Talk:PlanetPhysics/Minkowski's Four Dimensional Space World

Original TeX Content from PlanetPhysics Archive

edit
%%% This file is part of PlanetPhysics snapshot of 2011-09-01
%%% Primary Title: Minkowski's Four-Dimensional Space (``World")
%%% Primary Category Code: 03.30.+p
%%% Filename: MinkowskisFourDimensionalSpaceWorld.tex
%%% Version: 1
%%% Owner: bloftin
%%% Author(s): bloftin
%%% PlanetPhysics is released under the GNU Free Documentation License.
%%% You should have received a file called fdl.txt along with this file.        
%%% If not, please write to gnu@gnu.org.
\documentclass[12pt]{article}
\pagestyle{empty}
\setlength{\paperwidth}{8.5in}
\setlength{\paperheight}{11in}

\setlength{\topmargin}{0.00in}
\setlength{\headsep}{0.00in}
\setlength{\headheight}{0.00in}
\setlength{\evensidemargin}{0.00in}
\setlength{\oddsidemargin}{0.00in}
\setlength{\textwidth}{6.5in}
\setlength{\textheight}{9.00in}
\setlength{\voffset}{0.00in}
\setlength{\hoffset}{0.00in}
\setlength{\marginparwidth}{0.00in}
\setlength{\marginparsep}{0.00in}
\setlength{\parindent}{0.00in}
\setlength{\parskip}{0.15in}

\usepackage{html}

% this is the default PlanetMath preamble.  as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.

% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}

% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
%\usepackage{amsthm}
% making logically defined graphics
%\usepackage{xypic}

% there are many more packages, add them here as you need them

% define commands here

\begin{document}

 \subsection{Minkowski's Four-Dimensional Space (``World")
(Supplementary to Section 17)}
From \htmladdnormallink{Relativity: The Special and General Theory}{http://planetphysics.us/encyclopedia/SpecialTheoryOfRelativity.html} by Albert Einstein

We can characterise the Lorentz transformation still more simply if we
introduce the imaginary $\sqrt{-I} \cdot ct$ in place of $t$, as time-variable. If, in
accordance with this, we insert
\begin{eqnarray*}
x_1 & = & x \\
x_2 & = & y \\
x_3 & = & z \\
x_4 & = & \sqrt{-I} \cdot ct
\end{eqnarray*}
and similarly for the accented system $K^1$, then the condition which is
identically satisfied by the transformation can be expressed thus:

$${x'_1}^2 + {x'}_2^2 + {x'}_3^2 + {x'}_4^2 = x_1^2 + x_2^2 + x_3^2 + x_4^2   \quad . \quad . \quad . \quad \mbox{(12)}.$$

\noindent That is, by the afore-mentioned choice of ``coordinates," (11a) [see
the end of Appendix II] is transformed into this equation.

We see from (12) that the imaginary time co-ordinate $x_4$, enters into
the condition of transformation in exactly the same way as the space
co-ordinates $x_1, x_2, x_3$. It is due to this fact that, according
to the theory of relativity, the ``time'' $x_4$, enters into natural
laws in the same form as the space co ordinates $x_1, x_2, x_3$.

A four-dimensional continuum described by the ``co-ordinates" $x_1,
x_2, x_3, x_4$, was called ``world" by Minkowski, who also termed a
point-event a ``world-point." From a ``happening'' in three-dimensional
space, physics becomes, as it were, an ``existence ``in the
four-dimensional ``world."

This four-dimensional ``world'' bears a close similarity to the
three-dimensional ``space'' of (Euclidean) analytical geometry. If we
introduce into the latter a new Cartesian co-ordinate system ($x'_1,
x'_2, x'_3$) with the same origin, then $x'_1, x'_2, x'_3$, are
linear homogeneous functions of $x_1, x_2, x_3$ which identically
satisfy the equation

$${x'}_1^2 + {x'}_2^2 + {x'}_3^2 = x_1^2 + x_2^2 + x_3^2$$

The analogy with (12) is a complete one. We can regard Minkowski's ``world''
in a formal manner as a four-dimensional Euclidean space (with
an imaginary time coordinate); the Lorentz transformation corresponds
to a ``rotation'' of the co-ordinate system in the four-dimensional
``world."

\subsection{References}
This article is derived from the \htmladdnormallink{Einstein}{http://planetphysics.us/encyclopedia/AlbertEinstein.html} Reference Archive (marxists.org) 1999, 2002. \htmladdnormallink{Einstein Reference Archive}{http://www.marxists.org/reference/archive/einstein/index.htm} which is under the FDL copyright.

\end{document}
Return to "PlanetPhysics/Minkowski's Four Dimensional Space World" page.