A rocket with constant proper acceleration along its length. It could also represent three separate rockets with the same acceleration.

RocketConstantAcceleration.tex

\documentclass[crop=true, border=10pt]{standalone}
\usepackage{comment}
\begin{comment} 
:Title: Motion of a rocket with constant acceleration along its length
:Slug: Rocket motion
:Tags:  special relativity
:Author: J A Miller, UAH Physics & Astronomy, millerja@uah.edu, 2020/05/21

Worldlines for parts of a rocket (or three separate rockets), each with the same proper acceleration. 
We see that the middle and left parts of the rocket partially lie in the event horizon of the right 
end of the rocket, leading to causal disconnection and the rocket sections not being able to interact with each other.

This illustrates that proper acceleration must vary along the rocket length.

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.

This trajectory is a hyperbola in $x$-$t$ space, bounded by asymptotes passing through the hyperbola center at $x_c = x_0 - 1/\alpha$. For constant acceleration, each part of the rocket (or each rocket in the fleet) will have a different bounding hyperbola, which marks its even horizon. Eventually, some rockets will pass beyond the event horizon of others, and they will become causally disconnected. 

\end{comment}




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

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

% Define some colors
\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}




\begin{document}

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

% proper acceleration of rocket
\newcommand*{\accel}{0.2};

% initial rest/inertial frame length of rocket
\newcommand*{\length}{4.0};

% initial location of the left end
\newcommand*{\xL}{4.0};
	
% ----------------------- calculations ------------------------------
	
% initial location of right end
\pgfmathsetmacro\xR{\xL+\length};
    
% Initial location of middle
\pgfmathsetmacro\xM{\xL+\length/2};
    
% Find the hyperbola centers (where the asymptotes start
\pgfmathsetmacro\xLc{\xL-(1/\accel)};
\pgfmathsetmacro\xMc{\xM-(1/\accel)};
 \pgfmathsetmacro\xRc{\xR-(1/\accel)};


% -------------------------- figure -----------------------------------

\begin{tikzpicture}[scale=1,domain=-1:16]
	\tikzstyle{axisarrow} = [-{Latex[inset=0pt,length=9pt]}]
	
	% 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) {$t$};
	\draw[thick,axisarrow] (-1,0) -- (16,0);
	\node[inner sep=0pt] at (15.5,-0.4) {$x$};

	% Draw the asymptototes for the different points on the rocket
	\draw[chameleon,thick,dashed] (\xLc,0) -- (12+\xLc,12);
	\draw[scarlet,thick,dashed] (\xMc,0) -- (12+\xMc,12);
	\draw[plum,thick,dashed] (\xRc,0) -- (12+\xRc,12);

	% worldline for left of rocket
	\draw[domain=\xL:16,samples=500,variable=\x,chameleon,thick] plot ({\x},{ sqrt((\x-\xLc)^2 - (1/\accel)^2) });
	
	% wordline of the middle of the rocket
	\draw[domain=\xM:16,samples=500,variable=\x,scarlet,thick] plot ({\x},{ sqrt((\x-\xMc)^2 - (1/\accel)^2) });
	
	% worldline for front of rocket
   \draw[domain=\xR:16,samples=500,variable=\x,plum,thick] plot ({\x},{ sqrt((\x-\xRc)^2 - (1/\accel)^2) });
   
	% color the regions
	\draw[fill=plum!40,opacity=0.2] (-1,0)--(\xRc,0) --(12+\xRc,12) -- (-1,12) -- (-1,0);
	   
	% labels
	\node [] at (\xL,-0.5) {$x_L$};
	\node [] at (\xM,-0.5) {$x_M$};
	\node [] at (\xR,-0.5) {$x_R$};
	
\end{tikzpicture}
			

\end{document}

Output from the code.

\caption{Worldlines of the left, middle, and right points of a rocket of initial length $L_0 = 4$. The acceleration of each part is constant at $0.2$. The \textit{dashed lines} are the hyperbola asymptotes corresponding to each part of the rocket, and the \textit{shaded region} is beyond the event horizon for the right end of the rocket.}