1 §6 F §6 F our-dimensional our-dimensional Covariant form of Covariant form of special special theory of relativity theory of relativity • Seen before ， Relativity of time and space are inseparable interconnectedness. Three-dimensional space and one dimension of time and space constitute a unified four- dimensional space. In this section we further express to four-dimensional forms with theory of simple four- dimensional space-time , And then physical will be repres ented as four-dimensional Covariant form, Thus clearly shows that some of the intrinsic link between physical. • 1. 1. Orthogonal transform of Orthogonal transform of three-dimensional • Before discussing the four-dimensional space-time transformation, firstly,review the nature of two- dimensional (three-dimensional) spatial rotation transformation.
Transcript
1
§6 F§6 Four-dimensional our-dimensional Covariant Covariant form of form of special theory of relativityspecial theory of relativity
• Seen before， Relativity of time and space are inseparable
interconnectedness. Three-dimensional space and one dimension
of time and space constitute a unified four-dimensional space. In
this section we further express to four-dimensional forms with
theory of simple four-dimensional space-time , And then physical
will be represented as four-dimensional Covariant form, Thus
clearly shows that some of the intrinsic link between physical.
• 1. 1. Orthogonal transform of Orthogonal transform of three-dimensional
• Before discussing the four-dimensional space-time transformation,
firstly,review the nature of two-dimensional (three-dimensional)
spatial rotation transformation.
2
• For the two-dimensional coordinate rotation
transformation ， be shown as in the figure,
system of coordinates S turn θto S system of
coordinates . Assuming any point located on
the plane, P’s coordinates at the S system a
nd S system respectively is
)','(),( yxandyx
cossin'sincos'xxyyxx
yx
yx
cossinsincos
''
2222 yxyx
aaIaa
''
~~
)'(oo'SS
x
'x'y
y)','(
),(yxyxP
• Transformation relationship
between them is
• Expressed in matrix form is
• It is an orthogonal matrix ，Orthogonality condition is
a
3
• The three-dimensional rotation transformation of discussions with the same two-
dimensional
system coordinate ingcorrespondseries'series , ，SS
zayaxaz
zayaxay
zayaxax
333231
232221
131211
'
'
'
)3,2,1,(
offunction is
ji
andaij
321 ,, xzxyxx
)3,2,1(' ixax jiji
3 to1 from value the takeof behalfOn index freedom is
3 to1 fromsummation of behalfOn indicator same theis
，，
i
j
• Coordinate transformation between the old and new
can be written in general
• If using
• Concise form the equation can be expressed as
follows
4
• Matrix form
3
2
1
333231
232221
131211
3
2
1
'
'
'
x
x
x
aaa
aaa
aaa
x
x
x
jkikij
iiii
aa
Iaaaa
xxxx
xxxxxx
express element tomatrix theuse alsoCan
)matrix identity (~~matrix by the dRepresente
''as expressed beCan
''' 23
22
21
23
22
21
)3,2,1(' ixax jjii
• Rotation transformation distance remains unchanged ，that is have orthogonality
• According to which the inverse
transform to obtain the conversion
5
• Proof ： to multiplying both
• sides can get
)3,2,1(' ixax jiji
ika
kjkjjijikiik xxxaaxa 正交性'
)3,2,1(' kxax iikk即
)~(
'
'
'~ 1
3
2
1
3
2
1
aa
x
x
x
a
x
x
x
• This is inverse transform.Its matrix
form is
6
2 2 Physical classified according to transform the Physical classified according to transform the nature of spacenature of space
• The second chapter we have introduced the tensor
quantities by Category， can be summarized as
• 1） Scalar ： The same amount under coordinate rotation ， as
q'=q
• 2） Vector ： Its component transform by the following
coordinates transformation relationship under coordinate rotation)3,2,1(' iuau jiji
Operator
components Nine)3,2,1,(' jiTaaT kljlikij
• For example ： Speed, power, elect
ric field intensity,
• 3） Second-order tensor： Its components as
follows variations in the rotating coordinate
transformation
7
• Second-order tensor is two vector’s union-pr
oduct.
• Note： Repeat indicator represent sum ， this
operational is called indicators contraction 。 G
enerally ,how many free indictor it have, it is
Several tensor.
• Such as AiBi is scalar ， uiTij have a free indicto
r, it is A tensor （ is Vector ）， uiTjk have thre
e free indictors, represents Third-order tensor .
