# Open and Closed Models in Cosmology

1. Jul 31, 2011

### Anamitra

Let us consider the cosmological metric:
$${ds}^{2}{=}{dt}^{2}{-}{[}{a}{(}{t}{)}{]}^{2}{[}\frac{{dr}^{2}}{{1}{-}{k}{r}^{2}}{+}{r}^{2}{(}{d}{\theta}^{2}{+}{sin}^{2}{(}{\theta}{)}{d}{\phi}^{2}{]}$$ -------------- (1)
For closed models k is positive
We shall consider here a closed one:
We write:
$${ds}^{2}{=}{dt}^{2}{-}{d}{{X}_1}^{2}{-}{d}{{X}_{2}}^{2}{-}{d}{{X}_{3}}^{2}$$ --------------- (2)
Where:
$${d}{X}_{1}{=}{\int}\frac{{a}{(}{t}{)}}{\sqrt{{1}{-}{k}{r}^{2}}}{dr}$$--(3)
$${d}{X}_{2}{=}{\int}{a}{(}{t}{)}{r}{d}{\theta}$$
$${d}{X}_{3}{=}{\int}{a}{(}{t}{)}{r}{sin}{\theta}{d}{\phi}$$

When viewed in terms of physical variables equation (2) is a flat space time metric in terms of physical variables[X1,X2 and X3 are simply lengths] . The Special Relativity features of the light cone should be satisfied.
In a closed model let us consider a light cone at a point infinitesimally close to the boundary which is not expanding at a superluminally wrt the point in consideration.[at the current stage]..Rather the rate of expansion of the boundary is assumed to be quite slow [much less than c]with respect to the point in consideration. What would the forward motion of the light ray be like?
Incidentally it is important to understand closed models in view of certain figures[ apart from other reasons] for example the blue-cone diagram in the wiki picture:
http://en.wikipedia.org/wiki/Metric_expansion_of_space#Understanding_the_expansion_of_Universe
The circumference of the horizontal grid-line is finite at any particular instant of time -this is indicative of a closed model.

Last edited: Aug 1, 2011
2. Aug 1, 2011

### Anamitra

The post has been revised