212
B Bateman Waves
Substituting this into (B.3) gives
∗ F ab = B a u b − B b u a − η abcd u
c E
d .
(B.6)
From (B.5) and (B.6) we can write the invariants
1
2
F ab F
ab
= E a E
a
− B a B
a and
1
2
F ab
∗ F
ab
= 2 E a B
a .
(B.7)
Thus for pure electromagnetic radiation we must have
E a E
a
= B a B
a and E a B
a
= 0 .
(B.8)
Now define the vectors
e
a
=
E a
(−E b E b ) 1/2 , b
a
=
B a
(−E b E b ) 1/2 ,
p
a
=
η abcd E b u c B d
E f E f
= −η
abcd e b u c b d .
(B.9)
These are mutually orthogonal, unit space-like vectors (with e a e a = b a b a =
p a p a = −1) and each is orthogonal to u a . Checking that p a p a = −1 involves
the use of the identity
η
abcd η arst = −
δ b
r δ b
s δ b
t
δ c
r δ c
s δ c
t
δ d
r δ d
s δ d
t
.
(B.10)
Making use of the more general identity
η
apqr η clmn = −
δ a
c δ
a
l δ a
m δ a
n
δ
p
c δ
p
l δ
p
m δ
p
n
δ
q
c δ
q
l δ
q
m δ
q
n
δ r
c δ r
l δ r
m δ r
n
,
(B.11)
we find that
p
a p c = −δ
a
c − e
a e c + u
a u c − b
a b c ,
(B.12)
which we rewrite as
g ac = u a u c − e a e c − b a b c − p a p c .
(B.13)