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.Thus on our circular disc, or, to make the casemore general, in every gravitational field, a clock will go more quickly or less quickly, according tothe position in which the clock is situated (at rest).For this reason it is not possible to obtain areasonable definition of time with the aid of clocks which are arranged at rest with respect to thebody of reference.A similar difficulty presents itself when we attempt to apply our earlier definitionof simultaneity in such a case, but I do not wish to go any farther into this question.49Relativity: The Special and General TheoryMoreover, at this stage the definition of the space co-ordinates also presents insurmountabledifficulties.If the observer applies his standard measuring-rod (a rod which is short as comparedwith the radius of the disc) tangentially to the edge of the disc, then, as judged from the Galileiansystem, the length of this rod will be less than I, since, according to Section 12, moving bodiessuffer a shortening in the direction of the motion.On the other hand, the measaring-rod will notexperience a shortening in length, as judged from K, if it is applied to the disc in the direction of theradius.If, then, the observer first measures the circumference of the disc with his measuring-rodand then the diameter of the disc, on dividing the one by the other, he will not obtain as quotient thefamiliar number À = 3.14., but a larger number,2) whereas of course, for a disc which is at restwith respect to K, this operation would yield À exactly.This proves that the propositions ofEuclidean geometry cannot hold exactly on the rotating disc, nor in general in a gravitational field,at least if we attribute the length I to the rod in all positions and in every orientation.Hence the ideaof a straight line also loses its meaning.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 [ Pobierz caÅ‚ość w formacie PDF ]