Kruskal diagrams of Schwarzschild black holes

Kruskal diagram of Schwarzschild spacetime with light cones to illustrate the causal structure. Using Kruskal-Szekeres coordinates, lightlike worldlines remain at 45 degrees, i.e. the diagrams are conformal. The grid indicates lines of constant r and t. The horizon is at r = 2GM (V = ±U), singularity at r = 0. For more related figures, please see Relativity category.

Equations for the coordinate transformations:relativity_kruskal_diagram-001.png
Kurskal diagram of a Schwarzschild black holes:relativity_kruskal_diagram-002.png
Kurskal diagram of a Schwarzschild black holes with equidistant world lines:relativity_kruskal_diagram-003.png
Kurskal diagram with a particle falling inside the black hole (note the light cones remain at a 45° angle):relativity_kruskal_diagram-004.png

Edit and compile if you like:

% Author: Izaak Neutelings (October 2021)
% Inspiration:
% "Spacetime and Geometry: An Introduction to General Relativity", Sean M. Carroll
% "Gravity: An Introduction to Einstein's General Relativity", James B. Hartle
\documentclass[border=3pt,tikz]{standalone}
\usepackage{tikz}
\usepackage{amsmath} % for \text
\usepackage{mathrsfs} % for \mathscr
\usepackage{xfp} % higher precision (16 digits?)
\usepackage[outline]{contour} % glow around text
\usetikzlibrary{decorations.markings,decorations.pathmorphing}
\usetikzlibrary{arrows.meta} % for arrow size
\contourlength{1.2pt}
\newcommand{\calI}{\mathscr{I}} %\mathcal
\tikzset{>=latex} % for LaTeX arrow head
\colorlet{myred}{red!80!black}
\colorlet{myblue}{blue!80!black}
\colorlet{mygreen}{green!80!black}
\colorlet{mydarkred}{red!50!black}
\colorlet{mydarkblue}{blue!50!black}
\colorlet{mylightblue}{mydarkblue!6}
\colorlet{mypurple}{blue!40!red!80!black}
\colorlet{mydarkpurple}{blue!40!red!50!black}
\colorlet{mylightpurple}{mydarkpurple!80!red!6}
\colorlet{myorange}{orange!40!yellow!95!black}
\tikzstyle{cone}=[mydarkblue,line width=0.2,top color=blue!60!black!30,
bottom color=blue!60!black!50!red!30,shading angle=60,fill opacity=0.9]
\tikzstyle{cone back}=[mydarkblue,line width=0.1,dash pattern=on 1pt off 1pt]
\tikzstyle{world line}=[myblue!60,line width=0.4,shorten <=-2mm,shorten >=-2mm]
\tikzstyle{world line t}=[mypurple!60,line width=0.4,shorten <=-2mm,shorten >=-2mm]
\tikzstyle{particle}=[mygreen,line width=0.5]
\tikzstyle{photon}=[-{Latex[length=4,width=3]},myorange,line width=0.4,decorate,
decoration={snake,amplitude=0.9,segment length=4,post length=3.8}]
\tikzstyle{singularity}=[myred,line width=0.6,decorate,
decoration={zigzag,amplitude=1.8,segment length=5}]
\tikzset{declare function={% Kruskal-Szekeres coordinates
sing(\x) = {\fpeval{sqrt(\x*\x+1)}};%
rstar(\c) = {\fpeval{(\c/2-1)*exp(\c/2)}};%
kruskalu(\x,\c) = {\fpeval{sqrt(\x*\x+(\c/2-1)*exp(\c/2))}};%
kruskalv(\x,\c) = {\fpeval{sqrt(\x*\x-(\c/2-1)*exp(\c/2))}};%
 
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Click to download: relativity_kruskal_diagram.texrelativity_kruskal_diagram.pdf
Open in Overleaf: relativity_kruskal_diagram.tex.

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