# de Sitter and anti de Sitter metric

Is the following correct for the distance $d$ from the origin $(0,0)$ to point $(t,x)$ in the 2-dimensional de-Sitter and anti de-Sitter spaces? Here, $t$ is time and the distance may be called the interval if you prefer.

1) de-Sitter: $\sinh{d}=\frac{\sqrt{t^2-x^2}}{\sqrt{1-(t^2-x^2)}}$

2) anti de-Sitter: $\sin{d}=\frac{\sqrt{t^2-x^2}}{\sqrt{1+(t^2-x^2)}}$

If the above is correct, it implies that the de-Sitter space is akin to the hyperbolic space in terms of the distance measure, whereas anti de-Sitter space is more like the elliptic space. So, then the de-Sitter space has negative curvature, whereas it is positive for the anti de-Sitter space.

EDIT:

I derived (1) and (2) from a formula for $\Psi_{pq}$ given on page 439 of "Perspectives on projective geometry" by Jurgen Richter-Gebert, taking p=(0,0,1), q=(x,t,1) and A=B=diag(1,-1,+1) for the de-Sitter space and A=B=diag(1,-1,-1) for anti de-Sitter space. I've just checked my derivation and with the final amendments the formulas are correct as written. What I want to know is whether my identifications of the spaces is correct. Perhaps, my (1) should be for anti de-Sitter space and (2) for de-Sitter. However, I read somewhere that de-Sitter space has hyperbolic distance measure (in Beltrami-Klein model I'm using) in accord with my choice. Oh, and I haven't linked those wikipedia pages. I think it's confusing and unnecessary to view de-Sitter and anti de-Sitter spaces as sub-manifolds of a higher-dimensional Minkowski space.

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You should give the metric form to fix coordinates, your answer is unanswerable definitively the way you ask it. The correct curvatures are positive for deSitter, and negative for AdS, the proper analogy is that deSitter is a sphere and AdS is the hyperbolic space. But the Minkowski signature confuses this. – Ron Maimon Jul 4 '12 at 7:55
Should I? If I told you the distance was $\sqrt{t^2-x^2}$, you wouldn't hesitate to identify it as Minkowski. I did say that $t$ was time, doesn't that fix the coordinates for you? In any case, one should be able to identify the metric if the distance formula is given. – Andrey Sokolov Jul 4 '12 at 8:58
You mean that the coordinates are homogenous, so that "x" and "t" mean $\Delta x$ and $\Delta t$. Then your formulas are not right at all. I thought you meant that you are looking at the distance from the origin to position r,t in some fixed coordinate system for deSitter and AdS. – Ron Maimon Jul 4 '12 at 9:03
What would the correct formulas be? – Andrey Sokolov Jul 4 '12 at 9:33
I've adjusted the formulas a little to avoid ambiguity. – Andrey Sokolov Jul 4 '12 at 10:25