-
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
30 (Special Theory of Relativity)
: 3 .. 2558
( . ) 30 : 3 .. 2558 1 / 68
-
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
1 2 3 4 5 6 7
( . ) 30 : 3 .. 2558 2 / 68
-
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
, & &
( . ) 30 : 3 .. 2558 3 / 68
-
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
, & (Event) (Space) (Universe) 3
(Classical Physics) (Euclidean space) 3 E3( R3) (homogeneous) (isotropic)
+ / ( . ) 30 : 3 .. 2558 4 / 68
-
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
(Time) (dimension) (identical events) 2
1 13th CGPM, 1967/68
`` The second is the duration of 9 192 631 770 periods of theradiation corresponding to the transition between the twohyperne levels of the ground state of the caesium 133 atom.'' 9 192 631 770 133
( . ) 30 : 3 .. 2558 5 / 68
-
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
1 17th CGPM, 1983
`` The metre is the length of the path travelled by light invacuum during a time interval of 1/299 792 458 of a second.'' 1/299 792 458
(Speed of Light in Vacuum)c = 299 792 458 ms1 () (1)
+ c = 3 108 ms1 ( . ) 30 : 3 .. 2558 6 / 68
-
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
& (Frame of Reference) (coordinate system) (set of axes)
XY
Z
Oxy
z
x,y,z( )
1: 3 (E3) ( . ) 30 : 3 .. 2558 7 / 68
-
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
(Inertial Frame of Reference)
1 2
(uniform motion) (fixed star)A : (Newton's laws of motion)!
( . ) 30 : 3 .. 2558 8 / 68
-
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
O0 S0 ~u x O S( x ) t = 0 t0 = 0 ( S S0 )
2: 2 S S0 (event)
( . ) 30 : 3 .. 2558 9 / 68
-
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
3: 2 S S0 P (a) ( S0) P P(b) ( S) P
( . ) 30 : 3 .. 2558 10 / 68
-
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
4: P (x0; y0; z0) t0 S0 (x; y; z) t S ( 14 )
4 x0 = x ut ; y0 = y z0 = z
( . ) 30 : 3 .. 2558 11 / 68
-
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
S S0 (time scale) t = t0
P (Galilean coordinate transformation) (Galilean Coordinate Transformation)
x0 = x uty0 = y
z0 = z
t0 = t
9>>>>>=>>>>>;()
8>>>>>>>>>:
x = x0 + ut0
y = y0
z = z0
t = t0
( u c) (2)
( . ) 30 : 3 .. 2558 12 / 68
-
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
P x, y z S S0
vx =dxdt
, vy = dydt vz =dzdt
Sv0x =
dx0
dt0, v0y = dy
0
dt0 v0z = dz
0
dt0 S0
(Galilean velocity transformation) (Galilean Velocity Transformation)
v0x = vx uv0y = vy
v0z = vz
9>>=>>; ()8>>>:vx = v
0x + u
vy = v0y
vz = v0z
( u c) (3)
( . ) 30 : 3 .. 2558 13 / 68
-
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
P x
vS0/S = u ; vP/S = vx vP/S0 = v0x (4)
5: 2 S S0 P x
( . ) 30 : 3 .. 2558 14 / 68
-
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
P x x
vP/S0 = vP/S vS0/S vP/S = vP/S0 + vS0/S (5)( vS0/S; vP/S; vP/S0 c)
vS0/S S0 SvP/S0 P x S0vP/S P x S
( . ) 30 : 3 .. 2558 15 / 68
-
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
( S0) ( S) vS0/S = 1000 ms1 (missile) P vP/S0 = 2000 ms1 ( S0) 6
(5) vP/S = vP/S0 + vS0/S = 2000 ms1 + 1000 ms1
6: ( . ) 30 : 3 .. 2558 16 / 68
-
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
P x, y z S S0
ax =dvxdt
, ay = dvydt az =dvzdt
Sa0x =
dv0xdt0
, a0y = dv0y
dt0 a0z = dv
0z
dt0 S0
(Galilean acceleration transformation) (Galilean Acceleration Transformation)
a0x = ax
a0y = ay
a0z = az
9>>=>>; ()8>>>:ax = a
0x
ay = a0y
az = a0z
( u c) (6)
( . ) 30 : 3 .. 2558 17 / 68
-
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
m ~v ( v = j~vj) (Classical Momentum)
~p = m~v ( v c) (7) (Classical Kinetic Energy)
Ek =1
2mv2 ( v c) (8)
( . ) 30 : 3 .. 2558 18 / 68
-
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
~F m ~v ~a = d~v
dt ~F ~p
2 (Newtons 2nd law of motion) 2 (Newtons 2nd Law of Motion)
