Trying to plot a wormhole; getting bad results

Question

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

Answer 1

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]