8
3 FFour-dimensional our-dimensional form of form of Lorentz Lorentz transformationtransformation• The imaginary coordinates of the fourth
dimensional is introduced in formictx 4
formMatrix
144
33
22
411
'
'
'
'
xixx
xx
xx
xixx
4
3
2
1
4
3
2
1
0001000010
00
''''
xxxx
i
i
xxxx
返回
)4,3,2,1(' xax
• Then the form of Lorentz transformation translate into
9
• Appropriate interval constant can be expressed aselements)(matrix
matrixation transformLorentz theis)( aaitIn
Invariants'' xxxx
aaorIaaaa ~~
'xax
• It is equivalent to orthogonality condition of Lorentz
transformation
• Thus we can get Lorentz inverse transform
• Therefor， Lorentz transformation can be view as
a rotation transformation of four-dimensional space-time
in form ， the transformation have orthogonality .
10
• Note ： The index symbol of three-dimensional and
four-dimensional is different. Under the three-
dimensional case， The amount of each component of
the index with Latin alphabet as i ， j ， k. It can take
the values from 1 to 3, the corresponding indicators
represent the sum from 1-3. In order to distinguish
between the three-dimensional， In the space of four-
dimensional space-time， The amount of each
component of the index are used to represent the Greek
alphabet . It is the value from 1-4， Its the
same indicators represent the sum from 1-4.
,,,
11
4 4 Four dimensional covariantFour dimensional covariant
• In the form of four-dimensional， Unity of time and space in
the four-dimensional space-time space， Transform the
inertial reference system is equivalent to the four-dimensional
space-time space "rotation."Since the motion of matter in
space and time. Describe the motion of matter and physical pro
perties is bound to reflect the characteristics of space-time transfor
mation . In the promotion of three-dimensional form ，We can
classifiy physical in four-dimensional space "rotation" in
nature (Lorentz transformation) .
• 1） Lorentz invariant (scalar):In the same amount of Lorentz
transformation ， as： Interval, the inherent isochronous
12
• 2） Four-dimensional vector ： Has four components ， and
each component under the Lorentz transformation and transform
form of four-dimensional space-time coordinates are the same as ，that is
xax '
)4,3,2,1,(' TaaT
• 3） Second-order tensor： It has 16 components ， and
each component under the Lorentz transformation meet the
following form transformation
• Higher-order tensor can be defined similarly.• Then, we discuss on the four-dimensional vector
velocity and four-dimensional wave vector.
13
11 ） ） The four-dimensional vector velocityThe four-dimensional vector velocity • Speed usual sense is ui=dxi/dt. It does not represent the
components of the four-dimensional velocity vector.Because dxi
changes by vector ， and dt change under the Lorentz
transformation. In fact, this is something we can also be seen from
the Lorentz transformation formula of speed directly,
21
223
3
21
222
2
21
11
/1
/1'
/1
/1'
/1'
cu
cuu
cu
cuu
cu
uu
Obviously it is not Lorentz
transformation. It does’t change in
the form of four-dimensional
space-time coordinates under the
Lorentz transformation, So it does
not mean that the four-
dimensional components of the
velocity vector.
14
• Definition is four-dimensional velocity vector
d
dxU
),( ictxx i
d
dt
dt
dx
d
dx
22321 /1/1),,,,( cu
d
dticuuu
dt
dxu
而
),(),,,( 321 icuicuuud
dxU uu
It is apparent to a four-dimensional vector ， because dτis an
invariant ， but dxμ is a four-dimensional vector.xμIs a
four-dimensional coordinates
• So
15 UaU '
ii uU ,1
• Note：The first three components of Four-dimensional
velocity vector is not speed under common sense， Its contact
is when u<<c ，
• That is the first three components of Four-
dimensional velocity vector tend to speed under
common sense when u<<c. This is the reason it
is defined as a four-dimensional velocity vector.
