Draws the accelerated coordinates $\xi$ and $\eta$ in special relativity on the inertial $x$-$t$ system.

AcceleratedCoordinates.tex

\documentclass[crop=true, border=10pt]{standalone}
\usepackage{comment}
\begin{comment} 
:Title: Accelerated coordinate system
:Slug: Accelerated coordinates
:Tags:  special relativity
:Author: J A Miller, UAH Physics & Astronomy, millerja@uah.edu, 2020/05/12

Plot of accelerated coordinates $\xi$ and $\eta$ on an $x$-$t$ inertial spacetime.

Basic plot style and colors from:
    https://gist.github.com/mcnees/45b9f53ad371c38ba6f3759df5880fb1

---------------------------------------------------------------------
Background
---------------------------------------------------------------------

The equations of motion for an object undergoing uniform proper acceleration $\alpha$ are
\begin{equation}
    \begin{split}
		x &= \frac{1}{\alpha} \cosh(\alpha \tau) - \frac{1}{\alpha} + x_0 \\
		t &= \frac{1}{\alpha} \sinh(\alpha \tau),
    \end{split}
\end{equation}
where $\tau$ is the proper time, and $x_0$ is the starting location.

Accelerated coordinates are
\begin{equation}
    \begin{split}
		\xi &= \qty[ \qty( x - x_0 + \frac{1}{\alpha} )^2 - t^2  ]^{1/2} - \frac{1}{\alpha} \\
		\eta &= \frac{1}{\alpha} \tanh^{-1} \frac{t}{x - x_0 + 1/\alpha}.
    \end{split}
\end{equation}
When you put the equations of motion into these coordinates, you get $\xi = 0$ and $\eta = \tau$, as you would want. The object stays stationary at $\xi=0$ as the coordinate system rides with it. The quantity $\eta$ is the time coordinate, and equals the proper time.

The inverse transformation is
\begin{equation}
    \begin{split}
		x &= \qty( \frac{1}{\alpha} + \xi ) \cosh(\alpha \eta) - \frac{1}{\alpha} + x_0 \\
		t &= \qty( \frac{1}{\alpha} + \xi ) \sinh(\alpha \eta),
    \end{split}
\end{equation}
These are used to plot lines of constant $\xi$ and $\eta$ on an $x$-$t$ space.

\end{comment}



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

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

\usepackage{xcolor}
% 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}

% ----------------------- user specifications ------------------------

% maximum proper time for the lines of constant $\xi$
\newcommand{\etamax}{3.0}

% proper acceleration
\newcommand{\accel}{1.0}

% --------------some calculations and another spec -----------------

% inverse of acceleration
\pgfmathsetmacro{\accelinv}{1/\accel}

% initial starting point
\newcommand{\xinit}{\accelinv}  % $x_0$


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

\begin{tikzpicture}[scale=0.75,domain=-1:16]
	\tikzstyle{axisarrow} = [-{Latex[inset=0pt,length=7pt]}]

	% Draw the background grid.
	\draw [cornflower!30,step=0.2,thin] (-1,-12) grid (18,12);
	\draw [cornflower!60,step=1.0,thin] (-1,-12) grid (18,12);
	
	% Unprimed axes
	\draw[cornflower,thick,axisarrow] (0,-12) -- (0,12);
	\node[inner sep=0pt] at (0.5,11.7) {$t$};
	\draw[cornflower,thick,axisarrow] (-1,0) -- (16,0);
	\node[inner sep=0pt] at (15.5,-0.4) {$x$};

	% Clip everything that falls outside the grid
	\begin{scope}	% start SCOPE
	\clip(-1,-12) rectangle (16,12);
	
	% Light rays from the origin
	\draw[chameleon,thick] (0,0) -- (12,12);
	\draw[chameleon,thick] (0,0) -- (12,-12);
	
	% Plot lines of constant $\xi$
	\foreach \xi in {0,1,2,3,...,14}
	{
		\draw[domain=0:\etamax,smooth,variable=\eta,scarlet,thick,samples=20] 
		  plot ({(\accelinv+\xi)*cosh(\accel*\eta)-\accelinv+\xinit},{(\accelinv+\xi)*sinh(\accel*\eta)});
		\draw[domain=0:\etamax,smooth,variable=\eta,scarlet,thick,samples=100] 
		  plot ({(\accelinv+\xi)*cosh(\accel*\eta)-\accelinv+\xinit},{-(\accelinv+\xi)*sinh(\accel*\eta)});
	}
	
	% Plot lines of constant $\eta$
	\foreach \eta in {1/8,1/4,0.37,1/2,0.7,0.9,1.2}
	{
		\draw[domain=0:16,smooth,variable=\x,scarlet,thick,samples=50] 
		  plot ({\x},{(\x - \xinit + \accelinv)*tanh(\accel*\eta});
		  		\draw[domain=0:16,smooth,variable=\x,scarlet,thick,samples=50] 
		  plot ({\x},{-(\x - \xinit + \accelinv)*tanh(\accel*\eta});
	}
	
	\end{scope}	% end SCOPE
	
	
	% Labeling
	\node[inner sep=0pt,rotate=0,label={right:$\eta = 1/8$}] at (16.1,2.0) {};
	\node[inner sep=0pt,rotate=0,label={right:$\eta = 1/4$}] at (16.1,3.9) {};
	\node[inner sep=0pt,rotate=0,label={right:$\eta = 0.37$}] at (16.1,5.6) {};
	\node[inner sep=0pt,rotate=0,label={right:$\eta = 1/2$}] at (16.1,7.4) {};
	\node[inner sep=0pt,rotate=0,label={right:$\eta = 0.7$}] at (16.1,9.6) {};
	\node[inner sep=0pt,rotate=0,label={right:$\eta = 0.9$}] at (16.1,11.4) {};
	
	\node[inner sep=0pt,rotate=0,label={above right:$\alpha = 1, x_0 = 1/\alpha$}] at (2,10.0) {};
	
	
\end{tikzpicture}
			

\end{document}

An example.