# Trying to plot a wormhole; getting bad results

I have been attempting to plot a "drainhole" wormhole with the metric

$\qquad ds^2 = dt^2-dr^2-(r^2+a^2)(d\theta^2+Sin[\theta]^2d\phi^2)$

I set $a=0$, and after converting from spherical to cartesian coordinates, made the following plot:

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

Providing your math is correct, I see two problems:

1. You need to do the revolution with a single slice through your surface

-(((2 + x^2 + y^2) (1 + 2 x^2 + 2 y^2))/(1 + x^2 + y^2))

because RevolutionPlot3D plots the revolution of a curve (1-manifold) about a coordinate axis.

2. You need to revolve about the x-axis, not the z-axis (which is the default for RevolutionPlot3D

Making these correction gives

RevolutionPlot3D[
Evaluate @
Simplify[-(((2 + x^2 + y^2) (1 + 2 x^2 + 2 y^2))/(1 + x^2 + y^2)) /. y -> 0],
{x, -1, 1},
Mesh -> 20,
RevolutionAxis -> "X",
SphericalRegion -> True,
BoxRatios -> {1, 1, 1},
Boxed -> False,
Axes -> False,
PlotStyle -> Opacity[.1],
ImageSize -> Automatic] 