Since four-dimensional vector velocity is four-
dimensional vector ， It changes according to t
he four-dimensional vector under Lorentz tran
sformations. That is
16
22）） Four dimensional wave vectorFour dimensional wave vector• With a corner frequency ω, wave vector k plane
electromagnetic wave spread in a vacuum. Frequency and
propagation direction observed in another frame of reference of
the plane electromagnetic wave will change happen（ This is conf
irmed respectively by the Doppler effect and the aberration effect
). Now we use ω 'and k' said S ‘ to express frequency and wave
vector observed on the same plane electromagnetic wave. Then
what relationship will they meet ? To answer this question, we
first explain the phase
txk
vectorldimensiona-four
constitute),(c
ikand
• Is Lorentz invariant
17
• Let reference frame S and S '
coincide in the time origin t = t' = 0. At this time, the electromagnetic wave at
the origin is at crest (event 1), the phase is
0, that is
)'(SS
)'(xx)'(oo
S
)'(xxo 'o
'S)0',0'(
0''''
ttxx
txktxk
• After a period t0, S at the origin x =
0 is at the second peak, the phase
is-2π• （ This is event 2 ） , Its space-time coordinates is S(0 ，
t0)。 Event 2 was observed in the S ' ， Its space-time
coordinates is （ x' ， t' ）
18
• At the same time,the observed event 2 also should at crest (This is
a physical fact) ， so the phase is also -2π， So that we can see
under the reference frame transformation , Phase should be
constant, That phase is Lorentz invariant
Invariants'''' txktxk
• Note ： In this invariance is a physical fact (That is, from the viewpoint of
experimental ). About the phase invariance we can prove it by the Lorentz
transformation invariance; Can be used to prove by the transformation
field and the transformation relationship of four-photon momentum , The
following discussion of the issue in accordance with the phase change.
19
• According to phase invariant, have
Invariants'''' txktxk
Invariants''
'' ticc
ixkticc
ixk
Invariants),(),()','()'
,'( ticxc
ikticxc
ik
• Its transformation is like this
• Further can be written as
• We know ， (x ， ict) constitute a four-dimensional
coordinates of the vector ， but The dot product results of
(k ， iω/c) and four-dimensional vector is a scalar.Therefore
(k, iω / c) also constitute four-dimensional vector.
• Its first three components are wave vector in common
sence. At this time phase change can be expressed as
• Wave vector under the Lorentz transformation change
in accordance with the form of four-dimensional vector ，be
• To special Lorentz transformation,it
can be write to
21
• Seen, although from S 'series to see this wave remains a plane
wave, However, both the frequency and the direction of
propagation change .
)4()(')3(')2('
)1()('
1
33
22
211
kkkkk
ckk
thenisxxandkk ),'()'()'(vector
waveof axis ebetween th angle theassuming Now
'sin'
'
sin
'cos'
'
cos
1
1
ck
ck
ck
ck
22
• From (1) can get
)'4()/cos1(')4( c 得：由
)'1()/(cos)/cos('cos' cc
)'2(sin'sin'' kk
)/cos1('
)/(cos
sin'tan
c
c
)'4(
)'1/()'2( Aberration formula
Doppler effect
• From (2) and (3) can get
• So we have
23
• Assuming S 'is stationary relative to the light source ， the
n ω ' =ω0, Thereby obtaining the relativistic Doppler
effect.
)/cos1(0
c
）横向（ 2/
220 /1 c
220 /1/ cTT
c
)()/cos1(
0
经典多普勒效应c
Motion clock delay
Transverse Doppler effect
24
• Explanation： 1) aberration formula can also be
deduced by the speed of transformation formula;
2
• 2) aberration phenomenon was first
discovered by the Bradley (Brad Terai) in
1728 with astronomical observations . When
observing the stars on the earth, any star's
apparent position cyclical change in a year, o
r say observation telescope when tracking
star Barrel pointing will appear periodically
elliptical motion similar to a circle (Pictured).
This can be explained as follows
25
• Figure a, assuming that velocity of the earth isυ with
respect to S of the solar ， See the inclination of light
emitted by a star is α=π-θ in the S series (Stars are very
far away, can be considered a star hair is parallel
light).When using a telescope to observe the stars on
the earth(S' series ). inclination isα ' =π-θ ' ， becauseυ
<<c then
c/cos
sin'tan
'
'
图a
图b• As the Earth revolves around the sun ， Direction of movement change
a cycle in a year. Therefore, the apparent direction of the light emitted
by the same stellar changes a cycle,too. (Figure a) This has been
confirmed by experiments in astronomy.
sintan '
(cos / )c
26
• In the previous theory of relativity， These
exist aberration is interpreted as the Earth
campaign relativing to "ether" 。 But then
the Michelson - Morley experiment was denied
the Earth campaign relativing to "ether“.
It is this contradiction appears, that led to the
"ether" and absolute reference system is denied.
Thereby the special theory of relativity was
established.
27
5 5 covariance of physical lawscovariance of physical laws• When the reference system changes, invariant properties of
Equation form is called Covariance.The Physical that In the
covariance of the equation is called Covariates. Only each item i
n the equations that is Similar covariant equations have
covariance， Conversely, each item of the Covariant equation
must be Similar covariant equations. Like Fμ = G μ + T μboth
sides are four-dimensional vector. So this equation is covariant.
