\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
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
x_1, x_2,
x_3, x_4![{\displaystyle ,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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d40327e0a5701d9541bff0cb9b9c034703674c86)
. 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.
