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9 O y O ( O ) O (O ) 3 y O O v t = t = 0 ( ) O t = 0 t r = t P (, y, ) r = + y + (t,, y, ) (t) y = 0 () ( )O O t (t ) y = 0 () (t) y = (t ) y = 0 (3) O O v O O v O O O

y y O O v P(, y,, t) t (, y,, t ) t, O (,y,,t) t O O v (,y,,t ) t : t = 0 y O P r=t (, y,, t) : t = 0 t

y y O O vt v ' ='+vt P (, y,, t) t (, y,, t ) t, 3: t = t = 0 (t) y = (t ) y (4) 3 4 y t 4 4 4 O O v 4 (4) y O O v = vt (5) t = t (6) 3 (4) (t ) = (t) ( vt) = (t) + vt (vt) (t) (7) v = 0 3

v 0 osh θ sinh θ = (8) t = t osh θ + sinh θ (9) = t sinh θ osh θ (0) y = y () = () (t ) y = (osh θ sinh θ)(t) (osh θ sinh θ) y = (t) y (3) O ( = 0, y = 0, = 0) t O ( = vt, y = 0, = 0) 0 = t sinh θ vt osh θ (4) tanh θ = v (5) β v (6) α osh θ (7) sinh θ = ± α (8) (5) α ± = β (9) α α(= osh θ) sinh θ α α = β (0) ( β )α = () α = osh θ = ± β () sinh θ = ± β α = ± β (3) = t sinh θ osh θ = ± βt ) (± β β (4) 4

v 0(β 0) = vt osh θ = β (5) β sinh θ = β t = t osh θ + sinh θ = t β β = t sinh θ osh θ = βt + β (6) (7) (8) y = y (9) = (30) v v(β β) t = t + β β (3) = βt + β (3) y = y (33) = (34) γ β (35) t = γ(t β) = γ( βt + ) y = y = β = v t = γ(t + β ) = γ(βt + ) y = y = (36) (37) 4. O v L 0 O O 5

y y v O O L 0 =' -' ' ', 4: O v O O L 0 O = L 0 = γ( βt + ) (38) O O t L O = R = = + L 0 L = R L = γ( βt + L ) (39) + L 0 = γ( βt + R ) (40) L = R L = L 0 γ = β L 0, β v β L L 0 L L 0 v v L 0 (4) 4. O t = 0 t = τ τ O v v t = γ(t + β ) (4) 6

y y v O O, 5: O = 0 t = 0 t = τ O O τ = t f t i = γτ = t i = 0 (43) t f = γτ (44) τ β τ (45) O τ 4.3 O u O u 4 (t,, y, ) u = d (46) t = γ(t + β ) (47) = γ(βt + ) (48) u = d = + d ) γ(β γ( + βd ) = β + u + βu (49). β = 0 u = u 7

. v = (β = ) u = 3. v = u = u = 4. v = u = u = = (50) 5. u = u = β β = β = (5) β 5 4 4 t t = y, = y (5) Λ(v) γ βγ 0 0 βγ γ 0 0 Λ(v) = 0 0 0 0 0 0 (53) { = Λ(v) = Λ( v) (54) γ βγ 0 0 βγ γ 0 0 Λ( v) = 0 0 0 0 0 0 v v β β ) = Λ(v) = Λ(v)Λ( v) (56) (55) Λ(v)Λ( v) = () (57) Λ(v) Λ( v) Λ(v) t = Λ(v) t (A t ij = A ji ) 8

5. Λ(v) Λ( v) β = v γ = β ) 6 (5) 4 0 0 t = = (58) y (54) 3 3 α = Λ(v) α β β (59) β=0 (α = 0 3 ) Λ(v) α β Λ(v) α β 0 3 β) 0 3 α = Λ(v) α β β (60) Λ Λ(v) α β = Λ(v)β α 3 0 0 t = = = y 3 3 (6) t = γ(t β) = γ(t + β( )) = γ( βt + ) = γ(βt + ( )) y = y = (6) { = Λ( v) = Λ(v) (63) (54) 9

