Worldlines of Rindler observers in a particular Rindler reference frame (special relativity)

RindlerAccelFrame.tex

\documentclass[crop=true, border=10pt]{standalone}
\usepackage{comment}
\begin{comment} 
:Title: Rindler observers in a reference accelerated frame 
:Slug: Rindler observers
:Tags:  special relativity
:Author: J A Miller, UAH Physics & Astronomy, millerja@uah.edu, 2020/06/10

Worldlines of 9 accelerated observers in the accelerated coordinates $\xi$ and $\eta$ of a reference observer with proper acceleration of unity (in some inverse time units). These observers are called Rindler observers.

Figure caption:
\caption{
\textit{Purple arrows:} Worldlines of accelerated observers with $x_0 =  n/\alpha_R$ ($n = 1, 2, \ldots, 9$) in the accelerated reference frame $\xi$--$\eta$ of a reference observer (which has $n=0$). The fraction below each worldline is the acceleration of that observer in units of the reference observer. \textit{Filled circles:} The local proper time for an observer every $5$ units of time. \textit{Shaded area:} The region beyond the event horizon $\xi = -1/\alpha_R$ for all observers. Here, $\alpha_R = 1$ (in some inverse time unit).
        } 

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

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

\begin{enumerate}

\item Establish an accelerated $\xi$--$\eta$ frame for some reference acceleration $\alpha_R$. We will take the initial inertial frame location of this accelerated frame $x_{0R} = 0$. Hence, the transformations to this frame are given by
\begin{equation}\label{eq-xietaR}
    \begin{split}
		\xi &= \qty[ \qty( x + \frac{1}{\alpha_R} )^2 - t^2  ]^{1/2} - \frac{1}{\alpha_R} \\
		\eta &= \frac{1}{\alpha_R} \tanh^{-1} \frac{t}{x + 1/\alpha_R}.
    \end{split}
\end{equation}
The center of the hyperbola in the inertial frame is $-1/\alpha_R$, and this is also the $\xi$ location of the event horizon in the accelerated frame. (Note that the event horizon remains a constant distance away from the accelerated observer.)

\item Now establish another accelerated frame which starts at some $x_0 > 0$ and which has acceleration
\begin{equation}\label{eq-alphaRindler}
	\alpha = \frac{\alpha_R}{1+ x_0 \alpha_R}.
\end{equation}
This choice will guarantee that the proper distance between the reference observer and this observer will remain constant at $x_0$. Also note that the center of the hyperbola $x_0 - 1/\alpha$ for the inertial frame motion of this observer remains at $-1/\alpha_R$.

If we now substitute the inertial frame equations of motion for this observer
\begin{equation}\label{eq-xthypappen}
    \begin{split}
		x &= \frac{1}{\alpha} \cosh(\alpha \tau) - \frac{1}{\alpha} + x_0 \\
		   &= \frac{1}{\alpha} \cosh(\alpha \tau) - \frac{1}{\alpha_R} \\
		t &= \frac{1}{\alpha} \sinh(\alpha \tau)
    \end{split}
\end{equation}
into the tranformations \eqref{eq-xietaR) to the reference observer frame, we find that
\begin{equation}
	\begin{split}
		\xi &= x_0 \\
		\eta &= \frac{\alpha}{\alpha_R}\tau .
	\end{split}
\end{equation}
Thus, in the reference accelerated frame, the new, additional accelerated frame remains at location $x_0$ (a constant proper distance away) and has a local proper time $\tau$ related to the reference proper time $\eta$. For instance, taking $x_0 =  n/\alpha_R$, for some number $n$, we have $\eta = \tau/(1+n)$.

Also note that all these observers share the same event horizon as the reference observer.

\item We can now continue to add additional accelerated observers with progressively larger values of $x_0$, to build a network of related accelerated observers. These observers are called \emph{Rindler observers}, and all of their local times and locations are related.

\end{enumerate}

\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 of reference frame
\newcommand*{\accelR}{1.0}

% its inverse
\pgfmathsetmacro{\accelRinv}{1/\accelR}

% initial starting point for the reference frame in the inertial/lab system
\newcommand*{\xinitR}{0}  % this is $x_0$.

% maximum proper time of the reference frame
\newcommand*{\etamax}{7}


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

% Set the point style for the proper time ticks
\tikzset{
	taudot/.style={circle,draw=scarlet!70,fill=scarlet!20,inner sep=1.5pt} 
 }

\begin{document}

\begin{tikzpicture}[scale=1.0,domain=-2:10]
	\tikzstyle{axisarrow} = [-{Latex[inset=0pt,length=7pt]}]

	% Draw the background grid.
	\draw [cornflower!30,step=0.2,thin] (-2,-1) grid (10,8);
	\draw [cornflower!60,step=1.0,thin] (-2,-1) grid (10,8);

	% Clip everything that falls outside the grid
	\clip(-2,-1) rectangle (10,8);
	
	% Draw Axes
	\draw[thick,axisarrow] (0,-1) -- (0,8);
	\node[right,inner sep=0pt] at (0.2,7.7) {$\eta$ (Reference Frame Worldline)};
	\draw[thick,axisarrow] (-2,0) -- (10,0);
	\node[inner sep=0pt] at (9.7,-0.3) {$\xi$};
	
	% Event horizon
    \draw[fill=scarlet!40,opacity=0.2] (-2,-1)--(-\accelRinv,-1)--(-\accelRinv,8)--(-2,8)--(-2,-1);	

	\node[rotate=90,inner sep=0pt] at (-\accelRinv-0.2,4.0) {Event Horizon of all Frames};
	
	% Draw the worldlines of the other accelerated frames
	\foreach \xinit in {1,2,3,4,5,6,7,8,9}
	{
        \draw[thick,axisarrow,plum] (\xinit,0) -- (\xinit,7);		
	}
	
	% Add the locations of the proper time for the other Rindler observer		   
   \foreach \xinit in {1,2,3,4,...,9}
   {
        \pgfmathsetmacro{\accel}{\accelR/(1+\accelR*\xinit)};
    	\pgfmathsetmacro{\taumax}{\etamax*\accelR/\accel};
		\foreach \tau in {0,5,10,15,...,\taumax}		
			{
           		\pgfmathsetmacro{\eta}{\accel*\tau/\accelR};			
				\node[taudot] at ({\xinit},{\eta}) {};
			}
	}
	
	% Add the acceleration labels
	\foreach \xinit in {1,2,3,...,9}
	{
		\pgfmathsetmacro{\fac}{int(1+\xinit)};		
		\node[inner sep=0pt] at (\xinit,-0.3) {$1/\fac$};
	}

	
	% More labels	
	\node[inner sep=0pt] at (5,7.3) {Other Rindler Observer Worldlines};
	

	
\end{tikzpicture}
			

\end{document}

Output figure

Sample caption is given in the source code.