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Sgr.A 2 saida@daido-it.a.jp Sgr.A 1 3 1.1 2............................................. 3 1.2.............................. 4 2 1 6 2.1................................. 6 2.2................................... 7 3 2 8 3.1 4........................................ 8 3.2.............. 9 4 1 10 4.1.......................................... 10 4.2............................................... 12 4.3............................................. 13 5 2 14 5.1................ 14 5.2............................ 14 5.3................................. 16 5.4.............................. 16 5.5 Rapidit................................................. 17 5.6....................... 17 6 3 19 6.1........................................ 19 6.2....................................... 20 6.3................................... 21 6.4............ 22 6.5....................................... 22 6.6............................................. 24 1

7 4 25 7.1 4 4.................................... 25 7.2............................................. 26 8 5 27 8.1.................................... 27 8.2................................... 28 9 6 29 10 30 11 31 11.1............................... 31 11.2................ 33 11.3 35 11.4....................................... 35 11.5..................... 35 11.6. 35 A 36 A.1............................... 36 A.2 Imaginar Rapidit........................ 37 2

1 1.1 2 1 K A r a (t K K q(t v = d q(t = dt K A r a (t K K r a (t = r a (t q(t K a a(t = d2 r a (t dt 2 = a a (t d v dt = a a(t v F = m a = 2.99792485 10 8 [m/s] 2 2 3 6.4 A B [m/s] Light from B ( Newton Light [m/s] + v 2 [m/s]... NO!! v a [m/s] Light from A + v a [m/s]... NO!! Rest K 1 K 2 v 2 [m/s] ( Newton A B [m/s] Light from B ( Einstein Light [m/s] [m/s]... OK!! v a [m/s] Light from A [m/s]... OK!! Rest K 1 K 2 v 2 [m/s] ( Einstein 6.4 3

1.2 K (t,,, z K K v K (t,,, z v O K K O t K t K K t- K t t = v K t - K t t = v K K t- t - K t = K t = t:k t':k' t =1 Light = 1 O = v/ ( K-sstem v_ ' = - t t' t' = 1 Light O ' = 1 ( K'-sstem K O AO = BO K A P B K A P K P B K O P t B t 1 Light t 1 O P A -t 1 Light ( K-sstem K O K t- K O R P B O R 4

R K ( t r, r (1 P B t = + t 1 (2 (1 t E ( t 2, 2 = ( (3 t EO = DO D ( t 2, 2 v + t 1, v v + t 1 (4 D AP t = + 2 t 2 (5 (1 (4 R ( t r, r = ( v v + t 1, v + t 1 K t- t = v t K O K O t _ t' : t = v t 1 B t 2 E v _ t r R ' : t = - 2 O P t r = r t 1 D -t 1 A -t 2 2 ( K-sstem Light DO = EO v v + t 1 r = v + t 1 t 2 = v + t 1 v 2 = v + t 1 [se] [m] [m/se] [se] [m] [se] [m/se] [m/se] [m] or [se] [m/se] [se] or [m] 5 5

t m 2 3 1.1 4 5 6 7 4 0 8 9 10 11 2 1 2.1 2.1 - - A B - : ( a, a A - : ( a, a - : ( b, b B - : ( b, b - - a b Newton... Eulidean Spae r [m] A ' r ' a B ' b B ' b b a r 2 = ( a - b 2 + ( a - b 2 [m 2 ] = (' a - ' b 2 + (' a - ' b 2 A ' a ' 2 A B r [m] - - r [m] r 2 = ( a b 2 + ( a b 2 = ( a b 2 + ( a b 2 (2.1 3 6

2 2 (2.1 B r 2 2 + 2 - r = 2 + 2 - r ±r ' r r ' = r = -r ' = r -r ' = -r -r r 2 = 2 + 2 = ' 2 + ' 2 (2.1 2.2 1.2 4 t,,, z 4 4 2 P, Q 4 4 4 2 P Q s [m] 4 4 4 t 4 tp tq t t' P p (K-sstem s [m] Q q ' t t'q t'p 'p t' P s [m] 'q (K'-sstem 4 t t Q ' 7

