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.And yetthere is no more common-place statement than that the world in which we live is afour-dimensional space-time continuum.Space is a three-dimensional continuum.By this we mean that it is possible to describe theposition of a point (at rest) by means of three numbers (co-ordinales) x, y, z, and that there is anindefinite number of points in the neighbourhood of this one, the position of which can be describedby co-ordinates such as x1, y1, z1, which may be as near as we choose to the respective values ofthe co-ordinates x, y, z, of the first point.In virtue of the latter property we speak of a " continuum,"and owing to the fact that there are three co-ordinates we speak of it as being "three-dimensional."Similarly, the world of physical phenomena which was briefly called " world " by Minkowski isnaturally four dimensional in the space-time sense.For it is composed of individual events, each ofwhich is described by four numbers, namely, three space co-ordinates x, y, z, and a timeco-ordinate, the time value t.The" world" is in this sense also a continuum; for to every event thereare as many "neighbouring" events (realised or at least thinkable) as we care to choose, theco-ordinates x1, y1, z1, t1 of which differ by an indefinitely small amount from those of the event x,y, z, t originally considered.That we have not been accustomed to regard the world in this sense asa four-dimensional continuum is due to the fact that in physics, before the advent of the theory ofrelativity, time played a different and more independent role, as compared with the spacecoordinates.It is for this reason that we have been in the habit of treating time as an independentcontinuum.As a matter of fact, according to classical mechanics, time is absolute, i.e.it isindependent of the position and the condition of motion of the system of co-ordinates.We see thisexpressed in the last equation of the Galileian transformation (t1 = t)The four-dimensional mode of consideration of the "world" is natural on the theory of relativity,since according to this theory time is robbed of its independence.This is shown by the fourthequation of the Lorentz transformation:Moreover, according to this equation the time difference  t1 of two events with respect to K1 doesnot in general vanish, even when the time difference  t1 of the same events with reference toK vanishes.Pure " space-distance " of two events with respect to K results in " time-distance " ofthe same events with respect to K.But the discovery of Minkowski, which was of importance for theformal development of the theory of relativity, does not lie here.It is to be found rather in the fact ofhis recognition that the four-dimensional space-time continuum of the theory of relativity, in itsmost essential formal properties, shows a pronounced relationship to the three-dimensional37Relativity: The Special and General Theorycontinuum of Euclidean geometrical space.1) In order to give due prominence to this relationship,however, we must replace the usual time co-ordinate t by an imaginary magnitudeproportional to it.Under these conditions, the natural laws satisfying the demands of the(special) theory of relativity assume mathematical forms, in which the time co-ordinate playsexactly the same role as the three space co-ordinates.Formally, these four co-ordinatescorrespond exactly to the three space co-ordinates in Euclidean geometry.It must be clear even tothe non-mathematician that, as a consequence of this purely formal addition to our knowledge, thetheory perforce gained clearness in no mean measure.These inadequate remarks can give the reader only a vague notion of the important ideacontributed by Minkowski.Without it the general theory of relativity, of which the fundamental ideasare developed in the following pages, would perhaps have got no farther than its long clothes.Minkowski's work is doubtless difficult of access to anyone inexperienced in mathematics, but sinceit is not necessary to have a very exact grasp of this work in order to understand the fundamentalideas of either the special or the general theory of relativity, I shall leave it here at present, andrevert to it only towards the end of Part 2.Next: Part II: The General Theory of RelativityFootnotes1)Cf.the somewhat more detailed discussion in Appendix II [ Pobierz całość w formacie PDF ]