Constant Curvature Spaces

 

 

Mathematically, a Riemannian space that has constant curvature must have a Riemann Curvature Tensor of the following form.

 

 

(28.1)

 

 

 

In this equation K is a constant sometimes simply called the curvature of the space. In spaces of constant curvature, the spaces are qualitatively different depending on whether the curvature constant .

Let us now temporarily specialize to the 3-D subspace. We demand that

 

(28.2)

 

 

Contract this equation to get the 3-D Ricci Tensor

 

 

 

(28.3)  

which, upon simplification,  gives

 

 

(28.4)

Hence, in the constant curvature 3-D subspace, the Ricci Tensor is proportional to the 3-space metric.

 

Next we deal with the isotropy of the 3-D subspace. If it is to be globally isotropic then the 3-D subspace must be spherically symmetric about every point. Let the 3-D line element be represented as follows:

 

(28.5)

 

We write this line element in the most general 3-D spherically symmetric form:

 

(28.6)  

 

 

We now impose the constant curvature Ricci Tensor constraint (28.4) on the 3-D metric. We obtain:

 

 

(28.7)

 

 

(28.8)

 

 

 

The solution to both of these differential equations can be written in the following form:

 

 

 

(28.9)

 

 

 

Hence the line element for a 3-space of constant curvature is

 

 

(28.10)  

 

K is allowed to be positive, negative or zero.

 

A common alternative form for this line element can be found by letting the radial coordinate change as follows:

 

 

(28.11)

 

With this transformation the line element takes a conformally flat form

 

(28.12) 

We can use the 3-D forms of the line element that we have constructed to form the full 4-D cosmological line element. In 4-D we have

 

 

(28.13)

 

or, with the  coordinate form of the 3-D metric, we obtain

 

(28.14)  

It is conventional to let the curvature constant, which potentially can vary continuously over all of the real numbers from negative values to positive values, be specified so that it takes on only three integer values 1,0, or -1. The variable part of the curvature constant K is assumed absorbed into the radial coordinate. We set up this situation by introducing the integer constant k as follows:

 

(28.15)

.

Hence

(28.16)

.

It is now standard to once again change the radial variable, this time by a simple rescaling:

(28.17)

.

The 4-D line element then takes the form

 

 

 

Now define a new scale factor function:

 

 

(28.18)

Dropping the star notation we get the following

 

(28.19)                                                            

 

 

 

The alternative form where we use the  coordinate is now written:

 

(28.20)

 

 Either form of these two constant-curvature cosmological line elements is called the Robertson-Walker Line Element.

 

Note that at any epoch , the following relation gives the geometry of the 3-D time slice:

 

(28.21)

 

If we take the case , we can show that the  singularity is just a coordinate singularity by changing to the specially suited coordinates involving the new coordinate  defined in terms of r as 

 

(28.22)

 

The singularity is therefore eliminated since

 

 

(28.23)

 

The 3-D time slice then takes the form

 

(28.24)

 

 

It can be shown that this type of metric corresponds to a closed geometry that models a universe that will eventually contract back on itself. The curvature of such a  universe is said to be positive. The paths of two particles initially moving parallel to each other in such a universe will eventually diverge.

 

Next we consider the  case. The 3-D slice is just flat space in spherical polar coordinates:

 

(28.25)

 

 

 

In terms of Cartesian coordinates this line element can be written:

 

 

(28.26)

 

The geometry of such a universe model is said to be spatially flat. The curvature of such a  universe is said to be zero. The paths of two particles initially moving parallel to each other in such a universe will always remain parallel.

 

We next consider the remaining case where . It is then convenient to introduce the coordinate transformation involving the new coordinate  such that

 

(28.27)

 

 

 

such that

 

 

(28.28)

 

 

We then get the 3-D time slice to have the form

 

(28.29)  

It can be shown that this type of metric corresponds to an open geometry that models a universe that will always expand. The curvature of such a  universe is said to be negative. The paths of two particles initially moving parallel to each other in such a universe will eventually diverge.