Stereographic projection in de Sitter cosmological model

[ahmad2l]

We know stereographic projection is conformal but it isn't isometic and in general relativity it can not be used because in this theory general transformations must be isometric. But de sitter in his model (1917) used it (stereographic projection) to obtain metric in static coordinates. How can it be explained

 

[bcrowell]

in general relativity it can not be used because in this theory general transformations must be isometric.

Not true. In GR, any change of coordinates is legal as long as it's a diffeomorphism. There is no additional requirement.

In GR, when you do a change of coordinates, you change both the coordinates and the elements of the metric tensor itself. The result is that everything automatically still remains consistent. You're still describing the same spacetime, and it's still a solution to the field equations. All you've done is relabel everything. As a simple exampls, if you take Minkowski space and rescale all the coordinates by a factor of 1/2, and the metric was (+1,-1,-1,-1) in the original coordinates, then the metric in the new coordinates is (+4,-4,-4,-4).

When you talk about things like conformal transformations, you're not talking about a change of coordinates. It's more than a relabeling. For example, you can define a conformal transformation like $g\rightarrow \Omega^2 g$, while still leaving all the points fixed and describing them with the same coordinates. Or, alternatively, you can define conformal transformations that send points to other points, while leaving the metric the same. (This is how people typically think about conformal transformations in the complex plane.)

 

[ahmad2l]

thank you bcrowell. you saved me