In any inertial reference system it can be expressed in the same
form . Use inverse transform
'''
'
'
'
TGF
TaT
GaG
FaF
Form is unchanged ，
equation has
covariance.
28
§7 The §7 The Relativity Invariance of electrodynamicsRelativity Invariance of electrodynamics• According to the principle of relativity, the basic rule
of any inertial electromagnetic phenomena can be expressed
as the same form. The Maxwell equations summarize the
basic laws of macroscopic electromagnetic phenomena. A
series of inferences derived from the electromagnetic wave
in a vacuum to the speed of light c disseminate have been
proved by experiments. So Maxwell equations should be
applicable to any inertial, It can be expressed as a form of
four-dimensional relativistic covariant. Because there are
current density and charge density in Maxwell equations.
They are source of electromagnetic excitation .Here, we
discuss their transformation properties.
29
1 1 Four-dimensional current density vectorFour-dimensional current density vector• According to the law of conservation of charge , the total
charge of charged system should always remain the same, that
total charge Q does not change with the coordinate system , It is
the Lorentz scalar.
''
'''0' dVdV
dVQ
dVQQQQ
缩短纵向按横向不变
2222
/1/1'
cucudVdV
• Assuming charge system consolidation in S 'series, its
velocity is u relative to the S series, then
30
• Therefor, we can get
022
022 /1//1/' ucucu
)1(00 uuuJ uu
)2(04 icicJ u
UicuicJJ u 00 ),(),(
• Seen, charge density is the amount of a visible change in Lorentz t
ransformation .
• When Particle move at speed u, its current density is
• If we introduce The fourth component of current density
• Then according to four-dimensional velocity vector Previously
defined, Formula (1) and (2) expressed as a four-dimensional
current density vector
31
• At here, Current density J and the charge density ρ together
as four-dimensional vector. It shows both physical unity.
Changing the reference system, they can be transformed into each
other. But the law of conservation of charge is applicable in any
inertial reference system . Now it can be expressed as
0
x
J Obviously it is covariant, and is
a Lorentz scalar 。• From this fact speaks， Because of the unity of
relativistic space-time, showing unity between them
different physical quantities in the theory of relativity. Here we will see Electric and magnetic fields (vector
potential and scalar potential) Etc. also has this unity.
• When speaking electromagnetic radiation, we represent the
electromagnetic field with potential A and . Describesing the
electromagnetic field ,Maxwell equations turned it into a wave equation
satisfied by potential . In order to facilitate， we first discuss Dalem Burr
equation satisfied by potential that have covariant form in the Lorentz
gauge condition.
)01
(1
1
2
02
2
22
02
2
22
tcA
tc
Jt
A
cA
33
Using four-dimensional space-time coordinates,， The left
side of Dalem Burr equation can be written as
A
xxxx
A
tc
)()
1(
24
2
23
2
22
2
21
2
2
2
22
xx
这里ᑫ 24
2
23
2
22
2
21
2
xxxx
0/
)()( 4002
0 icJicicc Ɏ
• Be called Lorentz scalar operator. Further deformity is
AA
xx
34
• So have
• This formula and the right side of constitute a
four-dimensional vector
• So their left sides also should constitute four-dimensional vector.A
nd is Lorentz scalar operator , hen constitute four-dimensional
vector .
• Using to express, that is
40)( Jci
JA
0
J0
)/,( ciA
A
),(ciAA
Four-dimensional Four-dimensional
vector potentialvector potential
0
0
x
A
JA This is clearly This is clearly covariantcovariant
• At this time, Dalem Burr equation satisfied by potential and the
Lorentz gauge condition can be expressed as
35
• In the transformation of reference frame (Lorentz
transformation)， Four-dimensional vector
potential convert by four-dimensional vector,that is
)('
'
'
)('
1
33
22
211
A
AA
AAc
AA
)4,3,2,1(
'
AaA
36
3 3 Electromagnetic field tensorElectromagnetic field tensor • Electromagnetic field Respectively expressed by potential is Electromagnetic field Respectively expressed by potential is
t
AEAB
,
2
1
1
23
1
3
3
12
3
2
2
31
x
A
x
AB
x
A
x
AB
x
A
x
AB
)(
)(
)(
4
3
3
43
4
2
2
42
4
1
1
41
x
A
x
AicE
x
A
x
AicE
x
A
x
AicE
• Its components form are
37
• If we introduce a antisymmetric four-dimensional tensor
)(
AA
x
A
x
AF
123
132
231
FB
FB
FB
343
242
141
icFE
icFE
icFE
334224114
123213312
,,
,,
Ec
iFE
c
iFE
c
iF
BFBFBF
• The defined antisymmetric four-dimensional tensor can
be respectively expressed as Electric and magnetic
fields
• Thus was
38
• And this antisymmetric four-dimensional tensor can be expressed as
0
0
0
0
321
312
213
123
Ec
iE
c
iE
c
i
Ec
iBB
Ec
iBB
Ec
iBB
F
ElectromagElectromagnetic field netic field tensortensor
FaaF '
• Electromagnetic field tensor is Four-dimensional second-order
tensor. It changes according to the law of second-order tensor
under Lorentz transformations.That
39
4 4 four-dimensionalfour-dimensional Covariant form of Maxwell's Covariant form of Maxwell's equationsequations
• Electromagnetic fields can be expressed as electromagnetic tensor.
