Download Relativity

Electrodynamics
REN Xincheng, Postdoctoral , Associate Professor
Tel：2331505; 13310918078
Email:[email protected]
Chapter 6. Special theory
of relativity

In the nineteenth century, Newton’s theory has been developed quite perfect, in the
classical theory, it has been dominant. Physics community believe that no new work
to do, the remaining problems of physics are the application of Newton’s theory. It is
in this case, the clear sky in physics appeared two small black clouds (The zero results
of Michelson - Morley experiment and the blackbody radiation). However, in the
early of 20th century, the poly rain storm of physics community was caused by the
two small black clouds, major changes in physics have taken place by it, the theory of
relativity and quantum theory was developed. And the two theories have become the
two pillars of modern physics, they are the foundation of modern physics.

Theory of relativity is divided into special theory of relativity and general theory of
relativity. Special theory of relativity was created by Einstein in 1905, it is a theory
that limited to the inertial reference system; general theory of relativity was created by
Einstein in 1915, it applies to any reference system, and it is a theory that including
the gravitational field. In this course, only special theory of relativity is introduced.
Main content：
Historical background and experimental basis
The basic assumptions of special theory of relativity
Relative property of simultaneity
The relationship of Lorentz coordinate transformation
Moving clocks running slower and contraction in length
Lorentz velocity transformation
The four-dimensional covariant form of the relativity theory
Relativistic invariance of electrodynamics
Relativistic mechanics
§1 The difficulties of classical theory and the
experimental basis of theory special relativity
The same physical event is investigated in the two inertia systems

一. Galilean transformation
There are inertia system S and S’ moving relative to S
When t  t '  0, O coincides with O '
t time, the object reaches point P
y S y S 

r
o o'


r
P
x
S
S

r x, y, z, t 

r x, y, z, t 

 x, y, z, t  a


 ' x, y , z , t  a 

Direct
transformation
Inverse
transformation
x  x   t
y  y
z  z
x  x   t 
y  y
t  t
z  z
t  t
Velocity transformation and acceleration transformation
d
u ' x  u x   a' x  a x 
dt
Direct
transformation u '  u
y
y
a' y  a y
u' z  u z
a' z  a z
d
u x  u ' x  a x  a x 
Inverse
dt 
transformation
u y  u' y
a y  a y
u z  u' z
a  a
z
z
v is a
constant
Both are
inertial
systems
ax  a x
ay  a y
az  az
a x  a x
a y  a y
a z  a z
In the two inertial systems

 
a  a
二. Newton’s principle of relativity
S
S


F  ma


F   ma
 In Newtonian mechanics
 Force has nothing to do with the reference system,
mass has nothing to do with motion.
Mechanism of a macroscopic and low speed objects is in
the same form in any inertial system.
Or, Law of Newtonian mechanics is in the form of unchanged
under a Galilean transformation.
For example：law of conservation of momentum




S m1u1  m2 u 2  m1u10  m2 u 20




S  m1u '1  m2 u ' 2  m1u '10  m2 u ' 20
三、 The contradiction of principle of
relativity in electromagnetic theory


We have already discussed the basic content of electrodynamics. We
know that electromagnetic fields is in the form of waves in the case of
prompt varying, and propagate in space at the speed of light c. So far, we
have not related to the coordinate system problem. If considering the
reference system, under the Galilean transformation, that the
electromagnetic wave propagate at the speed of light c is tenable only to
a certain inertial system, light travels in all directions to differ relative to
other inertial system. Further, electromagnetic theory (Maxwell’s
equation) can only set up to this certain inertial system, so that relativity
principle in mechanics—all inertial systems are equivalent is no longer
applies to the electromagnetic phenomena, that is, the absolute resting
reference system is existent to electromagnetic theory.

How do we explain this? Electromagnetic theory (Maxwell’s equation) are
proved to be correct by a large number of experiments, then we are the
only two ways to explain:

1）Galilean transformation is applicable for electromagnetic theory, but there is
a favorably of reference frame in electromagnetic theory, that is, there is an
absolute resting reference frame. Relativity principle—all inertial frames are
equivalent is no longer applicable for electromagnetic theory;

2） Relativity principle (Covariance) is applicable for mechanical, and is also
applicable for electromagnetic theory, then the Galilean transformation need to
be modified, which means that laws of Newton's classical mechanics need to be
modified.