~F = d~pdt
( ~F 6= m~a) (9)
m v c ~F = d(m~v)dt
= md~vdt
2
~F = m~a ( v c m ) (10) ( . ) 30 : 3 .. 2558 19 / 68
-
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
(10) m v c ~F ~a
(6) (S0 S)
~a 0 = ~a (11) m ()
~F0 = ~F (12)
( . ) 30 : 3 .. 2558 20 / 68
-
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
m A ( ~vA) B ( ~vB) ~F (Work)
WA!B =Z BA
~F d~r (13)
( . ) 30 : 3 .. 2558 21 / 68
-
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
d~r (infinitesimal displacement) (infinitesimal time interval) dt ~v t
d~r = ~vdt (14) m v c
WA!B =Z BA
m~a ~vdt = mZ BA
d~vdt
~vdt = mZ BA~v d~v
= m
Z vBvA
v dv = m v2
2
vBv=vA
( . ) 30 : 3 .. 2558 22 / 68
-
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
(WorkEnergy Theorem)
WA!B =1
2mv2B
1
2mv2A (15)
( vA; vB c m )
WA!B = EkB EkA (16)
= (17)
( . ) 30 : 3 .. 2558 23 / 68
-
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
(postulates) 2
1: (Principle of Relativity)
`` ''
2:
`` ''
( . ) 30 : 3 .. 2558 24 / 68
-
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
7 1 (Electromotive Force emf)
7: :(a) (b)
( . ) 30 : 3 .. 2558 25 / 68
-
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
( S0) ( S) vS0/S = 1000 ms1 (light beam) P vP/S0 = c ( S0) 8
(5) vP/S = vP/S0 + vS0/S = c+ 1000 ms1 6= c
8: 2
( . ) 30 : 3 .. 2558 26 / 68
-
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
2 `` ''
9: (Einstein's speed limit) ( . ) 30 : 3 .. 2558 27 / 68
-
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
(proper time) (proper length)
( . ) 30 : 3 .. 2558 28 / 68
-
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
10: (a) S0 (source) (mirror) 2d t0 ( S0)(b) S 2` t ( S)
( . ) 30 : 3 .. 2558 29 / 68
-
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
S0 S
t0 =2d
c t = 2`
c(18)
` =
sd2 +
ut
2
2 (19) (19) (18)
t =2
c
sd2 +
ut
2
2 (20) ( . ) 30 : 3 .. 2558 30 / 68
-
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
(18) d = ct02
(20)
t =2
c
sct02
2+
ut
2
2 (21) (Time Dilation)
t =t0r1 u
2
c2
(22)
+ u < c t > t0 ! (Time dilation) t0
( . ) 30 : 3 .. 2558 31 / 68
-
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
1r1 u
2
c2
(Lorentz factor) (Lorentz Factor)
=1r
1 u2
c2
(23)
(22)
t = t0 (24)
( . ) 30 : 3 .. 2558 32 / 68
-
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
1: (muon) 2:2 0:99 S S0 S0 t0 = 2:2 s u = 0:99c (22)
t =t0r1 u
2
c2
=2:2 sr
1 (0:99c)2
c2
= 16 s
) 16
( . ) 30 : 3 .. 2558 33 / 68
-
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
11: (a) `0 S0 (b) S `
( . ) 30 : 3 .. 2558 34 / 68
-
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
(source) (mirror)
S0
t0 =2`0c
(25)
S ` u t1 ( S) ut1 `
d = `+ ut1 (26) ( . ) 30 : 3 .. 2558 35 / 68
-
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
d = ct1 (26)
ct1 = `+ ut1 (27) () S
t1 =`
c u (28)
A : ` c u c u S `
( . ) 30 : 3 .. 2558 36 / 68
-
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
() S
t2 =`
c+ u(29)
S t = t1 +t2
t =`
c u +`
c+ u=
2`
c
11 u
2
c2
(30)
S S0 (22) t = t0r
1 u2
c2
( . ) 30 : 3 .. 2558 37 / 68
-
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
(25) (30) (22)
2`
c
11 u
2
c2
=2`0c
r1 u
2
c2
(Length Contraction)
` = `0
r1 u
2
c2(31)
+ u < c ` < `0 ! ! S S0 (Length contraction) `0
( . ) 30 : 3 .. 2558 38 / 68
-
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
(31)
` =`0
(32)
12:
( . ) 30 : 3 .. 2558 39 / 68
-
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
13: (a) (b)
( . ) 30 : 3 .. 2558 40 / 68
-
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
2: 60 0:8
S S0 S0 `0 = 60 m u = 0:8c (31)
` = `0
r1 u
2
c2= (60 m)
r1 (0:8c)
2
c2= 36 m
) 36
( . ) 30 : 3 .. 2558 41 / 68
-
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
x0 P S0 ( 4)
S x0
14
14: P (x0; y0; z0) t0 S0 (x; y; z) t S ( 4 )
( . ) 30 : 3 .. 2558 42 / 68
-
