B
Bateman Waves in the Linear Approximation
B.1
Covariant Treatment on General Space-Time
Let u a with g ab u a u b = 1 be a unit time-like vector field in a space-time with
metric tensor components g ab in coordinates x a . The integral curves of u a constitute
a time-like congruence in the space-time, described in Chap. 1. A Maxwell field on
the space-time is described by a real bivector with components F ab = −F ba which
satisfies Maxwell’s source-free field equations:
F
ab ;b = 0 and
∗ F
ab ;b = 0 ,
(B.1)
with ∗ F ab =
1
2 η abcd F cd and η abcd =
√ −g g abcd as in Chap. 2. With respect to
the congruence tangent to u a we define the electric part of F ab by the space-like
covariant vector field
E a = F ab u
b (⇒ E a u
a
= 0) ,
(B.2)
and the magnetic part of F ab by the space-like covariant vector field
B a =
∗ F ab u
b (⇒ B a u
a
= 0) .
(B.3)
From (B.3) we have
B a η
apqr
=
1
2
η abcd η
apqr F
cd u
b
= u
p F
rq
+ u
q F
pr
+ u
r F
qp ,
(B.4)
and thus we can write F ab in terms of E a , H a as
F ab = E a u b − E b u a + η abcd u
c B
d .
(B.5)
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
P. A. Hogan, D. Puetzfeld, Frontiers in General Relativity, Lecture Notes
in Physics 984, https://doi.org/10.1007/978-3-030-69370-1
211
Bateman Waves in the Linear Approximation
B.1
Covariant Treatment on General Space-Time
Let u a with g ab u a u b = 1 be a unit time-like vector field in a space-time with
metric tensor components g ab in coordinates x a . The integral curves of u a constitute
a time-like congruence in the space-time, described in Chap. 1. A Maxwell field on
the space-time is described by a real bivector with components F ab = −F ba which
satisfies Maxwell’s source-free field equations:
F
ab ;b = 0 and
∗ F
ab ;b = 0 ,
(B.1)
with ∗ F ab =
1
2 η abcd F cd and η abcd =
√ −g g abcd as in Chap. 2. With respect to
the congruence tangent to u a we define the electric part of F ab by the space-like
covariant vector field
E a = F ab u
b (⇒ E a u
a
= 0) ,
(B.2)
and the magnetic part of F ab by the space-like covariant vector field
B a =
∗ F ab u
b (⇒ B a u
a
= 0) .
(B.3)
From (B.3) we have
B a η
apqr
=
1
2
η abcd η
apqr F
cd u
b
= u
p F
rq
+ u
q F
pr
+ u
r F
qp ,
(B.4)
and thus we can write F ab in terms of E a , H a as
F ab = E a u b − E b u a + η abcd u
c B
d .
(B.5)
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
P. A. Hogan, D. Puetzfeld, Frontiers in General Relativity, Lecture Notes
in Physics 984, https://doi.org/10.1007/978-3-030-69370-1
211