# PlanetPhysics/Space Time Continuum of the General Theory of Relativity Is Not a Euclidean Continuum

\subsection{The Space-Time Continuum of the General Theory of Relativity is Not a Euclidean Continuum} From Relativity: The Special and General Theory by Albert Einstein In the first part of this book we were able to make use of space-time co-ordinates which allowed of a simple and direct physical interpretation, and which, according to section 26, can be regarded as four-dimensional Cartesian co-ordinates. This was possible on the basis of the law of the constancy of the velocity of tight. But according to Section 21 the general theory of relativity cannot retain this law. On the contrary, we arrived at the result that according to this latter theory the velocity of light must always depend on the co-ordinates when a gravitational field is present. In connection with a specific illustration in Section 23, we found that the presence of a gravitational field invalidates the definition of the coordinates and the ifine, which led us to our objective in the special theory of relativity.
In view of the resuIts of these considerations we are led to the conviction that, according to the general 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 of temperature, and with which we made acquaintance as an example of a two-dimensional continuum. Just as it was there impossible to construct a Cartesian co-ordinate system from equal rods, 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 to one another, shaIll indicate position and time directly. Such was the essence of the difficulty with which we were confronted in Section 23.
But the considerations of Sections 25 and 26 show us the way to surmount this difficulty. We refer the fourdimensional space-time continuum in an arbitrary manner to Gauss co-ordinates. We assign to every point of the continuum (event) four numbers, </math>x_1, x_2, x_3, x_4$(co-ordinates),whichhavenottheleastdirectphysicalsignificance,butonlyservethepurposeofnumberingthepointsofthecontinuuminadefinitebutarbitrarymanner.Thisarrangementdoesnotevenneedtobeofsuchakindthatwemustregard$x_1, x_2, x_3Failed to parse (syntax error): {\displaystyle , as "space" co-ordinates and } x_4Failed to parse (syntax error): {\displaystyle , as a ``time" co-ordinate. The reader may think that such a description of the world would be quite inadequate. What does it mean to assign to an event the particular co-ordinates <math>x_1, x_2, x_3, x_4} , if in themselves these co-ordinates have no significance? More careful consideration shows, however, that this anxiety is unfounded. Let us consider, for instance, a material point with any kind of motion. If this point had only a momentary existence without duration, then it would to described in space-time by a single system of values </math>x_1, x_2, x_3, x_4$.Thusitspermanentexistencemustbecharacterisedbyaninfinitelylargenumberofsuchsystemsofvalues,theco-ordinatevaluesofwhicharesoclosetogetherastogivecontinuity;correspondingtothematerialpoint,wethushavea(uni-dimensional)lineinthefour-dimensionalcontinuum.Inthesameway,anysuchlinesinourcontinuumcorrespondtomanypointsinmotion.Theonlystatementshavingregardtothesepointswhichcanclaimaphysicalexistenceareinrealitythestatementsabouttheirencounters.Inourmathematicaltreatment,suchanencounterisexpressedinthefactthatthetwolineswhichrepresentthemotionsofthepointsinquestionhaveaparticularsystemofco-ordinatevalues,$x_1, x_2, x_3, x_4$,incommon.Aftermatureconsiderationthereaderwilldoubtlessadmitthatinrealitysuchencountersconstitutetheonlyactualevidenceofatime-spacenaturewithwhichwemeetinphysicalstatements.Whenweweredescribingthemotionofamaterialpointrelativetoabodyofreference,westatednothingmorethantheencountersofthispointwithparticularpointsofthereference-body.Wecanalsodeterminethecorrespondingvaluesofthetimebytheobservationofencountersofthebodywithclocks,inconjunctionwiththeobservationoftheencounterofthehandsofclockswithparticularpointsonthedials.Itisjustthesameinthecaseofspace-measurementsbymeansofmeasuring-rods,asalitttleconsiderationwillshow.Thefollowingstatementsholdgenerally:Everyphysicaldescriptionresolvesitselfintoanumberofstatements,eachofwhichreferstothespace-timecoincidenceoftwoeventsAandB.Intermsof[[../GaussianCoOrdinates/|GaussianCo-Ordinates]],everysuchstatementisexpressedbytheagreementoftheirfourco-ordinates<math>x_{1},x_{2},x_{3},x_{4}$. Thus in reality, the description of the time-space continuum by means of Gauss co-ordinates completely replaces the description with the aid of a body of reference, without suffering from the defects of the latter mode of description; it is not tied down to the Euclidean character of the continuum which has to be represented.