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
14 x0
= x ut ; y0 = y z0 = z
x0 =
x ut ; y0 = y z0 = z (33)
1 x =
x0 + ut0
; y = y0 z = z0 (34)
x0 (33) (34) x =
h
x ut+ ut0i
( . ) 30 : 3 .. 2558 43 / 68
-
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
t0 =
t ux
c2
(35) x (34) (33)
x0 = h
x0 + ut0
uti
t =
t0 +
ux0
c2
(36)
( . ) 30 : 3 .. 2558 44 / 68
-
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
P (Lorentz coordinate transformation) (Lorentz Coordinate Transformation)
x0 = x ut
y0 = y
z0 = z
t0 = t ux
c2
9>>>>>>=>>>>>>;()
8>>>>>>>>>>>>>:
x = x0 + ut0
y = y0
z = z0
t =
t0 +
ux0
c2
(37)
( . ) 30 : 3 .. 2558 45 / 68
-
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
dx0 =
dx udt = vx udt
dy0 = dy
dz0 = dz
dt0 = dt udx
c2
=
1 uvx
c2
dt
9>>>>>>>=>>>>>>>;(38)
dx =
dx0 + udt0
=
v0x + u
dt0
dy = dy0
dz = dz0
dt = dt0 + udx
0
c2
=
1 +
uv0xc2
dt0
9>>>>>>>=>>>>>>>;(39)
( . ) 30 : 3 .. 2558 46 / 68
-
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
P x, y z S S0 (Lorentz velocity transformation) (Lorentz Velocity Transformation)
v0x =vx u1 uvx
c2
v0y =
vy
1 uvx
c2
v0z =
vz
1 uvx
c2
9>>>>>>>>>>>>=>>>>>>>>>>>>;()
8>>>>>>>>>>>>>>>>>>>>>>>>>>>:
vx =v0x + u1 +
uv0xc2
vy =
v0y
1 +
uv0xc2
vz =
v0z
1 +
uv0xc2
(40)
( . ) 30 : 3 .. 2558 47 / 68
-
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
P x x
vP/S0 =vP/S vS0/S 1 vS0/S vP/S
c2
! vP/S = vP/S0 + vS0/S 1 +
vS0/S vP/S0
c2
! (41)
vS0/S S0 SvP/S0 P x S0vP/S P x S
( . ) 30 : 3 .. 2558 48 / 68
-
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
8 x (41)
vP/S =vP/S0 + vS0/S 1 +
vS0/S vP/S0
c2
! = c+ 1000 ms1 1 +
1000 ms1(c)
c2
! = c
15: 2
( . ) 30 : 3 .. 2558 49 / 68
-
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
3: 0:8c 0:9c() S, S0 P
vS0/S = 0:8c vP/S0 = 0:9c
( . ) 30 : 3 .. 2558 50 / 68
-
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
(41) vP/S =
vP/S0 + vS0/S 1 +
vS0/S vP/S0
c2
! = (0:9c) + (0:8c)1 +
(0:8c)(0:9c)
c2
= 0:988c < c
) 0:988c
A : (5)
vP/S = vP/S0 + vS0/S = 0:9c+ 0:8c = 1:7c > c
( . ) 30 : 3 .. 2558 51 / 68
-
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
m ~v ( v = j~vj) (Relativistic Momentum)
~p = m~vr1 v
2
c2
(42)
+ v c ~p = m~v ()
( . ) 30 : 3 .. 2558 52 / 68
-
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
S0 m ( S0) ~v S( S0 )
=1r
1 v2
c2
(43)
(42)
~p = m~v (44) p = j~pj
p = mv (45)
( . ) 30 : 3 .. 2558 53 / 68
-
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
Helios SpacecraftHelios spacecraft () 252 792 (0:000 234 )
v = 252 792 kmh1 0:000 234c
1:000 000 027
+ v c 1 http://en.wikipedia.org/wiki/
Helios (spacecraft)
( . ) 30 : 3 .. 2558 54 / 68
-
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
~F m ~F ~p 2 (9) ~F = d~p
dt
m ~F 6= m~a( ~F ~a )
~F = d(m~v)dt
= md~vdt
+ddt
m~v = m~a+ ddt
m~v
ddt
=
3
c2~a ~v ~F,
~v ~a 2
~F = m~a+
2
c2~a ~v~v ( m ) (46)
( . ) 30 : 3 .. 2558 55 / 68
-
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
ddt
=1
mc2~F ~v
~a = 1
m
~F 1
c2~F ~v~v ( m ) (47)
(46) (47) ~F ~a
( . ) 30 : 3 .. 2558 56 / 68
-
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
m A ( ~vA)
B ( ~vB) ~F (13)
WA!B =Z BA
~F d~r
2 (9) ~F = d~pdt
WA!B =Z BA
d~pdt
d~r =Z BA
d~rdt
d~p =Z BA~v dm~v
m
WA!B = mZ BA~v d~v (48)
( . ) 30 : 3 .. 2558 57 / 68
-
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
WA!B =mc2r1 v
2Bc2
mc2s
1 v2Ac2
( m ) (49)
m ~v ( v = j~vj) (Relativistic Energy)
E =mc2r1 v
2
c2
E = mc2 (50)
( . ) 30 : 3 .. 2558 58 / 68
-
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
(rest energy) E0
(Rest Energy)E0 = mc
2 (51)
Ek = E E0 (52)
(Relativistic Kinetic Energy)Ek = ( 1)mc2 (53)
( . ) 30 : 3 .. 2558 59 / 68
-
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
(Maclaurin series) (1 + x)n jxj < 1
(1 + x)n = 1 + nx+n(n 1)
2!x2 +
n(n 1)(n 2)3!
