Today in Physics 218: relativistic electrodynamics
in tensor form
‰ The dual field tensor
‰ Charge and current densities, the
Maxwell equations, and the
Lorentz force, in tensor form
‰ The four-potential and gauge
transformations
‰ The relativistic analogue of the
inhomogeneous wave equation
for potentials.
Henrik Lorentz, his blackboard
covered with what looks like E&M
equations in tensor form.
23 April 2004
Physics 218, Spring 2004
1
The dual electromagnetic field tensor
Last time we constructed a second-rank four-tensor for the
electric and magnetic fields:
⎛ F 00
⎜
10
⎜
F
F µν = ⎜
⎜ F 20
⎜ 30
⎜F
⎝
F 01
F 02
F 11
F 12
F 21
F 22
F 31
F 32
F 03 ⎞ ⎛ 0
⎟ ⎜
F 13 ⎟ ⎜ −Ex
⎟=
23 ⎟ ⎜ −Ey
F
⎜
⎟
33 ⎟ ⎜ −Ez
F ⎠ ⎝
Ex
Ey
0
Bz
− Bz
0
By
− Bx
Ez ⎞
⎟
− By ⎟
,
⎟
Bx ⎟
0 ⎟⎠
which we got by comparing the Lorentz transformation of the
elements of an antisymmetric second-rank four-tensor with
that of the electric and magnetic fields:
23 April 2004
Physics 218, Spring 2004
2
The dual electromagnetic field tensor (continued)
(
)
By = γ ( By + β Ez )
(
)
Bz = γ ( Bz − β Ey )
Ex = Ex
, Ey = γ Ey − β Bz
, Ez = γ Ez + β By
,
Bx = Bx
,
,
;
(
)
t 31 = γ ( t 31 + β t 03 ) ,
(
)
t 12 = γ ( t 12 − β t 02 )
t 01 = t 01 , t 02 = γ t 02 − β t 12 , t 03 = γ t 03 + β t 13
,
t 23 = t 23 ,
.
We picked
F 01 = Ex , F 02 = Ey , F 03 = Ez , F 12 = Bz , F 13 = − By , F 23 = Bx .
But we could just as well have picked
F 01 = Bx , F 02 = By , F 03 = Bz , F 12 = −Ez , F 13 = Ey , F 23 = −Ex .
23 April 2004
Physics 218, Spring 2004
3
The dual electromagnetic field tensor (continued)
This makes a different-looking tensor that is called the dual
of F:
Bx
By
Bz ⎞
⎛ 0
⎜
⎟
0
−Bx
−Ez Ey ⎟
µν ⎜
G =⎜
.
⎟
0
−Ex ⎟
⎜ −By Ez
⎜ − B −E
⎟
E
0
z
y
x
⎝
⎠
that, yet, embodies the same physics as F. Sometimes it’s
more convenient to use one than the other, so it’s handy to
have both around, as we’ll see in a minute.
23 April 2004
Physics 218, Spring 2004
4
The dual electromagnetic field tensor (continued)
Both F and G come in contravariant and covariant forms.
Elements with covariant indices differ in sign for the zeroth
component, compared to the contravariant form:
⎛ −F 00
⎜
⎜ −F 10
Fµν = ⎜
⎜ −F 20
⎜ 30
⎜ −F
⎝
23 April 2004
−F 01
−F 02
F 11
F 12
F 21
F 22
F 31
F 32
Physics 218, Spring 2004
−F 03 ⎞
⎟
F 13 ⎟
⎟ .
23 ⎟
F
⎟
33 ⎟
F ⎠
5
The four-current-density
Consider a cloud of electric
charge, flying by at velocity u.
In our frame the charge and
u
dQ , dV
current density represented
by an infinitesimal element
within the cloud is
dQ
ρ=
, J = ρu .
dV
In the element’s rest frame, though,
dQ
2 2 dQ
ρ0 =
= 1−u c
,
dV0
dV
because of the Lorentz contraction of one dimension of the
infinitesimal volume as seen from our frame.
23 April 2004
Physics 218, Spring 2004
6
The four-current-density (continued)
Thus, in terms of the “proper” charge density ρ0 = dQ dV0 ,
ρ=
ρ0
1 − u2 c 2
= ρ0
η0
c
, J=
ρ0 u
1 − u2 c 2
= ρ0η
,
where we have noted the presence of the four-velocity η µ .
Since proper charge density is invariant (there’s only one rest
frame for a given object), the charge and current density
above comprise a four-vector, just as the four-velocity does:
⎛ cρ ⎞
⎜J ⎟
x
Jµ = ⎜ ⎟ .