(2.1 4 s 2 s 2 (t 2 + 2 t- s = (t 2 + 2 t - s ±s D t- D d < s t - D d < s t- t D d > s 2 [ ] -s t' (or t s ' d (or d -s D s ' (or If s 2 = 2 t' 2 + ' 2 = 2 t 2 + 2 d < s, ' d < s t In K'-sstem ' d > s Contradition! t' ' d > s ' s Einstein... Non-Eulidean Spae-time! D 4 (2.1 3 2 3.1 4 K K ( t,,, z v K t = ± 1.2 K t = ± (t 2 + 2 = (t 2 + 2 = 0 ( t,,, z (t 2 + 2 + 2 + z 2 (t 2 + 2 + 2 + z 2 = 0 (t 2 + 2 + 2 + z 2 4 3.2 4 K ( t,,, z P ( t p, p, p, z p, Q ( t q, q, q, z q P Q 4 s 2 s 2 = 2 ( t p t q 2 + ( p q 2 + ( p q 2 + ( z p z q 2 (3.1 (3.1 s or 8

3.2 (3.1 s 2.2 4 (3.1 Q K K 3 v = ( v 1, v 2, v 3 K P ( t p, p, p, z p K P ( t p, p, p, z p K OP 4 2 s p 2 = (t p 2 + p 2 + p 2 + z p 2 (3.2a K OP 4 2 s p 2 = (t p 2 + p 2 + p 2 + z p 2 (3.2b s 2 p = s p 2 s 2 p = s p 2 v P 6 w 1 = p t p w 2 = p t p w 3 = z p t p (3.3a w 1 = p t p w 2 = p t p w 3 = z p t p (3.3b a µ µ = 0, 1, 2, 3 t p = a 0 t p + a 1 p + a 2 p + a 3 z p (3.4 (3.3 (3.4 5 s p 2 = s p 2 (3.3a (3.4 t p = (a 0 + a 1 w 1 + a 2 w 2 + a 3 w 3 t p (3.3b (3.2b s 2 p = (w 2 1 + w 2 2 + w 2 1 1 (t p 2 = (w 2 1 + w 2 2 + w 2 1 1 (a 0 + a 1 w 1 + a 2 w 2 + a 3 w 3 2 (t p 2 (3.3a (3.2a s 2 p = f s 2 p, where f = w 2 s 2 p = (w 2 1 + w 2 2 + w 2 1 1 (t p 2 1 + w 2 2 + w 2 w 2 1 + w 2 2 + w 2 1 1 1 1 (a 0 + a 1 w 1 + a 2 w 2 + a 3 w 3 2 (3.5 (3.5 2 v f v f = f( v v (3.5 s 2 p = f s 2 p (3.6 9

(3.5 (3.6 sp 2 = f s p 2 = f 2 sp 2 f = 1 or 1 f 3 f K K 3 v p = p, z p = z p (and z (and z P ( t p, p, p, z p = ( 0, 0, p, 0 ( t p, p, p, z p = ( 0, 0, p, 0 s 2 f s p 2 (3.6 p = 2 p = p 2 = s p 2 (3.2 f s p 2 = s p 2 f = +1 s p 2 = s p 2 [ ] ' t' O t p = ' p P. = ' 4 1 4.1 2.1 4 2 ( t,,, z ( 0, 0, 0, 0 4 s s 2 = onst. 3 (t 2 + 2 + 2 + z 2 = onst. K K K v [m/s] K ( t, K ( t, t T K ( t T, T = ( a, 0 OT 4 s T 2 = a 2 X K ( t X, X = ( 0, b OX 4 s X 2 = +b 2 = -b X': '= -b Hb T: t = a t t' T': t' = a Ha Light T': t' = -a Light Ha t = -a X': ' = b ' X: = b Hb Ha : - (t 2 + 2 = - a 2 ( = - (t' 2 + ' 2 Hb : - (t 2 + 2 = b 2 ( = - (t' 2 + ' 2 10

2 t T t OT 4 s T 2 = s T 2 (= a 2 X OX 4 s X 2 = s 2 X (= +b 2 t T t = a 2 X = b t T T 2 t- T H a (t 2 + 2 = a 2 (= s T 2, t t = v H a t- T K T ( t T, T s T 2 = s 2 T (t T 2 + T 2 = a 2 t T = 0 t T = a T K t = a K T ( t T, T = ( a, 0 X X 2 t- t X H b (t 2 + 2 = +b 2 (= s X 2, t = v H b t- X K X ( t X, X s X 2 = s 2 X (t X 2 + X 2 = b 2 t t X = 0 X = b X K = b K X ( t X, X = ( 0, b t t' T H a t = T t = a K T X H b t = X = b K X Light K K v Ha T T' ' X' X Hb ( ase : a < b 11