x3 + (54)
v < c
= 1 +1
2
v2
c2+
3
8
v4
c4+
5
16
v6
c6+ (55)
(53)
Ek =1
2mv2 +
3
8
mv4
c2+
5
16
mv6
c4+ (56)
+ v c Ek = 12mv2 (
) ( . ) 30 : 3 .. 2558 60 / 68
-
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
(50) E2 = m2c41 v
2
c2
E21 v
2
c2
= m2c4
E2 E2 v2
c2= m2c4
E2 mc22 v2c2
= m2c4
E2 mv2c2 = m2c4E2 p2c2 = m2c4
( . ) 30 : 3 .. 2558 61 / 68
-
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
(EnergyMomentum Relation)
E =qp2c2 +m2c4 (57)
(massless particle) (photon) (gluon) (stronginteraction)
E = pc (m = 0) (58)
( . ) 30 : 3 .. 2558 62 / 68
-
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
4: ( 9:109 1031 kg) MeV 1 MeV = 1:602 1013 J m = 9:109 1031 kg (51)
E0 = mc2 =
9:109 1031 kg3 108 ms12 = 8:198 1014 J
MeV 1:602 1013 JMeV1 E0 =
8:198 1014 J
1:602 1013 JMeV1 = 0:511 MeV) 0:511 MeV
( . ) 30 : 3 .. 2558 63 / 68
-
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
[1] Hugh D. Young and Roger A. Freedman, (),
3 , 1, , (2551).
[2] , (: ), 1, ., (2548).
[3] Hugh D. Young and Roger A. Freedman, Sears and ZemanskysUniversity Physics: With Modern Physics, 13th ed.,Pearson/Addison-Wesley, San Francisco (2012).
[4] Kenneth S. Krane, Modern Physics, 3rd ed., John Wiley & Sons,New Jersey (2012).
( . ) 30 : 3 .. 2558 64 / 68
-
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
[5] Raymond A. Serway and John W. Jewett, Jr., Physics for Scientistsand Engineers with Modern Physics, 8th ed., Brooks/Cole,Belmont, California (2010).
[6] Raymond A. Serway, Clement J. Moses and Curt A. Moyer, ModernPhysics, 3rd ed., Brooks/ColeThomson Learning, Belmont,California (2005).
( . ) 30 : 3 .. 2558 65 / 68
-
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
1 :
http://www.sciencecartoonsplus.com/gallery/einstein/ein15.gif
2 1: http://en.wikipedia.org/wiki/File:Coord system CA 0.svg
3 2: 2.1 26 Krane [4]4 3: 39.1 1146 Serway & Jewett [5]5 4: 2.2 27 Krane [4]6 5: 2.1 26 Krane [4]7 6: 37.2 (a) 1225 Young & Freedman [3]8 7: 37.1 1224 Young & Freedman [3]9 8: 37.2 (b) 1225 Young & Freedman [3]
( . ) 30 : 3 .. 2558 66 / 68
-
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
10 9: http://www.cartoonsidrew.com/2011/05/einsteins-speed-limit.html
11 10: 37.6 1229 Young & Freedman [3]12 11: 37.10 1234 Young & Freedman [3]13 12: 2.11 34 Krane [4]14 13: 2.12 35 Krane [4]15 14: 2.2 27 Krane [4]16 15: 37.2 (b) 1225 Young & Freedman [3]17 :
( . ) 30 : 3 .. 2558 67 / 68
-
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
30
( . ) 30 : 3 .. 2558 68 / 68
30