Review of Special Relativity
At the end of the 19th century it became clear that Maxwell’s formulation of
electrodynamics was hugely successful. The theory predicted the existence
of electromagnetic waves, which were eventually discovered by Hertz. In
Lecture 5, eqns 17 and 18, we see that in source free regions of space the
scalar and vector potential obey a wave equation. Wave equations were
already known to the classical physicists in, for example, sound waves. These
classical wave equations could be understood on the basis of Newtonian
mechanics. Some medium was disturbed from equilibrium and the resulting
disturbance propagates at a speed characteristic of the medium. If the
medium was in motion relative to an observer, then the apparent speed of the
disturbance to the observer was simply the vector sum of the velocity of the
medium plus the inherent velocity of propagation in the medium. The speed
of a sound wave relative to an observer, for example, depends on the speed of
sound in air and the wind velocity.
The Michelson-Morley experiment was an attempt to measure the motion of
the earth through the aether, a substance hypothesized to be the disturbed
medium for electromagnetic waves. The null result of the Michelson-Morley
experiment, and all its successors, forced physicists to come to terms with
the non-invariance of electromagnetic theory with Galilean Relativity.
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Galilean Relativity
Newtonian mechanics is invariant with respect to Galilean transformations. These
are transformations between reference frames O and O’ given by eqns (1).
P
(x,y,z,t) ,
(x’,y’,z’,t’)
O’
O
x
x’ = x – vt’
(1a)
t’ = t
(1b)
z’ = z
(1c)
y’ = y
(1d)
v
x’
Time is assumed to be a universal parameter,
independent of the reference frame.
The coordinates of point P transform according to equations 1. O’ moves to the right with
a velocity v with respect to O. Invariance of physical laws with respect to transformations
of inertial reference frames was a long held and justifiable assumption. We assume that
this invariance is a property of space and time. Observations by all competent observers
are equally valid. In the case of sound waves we could say that a reference frame
moving with the wind velocity is a preferred frame, for in this frame the equations are the
most simple.
2
In the absence of the aether there is no natural preferred reference frame for
electromagnetic theory. We still conclude that all inertial reference frames are
equally valid and hence the wave equations must have the same form in all inertial
reference frames. However, it is straightforward to show that the wave equation
does not satisfy Galilean relativity. Consider the transformation of the wave
equation for a one dimensional wave V(x,t). In the O system,
2
1 2
( 2  2 2 )V ( x, t )  0,
x c t
(2)
In general the transformation from one coordinate system to another is given by,
 x'  t ' 


, (3a)
x x x' x t '
 x'  t ' 


, (3b)
t t x' t t '
We will use eqns 1 to transform this into the O’ system.
3






, and
 v

x x'
t
x' t '
(4a) , 4(b)
Applying eqns 4 a second time gives
2
1 2
2
1  2 v 2  2 2v  
( 2  2 2 )V ( x, t )  0  ( 2  2 2  2
 2
)V ' ( x' , t ' )
2
x c t
x' c t ' c x'
c x' t '
(5)
So we see from eqn (5) that the wave equation is not a Galilean invariant.
Equations 1 must be modified so that the wave equation is invariant in transforming
from one inertial frame to another. The coordinates (y,z) perpendicular to v do not
change. We must consider a more general transformation for the x and t
coordinates. It makes sense to try a symmetrical representation of the
transformation.
x'  a x  b vt , (6a)
1
1
v
t '  a0t  b0 2 x,
c
(6b)
In eqns 6 we choose coefficients a’s and b’s to be dimensionless.
We now use equations 6 in equations 3 and derive for the wave equation
4
2
b02 2 a02  2
2
1 2
2
v
 
2
2 v


(
a

b
)

(
v

)

