A. Fresnel) 19 1900 (M. Planck) 1905 (A. Einstein) X (A. Ampère) (M. Faraday) 1864 (C. Maxwell) 1871 (H. R. Hertz) 1888 2.2 1 7 (G. Galilei) 1638 2

1 2012.8 e-mail: tatekawa (at) akane.waseda.jp 1 2005-2006 2 2009 1-2 3 x t x t 2 2.1 17 (I. Newton) C. Huygens) 19 (T. Young) 1

A. Fresnel) 19 1900 (M. Planck) 1905 (A. Einstein) X (A. Ampère) (M. Faraday) 1864 (C. Maxwell) 1871 (H. R. Hertz) 1888 2.2 1 7 (G. Galilei) 1638 2

1: (O. Roemer) (Io) 42 22 1 1676 2.14 10 8 [m/s] (J. Bladley) 1727 2 v c θ = v c, (1) 3

2: L 3: 1 c = 3.01 10 8 [m/s] (2) (A. Fizeau) 1849 3 720 L = 8.633 [km]) 1 sin θ θ 4

12.6 1 2 2 1 c = 2L t 2 8.633 103 = 1 12.6 1 720 1 2 = 3.13 10 8 [m/s]. (3) (J. L. Foucault) 1862 c = (2.980 ± 0.5) 10 8 [m/s], (4) 2.3 1875 1 4000 1 1 3 0.2 [µm] 2 10 7 1927 (Cd) 1 10 8 1 1960 86 Kr 4 10 9 2 3 5

(J. Harrison) 1761 10 6 2 1 1930 10 8 1956 1 1900 1 1 1966 133 ( 133 Cs) 10 13 10 14 133 1 2005 4 1000 1 1 1983 c = 2.99792458 10 8 [m/s]. (5) 1 1/299792458 1 3 3.1 19 (J. L. Lagrange) (W. R. Hamilton) (L. E. Boltzmann) 5 20 4 http://www2.nict.go.jp/w/w114/afs/index.html 5 6

S V u V v 4: S, S S S V 19 4[km/h] 40[km/h] 44[km/h] 4 S, S S S V S u S v = u + V c c 7

19 3.1.1 6 v = 30[km/s] β = v c 10 4, 10 4 (A. Michelson) 1881 5 7 L 1 c c v c + v t (1) = L 1 c v + L 1 c + v = 2L 1 1 c 1 β 2, (6) 8 c 2 v 2 6 7 8 β = v/c 8

L 2 L 1 5: t (2) = 2L 2 1 c, (7) 1 β 2 (0 ) = t (1) t (2), (8) 90 t (2) = 2L 2 c 1 1 β 2, (9) t (1) = 2L 1 1 c, (10) 1 β 2 9

(90 ) = t (2) t (1), (11) t(0 ) t(90 ) = t (1) + t (1) = 2(L 1 + L 2 ) c t(2) t (2) ( ) 1 1 β 2 1, (12) 1 β 2 β 1 9 L 1 = L 2 = L t(0 ) t(90 ) 2L c β2, (13) λ s s = 1 λ c ( t(0 ) t(90 )) 2L λ β2, (14) L = 1.2[m], λ = 6 10 7 [m], s = 0.04 (E. Morley) 1887 L = 11[m] s 0.4 s < 0.01 G. F. FitzGerald) (H. A. Lorentz) 9 (1 β 2 ) a 1 aβ 2 10

10 11 3.2 1905 3.2.1 ma = F, (15) r = r vt, (16) v ma = F, (17) 10 11 4% 20% 70% 11

3.2.2 3.3 3.3.1 (event) 3 1 4 4 4 2 1 P Q x P, x Q d d 2 = (x Q x P ) 2, (18) P, Q s 2 P Q c 2 (t Q t P ) 2 + (x Q x P ) 2. (19) 12

(19) (Minkowski) P Q ( s) 2 = c 2 ( t) 2 + ( x) 2, (20) 12 PQ x Q x P = c(t Q t P ), (21) s 2 P Q = 0, (22) (s P Q) 2 = c 2 ( t Q t P ) 2 + ( x Q x P ) 2 = 0, (23) 3.3.2 2 P Q 2 2 S P, Q (t p, x p ) (t q, x q ) 12 13

s 2 P Q c 2 (t Q t P ) 2 + (x Q x P ) 2. (24) S P Q (s P Q) 2 = c 2 (t Q t P ) 2 < 0 (25) s 2 P Q = (s P Q) 2 < 0, (26) 1 2 c (x Q x P ) 2 = { tq t P } 2 { tq } 2 v(t) dt < c dt = c 2 (t Q t P ) 2, (27) t P S P, Q (T p, X p ) (T q, X q ) s 2 P Q c 2 (T Q T P ) 2 + (X Q X P ) 2. (28) S P Q (s P Q) 2 = (X Q X P ) 2 > 0 (29) s 2 P Q = (s P Q) 2 > 0, (30) 14

ct O y x 6: (31) 3.3.3 x ct t = 0, x = 0 O P x 2 > c 2 t 2 Q x 2 < c 2 t 2 c 2 t 2 + x 2 = 0, x y c 2 t 2 = x 2 = x 2 + y 2, (31) t (31) 6 ct O ct O O 15

