6.3 Generalized Kerr–Schild Space–Times
137
null hyperplanes, (ii) κ < 0 ⇒ intersecting null hyperplanes, (iii) κ = 0 ⇒
intersecting null hyperplanes except when Re β = 0 or Im β = 0 or Re β = c 0 Im β.
6.3
Generalized Kerr–Schild Space-Times and Gravitational
Waves
The gravitational field of pure gravitational radiation is described by a Weyl tensor
which is algebraically degenerate, as we have seen in Chap. 2. However Trautman
[7] has pointed out that “there is another important property of waves, both linear
and gravitational: waves can propagate information. This means that wave-like
solutions depend on arbitrary functions, the shape of which contains the information
carried by the wave." A notable aspect of this statement is that it is a point of view,
which leaves the researcher free as to how it is implemented. Such statements are a
powerful stimulus for research. Trautman chose to illustrate this with what is now
referred to as a generalised Kerr–Schild [8, 9] metric required to satisfy Einstein’s
vacuum field equations. Such a metric, satisfying Einstein’s field equations with a
cosmological constant, is also useful in studying plane fronted gravitational waves
in the present context for the reasons we now describe. We assume the metric tensor
components, in coordinates x i , take the form
g ij = g
(0)
ij + 2 H k i k j ,
(6.64)
with
g
(0)
ij k i k j = k
j k j = 0 and k
i
= g
(0)
ij k j .
(6.65)
Some useful properties of such a metric tensor are derived in the Appendix D. If a
stroke denotes covariant differentiation with respect to the Riemannian connection
calculated with the metric tensor g ij and a semicolon denotes covariant differentiation with respect to the Riemannian connection calculated with the metric tensor
g
(0)
ij then
k i|j = k i;j + H
k i k j + H (k i;k k
k k j + k j ;k k
k k i ) ,
(6.66)
where H = H ,i k i . We will apply this to the case in which g
(0)
ij is the de Sitter
or anti-de Sitter metric tensor given via the line element (6.54) and k i = u ,i (⇔
k i dx i = du). Thus k i satisfies
k i;j = k j ;i ,
(6.67)
k
i ;i = 0 ,
(6.68)
k i;j = ξ i k j + ξ j k i ,
(6.69)
137
null hyperplanes, (ii) κ < 0 ⇒ intersecting null hyperplanes, (iii) κ = 0 ⇒
intersecting null hyperplanes except when Re β = 0 or Im β = 0 or Re β = c 0 Im β.
6.3
Generalized Kerr–Schild Space-Times and Gravitational
Waves
The gravitational field of pure gravitational radiation is described by a Weyl tensor
which is algebraically degenerate, as we have seen in Chap. 2. However Trautman
[7] has pointed out that “there is another important property of waves, both linear
and gravitational: waves can propagate information. This means that wave-like
solutions depend on arbitrary functions, the shape of which contains the information
carried by the wave." A notable aspect of this statement is that it is a point of view,
which leaves the researcher free as to how it is implemented. Such statements are a
powerful stimulus for research. Trautman chose to illustrate this with what is now
referred to as a generalised Kerr–Schild [8, 9] metric required to satisfy Einstein’s
vacuum field equations. Such a metric, satisfying Einstein’s field equations with a
cosmological constant, is also useful in studying plane fronted gravitational waves
in the present context for the reasons we now describe. We assume the metric tensor
components, in coordinates x i , take the form
g ij = g
(0)
ij + 2 H k i k j ,
(6.64)
with
g
(0)
ij k i k j = k
j k j = 0 and k
i
= g
(0)
ij k j .
(6.65)
Some useful properties of such a metric tensor are derived in the Appendix D. If a
stroke denotes covariant differentiation with respect to the Riemannian connection
calculated with the metric tensor g ij and a semicolon denotes covariant differentiation with respect to the Riemannian connection calculated with the metric tensor
g
(0)
ij then
k i|j = k i;j + H
k i k j + H (k i;k k
k k j + k j ;k k
k k i ) ,
(6.66)
where H = H ,i k i . We will apply this to the case in which g
(0)
ij is the de Sitter
or anti-de Sitter metric tensor given via the line element (6.54) and k i = u ,i (⇔
k i dx i = du). Thus k i satisfies
k i;j = k j ;i ,
(6.67)
k
i ;i = 0 ,
(6.68)
k i;j = ξ i k j + ξ j k i ,
(6.69)