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%%% Primary Title: Minkowski's Four-Dimensional Space (``World")
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\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}
