Can Flamm's Paraboloid be described by Cartesian equations?

[Victor Escudero]

I would like to know if there exist any equations in Cartesian coordinates that describe the shape in three dimensions of Flamm´s paraboloid and if you can write them to me because I have searched for them but I can’t find any specific equations of what I want. I suppose that this shape would depend to the mass and the density (or radius) of the spherical object that produces the deformation of space.

 

[Ibix]

If you work in cylindrical polars and identify the $r$ and $\phi$ coordinates with the Schwarzschild $r$ and $\phi$ coordinates then circles around the origin are the right size automatically. That leaves you to pick a function $z (r)$ so that the distance-squared in the surface associated with a small change in $r$ (Pythagoras) is the same as the distance-squared in the Schwarzschild metric associated with a small change in $r$ (read from the metric). A little algebra should get you an expression for $dz/dr$ which should integrate easily enough.