4.2 4.1 K K T K T A K T B K K K A = A t = t A K B B: t > a T: t = a A: t < a t K rest t = B t = t B K T = T t = t T = a? t' t' T': t' = a K' v Ha ' t A < a, t B > a K A T K K T B K K B t B K K (1 K ( t, K H a (t 2 + 2 = a 2 (= s 2 K T ( t T, T K t t = v ( a 2 T ( t T, T =, v a 1 (v/ 2 1 (v/ 2 T (2 K B ( t B, B = ( t T, 0 K B = K T a K B t B = (3 1 (v/ 2 t K T t T = t B 1 ( v T = a K T B K 2 1 1 (v/ 2 γ-fator γ = 1/ 1 (v/ 2 5 12

4.3 4.1 K K K K b [m] K K b t- = X K b X = b D K b K D = b K D < b b K < b K K K b K rest t(k v? b [m] K' t'(k' Hb ' X': ' = b X: = b D: < b K D D K K (1 K ( t, K H b (t 2 + 2 = b 2 (= sx 2 K X ( t X, X K t = v ( v b 2 X ( t X, X =, b 1 (v/ 2 1 (v/ 2 (2 X t /v t = v ( X + t X (3 D ( t D, D = ( 0, X v t X K b K = D = [ 1 ( v 2 ] X = b 1 ( v 2 K K K b 1 ( v 2 γ-fator γ = 1/ 1 (v/ 2 5 K K b K K b 1 (v/ 2 b t- K 13

5 2 5.1 4.2 4.3 { K ( t, 2 K ( t, ( t, ( t, K K v K ( t e, e [ ] E, z K ( t e, e t e = 1 1 (v/ 2 ( t e v e, e = 1 1 (v/ 2 ( v t e + e (5.1 K K' v rest (no aeleration t e t(k t e ' t'(k' E S E = S E ' ( t, ( t, ' (5.1 K K O e ' e K ( t e, e K ( t e, e (5.1 ( t e, e ( t e, e v K K v (5.1 t e = 1 1 (v/ 2 ( t e + v e, e = 1 1 (v/ 2 ( v t e + e (5.2 5.2 5.1 (5.1 E K ( t e, e E 4 (5.1 3 14

1 A B K t t = v A E t = v ( e + t e K A ( 1 ( te 1 (v/ 2 v v ( e, te 1 (v/ 2 v e t = v B E t t = v ( e + t e K B ( v ( v 1 (v/ 2 t 1 ( v e + e, 1 (v/ 2 t e + e t e O t(k A t e ' t'(k' 2 A B K A t B K A ( t e, 0 K B ( 0, e 3 4 3.1 (5.1 4 1 ( OA 2 te = 1 (v/ 2 v 2 e K (t e 2 K 1 ( v OB 2 = 1 (v/ 2 t 2 e + e K K e 2 2 2 t E t e 0 t e v e e 0 e vt e t e = 1 ( te v 1 (v/ 2 e t e e e = 1 ( v 1 (v/ 2 t e + e (5.1 [ ] e ' B e E ' (5.3a (5.3b 1 2 A.1 15

5.3 4.2 t(k t'(k' K B t B K ( t B, v X t B T = v t T = v t B t = vt K ( t T, 0 (5.1 t T 4.2 = t B (v/ 2 t B 1 (v/ 2 = t B 1 ( v K T B K 2 B O T' 1 1 (v/ 2 4.2 5.2 (5.3 5.3 4.2 ' 5.4 4.3 t(k t'(k' 4.3 (1 K X K ( t X, X X 4.3 K ( 0, b K (5.1 (5.2 t X = 0 + (v/ b 1 (v/ 2 ( v X ( t X, X = 0 + b X = 1 (v/ 2 b 1 (v/ 2, O D X' b 1 (v/ 2 ' 4.3 K X 4.3 (2, (3 K D K K b K = b 1 ( v K K K b 2 1 ( v 2 4.3 5.2 (5.3 5.4 4.3 16

