Lorentz Contraction

fig-lengthcontraction.tex

\documentclass[crop=true, border=10pt]{standalone}
\usepackage{comment}
\begin{comment} 
:Title: Length Contraction
:Author: J A Miller (millerja@uah.edu), 2021/02/13

Length contraction illustration

Plot style from
https://gist.github.com/mcnees/45b9f53ad371c38ba6f3759df5880fb1

---------------------------------------------------------------------
Figure caption
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\caption{
Spacetime diagram for a train with velocity in the unprimed frame $v = 3/5$. \textit{Red arrows:} The primed frame of the train. \textit{Blue arrows:} The unprimed frame of the ground. \textit{Thick lines:} The train length as measured in the two frames.
              } 

\end{comment}



\usepackage{xcolor}

\usepackage{tikz}
\usetikzlibrary{arrows.meta}
\usetikzlibrary{decorations.markings}

\usepackage{pgfplots}
\pgfplotsset{compat=1.13}


% 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}

% Define the boost
\newcommand*\vrel{0.6}           % velocity of primed frame
\newcommand*\gamrel{1.25}    % gamma of primed frame
\newcommand*\etarel{0.75}      % their product


\begin{tikzpicture}[scale=1,domain=0:3]
    \tikzstyle{axisarrow} = [-{Latex[inset=0pt,length=7pt]}]

    % Draw the grid.
    \draw [cornflower!30,step=0.2,thin] (-1,-1) grid (8,8);
    \draw [cornflower!60,step=1.0,thin] (-1,-1) grid (8,8);

    % Clip all lines that would fall outside the grid, if any
    %\clip(-1,-1) rectangle (10,8);
    
    % Draw the worldlines of the front and back of the train
    \draw[scarlet,thick,axisarrow] (0,0) -- (\vrel*8,8);
    \node[inner sep=0pt] at (4.9,7.6) {$t^\prime$};
    \draw[scarlet,thick,axisarrow] (3,0) -- (\vrel*8+3,8);
    
    % Draw x primed axes
    \draw[scarlet,axisarrow] (0,0) -- (8,\vrel*8);
    \node[inner sep=0pt] at (7.7,4.4) {$x^\prime$};
    
    % Draw train length in unprimed frame
    \draw[line width=2,cornflower] (0,0) -- (3,0);
     \draw[line width=2,cornflower] (3,5) -- (6,5);
     
    % Draw train length in primed frame
    \draw[line width=2,scarlet] (0,0) -- (5*15/16,3*15/16);
    \draw[line width=2,scarlet] (3,5) -- (5*15/16+3,3*15/16+5);
    
    % Draw unprimed axes. Do last to overprint.
    \draw[thick,cornflower,axisarrow] (0,-1) -- (0,8);
    \node[inner sep=0pt] at (0.25,7.5) {$t$};
    \draw[thick,cornflower,axisarrow] (-1,0) -- (8,0);
    \node[inner sep=0pt] at (7.5,-0.25) {$x$};
    
	% Labels    
    \node[inner sep=0pt,rotate=59] at (4.0,1.2) {Front of the train};
    \node[inner sep=0pt,rotate=59] at (1.4,3.0) {Back of the train};
    \node[inner sep=0pt,rotate=32] at (2.5,1.9) {Length in train frame};
     \node[inner sep=0pt] at (1.5,-0.3) {Length in track frame};
     
     \node[circle,draw=cornflower!70,fill=cornflower!20,inner sep=2pt,label={right:A}](A) at (\gamrel*\gamrel*3,\vrel*\gamrel*\gamrel*3)  {};
 
\end{tikzpicture}  

\end{document}