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

\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$,whichdetermineaneventor---inotherwords,apointofthefour-dimensionalcontinuum---aredefinedphysicallyinasimplemanner,assetforthindetailinthefirstpartofthisbook.ForthetransitionfromoneGalileiansystemtoanother,whichismovinguniformlywithreferencetothefirst,theequationsoftheLorentztransformationarevalid.Theselastformthebasisforthederivationofdeductionsfromthespecialtheoryofrelativity,andinthemselvestheyarenothingmorethantheexpressionoftheuniversalvalidityofthelawoftransmissionoflightforallGalileiansystemsofreference.MinkowskifoundthattheLorentztransformationssatisfythefollowingsimpleconditions.Letusconsidertwoneighboringevents,therelativepositionofwhichinthefour-dimensionalcontinuumisgivenwithrespecttoaGalileianreference-body<math>K$ 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.

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

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

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

\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 </math>x_1, x_2, x_3, x_4$,wealsoobtaintheresultthat<math>ds^{2}=dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}+dx_{4}^{2}$

\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.

References

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).}