Understanding de Sitter Space: FRW or Hyperboloid?

[ramparts]

Hey all - I'm trying to get a better qualitative understanding of de Sitter space, and I'm a bit confused about the de Sitter universe. I've always seen it mathematically as a hyperboloid embedded in a higher-dimensional Minkowski space (with some annoying metric I've forgotten) but I've read (alright, fine, on Wikipedia :P) that a patch of de Sitter universe can be expressed as an FRW cosmology with a(t) = exp(Ht) - So... under that description, why is the de Sitter metric not just an FRW metric with that scale factor?

 

[George Jones]

It's a matter of choosing the right coordinates. See

http://books.google.com/books?id=1TXO7GmwZFgC&pg=PP1&dq=mukhanov#PPA29,M1

from the bottom of page 29 to equation (1.98) on page 32.

 

[Mosis]

I always did understand de Sitter space through its cosmological description - a flat, vacuum filled universe that undergoes exponential expansion (equivalently, constant Hubble parameter). What about this description is unsatisfactory?

 

[ramparts]

Thanks George! Don't have my Wald or Carroll with me so I couldn't look it up...

The problem was that I was used to seeing the de Sitter metric in a form similar to eqn 1.94 in George's link, so I was a bit confused by the description of it as a FRW cosmology with a=exp(Ht). The coordinate transformation makes sense (or will, when I think about it some more .