Most people tend to accept the first way at that time, that the Galilean
transformation is applicable for the electromagnetic theory, but there is an
absolute resting and favorably of reference frame in electromagnetic theory.
In addition, mechanistic philosophy is in fashion in physics community at
that time, physics is considered that it can use a single image of classical
mechanics to describe, its outstanding performance is the "ether" hypothesis.
The hypothesis believe that the "ether" is a elastic medium delivering all the
electromagnetic waves including light passed, including all the
electromagnetic waves, it fills the whole universe. Electromagnetic wave is
the mechanical motion state of "ether" medium, vibration of charged
particles can cause deformation of the "ether", and propagation of this
deformation as the form of elastic wave is electromagnetic waves.

So people will think the absolutely stationary reference
frame relative to the light "ether" is stationary. And the
absolute reference system relative to the movement is
called the absolute motion. Earth walk through the
"ether" , measure the Earth relative to the "ether" of
absolute motion, naturally became a problem at that
time people first concern. This measurement is the
earliest known Michelson - Morley experiment.
However, a large number of experiments have not
measured the Earth's motion relative to this particular
reference frame, which results are the following
explanation:


A）The Earth is a better frame of reference than the
sun.
——But geocentric has been denied at that time.

B）Earth led movement of "ether" (or "ether" dragged
completely by the Earth);

——But it is Negated by flume experiment of Faissault.


C）There is no "ether" particular reference system；
——Can be attributed to a second route.

D） On the basis of the "ether", the Lorentz use electronic theory
and made some assumptions, at all levels effects. Proved that
electromagnetic interaction are nothing to do with the movement
of the system.

And proposed space-time transformation law - Lorentz
transformation. He took advantage of this transformation. From
the "ether" line changed to any other frame of reference, as long
as the speed is less than c, electron theory equations of motion
were held constant, the speed of light in all inertial frames are the
same. Thus explaining the zero result of Michelson - Morley
experiment , and he has got the right formula - relativistic spacetime transformation. Nevertheless, it remains that there are
fundamental flaws:

First, his theory attached to an Ethernet, so the principle of relativity does
not apply to electromagnetic phenomena. But Visibility from the results
discussed above, which itself leads "ether drift" can not detect ,and a logic
error in conclusionthe speed of light is the same in all inertial frames. We
can say that Lorentz theory drive away the "principle of relativity" from
the front door , but from the back door to put it in. It shook as the absolute
reference system of "ether" concept radically.

Second, in his transformation, each one have no specific physical
significance.

Thus it seems that the theory itself, its relying and the starting point is
wrong, it seems a bit far-fetched.

In summary, the first way is a dead end, then only take the second route.
Namely the principle of relativity (covariance) is suitable for mechanical, it
also applies to electromagnetic theory. Then Newton's laws of classical
mechanics (Galilean transformation) need to be modified.
§2 The basic assumption of special relativity

On the basis of the theory of Lorentz and others,
along the right path is a decisive step towards • Albert
Einstein (A. Einstein). He summed up the fact that a
large number of experiments, to get rid of the
influence of the old machinery and mechanical
concept of time and space. Fundamentally ,Bold
discarded "ether" speaking, critical inherited and
developed the results of Lorentz et al , proposed two
basic assumptions of relativity, thereby establishing a
new space-time theory - special relativity.
The basic assumption of Einstein's special theory of
relativity




1. Principle of Relativity：All the laws of physics is in the same form in any
inertial.
2. Constancy principle：The speed of light in vacuum for any inertial in
either direction are c, and is independent of the movement of the light
source.
The speed of light has nothing to do with sources Movement ,described by
binary trajectory distortion not occur
Discussion
Relativity theory of A.Einstein is the development of the theory of Newton
All the laws
of physics
Mechan
ical
laws
B.
Constant speed of light and the Galilean transformation
diametrically opposed to Galileo's principle of adding speed
C.
Conceptual change
Newtonian
mechanics
Special
relativity
mechanics
Time scale
Length scale
Measuring the
quality of
Revolutionary
Regardless of the
reference frame
Speed ​and reference
system related
(Relativity)
length time quality
The speed of light
is not change
related to reference
department
(Relativity)

