I have been attempting to plot a “drainhole” wormhole with the metric $ ds^2 = dt^2-dr^2-(r^2+a^2)(d\theta^2+Sin[\theta]^2d\phi^2)$ , I set $ a=0$ and plotted in mathematica:

RevolutionPlot3D[-1 - x^2 - y^2 -   1 - (1 + x^2 + y^2 + 1) (1 - 1/(x^2 + y^2 + 1)), {x, -1, 1},  {y, -\[Pi]/4, \[Pi]/4}, Boxed -> False, Axes -> False,   Ticks -> None, PlotStyle -> Opacity[.1], ImageSize -> {600, 600},   Mesh -> 20]   

After converting from spherical to cartesian coordinates.

Which lacks the characteristic narrow throat and dual open mouths, is this an issue with invalid assumptions with the metric, a plotting mistake, or is this the actual form of the wormhole?

I did set the redshift function equal to 1 and the shape function equal to 0 to get a simplified metric. This may violate the constraints set out here: https://arxiv.org/abs/1506.04685