3.4 t r = ( x) 2 + ( y) 2 + ( z) 2, S S x = y = z = 0, (32) ( s) 2 = c 2 ( t) 2 + ( r) 2 = c 2 ( t ) 2, (33) t = t 1 1 c 2 ( r t ) 2 = t v = r t, v = v, 1 v2 c 2, (34) (34) 1 13 13 16

τ2 τ 1 dt = t2 t 1 dt 1 v(t)2 c 2. (35) v S τ 2 τ 1 ( τ) 2 ( s)2 c 2, (36) 3.5 S(t, x, y, z) S (t, x, y, z ) S S x v t = t, x = x vt, y = y, z = z, (37) ( s) 2 = c 2 ( t) 2 + ( x) 2 + ( y) 2 + ( z) 2, (38) ( s ) 2 = (c 2 v 2 )( t) 2 2v( t)( x) + ( x) 2 + ( y) 2 + ( z) 2, (39) 1. 2. 3. β = v/c 1 1. 17

ct x y z = a 00 a 01 a 02 a 03 a 10 a 11 a 12 a 13 a 20 a 21 a 22 a 23 a 30 a 31 a 32 a 33 ct x y z. (40) 4 4 S S x v t = t = 0 S S y = y, z = z, (41) a 22 = a 33 = 1, (42) a 20 = a 21 = a 23 = a 30 = a 31 = a 32 = 0, (43) y = y, z = z a 02 = a 03 = a 12 = a 13 = 0, (44) ct = a 00 ct + a 01 x, (45) x = a 10 ct + a 11 x, (46) y = y, (47) z = z, (48) 4 4 4 2 2 2 S x = 0 S x v x = vt, (49) 18

(45) (46) ct = a 00 ct, x = a 10 ct, (50) a 10 ct = a 00 vt, (51) a 10 a 00 = v c, (52) S x = 0 S x v x = vt, (53) (46) a 10 ct + a 11 x = 0, (54) a 10 a 11 = v c, (55) (52) (55) a 00 = a 11, (56) γ a 00 = a 11 2. t = t = 0 x S x = ct, (57) (45) (46) ct = (a 00 + a 01 )ct, (58) x = (a 10 + a 11 )x, (59) S x = ct, (60) a 00 + a 01 = a 10 + a 11, (61) 19

a 01 = a 10 (52) a 01 = a 10 = v c γ, (62) γ ct = γ (ct v ) c x, (63) x = γ ( vt + x), (64) γ S S S S (63) (64) ct = x = 1 v/c γ[1 (v/c) 2 ] ct + γ[1 (v/c) 2 ] x, (65) v/c 1 γ[1 (v/c) 2 ] ct + γ[1 (v/c) 2 ] x, (66) S S v v ( ct = γ ct + v c x ), (67) x = γ (vt + x ), (68) γ = 1 1 (v/c) 2, (69) 14 β = v/c 0 (63) (64) γ 1 (63) ct = ct t = t, (70) (64) x = vt + x, (71) 14 γ γ 20

y V V v x 7: S S V x S v S v 3.6 3.6.1 c S S V x S v S v 7 v v x = γ( x + V t ), (72) y = y, (73) z = z, (74) t = γ ( t + Vc ) 2 x, (75) v x = x t = γ( x + V t ) γ ( t + V c x ) = V + v x 1 + V, (76) 2 c v 2 x v y = y t = v z = z t = v y γ ( ) 1 + V, (77) c v 2 x v z γ ( ) 1 + V, (78) c v 2 x V < c, v < c v < c 21

y v V x 8: S S V x S v S v c γ 1 v x = V + v x, v y = v y, v z = v z, 3.6.2 S S 8 x x θ θ v x = v cos θ, v y = v sin θ, (79) v x = v cos θ, v y = v sin θ, (80) v x = v cos θ = x t = V + v cos θ 1 + V c 2 v cos θ, (81) v y = v sin θ = y t = v sin θ γ ( 1 + V c v cos θ ), (82) 2 22

90 0 45 90 135 180-90 9: θ θ θ = 0 θ = 180 θ θ θ θ tan θ = v sin θ γ(v + v cos θ ), (83) v = v = c, tan θ = sin θ γ(β + cos θ ), (84) θ θ 9 θ θ β = 0 90 β = 0.8 140 β = 0.95 160 S S S 23

15 SF 3.6.3 S S x = 0 t 10 S t = t 2 t 1, (85) S S v S t = t 2 t 1 = γ (t 2 + v c 2 t 1 v ) c 2 = γ(t 2 t 1) = γ t > t, (86) S S 3.6.4 S x 11 S l = x = x 2 x 1, (87) 15 24