5.5 Rapidit 5.1 E K K (5.1 1 t e = t e osh ψ e sinh ψ osh ψ = γ := 1 (v/ 2 e = t e sinh ψ + e osh ψ sinh ψ = v (5.4 γ γ v γ osh2 ψ sinh 2 ψ = 1, 1 osh ψ <, < sinh ψ < ψ Rapidit 3 v Rapidit ψ v = tanh ψ ( < ψ < (5.5 5.6 (5.1 K K 3 K K K K 3 K K v = ( v, v, v z 5.1 3 (v, 0, 0 1 π 2 θ 2 z ϕ (5.6 z K z v ' z' K' ' π_ 2 θ θ 2nd rotation φ (v,v,v z θ z v.osθ φ v.sinθ v.sinθ.osφ (v,0,0 1st rotation v.sinθ z v.sinθ.sinφ 1st-rot. : about ais 2nd-rot. : about z ais 17

(5.6 1 (v, 0, 0, ( v, v, v z v os θ 0 sin θ v = 0 1 0 v z sin θ 0 os θ v 0 0 =: R 1 ( θ (5.6 2 ( v, v, v z (v, v, v z v os ϕ sin ϕ 0 v v = sin ϕ os ϕ 0 v 0 0 1 v z v z v v sin θ 0 = 0 0 v os θ v v = v z =: R 2 (ϕ 5.1 (5.1 t z = γ γ(v/ 0 0 γ(v/ γ 0 0 0 0 1 0 0 0 0 1 t z =: L (v t z, ( θ = π 2 θ (5.7 v sin θ os ϕ v sin θ sin ϕ v os θ (5.8, γ = 1 ( v (5.9 2 1 (5.6 L (v L gen ( v ( ( ( 1 0 1 0 L gen ( v = L (v 0 R 2 (ϕ 0 R 1 ( θ = γ γ v 1 + (γ 1 v2 v 2 γ v (γ 1 v v v 2 1 + (γ 1 v2 v 2 1 0 0 R 1 ( θ γ v z ( 1 0 0 R 2 ( ϕ (γ 1 v v z v 2 (γ 1 v v z v 2 1 + (γ 1 v2 z v 2 5.6 t z = L gen( v t z = γ ( t v + v + v z z [ γ t (γ 1 v + v + v z z v 2 [ γ t (γ 1 v + v + v z z v 2 [ γ t (γ 1 v + v + v z z v 2 v (5.8 ] v ] v ] v, γ = 1 1 v 2 2 (5.10 (5.11 γ 3 v v = ( v, v, v z r = (,, z v r = v + v + v z z (5.11 ( t v r = γ t t as [ r v r ] = r γ t (γ 1 v 2 v r t v as (5.12 18

(5.10 (5.11 K ( t, r 0 5.1 K ( t, r 0 (5.6 1, 2 R(θ, ϕ = ( 1 0 0 R 2 (ϕ ( 1 0 0 R 1 ( θ R1 1 ( θ = R 1 ( θ, R2 1 (ϕ = R 2( ϕ R(θ, ϕ ( 1 ( 1 ( ( R 1 1 0 1 0 1 0 1 0 (θ, ϕ = = 0 R 1 ( θ 0 R 2 (ϕ 0 R 1 ( θ 0 R 2 ( ϕ ( ( t t K = R(θ, ϕ r r 0 ( ( t t K = R(θ, ϕ r K K K K 5.6 (5.9 ( t, r 0 ( t, r 0 ( t r 0 r 0 ( t = L (v r 0 (5.9 (5.6 1, 2 ( ( ( t t R(θ, ϕ = R(θ, ϕ L (v = R(θ, ϕ L (v R 1 t (θ, ϕ R(θ, ϕ r 0 r 0 r 0 ( t r = L gen ( v ( t r L gen ( v = R(θ, ϕ L (v R 1 (θ, ϕ (5.13 (5.10 (5.11 6 3 6.1 K K v { K ( t, K ( t,,z K K u A v u v, u K' v K u' u =? A K A u 19

6.2 t(k t'(k' t- K K ( t, K ( t, K A u A t- u A t- K u u = u + v u < u + v t = (t = 1 t = v t' light ' = A v = 2 3, u = 2 3 [ ] u = u + v = (4/3 t- K t K K t' = (t' = 1 light A A t ' = ' = u' ' K A u < 6.3 u v, u, t = (t = 1 t(k t'(k' light ' = = _ 3 2 t' = t' light A (t' = 1 t ' t = ' = ' = 2 _ 3 - (t' 2 + ' 2 = - (t 2 + 2 = - 2 20

