Area preservation by the Lorentz transformations in special relativity

AreaPreservation.tex

\documentclass[crop=true, border=10pt]{standalone}
\usepackage{comment}
\begin{comment}
:Title:       Area preservation in the Lorentz Transformations
:Author: 	    J A Miller, UAH Physics & Astronomy, millerja@uah.edu
              2020/02/04

Plot to demonstrate how the Lorentz Transformation preserves area. Suppose the primed frame is moving in the $+x$ direction with speed $v$ relative to the unprimed frame.

Consider a unit square in the primed frame, with corners at $A^\prime =(x,t) =(0,0)$, $B^\prime =(0,1)$, $C^\prime =(1,1)$, and $D^\prime =(1,0)$. In the unprimed system, these points are transformed into $A = (0,0)$, $B = (\gamma v,\gamma)$, $C = (\gamma (1+v),\gamma (1+v))$, and $D = (\gamma,\gamma v)$, which outlines a parallelogram (in the unprimed coordinate system).

The area of the unit square is clearly one. The area of the parallelogram can be found by taking the magnitude of the cross product between the vectors $AD = (\gamma,\gamma v)$ and $AB = (\gamma v,\gamma)$. This area is also one, so that the Lorentz transformations preserve spacetime area (or volume or hypervolume for two- or three-dimensional motion, respectively). 

You could have concluded this right away by noting that the determinant of the Lorentz transformation matrix $L$ or $L^{-1}$ was unity. The area preservation property means that points in a given area of primed space are taken over to the same area in the unprimed space, or vice versa, although the shape of that area may be different in the two frames. That is, spacetime is not lost or gained when simply changing coordinates, which is certainly a desired property.

Now consider the eigenvalues and eigenvectors of $L$ (or its inverse). The eigenvalues are $\gamma \pm |\gamma v |$ and the corresponding (unnormalized) eigenvectors are $(\pm 1,1)$. 

Eigenvectors are special vectors that maintain their orientation (or ``span'') under a linear transformation, and the corresponding eigenvalues are the factor by which they are shrunk or expanded. So here, the eigenvector $(1,1)$ gets expanded by a factor $\gamma + |\gamma v|$, while the eigenvector $(-1,1)$ gets contracted by a factor $\gamma - |\gamma v|$. But the directions remain the same in the primed and unprimed spaces.

Now, these eigenvectors turn out to be the paths of light (also known as \emph{worldlines}) in spacetime! Thus, light worldlines have the same span in different frames, or their direction remains the same regardless of the coordinate system. And since light behaves the same in all frames, we certainly would not want its direction to change. The Lorentz transformations thus have two necessary properties, which in fact could have been the starting point for their derivation.

By the way, an outstanding discussion of eigenvectors and eigenvalues is given in a video by \href{https://www.youtube.com/watch?v=PFDu9oVAE-g}{3Blue1Brown}. Remember that Lorentz transformations are nothing more than a rather mundane linear transformation but applied (remarkably) to the fundamental physical quantities of space and time.

Basic plot style and colors from Robert McNees
https://gist.github.com/mcnees/45b9f53ad371c38ba6f3759df5880fb1
\end{comment}

\usepackage{xcolor}

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

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

% 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}

% Set the labeled point style
\tikzset{
    colordot/.style={circle,draw=cornflower!70,fill=cornflower!20,inner sep=1.5pt} 
 }
 


\begin{document}

% Specify the boost. Can be varied by user.
\newcommand*\vrel{0.6}
% Some calculations
\pgfmathparse{1/sqrt(1-\vrel*\vrel)} \let\gammarel\pgfmathresult
\pgfmathparse{\vrel*\gammarel} \let\etarel\pgfmathresult


\begin{tikzpicture}[scale=6,domain=-1:11]
    \tikzstyle{axisarrow} = [-{Latex[inset=0pt,length=8pt]}]

	% corners of the square in the primed system
	\coordinate (Apr) at (0,0);
	\coordinate (Bpr) at (0,1);
	\coordinate (Cpr) at (1,1);
	\coordinate (Dpr) at (1,0);
	
	% vertices of the parallelogram in the unprimed system
	\coordinate (A) at (0,0);
	\coordinate (B) at (\etarel,\gammarel);
	\coordinate (C) at ({\gammarel*(1+\vrel)},{\gammarel*(1+\vrel)});
	\coordinate (D) at (\gammarel,\etarel);
	
    % Draw the grid. Major grid lines only.
    \draw [cornflower!60,step=0.1,thin] (-0.1,-0.1) grid (2.1,2.1);

	% Draw axes
	\draw[cornflower,thick,axisarrow] (0,-0.1) -- (0,2.1) node (tup) {};
	\node[inner sep=0pt] at (0.07,2.05) {$t, t^\prime$};
	\draw[cornflower,thick,axisarrow] (-0.1,0) -- (2.1,0) node (xup) {};
	\node[inner sep=0pt] at (2.03,-0.04) {$x, x^\prime$};
	
	% Draw the square in the primed frame
	\draw [plum,thick] 
	(Apr) node[colordot,label={below left: A$^\prime$}]{} -- 
	(Bpr) node[colordot,label={above right: B$^\prime$}]{} --
	(Cpr) node[colordot,label={right: C$^\prime$}]{} -- 
	(Dpr) node[colordot,label={below right:D$^\prime$}]{} -- cycle;
	\draw[fill=plum!40,opacity=0.2] (Apr) -- (Bpr) -- (Cpr) -- (Dpr) -- cycle;
	
	% Transformed square in the unprimed frame	
	\draw [thick,scarlet] 
	(A) node[colordot,label={below right:A}]{} -- 
	(B) node[colordot,label={above: B}]{} --
	(C) node[colordot,label={C}]{} -- 
	(D) node[colordot,label={below:D}]{} -- cycle;
	\draw[fill=scarlet!40,opacity=0.2] (A) -- (B) -- (C) -- (D) -- cycle;
	
	% Draw eigenvectors
	\draw [thick, plum, dashed, axisarrow] (Dpr) -- (Bpr);
	\draw [thick, plum, dashed, axisarrow] (Apr) -- (Cpr);
	\draw [thick, scarlet, dashed, axisarrow] (D) -- (B);
	\draw [thick, scarlet, dashed, axisarrow] (Cpr) -- (C);

\end{tikzpicture}

	
\end{document}

The transformed square for a relative velocity of 3/5.

The transformed square for a relative velocity of 1/5