The worldlines of stars and a rocket in the inertial star frame (special relativity)

StarWorldlinesInInertialFrame.tex

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

Worldlines of stars and a rocket in the inertial $x$-$t$ frame, with light rays from one of
the stars to the rocket worldline.

This figure complements that of the view from the accelerated rocket frame:
https://gist.github.com/jamesamiller/b2609beb3b353f4dc50a3a42ad152d12

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.

\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}

% maximum proper time for the rocket motion
\newcommand*{\taumax}{2.95}

% 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}
\newcommand*{\xtwo}{5}
\newcommand*{\xthree}{2}

% Set the point style for the proper time ticks
\tikzset{
	taudot/.style={circle,draw=scarlet!70,fill=scarlet!20,inner sep=1.5pt} 
 }
 
 % Maximum t value of the plot
 \newcommand*{\tmax}{10}
 
% function to find the coordinates of the intersection of the light rays with the
% rocket worldline. \a is defined below.
\pgfkeys{/pgf/declare function={xint(\a) = 0.5*\a*\a*\accel/(1+\a*\accel);}}
\pgfkeys{/pgf/declare function={tint(\a) = 0.5*\a*(2+\a*\accel)/(1+\a*\accel);}}



% ------------------------ 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] (-1,-1) grid (9,\tmax);
	\draw [cornflower!60,step=1.0,thin] (-1,-1) grid (9,\tmax);

	% Clip everything that falls outside the grid
	\clip(-1,-1) rectangle (9,\tmax);
	
	% Draw Axes
	\draw[thick,axisarrow] (0,-1) -- (0,\tmax);
	\node[right,inner sep=0pt] at (0.2,9.7) {$t$};
	\draw[thick,axisarrow] (-1,0) -- (9,0);
	\node[inner sep=0pt] at (8.5,-0.3) {$x$};

	% Draw the worldline of the rocket	
	\draw[domain=0:\taumax,smooth,variable=\tau,plum,thick,samples=100,axisarrow] 
		   plot ({(\accelinv*cosh(\accel*\tau)-\accelinv+\xinit)},{\accelinv*sinh(\accel*\tau)});

	% Add the location at some proper times		   
		   \foreach \tau in {0,1,1.5,2.0,2.5,2.9}		
		{
			\node[taudot]  at ({(\accelinv*cosh(\accel*\tau)-\accelinv+\xinit)},{\accelinv*sinh(\accel*\tau)}) {};
		}
		
	% Worldlines of the stars
	\draw[thick,black,axisarrow] (\xone,-1) -- (\xone,\tmax);
	\node[rotate=90,inner sep=0pt] at (\xone-0.3,3.0) {Star 1};
	\draw[thick,black,axisarrow] (\xtwo,-1) -- (\xtwo,\tmax);
	\node[rotate=90,inner sep=0pt] at (\xtwo-0.3,7.0) {Star 2};
	\draw[thick,black,axisarrow] (\xthree,-1) -- (\xthree,\tmax);	
	\node[rotate=90,inner sep=0pt] at (\xthree-0.3,6.0) {Star 3};
	
	% Worldlines of the light reaching the rocket from a star
	\foreach \t in {-4,-3,-2,-1,0,1,2,3,4,5}
	{
	    \pgfmathsetmacro{\a}{\xtwo+\t}
		\draw[thick,scarlet,axisarrow] 
		   (\xtwo,\t) -- ({xint(\a)},{tint(\a)});
	}
	
	% Worldlines of the light from the star after the rocket has gone by
	\foreach \t in {6,7,8,9}
	{
	    \pgfmathsetmacro{\a}{\t-\xtwo}
		\draw[thick,scarlet,axisarrow] 
		   (\xtwo,\t) -- ({7},{7+\a});
	}
	
	% Asymptote
	\draw[thick,chameleon] (-\accelinv,0) -- (9,9+\accelinv);
	

\end{tikzpicture}

			

\end{document}

Output from the above code.