D
de Sitter Cosmology
D.1
Properties of Generalised Kerr–Schild Metrics
The generalised Kerr–Schild metric tensor has the algebraic form [1]
g ij = g
(0)
ij + 2 H k i k j ,
(D.1)
with
g
(0)
ij k i k j = k
j k j = 0 and k
i
= g
(0)
ij k j .
(D.2)
1. If the determinants of the matrices (g ij ) and ( g
(0)
ij ) are denoted det(g ij ) and
det( g
(0)
ij ) respectively then
det(g ij ) = det( g
(0)
ij ) .
(D.3)
The determinant of the matrix (g ij ) may be written, using a standard definition
of the determinant of a matrix, in the form
det(g ij ) = ( g
(0)
p0 + 2 H k p k 0 ) ( g
(0)
q1 + 2 H k q k 1 ) ( g
(0)
r2 + 2 H k r k 2 ) ×
( g
(0)
s3 + 2 H k s k 3 ) ) pqrs
= det( g
(0)
ij ) + 2 H
g
(0)
p0 g
(0)
q1 ( g
(0)
r2 k s k 3 + g
(0)
s3 k 2 k r )
© 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
237