6.3 6.1 6.2 K A u K A u A A P u v, u, t(k t'(k' A t' = t = light ' ' = t = t p t' = t' p t t' (A P ' = v = u ' = u' = = p ' = ' p ( t p, p K A P ( t p, p K ( t p, p ( t p, p K K v (5.2 ( t p = γ t p + v p 1 ( v γ = (6.1 p = γ t p + 1 (v/ 2 p P K A u u = p t p K A u u = p t p = p t p (6.2a (6.2b (6.2b (6.1 u = (6.2a vt p + p t p + (v/ = p v + p/t p 1 + (v p/( 2 t p u = u + v 1 + u v 2 (6.3 6.2 v = 2 3, u = 2 3 u = (4/3 1 + (2/32 2 2 = (4/3 1 + (4/9 = 12 (< 13 21

Rapidit K K v Rapidit ψ v = tanh ψ K A u Rapidit ϕ u = tanh ϕ (6.3 u = tan ϕ + tanh ψ 1 + tanh ϕ tanh ψ = tanh(ϕ + ψ (6.4 K K (6.3 Rapidit 6.4 (6.3 u = ( u ( v + (u v/ v = and/or u = 0 u Rapidit ϕ + ψ < tanh(ϕ + ψ < 1 [ ] = 2.99792485 10 8 [m/s] 1.1 3 1.1 (6.3 K A u = K v K A u = [ ] 6.5 6.1 A K K v { K ( t,,, z K ( t,,, z K' u' A u =? =, z = z v 6.1 K K 3 u A v = (v,0,0 K K u = ( u, u, u z K A 3 u = ( u, u, u z 22

K K u u 6.3 u = u + v 1 + (u v/ 2 K K, z u, u z z z t t- t t Projetion of A's world line onto t- plane t A t p A P K ( t p, p, p, z p K ( t p, p, p, z p P O p K K 6.1 (t p, p (t p, p (6.1 K K A O u = p A t- t p u = p A t - t p, z or, z (6.1 t p t p p = p Projetion of A's world line onto t'-' plane A t t' t' p P u = p t p γ p = t p + (v/ p u = p t p u z = γ u 1 + (u v/ 2 ' O = ' p = ' p K K ( u + v u 1 (v/ 2 K A 3 u =, 1 + u v 2 1 + u v 2 u =, u z 1 (v/ 2 1 + u v 2 1 [1 (v/2 ] [1 (u / 2 ] [ 1 + (u v/ 2 ] 2 u = u u u = 6.4 (6.5 23

Rapidit u Rapidit ϕ = ( ϕ, ϕ, ϕ z u = ( tanh ϕ, tanh ϕ, tanh ϕ z v Rapidit ψ v = tanh ψ u = ( sinh( ϕ + ψ osh( ϕ z + ψ, sinh ϕ osh( ϕ + ψ, sinh ϕ z osh( ϕ z + ψ (6.6 tanh α = sinh α osh α 6.6 (6.4 t light 6.4 P OK NO! P P 4 3, z t- z t-- t light t Q R Q R P R P R Q P R t 24

7 4 7.1 4 4 4 4 3 4 4 four vetor τ τ τ K ( t,,, z ( P (τ : t(τ, (τ, (τ, z(τ 4 v 4 P (τ v 4 := d OP(τ dτ = ( dt(τ dτ v 4 4 K 3 v = ( d( τ(t, dt df( τ(t dt, d(τ dτ d( τ(t dt,, d(τ dτ, d( τ(t dt dz(τ dτ t t(τ = dτ df ( dt 1 df dt dτ = dτ dτ 3 v = [ v 4 ] [ v 4 ] 4 p 4 m p 4 := m v 4 = ( m dt(τ, m d(τ, m d(τ, m dz(τ dτ dτ dτ dτ p 4 m dt(τ/dτ 7.2 K 3 p = [ p 4 ] [ p 4 ] P(τ (τ τ (7.1 (7.2 O t (K OP(τ+ τ τ OP(τ τ+ τ OP(τ τ v = lim 4 τ 0 t(τ O t (K v 4 1 τ OP(τ = d OP(τ dτ τ p = m 4 v 4 P(τ (τ 25

