A relativistic race between a tortoise and hare (special relativity)

TortoiseHareRace-I

\documentclass[crop=true, border=10pt]{standalone}
\usepackage{comment}
\begin{comment} 
:Title: Tortoise and Hare - Variation 1
:Author: J A Miller (millerja@uah.edu), 2020/03/03

A race between a tortoise and hare. 

The distance between neighboring grid lines is one.

Reference:
T. Takeuchi (https://www1.phys.vt.edu/~takeuchi/relativity/practice/problem03.html) and "An Illustrated Guide to Relativity" ISBN: 9780521141000

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

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Problem to accompany figure
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\emph{Tortoise and Hare Race - Version 1} Suppose a tortoise and a hare have a race, but in this race they run in opposite directions to finish lines 5 units away. The animals both race at a speed of $3/5$ toward their finish lines, as shown. Analyze the race from the perspective of the tortoise, the hare, and a referee on the ground.

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Solution to problem
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\emph{Totoise and Hare Race - Version 1} 
The referee on the ground will say both cross their finish lines at the same time of $t = 5/(3/5) = 25/3$, as both points T and H lie along a line of simultaneity in the unprimed system. However, the hare arrives at his finish line at H, and in his frame point A is simultaneous with that event. The hare will say, because he knows special relativity, that the tortoise is only at A and has not crossed the line. 

The tortoise is not ignorant though, and he knows that in his frame his finish is simultaneous with the hare being at point B. So, he says the hare has not crossed before he does. 

So, they both make the same claim about the other, and are both correct, as is the referee. It looks like rules will have to be clarified some in the realm of relativistic sports.

\end{comment}



\usepackage{xcolor}
\usepackage{tikz}
\usetikzlibrary{arrows.meta}

% 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 tortoise and hare. Tortoise velocity is negative.
\newcommand*\vrel{0.6}
\newcommand*\gamrel{1.25}

% finish time in the primed frame (same as unprimed) = 
% ground finish time/\gamrel = (25/3)/(5/4) = 20/3
\newcommand*\tpfinish{6.667}
% finish time in ground frame = 5/\vrel
\newcommand*\tfinish{8.333}


\begin{tikzpicture}[scale=1,domain=-6:6]
    \tikzstyle{axisarrow} = [-{Latex[inset=0pt,length=8pt]}]

    % Draw the grid.
    \draw [cornflower!30,step=0.2,thin] (-6,-3) grid (6,9);
    \draw [cornflower!60,step=1.0,thin] (-6,-3) grid (6,9);

    % Clip all lines that would fall outside the grid, if any
    \clip(-6,-3) rectangle (6,9);
    
    % Draw unprimed axes
    \draw[thick,cornflower,axisarrow] (0,-3) -- (0,9);
    \node[inner sep=0pt] at (0.25,8.5) {$t$};
    \draw[thick,cornflower,axisarrow] (-6,0) -- (6,0);
    \node[inner sep=0pt] at (5.5,-0.25) {$x$};
     
    % Draw the worldlines of the two finish lines
    \draw[plum,thick,axisarrow] (5,0) -- (5,9);
    \draw[plum,thick,axisarrow] (-5,0) -- (-5,9);
    
    % Draw the primed (hare) coordinate system
	\draw[thick,axisarrow,scarlet] (0,0) -- (5.3,5.3/\vrel) node (tp) {};
	\node[inner sep=0pt] at (5.5,8.7) {$t^\prime$};
	\draw[thick,axisarrow,scarlet] (0,0) -- (6,\vrel*6) node (xp) {};
	\node[inner sep=0pt] at (5.8,3.85) {$x^\prime$};
		
	% Draw the double primed (tortoise) coordinate system
	\draw[thick,axisarrow,chameleon] (0,0) -- (-5.3,5.3/\vrel) node (tpp) {};
	\node[inner sep=0pt] at (-5.5,8.7) {$t^{\prime\prime}$};
	\draw[thick,axisarrow,chameleon] (0,0) -- (5,-\vrel*5) node (xpp) {};
	\node[inner sep=0pt] at (5.0,-2.6) {$x^{\prime\prime}$};
	
	% Label some points
	\node[circle,draw=cornflower!70,fill=cornflower!20,inner sep=2pt,label={right:H}](H) at (5,5/\vrel)  {};
    \node[circle,draw=cornflower!70,fill=cornflower!20,inner sep=2pt,label={left:T}](T) at (-5,5/\vrel)  {};

	% Lines of simultaneity    
    \begin{scope}
    
    	\clip (0,0) -- (5,5/\vrel) -- (-5,5/\vrel) --(0,0);
          
        % hare finish in hare frame    	
    	\draw[dashed, scarlet] plot (\x,\vrel*\x + \tpfinish/\gamrel); 
        % tortoise finish in tortoise frame
    	\draw[dashed, chameleon] plot (\x,-\vrel*\x + \tpfinish/\gamrel);
    	% both finishes in ground frame
    	\draw[dashed, cornflower] plot (\x,\tfinish );
  
    \end{scope}
    
    % More labeling
	\node[circle,draw=cornflower!70,fill=cornflower!20,inner sep=2pt,label={left:B}](B) at (2.35,2.35/\vrel)  {};
    \node[circle,draw=cornflower!70,fill=cornflower!20,inner sep=2pt,label={right:A}](A) at (-2.35,2.35/\vrel)  {};
    
    \node[inner sep=0pt,rotate=59] at (1.5,2.2) {Hare worldline};
    \node[inner sep=0pt,rotate=-59] at (-1.5,2.2) {Tortoise worldline};
    
    \node[inner sep=0pt,rotate=90] at (4.7,5.0) {Finish worldline};
    \node[inner sep=0pt,rotate=90] at (-4.7,5.0) {Finish worldline};

\end{tikzpicture}  

\end{document}