§3 Relativity of simultaneity

一. Spacetime coordinates
1. Event
space-time coordinates

2. synchronized clocks

二. Relativity of simultaneity

-- A direct result of constancy principle

Einstein train for example
S  Einstein
S
train
S
Ground
A、B
Midpoint
t  t  0

A M  B
Reference System of
Experimental
device
On the train
S
M
M
Place signal receiver respectively
Place the optical signal generator
Send a light signal
S
Event1
A Received flash
Event 2
B Received flash
Research questions
The time interval between two
events
S
S
A M  B
S ？
M  Flash emitted Speed ​of light is
AM   BM  A
Event 1、Event 2

B
S ？
c
Simultaneously receive
the optical signal
Occur simultaneously
S S
how observer to see
it in S system ?
M  Flash
Speed ​of
light is also
A B  Move with
the
A Facing
light
Event 1、Event 2
c
S

A M  B
Received light earlier than
B
not Occur simultaneously Event 1 occurs first
Discussion
1）Relativity of simultaneity is a direct result of constancy principle
2）Relative effect
3）When the speed is much smaller than c, the result of two inertial is the
same.
§４Lorentz transformation
一.Export
the (special) Lorentz transformation
t  t  0
o o
Coincide
relative to the S series ,Sˊseries
speed is υ .
y S y S 
P
o o
x
x
Simultaneously ,issued a flash, the light reached the
point P after some time
S Px, y, z, t 
S  Px, y, z, t 
Search
The corresponding
relationship between
the two reference
values
By
the constancy principle：
x y z c t
(1)
2
2
2
2 2



x  y  z  c t
(2)
2
2
2
2 2
The
uniformity of space, t ' independent of y and z , it can be
assumed
y  y
z  z
Known,
(3)
the space homogeneous and isotropic of
transformation is linear, Namely:
x, y, z, t  and x, y, z, t
relationship is a linear relationship, Namely
x  a11 x  a12c t ct   a21 x  a22 c t (4)
x

0
 x ' x '  a;
1
(Or，if x'  ax 
2
1
x2  1

2
x1  1 
x2 ' x1 '  3a)

x2  2 

The next task is to determine the coefficient using the
above comparison coefficient Medley

Put (3), (4) into (2) and arrange, we have:
(a112  a212 ) x 2  2ctx(a11a12  a21a22 )  (a222  a122 )c 2 t 2 (5)

compare (5) and (1), can get
2
a112  a 21
1
 2
a11 0 , a22 0
2
a

a

1



 22
12
a a  a a  0 （6）
21 22
 11 12
a  1  a 2
21
 11

a 22  1  a122
(7 )

Substitute (7) into (6), we get
a12  a21


a11  a22
Further discussion of the movement of S 'system origin
S ': x'  0 
  a11 t  a12 ct  0
S : x   t
a12

2
则得

又 a11  1  a21
 1  a122
a11
c
1
Thereby have a11  a 22 
1 2 / c2
 / c
a12  a 21 
1 2 / c2
二.Result
Coordinate
transformation
(transformation)
x  t

 x 
2


1 2

c
 
y  y


z  z



t 2 x
t  
c

2


1 2

c

 
Let  
c
Then
1
1 
Transformation
x     x  t 
y  y
2
  

x    x   t  
Inverse
transformation
z  z
y  y
z  z
 

t   t  x
c 

 

t    t   x 
c 

Discussion
1) 时间和空间是相关的。t ' 与x, ,t 有关 ,它依
参考系的不同而不同，与物体的运动有关。
2）Ruler and clock of each lines
3) v  c,
 1
x   x  t
y  y
z  z
t  t
Back to the Galilean
transformation
Lorentz
transformation is the
development of the
Galileo
transformation
4)   c
transformation C is the speed of light limit
meaningless
5) By the Lorentz transformation, we see the relativity of
simultaneity
S
S
Event 1
( x1, t1)
( x1 , t1 )
Event 2
( x2 , t 2 )
( x2 , t 2 )
Two events occur
simultaneously
t1  t 2
t   t2  t1  0
t  t2  t1
？
t   t 2  t1 
t 

