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.This was possible on the basis of the law of theconstancy of the velocity of tight.But according to Section 21 the general theory of relativity cannotretain this law.On the contrary, we arrived at the result that according to this latter theory thevelocity of light must always depend on the co-ordinates when a gravitational field is present.Inconnection with a specific illustration in Section 23, we found that the presence of a gravitationalfield invalidates the definition of the coordinates and the ifine, which led us to our objective in thespecial theory of relativity.In view of the resuIts of these considerations we are led to the conviction that, according to thegeneral principle of relativity, the space-time continuum cannot be regarded as a Euclidean one,but that here we have the general case, corresponding to the marble slab with local variations oftemperature, and with which we made acquaintance as an example of a two-dimensionalcontinuum.Just as it was there impossible to construct a Cartesian co-ordinate system from equalrods, so here it is impossible to build up a system (reference-body) from rigid bodies and clocks,which shall be of such a nature that measuring-rods and clocks, arranged rigidly with respect toone another, shaIll indicate position and time directly.Such was the essence of the difficulty withwhich we were confronted in Section 23.But the considerations of Sections 25 and 26 show us the way to surmount this difficulty.We referthe fourdimensional space-time continuum in an arbitrary manner to Gauss co-ordinates.Weassign to every point of the continuum (event) four numbers, x1, x2, x3, x4 (co-ordinates), whichhave not the least direct physical significance, but only serve the purpose of numbering the pointsof the continuum in a definite but arbitrary manner.This arrangement does not even need to be ofsuch a kind that we must regard x1, x2, x3, as "space" co-ordinates and x4, as a " time "co-ordinate.The reader may think that such a description of the world would be quite inadequate.What does itmean to assign to an event the particular co-ordinates x1, x2, x3, x4, if in themselves theseco-ordinates have no significance ? More careful consideration shows, however, that this anxiety isunfounded.Let us consider, for instance, a material point with any kind of motion.If this point hadonly a momentary existence without duration, then it would to described in space-time by a singlesystem of values x1, x2, x3, x4.Thus its permanent existence must be characterised by an infinitelylarge number of such systems of values, the co-ordinate values of which are so close together asto give continuity; corresponding to the material point, we thus have a (uni-dimensional) line in thefour-dimensional continuum.In the same way, any such lines in our continuum correspond tomany points in motion.The only statements having regard to these points which can claim aphysical existence are in reality the statements about their encounters.In our mathematicaltreatment, such an encounter is expressed in the fact that the two lines which represent themotions of the points in question have a particular system of co-ordinate values, x1, x2, x3, x4, incommon.After mature consideration the reader will doubtless admit that in reality such encountersconstitute the only actual evidence of a time-space nature with which we meet in physicalstatements.57Relativity: The Special and General TheoryWhen we were describing the motion of a material point relative to a body of reference, we statednothing more than the encounters of this point with particular points of the reference-body.We canalso determine the corresponding values of the time by the observation of encounters of the bodywith clocks, in conjunction with the observation of the encounter of the hands of clocks withparticular points on the dials.It is just the same in the case of space-measurements by means ofmeasuring-rods, as a litttle consideration will show.The following statements hold generally : Every physical description resolves itself into a number ofstatements, each of which refers to the space-time coincidence of two events A and B.In terms ofGaussian co-ordinates, every such statement is expressed by the agreement of their fourco-ordinates x1, x2, x3, x4 [ Pobierz całość w formacie PDF ]