Review of Special Relativity

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Review of Special Relativity
At the end of the 19th century it became clear
that Maxwells 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 x vt (1a) t t
(1b) z z (1c) y y
(1d)
(x,y,z,t) , (x,y,z,t)
v
O
x
x
Time is assumed to be a universal parameter,
independent of the reference frame.
O
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.

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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,
In general the transformation from one coordinate
system to another is given by,
We will use eqns 1 to transform this into the O
system.
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4
(4a) , 4(b)
Applying eqns 4 a second time gives
(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.
In eqns 6 we choose coefficients as and bs to
be dimensionless. We now use equations 6 in
equations 3 and derive for the wave equation
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(7)
In order to ensure invariance w.r.t. coordinate
transformation we need
We can try to find a solution to 7 that is
symmetric, namely try a1b1 , and a0b0. Both 7a
and 7b give the same result.
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The counterpart to the Galilean transformations (
eqns 1), which makes the wave equation invariant
is called the Lorentz transformation.
The inverse transformation from x to x simply
requires changing the sign of v.

• Inherent in this derivation are two assumptions.
• 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.
• 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
Then we can show that
Eqn. 11 is true for macroscopic intervals too.
Time Dilation
Suppose that in frame O we keep a clock fixed in
space, dx0. We measure a time interval then.
This is called the proper time, dt. From eqn 11
we conclude
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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.
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
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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 rLA rLA
L
r

O
v
O
The cylinder has cross section area A and length
L in O, where it is at rest.
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The observer in O sees a modified charge density.
In fact, the charge density is increased by the
factor g, which is reminiscent of the time
dilation which also has the same factor g.
Moreover, the observer in O also sees a current
density J associated with the moving rod.
In eqn (16) we note that c2r2 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
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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.
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If we change our notation from (t,x,y,z) for the
time and space coordinates to
Then we can conclude for two separate inertial
reference frames O and O
The quantity xm 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