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.We are therefore not in a position to define exactly theco-ordinates x, y, z relative to the disc by means of the method used in discussing the specialtheory, and as long as the co- ordinates and times of events have not been defined, we cannotassign an exact meaning to the natural laws in which these occur.Thus all our previous conclusions based on general relativity would appear to be called in question.In reality we must make a subtle detour in order to be able to apply the postulate of generalrelativity exactly.I shall prepare the reader for this in the following paragraphs.Next: Euclidean and Non-Euclidean ContinuumFootnotes1)The field disappears at the centre of the disc and increases proportionally to the distance fromthe centre as we proceed outwards.2)Throughout this consideration we have to use the Galileian (non-rotating) system K asreference-body, since we may only assume the validity of the results of the special theory ofrelativity relative to K (relative to K1 a gravitational field prevails).Relativity: The Special and General Theory50 Relativity: The Special and General TheoryAlbert Einstein: RelativityPart II: The General Theory of RelativityEuclidean and Non-Euclidean ContinuumThe surface of a marble table is spread out in front of me.I can get from any one point on this tableto any other point by passing continuously from one point to a " neighbouring " one, and repeatingthis process a (large) number of times, or, in other words, by going from point to point withoutexecuting "jumps." I am sure the reader will appreciate with sufficient clearness what I mean hereby " neighbouring " and by " jumps " (if he is not too pedantic).We express this property of thesurface by describing the latter as a continuum.Let us now imagine that a large number of little rods of equal length have been made, their lengthsbeing small compared with the dimensions of the marble slab.When I say they are of equal length,I mean that one can be laid on any other without the ends overlapping.We next lay four of theselittle rods on the marble slab so that they constitute a quadrilateral figure (a square), the diagonalsof which are equally long.To ensure the equality of the diagonals, we make use of a littletesting-rod.To this square we add similar ones, each of which has one rod in common with thefirst.We proceed in like manner with each of these squares until finally the whole marble slab islaid out with squares.The arrangement is such, that each side of a square belongs to two squaresand each corner to four squares.It is a veritable wander that we can carry out this business without getting into the greatestdifficulties.We only need to think of the following.If at any moment three squares meet at a corner,then two sides of the fourth square are already laid, and, as a consequence, the arrangement ofthe remaining two sides of the square is already completely determined.But I am now no longerable to adjust the quadrilateral so that its diagonals may be equal.If they are equal of their ownaccord, then this is an especial favour of the marble slab and of the little rods, about which I canonly be thankfully surprised.We must experience many such surprises if the construction is to besuccessful.If everything has really gone smoothly, then I say that the points of the marble slab constitute aEuclidean continuum with respect to the little rod, which has been used as a " distance "(line-interval).By choosing one corner of a square as " origin" I can characterise every other cornerof a square with reference to this origin by means of two numbers.I only need state how many rodsI must pass over when, starting from the origin, I proceed towards the " right " and then " upwards,"in order to arrive at the corner of the square under consideration.These two numbers are then the "Cartesian co-ordinates " of this corner with reference to the " Cartesian co-ordinate system" whichis determined by the arrangement of little rods.By making use of the following modification of this abstract experiment, we recognise that theremust also be cases in which the experiment would be unsuccessful.We shall suppose that the rods" expand " by in amount proportional to the increase of temperature.We heat the central part of themarble slab, but not the periphery, in which case two of our little rods can still be brought intocoincidence at every position on the table.But our construction of squares must necessarily comeinto disorder during the heating, because the little rods on the central region of the table expand,whereas those on the outer part do not.With reference to our little rods defined as unit lengths the marble slab is no longer aEuclidean continuum, and we are also no longer in the position of defining Cartesian co-ordinates51 Relativity: The Special and General Theorydirectly with their aid, since the above construction can no longer be carried out.But since there areother things which are not influenced in a similar manner to the little rods (or perhaps not at all) bythe temperature of the table, it is possible quite naturally to maintain the point of view that themarble slab is a " Euclidean continuum." This can be done in a satisfactory manner by making amore subtle stipulation about the measurement or the comparison of lengths.But if rods of every kind (i.e.of every material) were to behave in the same way as regards theinfluence of temperature when they are on the variably heated marble slab, and if we had no othermeans of detecting the effect of temperature than the geometrical behaviour of our rods inexperiments analogous to the one described above, then our best plan would be to assign thedistance one to two points on the slab, provided that the ends of one of our rods could be made tocoincide with these two points ; for how else should we define the distance without our proceedingbeing in the highest measure grossly arbitrary ? The method of Cartesian coordinates must then bediscarded, and replaced by another which does not assume the validity of Euclidean geometry forrigid bodies.1) The reader will notice that the situation depicted here corresponds to the onebrought about by the general postitlate of relativity (Section 23).Next: Gaussian Co-ordinatesFootnotes1)Mathematicians have been confronted with our problem in the following form.If we are given asurface (e.g.an ellipsoid) in Euclidean three-dimensional space, then there exists for this surface atwo-dimensional geometry, just as much as for a plane surface.Gauss undertook the task oftreating this two-dimensional geometry from first principles, without making use of the fact that thesurface belongs to a Euclidean continuum of three dimensions.If we imagine constructions to bemade with rigid rods in the surface (similar to that above with the marble slab), we should find thatdifferent laws hold for these from those resulting on the basis of Euclidean plane geometry.Thesurface is not a Euclidean continuum with respect to the rods, and we cannot define Cartesianco-ordinates in the surface
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