Einstein’s special relativity and Lorentz
transformation and its consequences
1.
2.
3.
4.
5.
6.
Einstein’s special relativity
Events and space-time in Relativity
Proper time and the invariant interval
Lorentz transformation
Consequences of the Lorentz transformation
Velocity transformation
1
1. Principle of Relativity by Einstein (1905)
It is based on the following two postulates:
1) The laws of physics are the same for all observers in uniform motion
relative to one another (principle of relativity),
- need a transformation of coordinates which preserves the laws of
physics
2) The speed of light in a vacuum is the same for all observers,
regardless of their relative motion or of the motion of the source of the
light. http://en.wikipedia.org/wiki/Theory_of_relativity
V
V
A
B
A
B
Observer in the car: the light pulse
reaches A and B at the same time
Observer to whom the car is
moving with relative V: the light
pulse reaches A before B
Simultaneity breaks down  time cannot be regarded as a universal entity
- need a different transformation from Galileo’s but will converge to it
for V<<C
2
2. Events and space-time in Relativity
When and where is the object under our interest An Event in Relativity.
--An event is a point defined by (t, x, y, z), which describes the
precise location of a “happening” which occurs at a precise point in
space and at a precise time.
--“Space-time” is often
depicted as a “Minkowski
diagram”.
decelerated
Time (ct)
constant
accelerated
Space r
3
World lines
The world line of an object is the unique path of that object as it travels
through 4-dimensional space-time.
.World lines of particles/objects at constant speed are called
geodesics.
Time (ct)
decelerated
constant
accelerated
Space r
4
Events and space-time in Relativity
What is simultaneous in a moving frame is not simultaneous in the
stationary frame.
Here the signal has to be
sent later ( t > 0) from A …….
5
3 Proper time and the invariant interval
3.1 invariant interval (i.e. c is constant)
In 3-dimensional EUCLIDIAN space: P1
P2
In coordinate system O:
P1 ( x1 , y1 , z1 ), P2 ( x2 , y2 , z2 )
In coordinate system O’:
P1 ( x'1 , y'1 , z'1 ), P2 ( x'2 , y'2 , z'2 )
r 2  x 2  y 2  z 2  x'2 y'2 z'2
In relativity, we would like to find a similar quantity for pairs of events,
that is frame-independent, or the same for all observers, that is
invariant interval
(s)2  (ct )2  (r )2  (ct )2  (x)2  (y)2  (z)2
∆t is the difference in time between the events
∆r is the difference between the places of occurrence of the events.
6
3.2 Events, INTERVAL AND THE METRIC
A metric specifies the interval between two events
7
3.3 Proper time (length) the invariant interval
The proper time between two events is the time experienced by an
observer in whose frame the events take place at the same point.
According to the definition of the interval between two events:
c  ( s ) 2
i) If (s)2  (ct )2  (r )2  0 , the interval is said to be “timelike”
--there always is such a frame since positive interval means: | c |  r
so a frame moving at vector v
= (∆r) /(∆t), in which the events take
place at the same point, is moving at a speed < c
ii) If (s)2  (ct )2  (r )2  0 , the interval is said to be “spacelike”
--It is still invariant even though there is no frame in which both events
take place at the same point. (or (c∆t)2 < 0).
--There is no such frame because necessarily it would have to move
faster than the speed of light.
8
Sometimes the proper distance is defined to be the distance separating
two events in the frame in which they occur at the same time. It only
makes sense if the interval is negative, and it is related to the interval by
  S 
2
iii) if (s)2  (ct )2  (r )2  0 ,the interval is said to be “light-like” or
null…defining a null geodesic
This is the case in which
i.e., (ct )2  (r )2
Or, in which the two events lie on the worldline of a photon.
Because the speed of light is the same in all frames…….
…. an interval equal to zero in one frame must equal zero in all frames.
The three cases have different causal properties, which will be
discussed later.
9
4. A transformation formula – Lorentz Transformation
4.1 The formula fits into Einstein’s two postulates
We assume that relative transformation equation for x is the same as
the Galileo Trans. except for a constant multiplier on the right side, i.e,
x'   ( x  ut )
x   ( x'ut ' )
(1)
(2)
where  is a constant which can depend on u and c but not on the
coordinates. (based on Postulate 1)
How to find the  factor ?
By tracing the propagation of a light wave front in two different
frames, one of which is moving with a velocity of V along x-axis
w.r.t. the other.
10
Assume a light pulse that
starts at the origins of S
and S’ at t =t’=0
Y
’Y
After a time interval the
front of the wave moves
Y
S
u
Y
’
It is recorded as:
S
’
(X, t) in S
and
(X’, t’) in S’
O
’
z
Z’
O
XX’
z
O
’
X
X’
Z’
By Einstein’s postulates 2:
x = ct
x’=ct’
11
Substituting ct for x and ct’ for x’ in eqs. (1) and (2)
x'   ( x  ut) (1)
x   ( x'ut' ) (2)
ct '   (ct  ut)   (c  u)t
(3)
ct   (ct 'ut' )   (c  u)t ' (4)
t '  (c  u )
(3)  
(3' )
t
c
t'
c
(4)  
(4' )
t  (c  u )
Let (3’) = (4’)
u < c so  is
always > 1
2
If u~c,  
c
1
1
  2 2
 