7.2 K ( t,,, z K 3 v K ( t,,, z K E v = 4 K τ = t K' v m 2 1 (v/ 2 observed energ t (K t' = τ p 4 t(τ P(τ (τ light ' τ = K ( t(τ, (τ, 0, 0 K P (τ = ( τ, 0, 0, 0 K K K v (5.2 t(τ = γ ( τ + 0 = γ τ (τ = γ ( vτ + 0, γ = = γ v τ 1 1 (v/ 2 4 v 4 ( γ, γ v, 0, 0 K v 4 = (, 0, 0, 0 K (7.3 4 p 4 (7.2 (7.3 p 4 = m v 4 = ( m γ, m v γ, 0, 0 K ( m, 0, 0, 0 K (7.4 v < [ γ = 1 ( v ] 2 1/2 1( v 2 3 = 1 + + 2 8 K p 4 = m + m 2 K p 4 = m ( v ( v 4 (v 6 + O 2 3m( v + 8 4 (v 6 + m O 4 K p 4 = 1 [ m 2 + 1 2 m v2 + v 2 O ( v ] 2 26

v/ 0 mv 2 /2 4 p 4 = 1 (7.5 7.2 K (7.5 (7.4 (7.5 v E v = m 2 γ = m 2 1 (v/ 2 E 0 = m 2 (7.6 8 5 8.1 K S v, z K S S K u ( S u > v K T λ = u T v wave u u ( S T S : wave soure λ = u T S O P K S u u 6 or t P R O t':s K world sheet for one period of wave A light ' P λ' λ K : rest observer A ' B 27

K λ = PA S λ = PB P, A K ( t p, p [ P S ( T, 0 P S t p = γ T 1 (5.1 (5.2, γ = p = γ v T 1 (v/ 2 K A ( t a, a t a = t p t a = (/u a K u t a = γ T a = γ u T K λ = a p = γ ( u v T = γ ( u v λ (6.3 u = u v 1 (uv/ 2 λ = ut S K v λ = γ ( u v T = γ T = γ u v u u ( 1 uv 2 λ T (8.1a (8.1b f = 1/T, f = 1/T f = 1 γ u u v f T K R t r = T T S P t p = T (8.1 u = 8.2 K S 3 v = ( v, v, v z v α (u,0,0 K z S K K 3 ( u, 0, 0 u, T, λ = u T, u, T, λ = u T 8.1 S (8.1a (8.1b (8.1 28

S K S K v (8.1 5.6 (5.11 8.1 P γ v (8.1 γ v ( λ = γ ( u v T = γ 1 u v 2 λ T = γ u v T u f = 1 u f γ u v, γ = 1 1 v 2 / 2 (8.2 S v α v = v os α u = 9 6 6.4 K K v (< K ( t,, z K ( t, K A V A > V A A K A A K A t t t' light ' A: if V A > K-sstem t' Time Mahine!? ' R A A' K'-sstem A R K A 6.4 29

10 K K v K ( t,, z K ( t, K K K 4.2 K K K K K v K t r = β :R I S 2 = -α 2 Q light S 2 = -δ 2 t q = δ :Q K t(k O K' t' ' I: t i' = 2α t i = 2β t I ' < t I P: t p ' = α (< β K-sstem Contradit! t'(k' I O I: t i' = 2α t i = 2δ P: t r ' = α (>δ ' K'-sstem t i' > t i K K K O K K O I K [ A] K K K P (t p = α < β (= t r I K t i = 2β K t i = 2α [ B] K K K Q K (t p = α > δ (= t q I K t i = 2δ K t i = 2α A B K K K K 30

11 11.1 11.1 1 2 C (, λ ( (λ, (λ C v = ( d(λ dλ, d(λ dλ 7.1 4 v C 4 λ C (λ C p v O (λ Vetor on a flat Plane (, C λ ( d(λ C v = dλ, d(λ (11.1 dλ v C v p Tangent Plane v 4 4 p - surfae v Tangent Plane at p p p (λ v p (λ C Vetor on a Curved Surfae 31

2 2 p θ ative transformation θ passive transformation p θ v v' Vetor rotation (ative rotation ' p v No differene on the flat Plane θ ' Coordinate rotation (passive rotation p v v' ' p v ' Vetor rotation (ative rotation Coordinate rotation (passive rotation These are different on the Curved Surfae! 32

