8
Lorentz Invariance and Special Relativity
The principle of special relativity is the assertion that all laws of physics take the same form
as described by two observers moving with respect to each other at constant velocity
v
. If
the dynamical equations for a system preserve their form under such a change of coordinates,
then they must show a corresponding symmetry. Newtonian mechanics satisfied this principle
in the form of Galilei relativity, for which the relation between the coordinates was simply
r
′
(
t
) =
r
(
t
)
−
v
t
, and for which time in the two frames was identical. However, Maxwell’s
field equations do not preserve their form under this change of coordinates, but rather under
a modified transformation: the Lorentz transformations.
8.1
Spacetime symmetries of the wave equation
Let us first study the spacetime symmetries of the wave equation for a field component in
the absence of sources:
−
parenleftbigg
∇
2
−
1
c
2
∂
2
∂t
2
parenrightbigg
ψ
=
0
(404)
As we discussed last semester spatial rotations
x
′
k
=
R
kl
x
l
are realized by the field transfor
mation
ψ
′
(
x
′
,t
) =
ψ
(
x
,t
) =
ψ
(
R
−
1
x
′
,t
). Then
∇
′
k
ψ
′
=
R
−
1
lk
∇
l
ψ
=
R
kl
∇
l
ψ
, and the wave
equation is invariant under rotations because
∇
2
is a rotational scalar. If we have set up a
fixed Cartesian coordinate system we may build up any rotation by a sequence of rotations
about any of the three axes. Instead of specifying the axis of one of these basic rotations, it
is more convenient to specify the plane in which the coordinate axes rotate. For example,
we describe a rotation by angle
θ
about the
z
axis as a rotation in the
xy
plane.
∇
′
x
= cos
θ
∇
x
−
sin
θ
∇
y
,
∇
′
y
= sin
θ
∇
x
+ cos
θ
∇
y
.
(405)
Then it is easy to see from the properties of trig functions that
∇
′
2
x
+
∇
′
2
y
+
∇
′
2
z
=
∇
2
x
+
∇
2
y
+
∇
2
z
,
(406)
under a rotation in any of the three planes, and through composition under any spatial
rotation.
There must be a similar symmetry in the
xt
,
yt
, and
zt
planes.
But because of the
relative minus sign we have to use hyperbolic trig functions instead of trig functions:
∇
′
x
= cosh
λ
∇
x
+ sinh
λ
∂
c∂t
,
∂
c∂t
′
= + sinh
λ
∇
x
+ cosh
λ
∂
c∂t
(407)
The invariance of
∇
2
x
−
∂
2
/c
2
∂t
2
under this transformation then follows. Clearly there are
analogous symmetries in the
yt
 and
zt
planes. These transformations will replace the Galilei
boosts of Newtonian relativity.
83
c
circlecopyrt
2010 by Charles Thorn
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Now let us interpret this symmetry in terms of how the coordinates transform:
x
′
=
x
cosh
λ
−
ct
sinh
λ
≡
γ
(
x
−
vt
)
(408)
ct
′
=
ct
cosh
λ
−
x
sinh
λ
≡
γ
(
ct
−
vx/c
)
(409)
where
v
=
c
tanh
λ
.
From the identity cosh
−
2
= 1
−
tanh
2
, we see that
γ
≡
cosh
λ
=
1
/
radicalbig
1
−
v
2
/c
2
.
We see from the first equation that the origin of the primed coordinate
system
x
′
= 0 corresponds to
x
=
vt
which means that the origin of the primed system is
moving on the
x
axis at the speed
v
. We say that the primed system is boosted by velocity
v
=
v
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 Spring '08
 Staff
 Special Relativity, Charles Thorn

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