Using electromagnetic tensor, Maxwell's equations can be
expressed as four-dimensional Covariant form like this
Jt
EB
E
000
0
t
BE
B
0
)(0 AJx
F
)(0 Bx
F
x
F
x
F
40
• Then we deduce the above Covariant form of Maxwell's
equations
)(02
03
3
2
2
1
1
0
iciccx
E
x
E
x
EE
403
3
2
2
1
1
Jx
Ec
i
x
Ec
i
x
Ec
i
)1(4040
403
43
2
42
1
41 44 Jx
FJ
x
F
x
F
x
F F
41
• The electromagnetic field tensor is expressed as
101
001)( Jt
EB
101
23
2
2
3 1J
t
E
cx
B
x
B
)2(1010
4
1410
3
13
2
12 11 Jx
F
x
FJ
x
F
x
F F
)3()( 202
202
002 Jx
FJ
t
EB
)4()( 303
303
003 Jx
FJ
t
EB
• Similarly available
42
• The together of (1) to(4) is the formula (A)
)()4,3,2,1(0 AJx
F
003
3
2
2
1
1
x
B
x
B
x
BB
)6(0)(4
23
3
42
2
34
4
111
x
F
x
F
x
F
x
Bic
t
BE
)3,2,1(
)5(03
12
2
31
1
23
x
F
x
F
x
F
)4,3,2(
43
• Similarly too
)7(0)(1
34
4
13
3
4122
x
F
x
F
x
F
t
BE
)8(0)(2
41
1
24
4
1233
x
F
x
F
x
F
t
BE
)2,1,4(
)1,4,3(
)()4,3,2,1:(0 Bx
F
x
F
x
F
34
21
• The together of (5) to(8) is the formula (B)
44
5 5 Transform relations of electromagnetic fieldsTransform relations of electromagnetic fields
Research is： Two relative motion in the inertial
In determining the space-time point P
S series )( tzyx
S series )( tzyx
E B
E B
E D B H场量
E D B H场量
EB
已知
E
B
已知
45
• The electromagnetic field is expressed as tensor fields,and can be
deduced by Transform relations of electromagnetic fields , The
transform relations of electromagnetic fields are FaaF '
123332232231 '' BFaaFaaFB
FaaFB 31132 ''
)()( 3223343141311 Ec
BaFaFa
)(' 2233 Ec
BB
)('),(',' 23332211 BEEBEEEE
• According to this, According to this,
can getcan get
• Similarly
available
46
Direct transformationDirect transformation
11 EE
322 BEE
233 BEE
11 BB
3222 E
cBB
2233 E
cBB
（ υ→-υ）→ inverse transformation
11 EE
)( 322 BEE
)( 233 BEE )(
)(
2233
3222
11
Ec
BB
Ec
BB
BB
47
• The transformation can be expressed as vector form like these
''
'
'
||||
||||
BEE
EE
BEE
EE
''
'
'
'
2
||||
2
||||
Ec
BB
BB
Ec
BB
BB
48
1 ） In the direction of movement,the component of Electric field and
magnetic field is equal.
2） In the vertical direction of Movement ， Electric field and magnetic field
are related.
3 ） The unification of vector potential and scalar potential is four-
dimensional vector potential.The unification of Electric field and magnetic
field is Electromagnetic field tensor. This reflects the unity and relativity of
the electromagnetic field. Electric field and magnetic field are two aspects of
the same substance. In a given reference frame,Electric and magnetic fields
exhibit different properties； But when the reference system changes,they
can transform into each other.This is their unity.4)If the Charge that can produce field is stationary in a inertial, there is only
electrostatic field, no magnetic field in this system. But in another inertial that
have relative motion with it , it have both electric field and magnetic field.
