6
de Sitter Cosmology
Abstract
The equation u ≡ t −z = constant in Minkowskian space-time with line element
ds
2
= dt
2
− dx
2
− dy
2
− dz
2
= η ij dx
i dx
j ,
is an example of a null hyperplane. It represents the history of a 2-plane, parallel
to the x, y-plane in three dimensional Euclidean space, moving with the speed
of light in the positive z-direction. Thus the family of null hyperplanes u =
constant could be the histories of the wave fronts of plane electromagnetic waves
travelling in the positive z-direction in Euclidean space. The vector field normal
to the hyperplanes is k i = u ,i and with k i = η ij k j this has the properties that
k i k i = 0 and k i,j = 0. Thus k i is a null vector field (and therefore tangent
to u = constant) and covariantly constant (i.e. a constant vector field in the
coordinates x i = (t, x, y, z) with i = 0, 1, 2, 3). To generalise this notion of
null hyperplanes to space-times of non-zero constant curvature, the first obstacle
one encounters is the non-existence in such a space-time of covariantly constant
vector fields.
6.1
Null Hyperplanes in Space-Times of Constant Curvature
In a general space-time with metric tensor components g ab the components of the
Riemann curvature tensor are given in terms of the metric tensor and its derivatives
by
R ij kl =
1
2
(g ik,j l + g jl,ik − g il,j k − g jk,il ) + g ab ((
a
ik
b
jl −
a
il
b
jk ) ,
(6.1)
© 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_6
127
de Sitter Cosmology
Abstract
The equation u ≡ t −z = constant in Minkowskian space-time with line element
ds
2
= dt
2
− dx
2
− dy
2
− dz
2
= η ij dx
i dx
j ,
is an example of a null hyperplane. It represents the history of a 2-plane, parallel
to the x, y-plane in three dimensional Euclidean space, moving with the speed
of light in the positive z-direction. Thus the family of null hyperplanes u =
constant could be the histories of the wave fronts of plane electromagnetic waves
travelling in the positive z-direction in Euclidean space. The vector field normal
to the hyperplanes is k i = u ,i and with k i = η ij k j this has the properties that
k i k i = 0 and k i,j = 0. Thus k i is a null vector field (and therefore tangent
to u = constant) and covariantly constant (i.e. a constant vector field in the
coordinates x i = (t, x, y, z) with i = 0, 1, 2, 3). To generalise this notion of
null hyperplanes to space-times of non-zero constant curvature, the first obstacle
one encounters is the non-existence in such a space-time of covariantly constant
vector fields.
6.1
Null Hyperplanes in Space-Times of Constant Curvature
In a general space-time with metric tensor components g ab the components of the
Riemann curvature tensor are given in terms of the metric tensor and its derivatives
by
R ij kl =
1
2
(g ik,j l + g jl,ik − g il,j k − g jk,il ) + g ab ((
a
ik
b
jl −
a
il
b
jk ) ,
(6.1)
© 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_6
127