c
1
If
x  0
Known
t   0
2
x

2
c
2
t  0
Relativity of simultaneity

(Special) Lorentz transformation can be derived by
other methods
• Example 1(p.241)：In Figure Let the flash emitted from O points, S
system on the observation of a second after the optical signal are received
by
•
•
simultaneously,
P和
P
1
2
Let S ' system speed is 0.8c relative to the S system ,
Find P和P receiving the optical signal in time and position on the
1
2
system S '
S S'
  0.8c
解: S receive the optical signal
P1
P1 (c,0,0,1)；P2（-c,0,0,1) P 
t  0 , x  2c
2
S 2  c 2 t 2  r 2  4c 2
In S two events occur simultaneously
O

O'


S ' receive the optical signal
1
1
P1 ( c, 0 , 0, )
3
3
x  t
1
x' 
 c
2
2
3
1   /c
t   x/c 2
1
t' 

2
2
3
1   /c
P2 (-3c, 0 , 0 , 3 )
x' 
t' 
x  t
1   2 /c 2
t   x/c 2
1   /c
2
2
 3c
3
t '  8 / 3  0 , Namely not the same , x'  10c / 3
S '2  c 2 t '2 r 2  4c 2
Space - time interval is Lorentz invariant
Task（P290） ：4
§5 Space-time theory of relativity

1. Interval invariance

In the theory of relativity，Space interval and time
interval is changed due to different reference system.
But space-time interval
S 2  c 2 (t 2  t1 ) 2  [( x2  x1 ) 2  ( y 2  y1 ) 2  ( z 2  z1 ) 2 ]
 c 2 t 2  r 2
is Lorentz invariant. It can be proved by homogeneous
and isotropic of space and the basic assumptions of
relativity. Lorentz transformation can also be proved
that ,the previous example also illustrates this point.
2. Space-time structure
O(0,0,0,0) is
Interval of two events 
 P ( x, y , z , t )

S 2  c 2t 2  ( x 2  y 2  z 2 )  c 2t 2  r 2

Interval can be divided into the following three conditions:

1）Two events use optical signal contact
r  ct  s 2  0

2）Two events contact with less than the
speed of light signal
r  ct  s 2  0

3）Two events without contact (spatial distance exceeds light can
propagate in the time t distance)
r  ct  s 2  0

It is known by the interval
invariance. The above the division of
the space-time interval is absolute. It
is not changed by change of
reference （Although in different
reference lines, the same event at
which the point is not the same, but
in which the interval is
constant.).Here we analyze the
geometric meaning of this division：
Three-dimensional space and one
dimension time constitute a fourdimensional time and space . For
facilitate and intuitive of
discussion,as shown, we take the
discussion of Three-dimensional
space-time space, P stands for any
one events
ct
r  ct
 P ( x, y , z , t )
y
o
x
r  ct

The projection of P on the xy plane
represents the place of the incident；
projection of ct-axis means that the time
the event occurred multiplied by the
speed of light c.
ct
1) r  ct , s 2  0

Corresponding event point on the
vertex O of the cone surface
(called light cone), Any point on
the light cone can use light
signals to associate with the Opoint events, called class light
interval.
o
r  ct
y
x
r  ct
2) r  ct , s 2  0

P point is located outside the light cone, in
this region, the event P is not related to and
event O , they can not contact each other.
This area is called spacelike region
(interval).
ct
3) r  ct , s  0
o
2

Within the cone, the event link with less
than the speed of light signals. This area is
called timelike region (interval). It also
can be divided into absolute past (ct <0,
the lower cone) and the absolute future
(ct> 0, the inner cone)
r  ct
y
x
r  ct

3. Causal rate and the maximum transmission speed of interaction
• In the classic range, the causal rate is absolute .In an
inertial reference frame that event A precedes event B,
• event A is the cause of event B, event B is the result of
the event A .So in another inertial system the event A
must also precede event B. Within the scope of the
absolute nature of special relativity whether this causal
rate can exist?
• All things sports development, but there are certain
causal relationship of things the development. By
contacting the material movement,the first event as
cause leads to second event as a result .This causality is
absolute. It should not be transferred in different
reference systems.
Concept of time is abstracted out of development movement. Correct
time and space should reflect the absolute rate of causality. Here's to
analyze the causal rate how to embodied in relativistic spacetime .