Discussion
49
Inverse
transformation
33
22
11
EEEE
EE
223
322
11 0
Ec
B
Ec
B
BB
B
B
B
x
y
z
0
0
0
Special case ：
Only electrostatic field in a reference system
In the S series, not only has the electric field but also magnetic field.
Ec
B
2
1Easily obtained
50
6 6 InvariantsInvariants of of electromagnetic fieldelectromagnetic field• From the foregoing discussion, we seem to be able to see, we can alway
s make arbitrary values of E and B by Lorentz
transformation。 Actually, among electric and magnetic fields,
relativity also contains the absolute side， This is the invariants of
electromagnetic field. Because the electromagnetic field is expressed
with the electromagnetic field tensor Fμν, Therefore, if we want to find
the electromagnetic field invariants, just requires a variety of possible
electromagnetic tensor scalar product. Obviously we can not form two
independent variables
scalar ldimensiona-four all areThey
FF
FF
51
• Can prove
Invariants
Invariants)(2 222
2
EBc
iFF
BcEc
FF
Invariants
Invariants222
EB
BcE
Discussion
0 BEBE
• Therefore electromagnetic constitute two invariants
• 1） If the electric and magnetic fields in an inertial frame
are perpendicular to each other, then
• So it will be vertical in any inertial system.
52
• 2 ） If is equal in absolute value inertial ，
that is .They also is equal in any inertial (such as plane waves)
BcE
和
0222 BcE
makecan that
inertial a findnot could Weinertial in the If)3 ，BcE
• 4 ） In any inertial , The angle between E and B is an
obtuse angle (acute)
makecan that inertial a findnot
could weinertialan in if ,Conversely; BcEBcE
BcE
53
The angle is also an obtuse
angle (acute) in any inertial .
BandEbetween
222222 BcEorBcE
• 5 ） If in any inertial , E·B=0, we can always
find an inertial that make it only have electric
fields or magnetic fields. Specifically only
electric fields or magnetic fields depend on
• Thus, the problem of reference system of
electromagnetic phenomena is completely
resolved.
54
Case: Electromagnetic field amount of Uniform motion of a point charge
Known:point charge in laboratory reference frame
Request ： BE
Solution： Take charge of their stationary reference frame S 'series
Laboratory reference frame is the S series
E
q
rr
4 02
B 0
Eqx
r
Eqy
r
Eqz
r
x
y
z
4
4
4
03
03
03
x̂ Velocity
SIn the seriesComponent type
q
55
S
q
y y
x x
r P
E Eqx
rx x 4 0
3 zz
yy
EE
EE
yz
zy
xx
Ec
B
Ec
B
BB
2
2
0
By the amount of the converted field
56
Using Lorentz coordinate transformation, the results need to
represent with the amount of S. It should be noted that all
distances are measured simultaneously on the S series.
zzyyxx ',',
2/1222 zyxr
2
2
1
1
c
2/1222 zyxr
57
E E E Ex y z 2 2 2
2/322220
30 )(
1
4'4 zyx
rq
r
rq
E
qr
r
sin4
1
102
2
2 23
2
2c
EB
Result
2/32
222330 )1(
1
4rzy
r
rq
2322
2
20 sin1
1
4
r
q
58
1) Powerline
04
10
22E
q
r
2 4
1
102 2 1 2E
q
r
2) Gauss
theorem
In the two inertial same painted faces closure
0q
sdES
q 0
Nothing to do with sports 0
qSdE
S
Gauss theorem also applies to the movement of electric charge
Different field strengths But the total number of power lines are the same
Discussion
59
3) At low speed
E
qr
r
sin4
1
102
2
2 2 3 2
rr
qˆ
4 20
0
Electrostatic field
20
2 4
ˆ
r
rq
c
EB
q
Pr
60
§8 Kinetic basis of special relativity
• How the concept of high-speed movement dynamics?
• The basic starting point ：• 1) The form of basic law under the Lorentz
transformation is Changeless. Newtonian mechanics
need to be modified ；• 2） Back to Newtonian mechanics at low speed
• In this section we analyze the mechanics of a few basic
• Whenυ=0， the kinetic energy of an object is zero
and total energy is
• scilicet
65
• 1）Here, the rest energy is a constant m0c2， In the classical case,
we know that energy is pointless to attach a constant. However, in
the relativistic case, the emergence of the rest energy (m0c2 ) is
direct requested by covariance of special relativity and can not be
deleted.
• 2） From a physical point of view, one of the most fundamental
laws of nature is the law of conservation and energy conversion.
For additional term in this energy appearing, it have a physical
meaning only when it can be converted into other forms of energy.
