Penrose diagrams of Minkowski and Schwarzschild spacetime

Penrose diagram of Minkowski and Schwarzschild metrics to illustrate the causal structure of different spacetime geometries. Lightlike worldlines remain at 45 degrees as indicated by the photons and light cones, i.e. the diagrams are conformal. The grid indicates lines of constant r and t. Different types of infinities are indicated: lightlike, timelike, spacelike, postive/negative future and past. For more related figures, please see Relativity category.

Simple coordinate transformation to rotate the axes by 45°.

Transformation to Penrose coordinates for Minkowski space such that we can show infinity as a boundary:

Penrose diagram with a particle worldliness and a light cone with 45° angles. The lines of constant time or constant space are equidistant in spacetime:

Penrose diagram with full labeling (note the lines of constant time or constant space are not equidistant in xt spacetime, but equidistant in uv spacetime instead):

Penrose diagram for radius r between 0 and infinity (instead of x):

Transformation of Kruskal-Szekeres coordinates to show the geometry of a Schwarzschild black hole:

Penrose diagram for Schwarzschild black hole. It is derived via Kruskal-Szekeres coordinates above. The horizon is at r = 2GM (v = ±u), singularity at r = 0:

Extendend Penrose diagram for Schwarzschild black hole:

Edit and compile if you like:

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% Author: Izaak Neutelings (September 2021)
% Inspiration:
%   https://jila.colorado.edu/~ajsh/insidebh/penrose.html
%   https://tex.stackexchange.com/questions/99124/how-to-draw-penrose-diagrams-with-tikz
%   coordinates: https://arxiv.org/pdf/physics/0611033.pdf
%   https://arxiv.org/pdf/0711.0873.pdf
\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{angles,quotes} % for pic (angle labels)
\usetikzlibrary{arrows.meta} % for arrow size
\contourlength{1.4pt}
\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]
\tikzstyle{world line t}=[mypurple!60,line width=0.4]
\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=2,segment length=6.17}]
\tikzset{declare function={%
  penrose(\x,\c)  = {\fpeval{2/pi*atan( (sqrt((1+tan(\x)^2)^2+4*\c*\c*tan(\x)^2)-1-tan(\x)^2) /(2*\c*tan(\x)^2) )}};%
 
 
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