13.1 Basic Observations and Assumptions
207
metric of the universe on the largest cosmological scale is also homogeneous and
isotropic. This symmetry is a very strong constraint and will allow us to simplify
the mathematical problem and make it tractable. This assumption is basic and
powerful: it almost completely determines the general form of the metric as we
will show in the next section.
13.2 The Cosmological FLRW Metric
The observations and related assumptions in the preceding section place a rigid
constraint on the cosmological metric: it must represent a 3-dimensional space that
is homogeneous and isotropic—the same everywhere and the same when viewed in
any direction. We can obtain the general form of the metric of this 3-space if we first
consider the analogous 2-space problem, which is intuitive and can be visualized.
Time of course must be added to the space dimensions to give the final metric in
spacetime.
In two dimensions there are two spaces that immediately come to mind that are
homogeneous and isotropic: the Euclidean plane and the surface of a sphere. The
metric on the surface of a sphere with radius R was obtained in Chap. 4, but we will
repeat it here. In cylindrical coordinates (ρ, ϕ, z) the Euclidean 3-space metric and
the constraint equation for a sphere are
d
2
= dρ
2
+ ρ
2 dϕ
2
+ dz
2
, ρ
2
+ z
2
= R
2
.
(13.2)
In this section we will use d
2 for the spatial line elements, and reserve ds
2 for the
cosmological metric of space-time. We calculate dz from the constraint equation and
substitute it into the 3-space metric to get the 2-space metric for the surface of the
sphere
d
2
=
dρ
2
1 − ρ 2 /R 2 + ρ
2 dϕ
2
.
(13.3)
Figure 13.3 shows the polar coordinates on the sphere. For ρ R, near the north
pole, the coordinates are like plane polar coordinates. The radial coordinate ρ cannot
Fig. 13.3 The 2-spaces that are obviously homogeneous and isotropic—the plane and the sphere.
They have analogs in three dimensions, where they are called the hyperplane and the hypersphere
207
metric of the universe on the largest cosmological scale is also homogeneous and
isotropic. This symmetry is a very strong constraint and will allow us to simplify
the mathematical problem and make it tractable. This assumption is basic and
powerful: it almost completely determines the general form of the metric as we
will show in the next section.
13.2 The Cosmological FLRW Metric
The observations and related assumptions in the preceding section place a rigid
constraint on the cosmological metric: it must represent a 3-dimensional space that
is homogeneous and isotropic—the same everywhere and the same when viewed in
any direction. We can obtain the general form of the metric of this 3-space if we first
consider the analogous 2-space problem, which is intuitive and can be visualized.
Time of course must be added to the space dimensions to give the final metric in
spacetime.
In two dimensions there are two spaces that immediately come to mind that are
homogeneous and isotropic: the Euclidean plane and the surface of a sphere. The
metric on the surface of a sphere with radius R was obtained in Chap. 4, but we will
repeat it here. In cylindrical coordinates (ρ, ϕ, z) the Euclidean 3-space metric and
the constraint equation for a sphere are
d
2
= dρ
2
+ ρ
2 dϕ
2
+ dz
2
, ρ
2
+ z
2
= R
2
.
(13.2)
In this section we will use d
2 for the spatial line elements, and reserve ds
2 for the
cosmological metric of space-time. We calculate dz from the constraint equation and
substitute it into the 3-space metric to get the 2-space metric for the surface of the
sphere
d
2
=
dρ
2
1 − ρ 2 /R 2 + ρ
2 dϕ
2
.
(13.3)
Figure 13.3 shows the polar coordinates on the sphere. For ρ R, near the north
pole, the coordinates are like plane polar coordinates. The radial coordinate ρ cannot
Fig. 13.3 The 2-spaces that are obviously homogeneous and isotropic—the plane and the sphere.
They have analogs in three dimensions, where they are called the hyperplane and the hypersphere