S V t 10: t t S V 11: x S t S 16 x = x 2 x 1 = γ(x 2 x 1 ) = γ x, (88) δx = 1 γ l < l, (89) 17 16 17 25

3.7 3.7.1 3.6.3 (ct) (x) S S x ct = 0 ct = 0 S ct = v c x, tan θ = v c, S ct x = 0 x = 0 S x = v ct = vt, c 12 S x = 0 ct P 12 ct x ct B A 26

ct B A P O x 12: S S B S (ct, 0) S (ct, x B ) { ct = γ 0 = γ ( ct v ( c v c { ) } x B ) ct + x B, (90) }, (91) x B T T T = 1 γ T < T, (92) S S 3.7.2 13 (a) 27

(a) (b) 13: (a) (b) (b) 13 13 S S (x = 0) t = 0 t = 0 x = 0 14 t = 0 x = 0 Q P R 28

ct Q R P x 14: Q P R 3.7.3 8.6 8.6 17 29

ct B A O x 15: x = 0 ct ct A ct B 15 S ct A S ct A S ct B 0 τ A 30

2τ A τ A 15 A 15 A A A 15 4 4.1 i m x 2v 2m x 31

m 2m m 2m 2v v 16: m x 2v 2m x v 2m m 3v v v m 2v + 2m ( v) = 0, (93) 0 v = v + 2v 1 (v 2v/c 2 ) = 3v 1 (2v 2 /c 2 ), (94) mv 2 3mṽ v ṽ = v ṽ = v 1 (2v 2 /c 2 ) v, (95) 32

4.2 x v t t x v = x t, (96) x t t x τ s ( s) 2 = c 2 ( τ) 2 = c 2 ( t) 2 + ( x) 2 { ( x = c 2 + t = c 2 {1 )} ( t) 2 ( v c ) 2 } ( t) 2, (97) t x ct x, y, z 1 2 3 ct 0 18 4 u t (ct) τ u x x τ,, (98) u y y τ, u z z τ. (99) 18 4 33

4 u 4 4 p t mu t, (100) p x mu x, p y mu y, p z mu z, (101) 4 4 4 v v v = v, v 2 = v = v 2 x + v 2 y + v 2 z, (102) v 4 0 ( ) 2 (ct) (u t ) 2 + (u x ) 2 + (u y ) 2 + (u z ) 2 = + τ = = ( ) 2 x + τ ( ) 2 y + τ ( ) 2 z τ 1 { ( τ) 2 ( (ct)) 2 + ( x) 2 + ( y) 2 + ( z) 2} 1 ( τ) 2 4 4 ( (p t ) 2 + (p x ) 2 + (p y ) 2 + (p z ) 2 = m (ct) τ = = { c 2 ( τ) 2} = c 2, (103) ) 2 ( + m x ) 2 ( + m y ) 2 ( + m z ) 2 τ τ τ 1 { m 2 ( τ) 2 ( (ct)) 2 + ( x) 2 + ( y) 2 + ( z) 2} 1 m 2 ( τ) 2 { c 2 ( τ) 2} = (mc) 2, (104) 4 v p t = mu t = m (ct) τ p x = mu x = m x τ = m t τ = mc t = γmc, τ (105) x t = γmv x, (106) 34

p y = mu y = m y τ = m t y τ p z = mu z = m z τ = m t τ γ = 1 ( v c ) 2 ( = t τ t = γmv y, (107) z t = γmv z, (108) ), (109) 1 2 3 m γm, (110) 4 1 p x = m v x, (111) m m 19 m v c 4 4 0 4 0 c cp t = γmc 2. (112) γ (109) β = v/c 1 Taylor (1 + ε) (1 + ε) 2 = 1 + 2ε + ε 2, (113) (1 + ε) 3 = 1 + 3ε + 3ε 2 + ε 3, (114) (1 + ε) 4 = 1 + 4ε + 6ε 2 + 4ε 3 + ε 4, (115) ε 19 m 0 35

ε 2 2 3 3 4 4 n ε n ε (1 + ε) n 1 + nε, (116) n (116) n 20 1 + ε = (1 + ε) 1/2 1 + 1 2 ε, (117) (109) ( v ) 2 1 ( v ) ( 2 ( v ) ) 2 1 1, ε =, (118) c 2 c c 4 0 (112) cp t ( 1 1 ( v ) ) 2 mc 2, (119) 2 c cp t mc 2 + 1 2 mv2, (120) 2 1 1 v 0 0 E 0 = mc 2, (121) 21 E cp t, (122) 20 21 m 36

5 E = mc 2, (123) 1km 1km 1km 1km 5000km 5000km 5000km 5000km 37

6 1. 2. 3. 4., 5. ( ) 6. J.J. 1. 2. 3. 4. 5. 6. 38