Draw a spacetime diagram showing an accelerating rocket with constant proper length.

RigidRocket.tex

\documentclass[border=10pt]{standalone}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{mathrsfs}



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

\usepackage{tikz-3dplot}

\usepackage{pgfplots}

\usepackage{colortbl}

\usepackage{xcolor}
%--------------------------------------------------------------
% Define a few 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}


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

	%--------------------------------------------------------------
	% We place the origin so each point on the rocket is always
	% some constant proper distance from the event (0,0);
	%--------------------------------------------------------------
	
	%--------------------------------------------------------------
	% Set some constants we need.
	%--------------------------------------------------------------
	
	% Use units where c=1
    \pgfmathsetmacro\clight{1};
	% Set the acceleration of the back of the rocket in units of ls/s^2.
    \pgfmathsetmacro\alphaB{0.25};
	% Set the (constant) proper length of the rocket, from back to front
    \pgfmathsetmacro\Length{8};

	%--------------------------------------------------------------
	% Compute some constants we need.
	%--------------------------------------------------------------
	
	% Compute the acceleration at the front of the rocket, based on
	% the acceleration at the back \alphaB and the constant proper
	% length L.
    \pgfmathsetmacro\alphaF{\alphaB*\clight^2/(\clight^2 + \alphaB*\Length)};

	% Compute the acceleration at the midpoint of the rocket, based on
	% the acceleration at the back \alphaB and the constant proper
	% length L.
    \pgfmathsetmacro\alphaM{\alphaB*\clight^2/(\clight^2 + \alphaB*\Length/2)};

	%--------------------------------------------------------------
	% How much proper time to plot for each part of the rocket.
	%--------------------------------------------------------------

	% Plot for this amount of proper time according to an observer 
	% in the back of the rocket.
    \pgfmathsetmacro\tBmax{8};

	% The amount of proper time experienced by someone in the front
	% of the rocket, simultaneous with the back of the rocket at
	% \tBax. 
    \pgfmathsetmacro\tFmax{\tBmax*\alphaB/\alphaF};

	% The amount of proper time experienced by someone at the midpoint
	% of the rocket, simultaneous with the back of the rocket at
	% \tBax. 
    \pgfmathsetmacro\tMmax{\tBmax*\alphaB/\alphaM};

	%--------------------------------------------------------------
	% Draw things
	%--------------------------------------------------------------
	
	% Draw the background grid.
	\draw [cornflower!30,step=0.2,thin] (-1,-2) grid (16,12);
	\draw [cornflower!60,step=1.0,thin] (-1,-2) grid (16,12);

	% Clip everything that falls outside the grid
	\clip(-1,-2) rectangle (16,12);
	
	% Draw Axes
	\draw[thick,axisarrow] (0,-2) -- (0,12);
	\node[inner sep=0pt] at (0.5,11.7) {$t (s)$};
	\draw[thick,axisarrow] (-1,0) -- (16,0);
	\node[inner sep=0pt] at (15.5,-0.4) {$x (\ell s)$};

	% Light ray from the event (0,0) traveling in the positive
	% x-direction. The worldlines of all points of the rocket 
	% asymptote to this line.
	\draw[chameleon,thick] (0,0) -- (12,12);

	% Draw lines of simultaneity through N events happening every 1 
	% second, according to a clock at the back of the rocket.
	\foreach \tB in {1,2,...,4}
	{
		\draw[orange, dashed, thick] (0,0) -- (16,{16*tanh(\alphaB*\tB/\clight)});
	}

	%--------------------------------------------------------------
    % Back of the rocket
	%--------------------------------------------------------------
	
	% Plot the wordline of the back of the rocket, as a function of
	% its proper time.
	\draw[domain=0:\tBmax,smooth,variable=\tB,scarlet,thick] plot ({(\clight^2/\alphaB)*cosh(\tB*\alphaB/\clight)},{(\clight/\alphaB)*sinh(\tB*\alphaB/\clight)});

	% Add a node at events spaced 1 second of proper time apart
	% along the back of the rocket's worldline.
    \foreach \tB in {0,1,2,...,\tBmax}
	{
			\node[circle,draw=scarlet!70,fill=scarlet!20,inner sep=2pt]  at ({(\clight^2/\alphaB)*cosh(\alphaB*\tB/\clight)},{(\clight/\alphaB)*sinh(\alphaB*\tB/\clight)}) {};
	}	

	%--------------------------------------------------------------
	% Middle of the rocket
	%--------------------------------------------------------------

	% Plot the wordline of the midpoint of the rocket, as a function of
	% its proper time.
	\draw[domain=0:\tMmax,smooth,variable=\tM,cornflower,thick] plot ({(\clight^2/\alphaM)*cosh(\tM*\alphaM/\clight)},{(\clight/\alphaM)*sinh(\tM*\alphaM/\clight)});

	% Add a node at events spaced 1 second of proper time apart
	% along the midpoint of the rocket's worldline.
    \foreach \tM in {0,1,2,...,\tMmax}
	{
			\node[circle,draw=cornflower!70,fill=cornflower!20,inner sep=2pt]  at ({(\clight^2/\alphaM)*cosh(\alphaM*\tM/\clight)},{(\clight/\alphaM)*sinh(\alphaM*\tM/\clight)}) {};
	}	

	%--------------------------------------------------------------
	% Front of the rocket
	%--------------------------------------------------------------

	% Plot the wordline of the front of the rocket, as a function of
	% its proper time.
	\draw[domain=0:\tFmax,smooth,variable=\tF,plum,thick] plot ({(\clight^2/\alphaF)*cosh(\tF*\alphaF/\clight)},{(\clight/\alphaF)*sinh(\tF*\alphaF/\clight)});

	% Add a node at events spaced 1 second of proper time apart
	% along the front of the rocket's worldline.
    \foreach \tF in {0,1,2,...,\tFmax}
	{
			\node[circle,draw=plum!70,fill=plum!20,inner sep=2pt]  at ({(\clight^2/\alphaF)*cosh(\alphaF*\tF/\clight)},{(\clight/\alphaF)*sinh(\alphaF*\tF/\clight)}) {};
	}

	%--------------------------------------------------------------
	% Add some labels for each worldline.
	%--------------------------------------------------------------

	\node [] at ({\clight^2/\alphaB},-0.5) {$x_{B}$};
	\node [] at ({\clight^2/\alphaM},-0.5) {$x_{M}$};
	\node [] at ({\clight^2/\alphaF},-0.5) {$x_{F}$};
	
\end{tikzpicture}
			

\end{document}