Use TikZ/PGF to programmatically draw spacetime diagrams for uniformly accelerated observers. Set the acceleration, initial conditions, and other parameters.
Questions/Comments to Robert McNees at rmcnees@luc.edu, http://jacobi.luc.edu January 2015
%-----------------------------------------------------------------------
% Use TikZ/PGF to programmatically draw spacetime diagrams for
% uniformly accelerated observers. Set the acceleration, initial
% conditions, and other paramters below.
%
% Questions/Comments to Robert McNees at rmcnees@luc.edu
% http://jacobi.luc.edu
% January 2015
%-----------------------------------------------------------------------
\documentclass[11pt]{article}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{mathrsfs}
\usepackage{tikz}
\usetikzlibrary{arrows.meta,decorations.pathmorphing,math}
%-----------------------------------------------------------------------
% Custom colors used in links and graphics
%-----------------------------------------------------------------------
\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}{0.937, 0.161, 0.161}
\definecolor{brick}{rgb}{0.64314, 0, 0}
\definecolor{sunrise}{rgb}{0.80784, 0.36078, 0}
\newcommand{\scri}{\mathscr{I}}
\begin{document}
\pagestyle{empty}
%-----------------------------------------------------------------------
% Set constants (speed of light, acceleration)
%-----------------------------------------------------------------------
% The speed of light, c=1
\newcommand*{\sol}{1.0}
% The acceleration a
\newcommand*{\accel}{0.8}%
%-----------------------------------------------------------------------
% Set initial values (initial t, x, and v)
%-----------------------------------------------------------------------
% The initial time t_0
\newcommand*{\tinit}{-0.5}%
% The initial position x_0
\newcommand*{\xinit}{0.4}%
% The initial velocity. Obviously the magnitude must be less than \sol!
\newcommand*{\vinit}{-0.2}%
%-----------------------------------------------------------------------
% Set the maximum value of t
%-----------------------------------------------------------------------
\newcommand*{\tmax}{5.0}%
%-----------------------------------------------------------------------
% Set the number of time intervals
%-----------------------------------------------------------------------
\newcommand*{\numintervals}{4}
%-----------------------------------------------------------------------
% Calculate the intervals \Delta t.
%-----------------------------------------------------------------------
\pgfmathsetmacro{\tinterval}{divide(\tmax-\tinit,\numintervals+1)}
%-----------------------------------------------------------------------
% Calculate the initial proper velocity.
%-----------------------------------------------------------------------
\pgfmathsetmacro{\uinit}{divide(\vinit,sqrt(1-pow(divide(\vinit,\sol),2)))}%
%-----------------------------------------------------------------------
% Calculate the initial value of the relativistic parameter gamma.
%-----------------------------------------------------------------------
\pgfmathsetmacro{\gammainit}{divide(1,sqrt(1-divide(pow(\vinit,2),1)))}%
%-----------------------------------------------------------------------
% Now define the position x(t) of the accelerated observer.
%-----------------------------------------------------------------------
\pgfmathdeclarefunction{pos}{1}{%
\pgfmathparse{divide(pow(\sol,2),\accel)*(sqrt(1+pow(divide(\accel,\sol)*(#1-\tinit) + divide(\uinit,\sol),2)) - sqrt(1 + pow(divide(\uinit,\sol),2)) ) + \xinit}
}
%-----------------------------------------------------------------------
% Points on the observer's worldline are a constant proper distance
% from the point (\xstar,\tstar).
%-----------------------------------------------------------------------
% Define \xstar
\pgfmathsetmacro{\xstar}{\xinit - divide(pow(\sol,2),\accel)*sqrt(1+pow(divide(\uinit,\sol),2))}%
% Define \tstar
\pgfmathsetmacro{\tstar}{\tinit - divide(\vinit,\accel)*\gammainit}%
%-----------------------------------------------------------------------
% Find the maximum value of x(t) so we know how big the plot should be.
