tikz-3d/pgfplots：如何绘制/绘制弯曲时空？

Question

由于我尝试为物理专业的学生写一篇关于张量的简短介绍，我提到了爱因斯坦的场方程（作为动机）。因此，我想创建一个弯曲时空的 tikzpicture/pgfplots。首先，我创建了一个 3d 图，它看起来很像最常见的弯曲时空图

\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}

这让我

但实际上，这并不是弯曲时空的最佳表示，因为这幅图只显示了一个方向的曲率（这里是沿着 z 轴）。更好的可视化应该是 像这样（YouTube 动画）或这。

我发现类似的东西\usepgfmodule{nonlineartransformations}可能会有帮助，但是当谈到 tikz/pgfplots 时，我仍然是个初学者。

提前感谢您的所有回答！

Answer 1

两种可能性：

• 几乎纯钛钾Z one（只需要perspective的库3d view）：

\documentclass[tikz,border=1.618mm]{standalone}
\usetikzlibrary{perspective}

\begin{document}
\begin{tikzpicture}[3d view={120}{30},samples=51,domain=-2.5:2.5]
\foreach\i in {-2.5,-2.25,...,2.5}
{% curved spacetime
  \draw[gray] plot (\x,\i,{2.5-0.5*exp(-0.75*(\x*\x+\i*\i))});
  \draw[gray] plot (\i,\x,{2.5-0.5*exp(-0.75*(\x*\x+\i*\i))});
  \draw[gray] plot (\x,{2.5-0.5*exp(-0.75*(\x*\x+\i*\i))},\i);
  \draw[gray] plot (\i,{2.5-0.5*exp(-0.75*(\x*\x+\i*\i))},\x);
  \draw[gray] plot ({2.5-0.5*exp(-0.75*(\x*\x+\i*\i))},\x,\i);
  \draw[gray] plot ({2.5-0.5*exp(-0.75*(\x*\x+\i*\i))},\i,\x);
}% axes
\draw[thick,-latex] (2,0,0) -- (5,0,0) node[left]  {$x$};
\draw[thick,-latex] (0,2,0) -- (0,5,0) node[right] {$y$};
\draw[thick,-latex] (0,0,2) -- (0,0,5) node[above] {$z$};
\end{tikzpicture}
\end{document}

• 还有一个pgfplots版本：

\documentclass[border=1.618mm]{standalone}
\usepackage{pgfplots}
\pgfplotsset{compat=1.18}

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