5Electromagnetism and relativity

IB Electromagnetism

5.2 Conserved currents
Recall the continuity equation
ρ
t
+ · J = 0.
An implication of this was that we cannot have a charge disappear on Earth
and appear on Moon. One way to explain this impossibility would be that the
charge is travelling faster than the speed of light, which is something related to
relativity. This suggests that we might be able to write it relativistically.
We define
J
µ
=
ρc
J
Before we do anything with it, we must show that this is a 4-vector, i.e. it
transforms via left multiplication by Λ
µ
ν
.
The full verification is left as an exercise. We work out a special case to
justify why this is a sensible thing to believe in. Suppose we have a static charge
density
ρ
0
(
x
) with
J
= 0. In a frame boosted by
v
, we want to show that the
new current is
J
0µ
= Λ
µ
ν
J
ν
=
γρ
0
c
γρ
0
v
.
The new charge is now
γρ
0
ρ
0
. This is correct since we have Lorentz
contraction. As space is contracted, we get more charge per unit volume. Then
the new current is velocity times charge density, which is J
0
= γρ
0
v.
With this definition of J
µ
, local charge conservation is just
µ
J
µ
= 0.
This is invariant under Lorentz transformation, simply because the indices work
out (i.e. we match indices up with indices down all the time).
We see that once we have the right notation, the laws of physics are so short
and simple!