PH217: Aug-Dec 2003 1

Special Relativity
This is a reminder of the basic results of Special Relativity that we need to use
during this course.

Lorentz Transformation

The four-dimensional coordinate transformation that links two frames moving

with a velocity relative to each other is called the Lorentz transformation. If the




frame K (x y z t ) moves with relative velocity V in the positive x direction




with respect to the frame K (x y z t), then the transformation laws between the
two sets of coordinates are

x  (x Vt )
y  y
z  z
t  (t V

x)
c2
where c is the speed of light in vacuum and

 1
1 V 2 c2

is called the Lorentz Factor.

The inverse transformations can be obtained by replacing V with V.

From the above transformations follow the Lorentz contraction of length in the
direction of the relative velocity:

L  L 0
where L0 is the length in the rest frame of the object, and time dilation
 t   t0

where t0 is the time interval in the rest frame of the clock.
PH217: Aug-Dec 2003 2

Transformation of velocities






 
if u(ux uy uz ) and u (u x uy uz ) represent the velocity of the same particle in the
frames K and K respectively, then noting that u x dx dt, u x dx dt and so on

for other components, one finds the transformation rule


ux  

  
  
ux V uy uz
u x V
c2 )

uy uz
1 u x V c2 (1 (1 u x V c2 )

In the case of motion parallel to the x-axis this reduces to


u  u V
1 u V c2

It is easily seen that if u equals c, then it does so in all frames irrespective of their
relative velocities.

Let us choose the coordinate system such that the velocity u (u ) of the particle
lies in the xy (x y ) plane in frame K (K ). If and denote the angle with respect
to the x (x ) axis in the frames K and K respectively, then from the velocity
transformation law one obtains

  u sin 
tan
(u cos V)
which describes the change in the direction of the velocity on transformation be-
tween the two reference systems.

Setting u  c one then obtains the law of aberration:

  sin 
tan
(cos V c)

In terms of sines and cosines, the law of aberration is



sin   

c)
cos 
sin cos Vc
(1 V cos

1 V cos c
PH217: Aug-Dec 2003 3

Energy and Momentum

The momentum of a particle in special relativity is given by

 
p u m0 u 
where m0 is the rest mass of the particle, and
 u  1
1

u2 c2

The energy E is given by

E2  p2 c2
 m20 c4
one can construct a four-vector out of the two: the four momentum

pi 


(E c p)

In case of electromagnetic waves, the analogous quantity is the 4-dimensional

 


wave vector
ki ( c k)
Transformation of this four vector gives the Doppler shift relation

   

c)
0
(1 V cos
where V is the velocity of the source which emits radiation at a frequency 0 as 

measured in the rest frame of the source. is the angle (as measured by the ob-
server) between the direction of emission of the wave and the direction of motion
of the source.

Reference

Landau, L. D. and Lifshitz, E.M., The Classical Thoery of Fields, chapters 1, 6.