Consider the Poincare half plane model for the n-dimensional hyperbolic space $\mathbb{H}^n$. 
$\mathbb{H}^n$ can be constructed out of $\mathbb{R}^{n-1}$ by crossing it with $(0;\infty)$ and equpping the product with the following metric:


Let $\gamma=(\gamma\_1,\gamma\_2)$ be a path $[0;t]\rightarrow \mathbb{R}^{n-1}\times (0;\infty)$ such that $\gamma\_1$ is parametrized by arclength. Then define its length to be 


$\int\_0^t\frac{1}{\gamma\_2(t')}\sqrt{1+\dot\gamma\_2(t')}$.


Then define the distance of two points as the infimum over the length of all paths connecting them. (Hopefully it is really a metric). 


So one could perform this construction on any geodesic metric space. Has this construction already been studied before?


Does this construction turn CAT(0) spaces into CAT(-1) spaces ?



Hyperbolizing geodesic spaces
This construction is called "parabolic cone" and indeed it turns CAT(0) spaces into CAT(-1). See [this paper](http://www.math.uiuc.edu/~sba/wp.pdf) of Alexander and Bishop.