%-----------------------------------------------------------------------
\pgfmathsetmacro{\xmax}{pos(\tmax)}%
%-----------------------------------------------------------------------
% Draw everything
%-----------------------------------------------------------------------
\begin{center}
\begin{tikzpicture}[>=LaTeX]
%-----------------------------------------------------------------------
% Clip a rectangular region so we can draw things later without
% worrying about them poking out the sides of the diagram.
%-----------------------------------------------------------------------
% First, set a lower value of t depending on whether \tinit is positive or negative.
\tikzmath{
if \tinit>0.0 then {let \tlower = -0.5*\tmax;} else {let \tlower = \tinit-0.5*\tmax;};
}
% Now clip the region. The +0.1 and +0.3 add padding for x and t labels and make sure the clip region doesn't end on a grid line.
\clip (-\xmax+0.1,\tlower) -- (\xmax+0.3,\tlower) -- (\xmax+0.3,\tmax+0.3) -- (-\xmax+0.1,\tmax+0.3) -- (-\xmax+0.1,\tlower);
%-----------------------------------------------------------------------
% Draw a background grid.
%-----------------------------------------------------------------------
\draw[step=.25,blue!15] (-\xmax,-\tmax) grid (\xmax+0.3,\tmax+0.3);
%-----------------------------------------------------------------------
% Draw the x and t axes.
%-----------------------------------------------------------------------
\draw [-{Stealth[length=2.5mm]},thick] (-\xmax,0) -- (\xmax,0) node[above] {\large$x$};
\draw [-{Stealth[length=2.5mm]},thick] (0,-\tmax) -- (0,\tmax) node[right] {\large $t$};
%-----------------------------------------------------------------------
% Draw lines from (\xstar,\tstar) to \numintervals evenly-spaced
% points on the worldline.
%-----------------------------------------------------------------------
\foreach \x in {0,1,...,\numintervals} {
\draw[cornflower] (\xstar,\tstar) -- ({pos(\tinit+\x*\tinterval)},{\tinit+\x*\tinterval});
}
%-----------------------------------------------------------------------
% Plot the worldline of the observer as a smooth curve with no
% arrows on it.
%-----------------------------------------------------------------------
% \draw[thick, chameleon, domain=\tinit:\tmax,smooth,variable=\t,->]
% plot[] ({pos(\t)},{\t});
%-----------------------------------------------------------------------
% Plot the worldline as \numintervals equally spaced (in t)
% segments with an arrow on each one.
%-----------------------------------------------------------------------
\foreach \i in {0,1,...,\numintervals} {
\pgfmathsetmacro{\tstart}{\tinit+\i*\tinterval-0.05};
\pgfmathsetmacro{\tfin}{\tinit+(\i+1)*\tinterval};
\draw[thick, chameleon, domain=\tstart:\tfin,smooth,variable=\t,->]
plot[] ({pos(\t)},{\t});
}
%-----------------------------------------------------------------------
% Make a small, filled circle at (\xinit,\tinit).
%-----------------------------------------------------------------------
\fill[chameleon,opacity=1] (\xinit,\tinit) circle (2pt);
% Optionally add a label next to the point.
%\node at (\xinit+.75,\tinit) {\small $(x_0,t_0)$};
%-----------------------------------------------------------------------
% Draw a small, filled circle at (\xstar, \tstar).
%-----------------------------------------------------------------------
\fill[scarlet] (\xstar,\tstar) circle (2pt);
%-----------------------------------------------------------------------
%Draw the light cone of (\xstar,\tstar).
%-----------------------------------------------------------------------
\draw[scarlet, thick, dashed, domain=-\tmax:\tmax,smooth,variable=\t]
plot ({\sol*\t+\xstar},{\t+\tstar});
\draw[scarlet, thick, dashed, domain=-\tmax:\tmax,smooth,variable=\t]
plot ({-\sol*\t+\xstar},{\t+\tstar});
\end{tikzpicture}
\end{center}
\end{document}