Example: Excised, Horizon-Penetrating Coordinates for Black Hole Spacetime

Published 2018-03-20 | Author: Jonah Miller

This is a "Penrose" or "conformal" diagram for a black hole spacetime in general relativity. In these diagrams, the top of the plot is the future and the bottom of the plot is the past. Light travels along lines at 45 degrees off of the y-axis.

In general relativity, spacetime can be cut up into slices in many different ways. This plot shows two popular ones: Schwarzschild coordinates and Kerr-Schild coordinates. It also shows a popular way of avoiding the black hole singularity via "excision" where the spacetime is terminated at some finite radius behind the event horizon.

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Excised, Horizon-Penetrating Coordinates for Black Hole Spacetime

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% Horizon penetrating coordinates (vs. Schwarzschild coordinates)
% for a black hole spacetime, with excision
% Author: Jonah Miller
\documentclass[tikz,border=10pt]{standalone}
\usetikzlibrary{decorations.pathmorphing}

\tikzset{zigzag/.style={decorate, decoration=zigzag}}
\def \L {2.}

% fix for bug in color.sty
% see: http://tex.stackexchange.com/questions/274524/definecolorset-of-xcolor-problem-with-color-values-starting-with-f
\makeatletter
\def\@hex@@Hex#1%
 {\if a#1A\else \if b#1B\else \if c#1C\else \if d#1D\else
  \if e#1E\else \if f#1F\else #1\fi\fi\fi\fi\fi\fi \@hex@Hex}
\makeatother

% Define a prettier green
\definecolor{darkgreen}{HTML}{006622}

\begin{document}
\begin{tikzpicture}

  % causal diamond
  \draw[thick,red,zigzag] (-\L,\L) coordinate(stl) -- (\L,\L) coordinate (str);
  \draw[thick,black] (\L,-\L) coordinate (sbr)
    -- (0,0) coordinate (bif) -- (stl);
  \draw[thick,black,fill=blue, fill opacity=0.2,text opacity=1] 
    (bif) -- (str) -- (2*\L,0) node[right] (io) {$i^0$} -- (sbr);

  % null labels
  \draw[black] (1.4*\L,0.7*\L) node[right]  (scrip) {$\mathcal{I}^+$}
               (1.5*\L,-0.6*\L) node[right] (scrip) {$\mathcal{I}^-$}
               (0.2*\L,-0.6*\L) node[right] (scrip) {$\mathcal{H}^-$}
               (0.5*\L,0.85*\L) node[right] (scrip) {$\mathcal{H}^+$};

  % singularity label
  \draw[thick,red,<-] (0,1.05*\L) 
    -- (0,1.2*\L) node[above] {\color{red} singularity};
  % Scwharzschild surface
  \draw[thick,blue] (bif) .. controls (1.*\L,-0.35*\L) .. (2*\L,0);
  \draw[thick,blue,<-] (1.75*\L,-0.1*\L)  -- (1.9*\L,-0.5*\L)
    -- (2*\L,-0.5*\L) node[right,align=left]
    {$t=$ constant\\in Schwarzschild\\coordinates};
  % excision surface
  \draw[thick,dashed,red] (-0.3*\L,0.3*\L) -- (0.4*\L,\L);
  \draw[thick,red,<-] (-0.33*\L,0.3*\L) 
    -- (-0.5*\L,0.26*\L) node[left,align=right] {excision\\surface};
  % Kerr-Schild surface
  \draw[darkgreen,thick] (0.325*\L,0.325*\L) .. controls (\L,0) .. (2*\L,0);
  \draw[darkgreen,dashed,thick] (0.325*\L,0.325*\L) -- (-0.051*\L,0.5*\L);
  % Kerr-Schild label
  \draw[darkgreen,thick,<-] (0.95*\L,0.15*\L) -- (1.2*\L,0.5*\L)
    -- (2*\L,0.5*\L) node[right,align=left]
    {$\tau=$ constant\\in Kerr-Schild\\coordinates};
\end{tikzpicture}
\end{document}

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