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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]  

Wormhole

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

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1 Answer 1

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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]

wormhole

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    $\begingroup$ Physically speaking, you cannot set the shape function as 0, since b(r) essentially determines the shape of the surface defined by the wormhole metric. If you do not explicitly specify the shape function, you are essentially plotting a trivial spherically symmetric geometry discarding coordinate singularities. Regarding the plotting end of the problem, I agree with the answer provided. $\endgroup$
    – madmiKe
    Dec 13, 2021 at 11:12

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