2
(
a
b

a
b
)
1
1
1 0
0 1
x 2 c 2 t 2
c 2 x'2 c 4
c 2 t '2
c2
t ' x'
(7)
In order to ensure invariance w.r.t. coordinate transformation we need
v2
(a  b 2 )  1,
(7a)
c
b02 2 a02
1
( 4 v  2 )   2 , (7b)
c
c
c
(a1b0  a0b1 )  0,
(7c)
2
1
2
1
We can try to find a solution to 7 that is symmetric, namely try a1=b1 , and a0=b0.
Both 7a and 7b give the same result.
a02  a12   2 
v
 ,
c
1
,
2
1 
(8a)
(8b)
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The counterpart to the Galilean transformations ( eqns 1), which makes the wave
equation invariant is called the Lorentz transformation.
x'   ( x  vt ),
(9a)
v
t '   (t  2 x), (9b)
c
y '  y,
(9c)
z '  z,
The inverse transformation
from x’ to x simply requires
changing the sign of v.
(9d)
Inherent in this derivation are two assumptions.
1) The first is that the speed of light, c, is the same in the O’ and O reference
frames. This is actually an experimental fact.
2) The second is that the laws of physics have the same form in all inertial
reference frames.
There is, in fact, nothing special about electromagnetism other than in the vacuum
the waves propagate at a universal speed, c. Any wave disturbance that travels at
this speed will also require the Lorentz transformation. One point of view is that the
Lorentz transformation says something about how space-time is constructed.
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We would also have discovered the inadequacy of the Galilean transformation if
physicists had had access to high speeds before the discovery of electromagnetism.
Lorentz Invariants
If we can frame our laws in such a way that they are Lorentz invariant then we
have satisfied the requirements of Special Relativity. Consider the following
Invariant interval, ds2
From eqns. 9
dx'   (dx  vdt ), and
cdt '   (cdt 
v
dx),
c
(10)
Then we can show that
ds 2  c 2 dt 2  dx 2  c 2 dt '2 dx'2 ,
(11)
Eqn. 11 is true for macroscopic intervals too.
Time Dilation
Suppose that in frame O’ we keep a clock fixed in space, dx’=0. We measure a
time interval then. This is called the ‘proper’ time, dt. From eqn 11 we conclude
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dx 2
c dt  c dt  dx , or , dt  dt  2 ,
c
2
2
2
2
2
2
2
(12)
The observer in frame O will see the time interval dt to be larger than dt in O’.
We can solve for dx in eqn 12 using eqns 10.
v2
dx  vdt  0, so , c dt  c dt  v dt  c (1  2 )dt 2 ,
c
dt  dt , (13)
2
2
2
2
2
2
2
Equation 13 expresses the time dilation phenomenon.
Length Contraction
Suppose the observer O wants to measure the length of an object, which he
knows in the O’ frame has a length dx’. In order for O to make the measurement
of length he must do so at a fixed time, so that dt = 0. From eqn 11
ds 2  dx 2  c 2 dt '2 dx'2 , but from eqns 10, cdt '   dx,
 dx 2  ( dx) 2  dx'2  (1   2 2 )dx 2  dx'2 ,
dx 2  (1   2 )dx'2  dx'2 /  2 ,
(14)
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Equation 14 expresses the length contraction phenomenon. There are
some quantities that do not depend on space-time, like the total charge on
an object. The total charge should be invariant. However, the charge
density is not an invariant quantity. Consider the cylinder of uniform
charge below as observed by observers O and O’.
Q = r’L’A = rLA
L’
r‘
O’
v
O
The cylinder has cross section area A and length L’ in O’, where it is at rest.
r ' L'  rL 
L'
r  r '  r ' , (15)
L
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The observer in O sees a modified charge density. In fact, the charge density is
increased by the factor , which is reminiscent of the time dilation which also has
the same factor . Moreover, the observer in O also sees a current density J
associated with the moving rod.

 


J  rv  cr , recall   v . But this can be rewritten
c


J  cr '  . Form the quantity
 
2
(cr )  J  J  c 2 ( r 2  r 2  2 )  c 2 r 2 (1   2 )  c 2 r '2  2 (1   2 )
 
c r '  (cr )  J  J
2
2
2
(16)
In eqn (16) we note that c2r’2 must be an invariant quantity with respect to Lorentz
transformations. The observer O is completely arbitrary. Another observer with a
different relative velocity with respect to O’ would come to the same conclusion if the
quantity on the right hand side of (16) were formed. Thus if there are two observers in
reference frames 1 and 2 we can write
10
 
 
2
c r '  (cr1 )  J1  J1  (cr 2 )  J 2  J 2
2
2
2
(17)
Notice the close parallels between eqns (11), (12), and (17). The charge
density in the frame in which the charge distribution is at rest, O’, is the
counterpart to the proper time in that frame. The current density is the
counterpart to the position x. We can rewrite equations (9) by
multiplying eqn (9b) by c.
x'   ( x  ct ),
(18a)
v
ct '   (ct  x)   (ct  x),
c
(18b)
x'   ( x  x ),
(19a)
x 0 '   ( x 0  x),
(19b)
0
x  ct , x '  ct ' , (19c)
0
0
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If we change our notation from (t,x,y,z) for the time and space coordinates to
(ct , x, y, z )  ( x 0 , x1 , x 2 , x 3 )  x  , and likewise define
(cr , J x , J y , J z )  ( J 0 , J 1 , J 2 , J 3 )  J 
Then we can conclude for two separate inertial reference frames O and O’
 
 
0 2
( x ' )  x 'x '  ( x )  x  x, (20)
 
 
0 2
0 2
( J ' )  J 'J '  ( J )  J  J , (21)
0
2
The quantity x is called the space-time four-vector. Its Lorentz transformation
properties are given by eqns (19).
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REFERENCES
1) “Classical Electrodynamics”, 2nd Edition, John David Jackson, John
Wiley and Sons, 1975
2) “Electrodynamics”, Fulvio Melia, University of Chicago Press, 2001
3) “Introduction to Electrodynamics”, 2nd Edition, David J. Griffiths,
Prentice Hall, 1989