⎜ Jy ⎟
⎜ ⎟
⎜J ⎟
⎝ z⎠
23 April 2004
Physics 218, Spring 2004
7
The four-current-density (continued)
Charge and current density are related (in a given reference
frame) by
∂ρ
=0
—◊ J +
∂t
3
3
∂J i ∂J 0
1 ∂J 0
=
+
=
+ 0
i
i
c ∂t i =0 ∂x ∂x
i = 0 ∂x
∑
∂J µ
∂x
23 April 2004
µ
∂J i
∑
, or
= 0 . Relativistic continuity equation
Physics 218, Spring 2004
8
Aside: the four-gradient
The four-gradient with respect to contravariant coordinates,
∂ ∂x µ , turns out to be a covariant four-vector operator, as
might be worth showing here. To show that it is covariant, it
suffices to show that, for any scalar function f, ∂f ∂x µ
transforms like a covariant four vector. This just requires the
chain rule and the Lorentz transformation for relative motion
along the x direction:
∂f
1 ∂f 1 ⎛ ∂f ∂t ∂f ∂x ∂f ∂y ∂f ∂z ⎞
=
= ⎜
+
+
+
⎟ .
0
c ∂ t c ⎝ ∂t ∂ t ∂x ∂ t ∂y ∂ t ∂z ∂ t ⎠
∂x
(
)
But t = γ t + vx c 2 and x = γ ( x + vt ) , so
∂y ∂z
∂t
∂x
=γ ,
=γv ,
=
=0 .
∂t
∂t
∂t ∂t
23 April 2004
Physics 218, Spring 2004
9
Aside: the four-gradient (continued)
Thus
∂f
∂f ⎞
∂f ⎞
1 ⎛ ∂f
⎛ ∂f
= γ ⎜ +v ⎟=γ ⎜ 0 +β
⎟ .
0
c ⎝ ∂t
∂x ⎠
∂x ⎠
∂x
⎝ ∂x
Similarly, ∂f
∂f ∂f ∂t ∂f ∂x ∂f ∂y ∂f ∂z
=
=
+
+
+
1
∂x ∂t ∂x ∂x ∂x ∂y ∂x ∂z ∂x
∂x
∂f γ v ∂f
∂f ⎞
⎛ ∂f
=
+ γ =γ ⎜ 1 +β 0 ⎟ ,
2
∂t c
∂x
∂x ⎠
⎝ ∂x
∂f
∂f ∂f ∂t ∂f ∂x ∂f ∂y ∂f ∂z ∂f
∂f
=
=
+
+
+
=
= 2,
2
∂y ∂t ∂y ∂x ∂y ∂y ∂y ∂z ∂y ∂y ∂x
∂x
∂f
∂f
∂f
=
= 3 .
3
∂z ∂x
∂x
23 April 2004
Physics 218, Spring 2004
10
Aside: the four-gradient (continued)
But this is how a covariant vector is supposed to transform.
Contravariant:
Covariant:
a 0 = γ a0 − β a1
− a0 = γ ( − a0 − β a1 )
(
)
a 1 = γ ( a1 − β a0 )
a 2 = a2
a1 = γ ( a1 + β a0 )
Remember to change
signs for your covariant
zeroth components!
a2 = a2
a 3 = a3
a3 = a3
∂
So ∂ µ ≡
is covariant. Similarly (Griffiths problem 12.55),
µ
∂x
∂
µ
the other form, ∂ ≡
, is contravariant.
∂xµ
$
23 April 2004
Physics 218, Spring 2004
$
$
11
The Maxwell equations in tensor form
Note that
∂Ex ∂Ey ∂Ez
4π 0 ⎞
⎛
∂ν F =
+
+
= — ◊ E ⎜ = 4πρ =
J ⎟ ,
∂x
∂y
∂z
c
⎝
⎠
1 ∂Ex ∂Bz ∂By ⎡ 1 ∂E
⎤
⎛ 4π ⎞
1ν
Jx ⎟ .
∂ν F = −
+
−
= ⎢−
+ — ¥ B⎥
⎜=
c ∂t
∂y
∂z
⎣ c ∂t
⎦x ⎝ c
⎠
0ν
Similarly,
⎡ 1 ∂E
⎤
∂ν F 2ν = ⎢ −
+ — ¥ B⎥
⎣ c ∂t
⎦y
⎛ 4π ⎞
Jy ⎟ ,
⎜=
⎝ c
⎠
⎡ 1 ∂E
⎤
= ⎢−
+ — ¥ B⎥
⎣ c ∂t
⎦z
⎛ 4π ⎞
Jz ⎟ .
