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Re: [Phys-L] math +- visualization etc. etc.



The plots Joel mentions can be generated in LaTeX also. I can't access my website right now (IT messed up and removed my access during their end-of-semester cleanup) to give you a link, so here's a LaTeX document that lets you change the speed by changing the \betaval value near the top of the document. (All you people who don't use LaTeX yet, you really should.) The "cm= ..." item in the \draw ... grid..." command operates exactly like the transformation matrix in Inkscape which John Denker mentions in his SR web docs.


-----Original Message-----
Subject: Re: [Phys-L] math +- visualization etc. etc.

I found the program for plotting hyperbolic coordinates superposed on Cartesian to be immensely useful for simplifying the drawing of x-t space-time diagrams.

It can be found at:

http://www.physics.pomona.edu/sixideas/sicpr.html

This is from Moore's "Six Ideas . . ." website, the program can be downloaded about halfway down the page in the section "For Use With Unit R". It easily saves 60-80% of your time in drawing simple 2-D spacetime diagrams.

Joel

\documentclass[portrait,letterpaper,11pt,notitlepage]{article}
\usepackage{fouriernc}
\usepackage[margin=0.75in,top=0.5in,bottom=0.5in]{geometry}
\usepackage[units,arrowvectors,braket]{physymb}
\usepackage{siunitx}

\usepackage{graphicx}

\usepackage{wrapfig}
\usepackage{tikz}
\usetikzlibrary{calc}
\newcommand{\betaval}{0.4}

\begin{document}
\pagestyle{empty}
Consider 4 events (the {\color{red}$\circ$}s) in a certain reference frame. Three of them are simultaneous, but at different locations. The fourth is a the same location as the first, but occurs earlier. Here they are plotted on a space-time diagram. (There are some black $\times$s also; ignore them for right now.)

\begin{tikzpicture}
\draw[step=5mm,dashed,cm={1,0,0,1,(0,0)}](0,0) grid (6,6);
\node[red] at (1,1) {$\circ$};
\node[red] at (5,2) {$\circ$};
\node[red] at (1,2) {$\circ$} node[red] at (2,2) {$\circ$};
\draw[->](-.2,-.2) -- (-.2,3) node[above] {ct};
\draw[->](-.2,-.2) -- (3,-.2) node[right] {x};

\draw
let \n1={\betaval},
\n2={0.5*ln((1+\n1)/(1-\n1)))},
\n3={cosh(\n2)},
\n4={sinh(\n2)} in
[cm={\n3,\n4,\n4,\n3,(0,0)}] node at (1,2) {$\times$} node at (1,3) {$\times$} node at (2,3) {$\times$} node at (4,3) {$\times$};
\end{tikzpicture}

At time, t=0, a moving observer with $\beta$ = \betaval \ passes by such that the moving origin is at x=0 (=x'). The time for the observer t', is immediately set to zero, too. This is a space-time diagram of how the moving observer views the {\color{red}$\circ$} events.

\begin{tikzpicture}

\node[red] at (5,2) {$\circ$};
\node[red] at (1,1) {$\circ$};
\node[red] at (1,2) {$\circ$} node[red] at (2,2) {$\circ$};
\draw
let \n1={\betaval},
\n2={0.5*ln((1+\n1)/(1-\n1)))},
\n3={cosh(\n2)},
\n4={sinh(\n2)} in [step=5mm,blue,cm={\n3,\n4,\n4,\n3,(0,0)}] grid (6,6)
node[black] at (1,2) {$\times$} node[black] at (1,3) {$\times$} node[black] at (2,3) {$\times$} node[black] at (4,3) {$\times$};
\draw
let \n1={atan(\betaval)} in [->,blue,rotate=-\n1 ](-.3,-.3) -- (-.3,3) node[above] {ct};
\draw
let \n1={atan(\betaval)} in [->,blue,rotate=\n1 ](-.3,-.3) -- (3,-.3) node[right] {x};

\end{tikzpicture}

Not only are the "simultaneous events" of the first frame at different positions, they are no longer simultaneous. The event that happened farther from the origin now occurs first. The events that happened at a common location in the first frame are now displaced from one another.

The black $\times$s are events in the moving frame which either occur simultaneously (the three topmost events) or at a common location (the two left most events). These are also plotted in the top (non-moving) frame. They are non-simultaneous and at differing locations in the "non-moving" frame.

Now let us draw both coordinates overlaid to see how lengths and time intervals align.

\begin{tikzpicture}[scale=2]
\draw[step=5mm,dashed,cm={1,0,0,1,(0,0)}](0,0) grid (6,6);

\node[red] at (5,2) {$\circ$};
\node[red] at (1,1) {$\circ$};
\node[red] at (1,2) {$\circ$} node[red] at (2,2) {$\circ$};
\draw
let \n1={\betaval},
\n2={0.5*ln((1+\n1)/(1-\n1)))},
\n3={cosh(\n2)},
\n4={sinh(\n2)} in [step=5mm,blue,cm={\n3,\n4,\n4,\n3,(0,0)}] grid (6,6)
node[black] at (1,2) {$\times$} node[black] at (1,3) {$\times$} node[black] at (2,3) {$\times$} node[black] at (4,3) {$\times$};

\end{tikzpicture}

\end{document}