Two coordinate systems with a 1-D boost in special relativity.

Coordinate Systems.tex

\documentclass[crop=true, border=10pt]{standalone}
\usepackage{comment}
\begin{comment} 
:Title: Standard coordinate systems
:Author: 	    J A Miller, UAH Physics & Astronomy, millerja@uah.edu
                   2020/07/01

The basic scenario of a primed coordinate system moving along the $x$ axis with a speed $v$

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Figure caption
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\caption{Our standard setup for the unprimed and primed reference frames. The primed frame moves along the $x$ axis with a velocity $v$ (which can be $>0$ or $<0$) relative to the unprimed frame, and the orientation of all axes is the same as in the unprimed frame.} 

\end{comment}



\usepackage{tikz}
\usetikzlibrary{arrows.meta,calc,intersections}

\usepackage{pgfplots}
\pgfplotsset{compat=1.16}

\usepackage{xcolor}
\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}


% ----------------------- parameter specifications ------------------------

% graph boundaries
\newcommand*{\xa}{-1}  
\newcommand*{\ya}{-1}
\newcommand*{\xb}{7}  
\newcommand*{\yb}{7}
\newcommand*{\spacing}{1}

% x and y scales
\newcommand*{\xscale}{1cm} 
\newcommand*{\yscale}{1cm}  

% some calculations now ...


% ------------------------ other functions ------------------------------



% ------------------------ begin figure ------------------------------

\begin{document}

\begin{tikzpicture}[
	scale=1,
	%domain=\xa:\xb,
	x=\xscale,y=\yscale, 
	axisarrow/.style=-{Latex[inset=0pt,length=10pt]},
	velocityarrow/.style=-{Latex[inset=0pt,length=20pt,fill=lightblue]},
	minor gridlines/.style={cornflower!30,step=0.2,thin},
	major gridlines/.style={cornflower!60,step=1.0,thin},
	axes/.style={cornflower,thick,axisarrow},      % unprimed axes
	primed axes/.style={scarlet,thick,axisarrow},  % primed axes
	primed major gridlines/.style={scarlet!60,thin},
	double primed axes/.style={chameleon,thick,axisarrow},  % double primed axes
	double primed major gridlines/.style={chameleon!60,thin},
	LOS/.style={cornflower,thick,dashed},  % line of simultaneity in unprimed frame
	LOS primed/.style={scarlet,thick,dashed},  % line of simultaneity in primed frame
	LOS double primed/.style={chameleon,thick,dashed},  % line of simultaneity in primed frame
	light ray/.style={sunrise,thick,axisarrow}, 
	textnode/.style={inner sep=0pt},  % for a text only node
	dot/.style={circle,draw=scarlet!70,fill=scarlet!20,inner sep=1.5pt} 
    ]
    
   % In case we need to increase the font size because of a large diagram
   \tikzset{every node/.style={scale=1.2}};
 
   % some coordinates   
   \coordinate (O) at (0,0); 
    
    % Draw the grid for the unprimed frame
    %\draw [minor gridlines] (\xa,\ya) grid (\xb,\yb);
    \draw [major gridlines] (\xa,\ya) grid (\xb,\yb);
    
    % Draw unprimed axes
    \draw[axes,name path=taxis] (0,\ya) -- (0,\yb);
    \node[textnode] at (0.3,\yb-0.5) {$t$};
    \draw[axes,name path=xaxis] (\xa,0) -- (\xb,0);
    \node[textnode] at (\xb-0.5,-0.3) {$x$};
    
    % Draw the grid for the primed frame
    %\draw [minor gridlines] (\xa,\ya) grid (\xb,\yb);
    \draw [primed major gridlines] (\xb+\spacing,\ya) grid (\xb+\spacing +\xb-\xa,\yb);
    
    % Draw primed axes
    \draw[primed axes,name path=tpaxis] (\xb+\spacing+1,\ya) -- (\xb+\spacing+1,\yb);
    \node[textnode] at (\xb+\spacing+1+0.3,\yb-0.5) {$t^\prime$};
    \draw[primed axes,name path=xpaxis] (\xb+\spacing,0) -- (\xb+\spacing +\xb-\xa,0);
    \node[textnode] at (\xb+\spacing +\xb-\xa-0.5,-0.3) {$x^\prime$};  
    
    \draw[velocityarrow,ultra thick] ( \xb+\spacing+2,1.5)-- ( \xb+\spacing+5,1.5) node[textnode,label={right:\Large $v$}] (v) {};
    \node[textnode,label={right: Unprimed frame}] at (2.2,6.2) {};
    \node[textnode,label={right: Primed frame}] at (11.2,6.2) {};
    

\end{tikzpicture}  

\end{document}