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:
Kurskal diagram of a Schwarzschild black holes:
Kurskal diagram of a Schwarzschild black holes with equidistant world lines:
Kurskal diagram with a particle falling inside the black hole (note the light cones remain at a 45° angle):
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 coordinatessing(\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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