B Bateman Waves
215
and thus (B.29) yields
E c E
c k
a ;b k
b
+
(E c E
c ) ,b k
b
+ E c E
c k
b ;b
k
a
= 0 .
(B.31)
Using (B.27) this simplifies to
k
a ;b k
b
= −(u b;c k
b k
c ) k
a ,
(B.32)
and we have recovered Mariot’s [1] result, namely, the propagation direction of
electromagnetic radiation in space-time is geodesic.
Expanding k a;b + k b;a on the orthonormal basis u a , e a , b a , p a and using (B.18),
(B.20), (B.22) and (B.32) we find that
k a;b + k b;a = λ g ab + ξ a k b + ξ b k a ,
(B.33)
with
λ = k
a ;a + u a;b k
a k
b ,
(B.34)
and
ξ a = (A + B) u a − k b;c (e
b p
c
+ e
c p
b ) e a − k b;c (b
b p
c
+ b
c p
b ) b a + (A − B) p a ,
(B.35)
with
A = −
1
2
k
a ;a − u a;b k
b k
c ,
(B.36)
B =
1
2
k a;b (u
a p
b
+ u
b p
a ) .
(B.37)
As a result of establishing (B.33) we have recovered, using the Robinson–Trautman
[2] test, Robinson’s [3] result that the propagation direction of electromagnetic
radiation in space-time is not only geodesic but is shear-free.
B.2
Ricci Identities and Bianchi Identities
We are concerned here only with vacuum space-times for which R ab = 0, where
R ab are the components of the Ricci tensor. In this case the Weyl tensor coincides
with the Riemann tensor and thus C abcd = R abcd in Chap. 2. We are also interested
in identifying a time-like congruence in the space-time consisting of the integral
curves of a unit time-like vector field u a (with u a u a = 1) as in Chap. 1, but one
which is geodesic ( ˙
u a = u a ;b u b = 0), twist-free (ω ab = 0) and expansion-free
215
and thus (B.29) yields
E c E
c k
a ;b k
b
+
(E c E
c ) ,b k
b
+ E c E
c k
b ;b
k
a
= 0 .
(B.31)
Using (B.27) this simplifies to
k
a ;b k
b
= −(u b;c k
b k
c ) k
a ,
(B.32)
and we have recovered Mariot’s [1] result, namely, the propagation direction of
electromagnetic radiation in space-time is geodesic.
Expanding k a;b + k b;a on the orthonormal basis u a , e a , b a , p a and using (B.18),
(B.20), (B.22) and (B.32) we find that
k a;b + k b;a = λ g ab + ξ a k b + ξ b k a ,
(B.33)
with
λ = k
a ;a + u a;b k
a k
b ,
(B.34)
and
ξ a = (A + B) u a − k b;c (e
b p
c
+ e
c p
b ) e a − k b;c (b
b p
c
+ b
c p
b ) b a + (A − B) p a ,
(B.35)
with
A = −
1
2
k
a ;a − u a;b k
b k
c ,
(B.36)
B =
1
2
k a;b (u
a p
b
+ u
b p
a ) .
(B.37)
As a result of establishing (B.33) we have recovered, using the Robinson–Trautman
[2] test, Robinson’s [3] result that the propagation direction of electromagnetic
radiation in space-time is not only geodesic but is shear-free.
B.2
Ricci Identities and Bianchi Identities
We are concerned here only with vacuum space-times for which R ab = 0, where
R ab are the components of the Ricci tensor. In this case the Weyl tensor coincides
with the Riemann tensor and thus C abcd = R abcd in Chap. 2. We are also interested
in identifying a time-like congruence in the space-time consisting of the integral
curves of a unit time-like vector field u a (with u a u a = 1) as in Chap. 1, but one
which is geodesic ( ˙
u a = u a ;b u b = 0), twist-free (ω ab = 0) and expansion-free