62 4. PROBLEMS IN ELECTROSTATICS AND ELECTRODYNAMICS
By concatenation, we recover
E.x/ D
Z
d e.x; / D
Z
d
2
".z/'
t C
j
z
j
c
Oz D
2
".z/ Oz
in agreement with the Maxwell field from a charged sheet. We notice that, as expected, the
space part of the electric fields change sign at the plane of the sheet, pointing out at each side.
Consequently, an event passing through a charged sheet of equal sign will decelerate in space
on its approach and then accelerate as it retreats. However, unlike the field of a point event, the
temporal part
0
is an even function of spatial distance and so the event may accelerate along
the time axis on both its approach to the charged sheet and its retreat. In such a case, the spatial
motion will asymptotically return to its initial condition, while the event acquires a net temporal
acceleration, corresponding to a shift in energy and mass.
4.4 PLANE WAVES
From the wave equation (3.22) for j
˛
.x; / D 0 we may write the field in terms of the Fourier
transform [6]
f
˛ˇ
.x; / D
1
.2/
5
Z
d
5
k e
ikx
f
˛ˇ
.k/ D
1
.2/
5
Z
d
4
k d e
i.kxCk
0
x
0
C
55
c
5
/
f
˛ˇ
.k; /;
where
D k
5
D
55
k
5
is understood to represent the mass carried by the plane wave, much as k
0
and k represent energy
and 3-momentum. is interpretation is supported by the wave equation which imposes the 5D
constraint
k
˛
k
˛
D k
2
.k
0
/
2
C
55
2
D 0 H)
55
2
D .k
0
/
2
k
2
(4.21)
expressing in terms of the difference between energy and momentum. Under concatenation,
the field becomes
F
˛ˇ
.x/ D
Z
d
f
˛ˇ
.x; / D
Z
d
4
k
.2/
4
e
ik
x
1
c
5
f
˛ˇ
.k; 0/ D
Z
d
4
k
.2/
4
e
ik
x
F
˛ˇ
.k/
and recovers the 4D mass-shell constraint k
k
D 0 for the Maxwell field. In the transform
domain, the sourceless pre-Maxwell equations take the form
k e
55
0
D 0 k b D 0 k k
0
0
D 0
k e k
0
b D 0 k b C k
0
e
55
D 0
k b D 0 e C k
0
k
0
D 0
4.4. PLANE WAVES 63
which can be solved by taking
k
and e
?
as independent 3-vector polarizations, and writing
e
k
D
55
k
0
k
?
D
k
0
e
?
0
D
1
k
0
k
k
b D
1
k
0
k e
?
for the remaining fields. Unlike Maxwell plane waves, for which E, B, and k are mutually orthog-
onal, the pre-Maxwell electric fields e and have both transverse and longitudinal components.
When ! 0, we find that e, b, and k become mutually orthogonal and becomes a decoupled
longitudinal polarization parallel to k.
We use (3.11) to write the convolved field as
f
˛ˇ
ˆ
.x; / D
Z
ds
ˆ. s/f
˛ˇ
.
x; s
/
D
1
.2/
5
Z
d
4
k d e
i.kxk
0
x
0
C
55
c
5
/
f
˛ˇ
ˆ
.k; /;
where
f
˛ˇ
ˆ
.k; / D
1 C
.
c
5
/
2
f
˛ˇ
.k; /
introduces a multiplicative factor that will appear once in each field bilinear of T
˛ˇ
ˆ
. In terms of
the 3-vector fields, the mass-energy-momentum tensor components are
T
00
ˆ
D
1
2c
e e
ˆ
C b b
ˆ
C
55
ˆ
C
0
0
ˆ
T
0i
ˆ
D
1
c
e b
ˆ
C
55
0
ˆ
i
T
50
ˆ
D
1
c
e
ˆ
T
5i
ˆ
D
1
c
b
ˆ
C
0
e
ˆ
i
T
55
ˆ
D
1
2c
ˆ
0
0
ˆ
C
55
.
e e
ˆ
b b
ˆ
/
:
For the plane wave, the energy density is
T
00
ˆ
D
1
c
e
2
?
C
55
2
k
1 C
.
/
2
which, since e
2
?
D
1
2
e
2
?
C b
2
, is equivalent in form to the energy density in Maxwell theory
00
D
1
2c
E
2
C B
2
with the addition of the independent polarization
k
. e mass density is found to be
T
55
ˆ
D
2
ck
2
0
e
2
?
C
55
2
k
1 C
.
/
2
D
2
k
2
0
T
00
ˆ
64 4. PROBLEMS IN ELECTROSTATICS AND ELECTRODYNAMICS
expressing energy density scaled by the squared mass-to-energy ratio for the field. e energy
flux—the standard Poynting 3-vector—is
T
0i
ˆ
! T
0
ˆ
D
k
k
0
T
00
ˆ
expressing the energy density T
00
ˆ
flowing uniformly in the direction of the momentum nor-
malized to energy. Comparing the proportionality factor to that for a free particle
k
k
0
!
p
E=c
D
1
c
M d x=d
M dt =d
D
v
c
which will not generally be a unit vector unless D 0, as it must be for Maxwell plane waves.
e mass flux vector—a second Poynting 3-vector—can be written
T
5i
ˆ
! T
5
ˆ
D
k
T
55
ˆ
expressing the mass density T
55
ˆ
flowing uniformly in the direction of the momentum normal-
ized to mass. Finally,
T
50
ˆ
D
k
0
T
55
ˆ
D
k
0
T
00
ˆ
so that T
5
ˆ
can be written as
T
5
ˆ
D
k
T
55
ˆ
D
k
k
2
0
T
00
ˆ
expressing the mass density T
55
ˆ
flowing in the direction of the 4-momentum. In this sense, T
50
ˆ
represents the flow of mass into the time direction. We notice that when ! 0, as is the case
for Maxwell plane waves, k=k
0
becomes a unit vector and T
5˛
ˆ
D 0, so that mass density and
flow vanish. e interpretation of plane waves carrying energy and momentum (energy flux)
uniformly to infinity is thus seen to generalize to mass flow, where mass is best understood
through (4.21) as the non-identity of energy and momentum.
Suppose that a plane wave of this type impinges on a test particle in its rest frame, de-
scribed by x
˛
./ D
.
c; 0; c
5
/
. Since Px D 0, the wave will interact with the event through the
Lorentz force (3.6) and (3.7) as
M Rx
.
/
D
e
c
f
0
.x; / Px
0
.
/
C c
5
f
5
.x; /
d
d
.
1
2
M Px
2
/ D
ec
5
c
55
0
Px
0
which for
k
¤ 0 ) k
k
¤ 0 becomes
R
t D
55
e
c
5
Mc
2
1
k
0
k
k
Rx D
e
M
h
e
?
1 C
c
5
c
k
0
C
k
c
5
c
C
55
k
0
i
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