Talk:PlanetPhysics/Space Time Continuum of the Special Theory of Relativity Considered As a Euclidean Continuum

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%%% Primary Title: The Space-Time Continuum of the Special Theory of Relativity Considered as a Euclidean Continuum
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 \subsection{The Space-Time Continuum of the Special Theory of Relativity Considered as a
Euclidean Continuum}
From \htmladdnormallink{Relativity: The Special and General Theory}{http://planetphysics.us/encyclopedia/SpecialTheoryOfRelativity.html} by \htmladdnormallink{Albert Einstein}{http://planetphysics.us/encyclopedia/AlbertEinstein.html}

We are now in a \htmladdnormallink{position}{http://planetphysics.us/encyclopedia/Position.html} to formulate more exactly the idea of
Minkowski, which was only vaguely indicated in \htmladdnormallink{section}{http://planetphysics.us/encyclopedia/IsomorphicObjectsUnderAnIsomorphism.html} 17. In
accordance with the special theory of relativity, certain co-ordinate
\htmladdnormallink{systems}{http://planetphysics.us/encyclopedia/SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence.html} are given preference for the description of the
four-dimensional, \htmladdnormallink{space-time}{http://planetphysics.us/encyclopedia/SR.html} continuum. We called these ``Galileian
co-ordinate systems." For these systems, the four co-ordinates $x, y,
z, t$, which determine an event or---in other words, a point of the
four-dimensional continuum---are defined physically in a simple
manner, as set forth in detail in the first part of this book. For the
transition from one Galileian system to another, which is moving
uniformly with reference to the first, the equations of the Lorentz
transformation are valid. These last form the basis for the derivation
of deductions from the special theory of relativity, and in themselves
they are nothing more than the expression of the universal validity of
the law of transmission of light for all Galileian systems of
reference.

Minkowski found that the Lorentz transformations satisfy the following
simple conditions. Let us consider two neighboring events, the
relative position of which in the four-dimensional continuum is given
with respect to a Galileian reference-body $K$ by the space co-ordinate
differences $dx, dy, dz$ and the time-difference $dt$. With reference to a
second Galileian system we shall suppose that the corresponding
differences for these two events are $dx', dy', dz', dt'$. Then these
magnitudes always fulfill the condition \footnotemark.

$$dx^2 + dy^2 + dz^2 - c^2dt^2 = dx' 2 + dy' 2 + dz' 2 - c^2dt'^2$$

The validity of the Lorentz transformation follows from this
condition. We can express this as follows: The magnitude

$$ds^2 = dx^2 + dy^2 + dz^2 - c^2dt^2$$

\noindent which belongs to two adjacent points of the four-dimensional
space-time continuum, has the same value for all selected (Galileian)
reference-bodies. If we replace $x, y, z$, $\sqrt{-I} \cdot ct$ , by $x_1,
x_2, x_3, x_4$, we also obtain the result that

$$ds^2 = dx_1^2 + dx_2^2 + dx_3^2 + dx_4^2$$

\noindent is independent of the choice of the body of reference. We call the
magnitude ds the ``distance'' apart of the two events or
four-dimensional points.

Thus, if we choose as time-variable the imaginary variable $\sqrt{-I} \cdot ct$
instead of the real quantity $t$, we can regard the space-time
continuum---accordance with the special theory of relativity---as a
``Euclidean'' four-dimensional continuum, a result which follows from
the considerations of the preceding section.

\subsection{References}
This article is derived from the Einstein 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.


\footnotetext{Cf. Appendixes I and 2. The relations which are derived
there for the co-ordinates themselves are valid also for co-ordinate
differences, and thus also for co-ordinate differentials (indefinitely
small differences).}

\end{document}