10 ve Σ E = ex 4πε 0 r 3, B = 0 25

Σ v x(20) E x = ex B 4πε 0 r 3 x = 0 ey E y = γ B 4πε 0 r 3 y = γ v ez 4πε 0 r 3 ez E z = γ B 4πε 0 r 3 z = γ v ey 4πε 0 r 3 Σt = 0Σ x = γx, y = y, z = z x e 1 + γy e 2 + γz e 3 = γx r 2 = γ 2 x 2 + y 2 + z 2 = x2 1 β 2 + y2 + z 2 = 1 1 β 2 ( r 2 β 2 y 2 β 2 z 2) = 1 1 β 2 [ (1 β 2 )r 2 + β 2 x 2] ) E = ( ex [( ) 4πε 0 r 2 + ( v x ) 2 ] 3 2 B = v E v ( v ) 2 E = ex 4πε 0 r 3 = E 0 B = v E 2 0 = µ 0ev x = µ 0 J x 4πr 3 4π r 3 v v v x = 0 ex E = γ 4πε 0 r E 3 0 v v y = z = 0 E = E ) E = ( ex 4πε 0 r E 3 0 26

B = v E 1 e x E (v ) v B EB E 5.11 Σ E Σ E vv E = 0 E = E B = B = 0 E = γ (E + v B) = 0 B = γ ( B v E ) = 0 Σ Σ E v E = E = 0 B = B = 0 E = γ (E + v B) = γe B = γ ( B v E ) = γ v E E = γe, B = γ v E 2 Σ E B E B v B 2 1 E 2 = 1 E 2 < 0 1 B E = 0 27

5.12 B 2 1 E 2 1 B E Page 292: 12,13,14,15 v 6.1 p = γm 0 v = p µ = m 0 U µ m 0v, p 4 = iγm 0 = W = m 0 2 = γm 0 i m 0 = p µ = (p, i W ) p µp µ = m 2 0 2 = onst i W 6.2 m 0 W 2 = p 2 + m 2 0 4 m 0 28

m m 0 6.3 m = γm 0 = m 0 p = mv, W = m K µ K µ = dp µ dτ = (K, K 4) = (K, i ) K v ik 4 = dw dτ F = K = dp dτ = 2 W p dp dτ = v dp dτ = K v, K v = dw dτ K K = γf F = dp dt, F v = dw dt F F p W F F Page 293: 16,17,18,19 29