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TikZ plot of the Andromeda "paradox" in special relativity
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\documentclass[crop=true, border=10pt]{standalone} | |
\usepackage{comment} | |
\begin{comment} | |
:Title: Attack from Andromeda | |
:Author: J A Miller (millerja@uah.edu), 2020/02/04 | |
The Andromeda "paradox". Spacetime diagram for aliens launching an attack on Earth, according to three observers: one on the ground in the same frame as the aliens, and two walking past the first in opposite directions with the same speed. | |
References: | |
1. https://en.wikipedia.org/wiki/Rietdijk%E2%80%93Putnam_argument | |
2. https://en.wikipedia.org/wiki/Eternalism_(philosophy_of_time) | |
Plot style and colors from McNees | |
https://gist.github.com/mcnees/45b9f53ad371c38ba6f3759df5880fb1 | |
\end{comment} | |
\usepackage{xcolor} | |
\usepackage{tikz} | |
\usetikzlibrary{arrows.meta} | |
\usetikzlibrary{decorations.markings} | |
\usepackage{pgfplots} | |
\pgfplotsset{compat=1.16} | |
% colors | |
\definecolor{plum}{rgb}{0.36078, 0.20784, 0.4} | |
\definecolor{chameleon}{rgb}{0.30588, 0.60392, 0.023529} | |
\definecolor{cornflower}{rgb}{0.12549, 0.29020, 0.52941} | |
\definecolor{scarlet}{rgb}{0.8, 0, 0} | |
\definecolor{brick}{rgb}{0.64314, 0, 0} | |
\definecolor{sunrise}{rgb}{0.80784, 0.36078, 0} | |
\definecolor{lightblue}{rgb}{0.15,0.35,0.75} | |
\begin{document} | |
% the speeds of the walking people | |
\newcommand*\vrel{0.25} | |
\pgfmathparse{1/sqrt(1-\vrel*\vrel)} \let\gammarel\pgfmathresult | |
\pgfmathparse{\vrel*\gammarel} \let\etarel\pgfmathresult | |
% some ancillary calculations | |
\pgfmathparse{1/\vrel - \vrel} \let\fac\pgfmathresult | |
\begin{tikzpicture}[scale=1,domain=-3:9] | |
\tikzstyle{axisarrow} = [-{Latex[inset=0pt,length=8pt]}] | |
% Draw the grid. | |
\draw [cornflower!30,step=0.2,thin] (-3,-3) grid (9,9); | |
\draw [cornflower!60,step=1.0,thin] (-3,-3) grid (9,9); | |
% Clip all lines that would fall outside the grid, if any | |
\clip(-3,-3) rectangle (9,9); | |
% Draw Axes | |
\draw[thick,cornflower,axisarrow] (0,-3) -- (0,9); | |
\node[inner sep=0pt] at (0.25,8.7) {$t$}; | |
\draw[thick,cornflower,axisarrow] (-3,0) -- (9,0); | |
\node[inner sep=0pt] at (8.7,-0.25) {$x$}; | |
% Draw the worldline of Andromeda | |
\draw[plum,thick,axisarrow] (8,-3) -- (8,9); | |
% Draw the primed coordinate system | |
\draw[thick,axisarrow,scarlet] (-3*\vrel,-3) -- (9*\vrel,9) node (tp) {}; | |
\node[inner sep=0pt] at (2.5,8.7) {$t^\prime$}; | |
\draw[thick,axisarrow,scarlet] (0,0) -- (9,\vrel*9) node (xp) {}; | |
\node[inner sep=0pt] at (8.7,1.9) {$x^\prime$}; | |
% Draw the double primed coordinate system (make \vrel negative) | |
\draw[thick,axisarrow,chameleon] (0,0) -- (-9*\vrel,9) node (tpp) {}; | |
\node[inner sep=0pt] at (-1.8,8.7) {$t^{\prime\prime}$}; | |
\draw[thick,axisarrow,chameleon] (0,0) -- (9,-9*\vrel) node (xpp) {}; | |
\node[inner sep=0pt] at (8.7,-2.5) {$x^{\prime\prime}$}; | |
% Lines of simultaneity | |
\draw[dashed,thick,scarlet] (8,0) -- (-8*\vrel/\fac,-8/\fac) {}; | |
\draw[dashed,thick,chameleon] (8,0) -- (-8*\vrel/\fac,8/\fac) {}; | |
% Label some points | |
\node[circle,draw=cornflower!70,fill=cornflower!20,inner sep=2pt,label={left:T}](T) at (-8*\vrel/\fac,-8/\fac) {}; | |
\node[circle,draw=cornflower!70,fill=cornflower!20,inner sep=2pt,label={left:A}](A) at (-8*\vrel/\fac,8/\fac) {}; | |
\node[circle,draw=cornflower!70,fill=cornflower!20,inner sep=2pt,label={above left:S}](S) at (0,0) {}; | |
\node[circle,draw=cornflower!70,fill=cornflower!20,inner sep=2pt,label={above right:L}](L) at (8,0) {}; | |
% Label the worldlines | |
\node[inner sep=0pt,rotate=90] at (7.7,5.2) {Andromeda worldline}; | |
\node[inner sep=0pt,rotate=90] at (-0.3,5.2) {Stationary observer worldline}; | |
\node[inner sep=0pt,rotate=76] at (1.6,5.2) {Toward Andromeda}; | |
\node[inner sep=0pt,rotate=-76] at (-1.6,5.2) {Away from Andromeda}; | |
\end{tikzpicture} | |
\end{document} |
Author
jamesamiller
commented
Feb 4, 2020
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