2
2 2
c u
u
1

u
/
c
1 2
c
2
When u << c
~1
12
The relativistic transformation for x and and x’ is
x'   ( x  ut) (1)
x   ( x'ut' ) (2)
Lorentz transf.
If u << c
 ~1
x'  x  ut;
x  x'ut
Galileo transf.
13
The transformation between t and t’ can be derived:
For the wave front of light, x=ct, x’=ct’
Divide c into Eq.(1) x'  ct '   ( x  ut)   (c  u)t
t' 

c
(ct  ut )   (t  uct
)   [t 
2
Divide c into Eq(2)
t
c
(1)
ux
]
2
c
x  ct   (c  u)t '
(2)

'
(ct 'ut ' )   (t ' uct ' )   [t ' ux
]
2
2
c
c
c
The complete relativistic transformation (L.T.) is
ux'
x   ( x'ut ' ), y  y' , z  z ' , t   [t ' 2 ]
(5)
c
ux
x'   ( x  ut ), y'  y, z '  z, t '   [t  2 ]
(6)
c
14
4. 2 The interval of two events under Lorentz transformation.
For two events, (t1, x1,y1,z1) and (t2,x2,y2,z2), we define:
(T, X, Y, Z) = (t1-t2,x1-x2,y1-y2,z1-z2)
cT '   (cT 
then Lorentz transformation becomes
X '  (X 
2
c T
'2
v2
 X '   [c T  2 X 2  2vXT
c
 X 2  v 2T 2  2vXT ]
2
2

2
2
c 2T '
2
2
(c T
2
 X
2
 X
'2
2
v
X)
c
v
cT )
c
v2
)(1  2 ), i.e.,
c
 c 2T 2  X
2
i.e., (S ' )2  (S )2
The interval of two events is an invariant under Lorentz Transformation.
15
For short: the interval is a Lorentz scalar.
4.3 Lorentz transfermation in 4-dimensional formula
The L-T could be formally defined as a genernal linear transformation
that leaves all intervals between any pair of events unaltered.
Introduce 4-D vector
 ct1

x
x   1
 y1


 z1







x  ct1 , x1 , y1 , z1 
S 212  (x1  x 2 )g (x1  x 2 )
1 0 0 0 
0  1 0 0 
Here we have introduced:

g
0 0  1 0 


0 0 0  1
16
L-T can be expressed as
ct '
 x' 
 
 y' 
 
z' 
X'  LX
v



c

v


  
c

0
0
0
0

0 0 

0 0


1 0
0 1
ct 
x 
 
y 
 
z 
17