From the above discussion illustrates the structure of space-time, if
the events light cone P is in the upper half of the O.（Including the
cone of light cone），In any inertial, P held within the upper half of
the light cone O, namely P is the absolute future of O . This interval
feature is that P and O are linked by interacting of available speed
of light or below the speed of light. So if superluminal interaction
does not exist, a necessary condition for the occurrence of two events
P and O is P within O cone of light. Causal rate in this region is
absolute.
Since outside the light cone (namely class space separated),
two events can not be linked to below the speed of light
signals, without the existence of superluminal effect. So,
any two events are not related in spacelike interval .
Therefore, a causal relationship does not exist in spacelike
interval. The above conclusions can be directly illustrate
through the Lorentz transformation:

For two events
t 2 't1 ' 
(t 2  t1 )   ( x 2  x1 ) / c 2
1 2 / c2
  (t 2  t1 )(1   u / c )
2
x2  x1
In it ：u 
t 2  t1

If we discuss is timelike (lightlike) region, Then u
represents the signal transmitted from event 1 to event
2, it can only be less than or equal to the speed of light.
υ represents relative velocity of an inertial frame with
another inertial frame.It is also the velocity of matter,
from the existing experiments have proved that it can
not be equal to the speed of light, namely

u  c ,  c
So, if t 2  t1
have
t 2 '  t1 '
This proves that causality rate is absolute for the timelike
space .Conversely, if the causal rate is absolute, then
naturally introduced that interaction propagation
speed of light c is the speed limit.

4 . Relativity of simultaneity

We've already discussed the relative simultaneously,
and are described by the Lorentz transformation.

Relativity of simultaneity is two events occur
simultaneously in a reference system in different
locations. In another reference frame is not
simultaneous. Obviously, spacelike interval, can not
occur causality .The temporal order of successively or
simultaneously has no absolute meaning, because it is
different in different frames of reference.

5. Time dilation (slow motion clock)
As the specter of life?

Mentioned Bell actually represent a specific
physical processes. It as timing standard.
 Research is：
And consolidation process of the measurement time of
reference（namely the time between two events have occurred
in the same place at intervals, with the same bell measuring
only).With the process and other relative movement of the
inertial frame,measuring the time relationship （Were used to
measure with two different locations in the bell) between the
process.

Proper time：In a reference frame, the time between
two events have occurred in the same place at intervals.
That is the process of consolidation of the time reference
system of the measurement process, also called proper time.
 Time in two places: different locations clock.
 Different reference systems can only point on the clock in
the same space. In the course of inertial relative motion, the
time measurement process to use a range of different
locations in the alignment of the clock.
Investigation of a bell S 
x  0 Two events occur at the same location
t   
t
Time the original
Time in two places
By Inverse transform of Lorentz
t 
t  

c
1
t 
x 
2

2
c
2
1
1

1
x  0
t   

2
c
2
t
2
c

1
2

Original time is the shortest
Discussion
• 1） Motion clock slow
effect is space-time itself
objective characteristics. Caused no sense of people.It
has been confirmed by many experiments.

With μ sub in a limited lifespan through the atmosphere
to illustrate .
The average life expectancy still μ Son 2.2 10 6 s,
Atmosphere (Much larger tha n the thickness 660m)
generated on

2）Clock delay effect is relative . This can be explained
by the principle of inertial equivalence. In order to
deepen our understanding can be analyzed from the
perspective of measurement.
• 3）In the case of acceleration, the clock delay will cause
the absolute physical effects. If you start the alignment
of the clock, One does uniformly accelerated motion
around closed path . Another does not move in the
same place. When the former motion one week with the
latter meet, the total time it has experienced is less than
the original location stationary clock elapsed time .
If the above two bell instead of a pair of twins, will appear
people travelling and returning along a closed path one
week absolutely more younger than Stagnant man.
There is a saying that the sky day on the ground for a
year.
This is the absolute effect of the time delay. It involves
accelerating reference frame, are of general relativity.
Introduction twin paradox.