(t) y t = t (64) t t = 0 0 + + + 3 3 = (t,, y, ) (65) y = (t) y = (t ) y (66) = t (67) 4 = Λ( v) (68) = Λ(v) (69) t = t Λ( v) t Λ(v) = t Λ( v) Λ(v) = (70) Λ( v) Λ(v) Λ(v) Λ( v) Ã B = Ã B (7) 4 B 4 Ã 4 Ã (7) Ã 4 B (54) (63) 4 0 t = = (7) y 3 0

= 0 t = y 3 (73) 7 4 (t) y (τ) (τ) = (t) y (74) t = 0 (d, dy, d) () = () d dy d (75) = ( ) ( ) d ( ) dy ( ) d (76) u ũ u d dy d, ũ d dy d (77) ũ u = 4 u = {u α } ũ = {u α } (75) = ( ) = d () d dy d (78) ( ) d ( ) d = v (79) = ( ) = v d = d = γ d β = γ (80) (8)

u γ d dy d = γ v v, y v ũ γ d dy d = γ v v y v (8) v i (i =, y, ) 8 4 4 ( ) m 0 p p p 0 p p = m 0 p 3 p 0 p p = m 0 p 3 d dy d d dy d = γm 0 = γm 0 d dy d, (83) d dy d (84) p p 3 v 0 (γ ) 4 p t p p t p = (p 0 ) (p ) (p ) (p 3 ) (85) ( ) ( ) ( ) d dy d = m 0γ (86) = m 0γ ( v ) = m 0 = (87) (p 0 ) = (m 0 ) + (p ) + (p ) + (p 3 ) (88) τ [ (p 0 ) (p ) (p ) (p 3 ) ] = 0 (89) p 0 dp0 = p dp + p dp d = γ d + p3 dp3 (90) (9)

p γ m 0 dp0 = γ m 0 [ d dp + dy dp + d dp 3 ] (9) γ m 0 dp 0 = dp dp dp3 d + dy + d = P d r (93) P dp 0 p 0 = γm 0 p 0 = E m = γm 0 E = m (94) (88) E = (m 0 ) + (p ) + (p ) + (p 3 ) = (m 0 ) + (p) (95) p = (p, p, p 3 ),,3 p t p = (p 0 ) (p ) (p ) (p 3 ) = m 0 (96) p = p = p 3 = 0 p 0 = m 0 (97) m 0 m = γm 0 m v v 9 4 P (P, P, P 3 ) F i F i = dp i d i = m 0, (i =...3) (98) 4 F i = dp i d = m 0 γv (99) f α = dpα = γ dpα (00) 3

(p 4 ) (8) τ f f α = m 0 d α = m 0γ d d α = m 0γ d γ α (0) v 0 τ t f i F i f 0 = dγm 0 de (0) f 0 = 0 0 0 t = = y 3 (03) = β (04) α α β = Λ(v) (05) β = α Λ(v)β α = Λ(v) α β (06) Λ = Λ(v) α α β (07) β A = 0 3 (08) A = Λ(v)A (09) A = Λ( v)a (0) 4

A A α α 0 0 t = = = () y 3 3 3 0 = = 3 t = 0 3 t = = y ( ), 0,, 3 0 3 0 3 () (3) = t y (4) 4 A B 6 4 4 T T αβ = A α B β (5) O T T αβ A α B β = Λ(v) α µ Aµ Λ(v) β ν Bν (6) = Λ(v) α µ Λ(v)β ν T µν (7) Λ(v) 3 4... C αβ = Λ( v) α µ Λ( v) β ν C µν (8) 5

ϕ A 4 A = {A α } = ϕ A A y A (9) A + ϕ t = 0 (0) A = 0 () 4 A F αβ = α A β β A α () F αβ = F βα (3) F αα = 0 (4) 6 B = A (5) E = ϕ t A (6) ( B E) F αβ F 0 = 0 A A 0 = ( (t) A ) ϕ = ( ϕ + A ) t F 0 = 0 A A 0 = ( (t) A y ) ϕ y = ( ϕ y + A ) y t F 03 = 0 A 3 3 A 0 = ( (t) A ) ϕ = ( ϕ + A ) t F = A A = ( A y ) y A F 3 = A 3 3 A = A F = A 3 3 A = y A ( ( A A y ) ) = E = E y = E (7) (8) (9) = ( A ) = B (30) = ( A )y = B y (3) = ( A ) = B (3) 6

0 E F = E y E E 0 B B y E y B 0 B E B y B 0 (33) 7

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