Plot the worldlines of stars in an accelerated rocket frame (special relativity)

StarWorldlinesInAccelFrame.tex

\documentclass[crop=true, border=10pt]{standalone}
\usepackage{comment}
\begin{comment} 
:Title: Motion of stars in the accelerated rocket frame
:Slug: Accelerated frame
:Tags:  special relativity
:Author: J A Miller, UAH Physics & Astronomy, millerja@uah.edu, 2020/05/12

Worldlines of stars in the accelerated rocket coordinates $\xi$ and $\eta$.

The view from the inertial frame of the stars is given here:
https://gist.github.com/jamesamiller/95ed21a498e4ebd797cf309350d7a188

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 can be defined by
\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 transformation is only defined for $|t| < x - x_0 + \frac{1}{\alpha}$.

We can plot the motion of objects in this frame. For example, suppose a rocket has uniform acceleration $\alpha$ starting from $x_0 = 0$ in the inertial frame. If a star is at some inertial location $x$, we can plot its worldline in the rocket frame.

We will observe it to approach and fall behind the rocket, and tend toward an event horizon at $-1/\alpha$.

\end{comment}




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

\usepackage{pgfplots}
\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}


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

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

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

% initial starting point for the rocket
\newcommand*{\xinit}{0}  % this is $x_0$

% inertial frame location of stars
\newcommand*{\xone}{8}
\pgfmathsetmacro{\tmaxone}{(\xone-\xinit+\accelinv)}

\newcommand*{\xtwo}{5}
\pgfmathsetmacro{\tmaxtwo}{(\xtwo-\xinit+\accelinv)}

\newcommand*{\xthree}{2}
\pgfmathsetmacro{\tmaxthree}{(\xthree-\xinit+\accelinv)}

% define the arctanh
\pgfkeys{/pgf/declare function={arctanh(\x) = 0.5*(ln(1+\x)-ln(1-\x));}}


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

\begin{document}

\begin{tikzpicture}[scale=1,domain=-2:9]
	\tikzstyle{axisarrow} = [-{Latex[inset=0pt,length=7pt]}]

	% Draw the background grid.
	\draw [cornflower!30,step=0.2,thin] (-2,-1) grid (9,5);
	\draw [cornflower!60,step=1.0,thin] (-2,-1) grid (9,5);

	% Clip everything that falls outside the grid
	\clip(-2,-1) rectangle (9,5);
	
	% Draw Axes
	\draw[thick,axisarrow] (0,-1) -- (0,5);
	\node[right,inner sep=0pt] at (0.2,4.7) {$\eta$ (Worldline of the Rocket)};
	\draw[thick,axisarrow] (-2,0) -- (9,0);
	\node[inner sep=0pt] at (8.5,-0.3) {$\xi$};
	
	% Event horizon
	\draw[thick,dashed,scarlet] (-\accelinv,-1) -- (-\accelinv,5);
	\node[rotate=90,inner sep=0pt] at (-\accelinv-0.2,2.0) {Event Horizon};
	
	% Draw the worldlines of the stars	
	\foreach \tmax in {\tmaxone,\tmaxtwo,\tmaxthree}
	{
		\draw[domain=0:\tmax-0.0001,smooth,variable=\t,plum,thick,samples=1000,axisarrow] 
		   plot ({(sqrt(\tmax*\tmax-\t*\t)-\accelinv)},{(\accelinv*arctanh(\t/\tmax))});
	}
	
	% Some more labels	
	\node[right,inner sep=0pt] at (4,1.4) {Worldlines of the Stars};
	\node[above right,inner sep=0pt] at (6,4) {$\alpha = 1, x_0 = 0$};

	
\end{tikzpicture}
			

\end{document}

The plot using the above parameters.