Then it can convert into other forms of energy under certain
conditions. This has been proved by experiments (use of atomic
energy, etc.)
Discussions
66
2 The relationship of Momentum mass and energy2 The relationship of Momentum mass and energy
• By four-dimensional momentum may constitute invariants
Invariants2
22
c
WPPP
200,0 cmWWP
2202
22 0 cm
c
WP
420
222 cmPcW
The relationship of relativistic The relationship of relativistic momentum mass and energymomentum mass and energy
W20cm
Pc
• In stationary object
• scilicet
67
3 3 Mass-energy relationMass-energy relation
• 1 ） Mass-energy relationMass-energy relation
• The aforementioned m0c2 is required by relativistic
covariance. It represents the internal energy of the object
when it is stationary. This shows that there is a movement of
internal tationary objects, a certain quality of particle
corresponds a certain internal energy. Conversely, particles
with a certain kinetic energy inside performance of a certain
inertial mass.
200 cmW
• It is an important corollary of relativity 。
Mass-energy relationMass-energy relation
68
22）） Binding energyBinding energy and and quality lossquality loss
• Due to the specific structure of the object is independent of
covariance. So for complex objects, these mass-energy relation is still
valid, scilicet
200 cMW
The total internal energy in the rest of the composite object(Stationary centroid)
The rest mass of the composite object
• When a group of objects form composite objects, Due to the
interaction energy between the particles
and the kinetic energy of the relative motion.So when the whole
body is stationary, the total energy is generally not equal to
stationary energy of constitutes particles. Scilicet
69
• Difference between the two is called the binding energy of the object, scilicet
200 cmW i
The mass of particle i th
02
0 WcmW i
00 MmM i
200 / cWM
• Correspondingly, mass of an object M0= W0/c2 does no
t mean that the sum of stationary quality composed of v
arious particles , difference between the two is called difference between the two is called
the Quality loss.the Quality loss.
• ExplanationExplanation ： 1）Mass-energy relation has been
proved quite a lot of good experimental . In turn, it
reinforces the validity of the special theory of relativity.
70
• The use of atomic energy is mainly based on mass-energy relationship. We
can say the quality of the object loss（ Such as fusion and fission process )，
Thereby allowing the small inertial mass of the object into the direction to
release energy. to release energy. Then ,can make objects transformed to
little direction to release the energy.
Decay0
)photon theof massrest No(energyPhoton 2 cm
• 2）Mass-energy relationship is reflected the relationship between mass
as a measure of the inertial and energy as a measure of physical. In the
processes of material response and conversion, In the form of material
changes, and the form of movement changes ,too. But it’s not that
matter convert into energy. Substances does not eliminate in the
transformation process, but only convert from one form to another
form.
71
(like ). The energy remains conserved in the
conversion process. In the theory of relativity, the conservation of
energy and momentum are still fundamental laws in nature.
0
VMmm
2211
20
22 cMmc
0V
0
2
2
00 2
1
22 m
c
mmM
• Example: Two identical particles move relatively at the same rate,
complex after the collision. Calculating speed and quality of
composite particles.
• Solution: Assuming the mass of composite particles is M, speed is V ，in the collision course， obtained by the momentum and energy
conservation
72
4 4 Equations of relativistic mechanicsEquations of relativistic mechanics• To make the equations of Newtonian mechanics in the
concept of new space under the Lorentz transformation is
covariant， we have constitutes a four-dimensional
momentum at above —The energy-momentum four-
vector is Pμ. If we use intrinsic time to measure the rate
of change of the energy momentum, then the change rate
d
dP
d
dPK
• Therefore, if the external effects on the object can be
described by a four-dimensional force vector K μ, The
basic equations of mechanics can be written as
covariant form
• is also a four-dimensional vector.
73
• In low-speed approximation， The equation should transition to the
classical Newtonian laws. ’s space components should also be
translated into the classic of the force . We can see that the above equation
is satisfied covariance mechanics above. In addition to, its spatial
component and the fourth component has a certain relationship
K
4K K
dPd
K
d
PdP
W
c
c
icmcP
d
d
c
i
d
dW
c
i
d
dPK
242
0224
4
Kc
imPcmW 0
20 ,
K
c
iK 4
74
• Thus, the four-dimensional force vector that
acts on the object at the speed of is ),( K
c
iKK
d
dWK
d
PdK
ddt
KcF
22 /1
dt
dWF
dt
PdF
• Relativistic mechanics equations
will include the following two
equations
• Time rate of change of momentum and energy of the above are
measured by the intrinsic time. For convenience, we can Expressed t
he equation by time change rate of reference. Due to dt=γdτ， so
long as the form is
Then, get
75
• It is identical in form of non-relativistic mechanics equations.