⎜=
⎝ c
⎠
∂ν F
23 April 2004
3ν
Physics 218, Spring 2004
12
The Maxwell equations in tensor form (continued)
In other words, both Gauss’s and Ampère’s laws are
contained in the expression
4π µ
µν
∂ν F =
J
.
c
Similarly,
∂ν G 0ν = — ◊ B
(= 0)
,
⎡ 1 ∂B
⎤
∂ν G = − ⎢
+ — ¥ E⎥ (= 0) ,
⎣ c ∂t
⎦i
so Faraday’s law and the “no monopoles” law are
iν
∂ν G µν = 0 .
One can see how this notation might save some writing.
23 April 2004
Physics 218, Spring 2004
13
The Lorentz force in tensor form
This turns out simply to be
qην µν
K =
F
c
µ
.
To wit:
qην 1ν q
1
K =
F = −η 0 F 10 + η 1 F 11 + η 2 F 12 + η 3 F 13
c
c
q⎡
u
⎛
⎞
⎤
= ( −cγ u )( −Ex ) + uy γ u Bz + uzγ u − By = qγ u ⎜ E + ¥ B ⎟
⎦
c⎣
c
⎝
⎠x
(
)
(
)
Similarly,
u
⎛
⎞
K 2 = qγ u ⎜ E + ¥ B ⎟
c
⎝
⎠y
23 April 2004
u
⎛
⎞
, K 3 = qγ u ⎜ E + ¥ B ⎟
c
⎝
⎠z
Physics 218, Spring 2004
.
14
.
The Lorentz force in tensor form (continued)
u
⎛
⎞
K = qγ u ⎜ E + ¥ B ⎟ ,
c
⎝
⎠
or, since K = γ u F , as we saw in lecture on 16 April,
u
⎛
⎞
F = q⎜E + ¥ B⎟ ,
c
⎝
⎠
as usual.
‰ The zeroth component of K turns out to give (Griffiths
problem 12.54):
q
0
K = 2 γ uu ◊ E .
c
0
dp
d W γ u dW
0
Since K =
=
=
, (lecture, 16 April), this
dτ
dτ c
c dt
Thus
$
$
$
just means that power is force times velocity.
23 April 2004
Physics 218, Spring 2004
15
The four-potentials and gauge transformations
Unlike the fields, and as you might expect – given the
relationship of V and A to the energy and momentum per
unit charge – the potentials can be combined into a fourvector:
Aµ = (V , A ) .
It turns out that, just as the fields are easily expressed as
vector derivatives of the potentials in “nonrelativistic” E&M,
the field tensor is easily expressed in terms of derivatives of
the four-potential.
‰ Consider simply the antisymmetric combination:
∂ µ Aν − ∂ν Aµ
23 April 2004
Physics 218, Spring 2004
16
The four-potentials and gauge transformations
(continued)
‰ The 01 element of this is
∂Ax ∂A0
1 ∂Ax ∂V
0 1
1 0
∂ A −∂ A =
−
=−
−
∂x0 ∂x1
∂x
c ∂t
⎛ 1 ∂A
⎞
= ⎜−
− —V ⎟ = Ex
⎝ c ∂t
⎠x
.
Similarly, the 02 and 03 elements give the other two
components of E.
‰ The 12 element is
∂A2 ∂A1 ∂Ay ∂Ax
1 2
2 1
∂ A −∂ A =
−
=
−
∂x1 ∂x2
∂x
∂y
= ( — ¥ A )z = Bz
23 April 2004
Physics 218, Spring 2004
.
17
The four-potentials and gauge transformations
(continued)
‰ Similarly, the 13 and 23 elements give the other two
components of B. Evidently,
F µν = ∂ µ Aν − ∂ν Aµ
.
Note that the Lorentz gauge condition – the one that’s the
most use in wave equations for the potentials – can also be
written easily in these terms:
∂Ax ∂Ay ∂Az
∂V
1 ∂V
=0=
+
+
+
= ∂ν Aν
—◊ A +
c ∂t
∂x
∂y
∂z ∂ ( ct )
23 April 2004
Physics 218, Spring 2004
.
18
The relativistic analogue of the inhomogeneous
wave equation for potentials
Earlier this semester, we found that elimination of the fields
in favor of the potentials turned some of the Maxwell
equations into inhomogeneous wave equations in the
components of the potential. That works here too:
4π µ
µν
J
∂ν F =
c
∂ν ∂ µ Aν − ∂ν ∂ν Aµ =
In the first term, switch the order of differentiation and apply
the Lorentz gauge condition:
(
)
∂ µ ∂ν Aν − ∂ν ∂ν Aµ = −∂ν ∂ν Aµ
,
2
23 April 2004
4π µ
A ≡ ∂ν ∂ A = −
J
c
µ
ν
µ
Physics 218, Spring 2004
, or
.
19
We’re done.
That’s it for new material. The final two class
meetings will be reviews.
23 April 2004
Physics 218, Spring 2004
20