tikz-3d/pgfplots: How to draw/plot curved spacetime?

Since I try to write a short introduction to tensors for physics students, I mentioned the Einstein’s field equations (as a motivation).
Therefore, I wanted to create a tikzpicture/pgfplots of curved spacetime.
First, I created a 3d plot that looks familiar with the most common picture of curved spacetime

documentclass[tikz]{standalone}
usepackage{pgfplots}
pgfplotsset{compat = newest}

begin{document}

begin{tikzpicture}
  begin{axis}[
    unit vector ratio*=1 1 1,
    hide axis = true,
    x domain = -1.5:1.5,
    zmin = -2.5,
    zmax = 2.5
    ]    
    addplot3 [
        surf, 
        faceted color = lightgray,
        fill = white, 
        line width = 0.1pt,
        samples = 50] {-1.5*exp(-0.5*(x^2+y^2))};

  end{axis}
end{tikzpicture}

end{document}

which gives me

But actually, this is not the best representation for curved spacetime, since this picture only shows the curvature in one direction (here, along the z-axis).
A much better visualization would be something
like this (YouTube Animation) or this.

I found that something like usepgfmodule{nonlineartransformations} might help here, but I am still a beginner when it comes to tikz/pgfplots.

Thank you in advance for all your answers!