• Note ： 1） and W are the relativistic momentum and
energy in the formulas, scilicet
2200 /1/ cmmP
22
202
0/1 c
cmcmW
P
• 2) F is not a component of dimensional force vector . Its transformation relations should be exported by four-
dimensional force K μ. Only in low-speed
approximation,F behalf of the classical force .
76
5 5 Lorentz forceLorentz force• An important application of relativistic mechanics are research on
motion of charged particles in electromagnetic field. For this reason we
need to be discussed Lorentz force. The force acting of charged particle in
an electromagnetic field is
)()( ABEqF
)( BEqF
dt
PdF
KcF
22 /1
• Whether the form of the force in electromagnetic field in the theory of relativity remains unchanged it?This is a problem to be described below, i.e., to prove
is covariant in the theory of relativity.
• Above, we get equation of covariant relativistic
mechanics• In it K is the spatial component of
the four-dimensional force.
77
• So long as the right of formula (A) is written as22 /1)( cKBEq
particle of charge theConcerning )3
)charge moving aon force theDescribes(
ldimensiona-four of speed the torelated be Should)2
nsinteractio neticelectromag of Because(
tensor neticelectromag Includes)1
）F
UqFK
• And K is a component of four-dimensional spatial force , is
covariates。 This would explain formula (A) is covariant in the
relativistic.• First, to constitute a four-dimensional force vector K μ, must
•For this reason we use electromagnetic field tensor Fμν and the velocity vector U ν constitute a four-dimensional vector
78
)( 41431321211 UFUFUFqUqFK
])([ 11 BEq
])([ 222 BEqK
])([ 333 BEqK
)]([/1
1)]([
22BEq
cBEqK
KcBEq
22 /1)]([
Then
Similarly too
Together is
Scilicet
79
• Therefore, the Lorentz force equation (A) meet
the requirements of relativistic covariance.
)]([ BEqdt
Pd
• Equations of motion of parti
cles in the field is:
It applies to all inertial frames.Thus,it be able to describe the movement of high-
speed particles.About the correctness of relativistic mechanics
has been confirmed by experiments. So far we have clarified the basic laws of electrodynamics（Maxwell's equations and the Lorentz force） are applicable to all inertial frames of physical laws.
80
• Example: Discuss the charged particles move in a constant
homogeneous magnetic field
• Solution: In uniform constant magnetic field B, the motion equations of
charged particles is
)1()( Bqdt
Pd
)2(0 Bq
dt
dW
Bqdt
d
c
m
c
m
dt
d
22
0
22
0
/1/1
Bm
q
0
• So the energy of the particles is constant， and the
value of velocityυ is also constant. From (1) we obtain
Scilicet
81
• Thus have
Bm
q
0
|| 0
above The.constant a oconstant a is Since || alsisthen ，
qB
P
qB
maBq
a
m
0
20
•Solutions of this equation is the circular motion. The
radius of a circle can be obtained by the centripetal force
equal to the force. Scilicet
0 particle of role in the is lcentripeta
itst motion tha ofequation icrelativist-non toequivalent is
mmBq
82
• The angular frequency of the circular motion is
0m
qB
a
0/mqB• In the non-relativistic case,
• Regardless of the particle velocity.
• In the relativistic case ， increases with the
energy of the particles , thus frequency
decreases 。
83
• We now know that there are four fundamental interactions of
nature: electromagnetism, gravity, strong and weak interactions.
Electromagnetic and gravitational interactions are long-range.
The strong and weak interactions are short-range force
（ There are only 10-15m in the range ）
• As mentioned above, the electromagnetic interaction
can fully included in the scope of special relativity. Und
er certain conditions, non-relativistic quantum mechani
cal equations can correctly describe the motion of char
ged particles.
• About gravitational interaction, We will learn later
what must be further promoted general relativity to sp
ecial relativity to make it a relativistic theory.
![arXiv:1308.4685v2 [gr-qc] 7 Feb 2014 - HEP - INSPIRE-HEPinspirehep.net/record/1250251/files/arXiv:1308.4685.pdf · · 2014-02-10four dimensional, Lorentz-covariant local theory](https://static.documents.pub/doc/80x56/5af473de7f8b9a8d1c8c32ee/arxiv13084685v2-gr-qc-7-feb-2014-hep-inspire-13084685pdf2014-02-10four.jpg)