( v, v - 2 v = ( v, v - 10 11.2 11.2 11.1 11.2 11.1 11.1 2 p ' = g(, v C ' = f(, Coordinate Transformation (passive transformation (, (, = (,, (, (11.2 = (, v p C v λ C 33

11.1 (11.1 C (, C ( (λ, (λ (11.2 (, C ( (λ, (λ ( := ( (λ, (λ, ( (λ, (λ (11.3 v (11.1 ( d(λ ( v, v = dλ, d(λ (, dλ v = ( d ( v, v (λ = dλ, d (λ (, dλ (11.3 d (λ = d(λ (, dλ dλ + d(λ (, at C dλ at C d (λ = d(λ (, dλ dλ + d(λ (, at C dλ at C (11.4 (11.5a (11.5b (, / at C (, = (λ, = (λ v = ( v v = Λ ( v v Λ = at C (11.6 v - (v, v - (v, v 1 (11.6 2 3 v (, 2 (, (v, v (11.5 2 (11.6 (, (v, v (, (v, v Λ 2 n (11.6 n n n 1 1 1 1 2 n n v = v 1 v 2. v n = Λ v 1 v 2. v n, Λ = 2 2 2 1 2 n...... n n n 1 2 n at C (11.7 34

11.3 11.3 3 4 4 4 4 4 4 4 4 4 4 11.4 11.4 11.5 11.5 11.6 11.6 35

A A.1 5.2 4 (5.1 5.1 v v v K v E a(v, b(v, f(v, g(v ansatz t e = a(v t e + b(v e (A.1a e = f(v t e + g(v e (A.1b a(v, b(v, f(v, g(v (A.1 (5.1 a(v, b(v, f(v, g(v (A.1 E 4 2 2.2 (t e 2 + e 2 = (t e 2 + e 2 ( s e 2 = s e 2 (A.2 t-, t - t t = (/v, = 0 (A.3a t = (v/, t = 0 (A.3b t =, t = (A.3 (5.1 (A.1 E E t 5 36

1 E t t e = (A.3a v e (A.1b e = [ v f(v + g(v ] e e = 0 2 E t e = v (A.3b e (A.1a t e = [ v a(v + b(v ] e t e = 0 f(v = v g(v b(v = v a(v (A.4 (A.5 3 E t e = e (A.3 t e = e (A.1 t e = [ a(v + b(v ] e e = [ f(v + g(v ] e a(v + b(v = f(v + g(v (A.4 (A.5 4 ( 1 v (A.1 (A.4 (A.5 (A.6 t e = a(v ( t e v e e = g(v ( v t ( v e + e = a(v t e + e ( v a(v = 1 g(v a(v = g(v (A.6 a(v [ (A.2 (A.7 (t e 2 + 2 e = a(v 2 1 ( v ] 2 ( (te 2 + 2 e [ a(v 2 1 ( v ] 2 5 a(v = 1 a(v = 1 1 (v/ 2 or 1 1 (v/ 2 K v = 0 t = t, = v = 0 (A.1a t e = t e a(0 = 1 > 0 b(0 = 0 1 (A.8 2 a(v a(v = 1 (v/ 2 a(v (A.7 (5.1 a(v, b(v, f(g, g(v (A.1 [ ] (A.7 (A.8 A.2 Imaginar Rapidit 5.5 Rapidit (5.5 A.2 5.1 K K 3 v 37

K Rapidit ψ v = tanh ψ (5.5 K ( t, K ( t, =, z = z -z t- t - (5.1 Rapidt ψ (5.4 K K' v rest (no aeleration Imaginar Time Imaginar Rapidit Rapidit (5.4 t or τ := it Imaginar Time i = 1 t or τ := it 4 s 2 = (t 2 + 2 + 2 + z 2 = (τ 2 + 2 + 2 + z 2 0 t = iτ ( τ, ( τ, Imarinar Rapidit Imaginar Rapidit ψ or ζ := iψ 2 sinh ψ = e iζ e iζ = 2 i sin ζ 2 osh ψ = e iζ + e iζ = 2 os ζ τ Rapidit (5.4 = τ os ζ sin ζ = τ sin ζ + os ζ ζ (A.9 τ Imaginar angle: ζ τ' ' Invariant Cirle Coneived Eulidean Spae Mathematial Transformation τ = -it ( ζ = -iψ t(k t'(k' ψ Invariant Hperbora light (ψ ' Rapidit: ψ Real Minkowski Spaetime (seen from K 38

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