6.length contraction
 How
to measure the length of motion
?——Measured simultaneously at
both ends of the ruler
 Original length (intrinsic length):
with the rod relative to the rest frame
of the measured length
 The
baseline of measuring the length
also uses natural phenomena
characteristic length, such as
Stick
rest in S 'system, l0 is static length
S S

l0
Stick motion relative to S line with a high
speed .What is the length of the rod S
system ?
S S
l0
Conditions of simultaneous
measurement are necessary
Corresponding space-time
coordinates
Event1：Measure the
left end of the rod
Event2：Measure the
right end of the rod

S
S'
x1 , t1
x1, t1
x2 , t2
x2 , t2
(t 2  t1 )
Inherent length is the longest
Event1：Measure the left
end of the rod
Event2：Measure the
right end of the rod
S
x1 , t1
x2 , t2
S
x1, t1
x2 , t2
l  x2  x1 l0  x2  x1
t  0
By the Lorentz transformation
x  
x  t
1

2
c
2
l  l0 1 

2
c
2
Discussion
 1）Scale shortening effect is decided by the basic
properties of space-time. Regardless of the specific
structure of the interior of the object;
 2）Relative effect; longitudinal effect;
 3）at low speed Galilean transformation
 4）Direct result of simultaneous relativity
 5）Clock delaying is relevant. For example, explain
the problem μ sub penetrate the atmosphere .
Task（P291） ：7

7 Relativistic velocity transformation
Definitions
By the Lorentz
Coordinate
transformation
Ratio of the above
two equations
dx
ux 
dt
dx  u x  

2
dt

1 2
c
dx 
u' x 
dt 
dt 

dt
1
ux 
u' x 

1 2 ux
c

c
1
2
ux

2
c
2
Known by the Lorentz
transformation
dy
dy dy
dt


dt 
d t  dt 
dt
Obtained by the
above two formulas
The same get
dt 

dt
1

c
1
2
uy

u' y 
1 2

c
1  2 ux
c
uz
2
u' z 
1 2

c
1 2 ux
c
2
ux

2
c2
Lorentz velocity transformation formula
  
Transformation
u' x 
u' y 
ux 
1

c
uy

1  2 ux
c
2
Inverse transformation
ux 
ux
2
1 2
c
uz
2
u' z 
1 2

c
1 2 ux
c
uy 
u ' x 
1

c
2
u' y
1

2
u' x
1
u' x
2
c2
c
u' z
2
uz 
1 2

c
1  2 u' x
c
Non - relativist ic limit   c, u  c,
Relationsh ip transitio n to Galilean v elocity tr ansformati on relations

Case ：Consider a spacecraft fly at a speed of 0.80c above the Earth.
If an object is then emitted from the spacecraft along the direction.The
relative velocity of the spacecraft is 0.90c。


Q: From the ground, how much is the object speed ?
Solution: Select the spacecraft
reference system for the S 'series
Terrestrial
S
S

  0.80c
reference frame is
the S series，
u ' x  0.90c by the inverse transform speed, get
u ' x 
0.90c  0.80c

 0.99c
ux 
1  0.80  0.90

1  2 u' x
c
u' x
x x

In the book

Case 1（P255）（Use constant interval）
u  c  u'  c

Case 2（P255）Seeking speed of light in uniform motion
medium
Solution: Take S ‘series consolidation on media, Let speed
is Υ along the x axis relative to the S series.Then in S ', light
in each direction are c / n of the medium, in which n is the
refractive index of the medium. Utilization rate inverse
transform , get, x as the positive direction,the speed of light
c
is

c

c
1
 c
ux  n

(   )(1  )   (1  2 )
c / n
n
nc
n
n
1 2
c

Also available on the direction of inverse media movement (negative xaxis) propagation speed of light

c
 
c
1
n
ux 
   (1  2 )
c / n
n
n
1 2
c
These results have been confirmed by the experiment of

Fizeau water. Light propagating in the other direction
u y can
?
be similarly obtained, as
ux 
c 2
u' x 
  , u ' y  ( )   2
S S' 
 ux
n
u' x
y
1 2
c
c / n u' y
uy 
u' y 1   2 / c 2
1

c
2
u' x
c 2
 ( )  2 / 1 2 / c2
n
Task（P290） ：2、5
x(x' )