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PlanetPhysics/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 Relativity: The Special and General Theory by Albert Einstein

We are now in a position to formulate more exactly the idea of Minkowski, which was only vaguely indicated in section 17. In accordance with the special theory of relativity, certain co-ordinate systems are given preference for the description of the four-dimensional, space-time continuum. We called these ``Galileian co-ordinate systems." For these systems, the four co-ordinates </math>x, y, z, t by the space co-ordinate differences and the time-difference . With reference to a second Galileian system we shall suppose that the corresponding differences for these two events are . Then these magnitudes always fulfill the condition \footnotemark.

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

\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 , , by </math>x_1, x_2, x_3, x_4

\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 instead of the real quantity , 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.

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This article is derived from the Einstein Reference Archive (marxists.org) 1999, 2002. Einstein Reference Archive 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).}