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

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%%% Primary Title: The Space-Time Continuum of the General Theory of Relativity is Not a Euclidean Continuum
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 \subsection{The Space-Time Continuum of the General Theory of Relativity is Not a
Euclidean Continuum}
From \htmladdnormallink{Relativity: The Special and General Theory}{http://planetphysics.us/encyclopedia/SpecialTheoryOfRelativity.html} by \htmladdnormallink{Albert Einstein}{http://planetphysics.us/encyclopedia/AlbertEinstein.html}
In the first part of this book we were able to make use of \htmladdnormallink{space-time}{http://planetphysics.us/encyclopedia/SR.html} co-ordinates which allowed of a simple and direct physical
interpretation, and which, according to \htmladdnormallink{section}{http://planetphysics.us/encyclopedia/IsomorphicObjectsUnderAnIsomorphism.html} 26, can be regarded
as four-dimensional Cartesian co-ordinates. This was possible on the
basis of the law of the constancy of the \htmladdnormallink{velocity}{http://planetphysics.us/encyclopedia/Velocity.html} of tight. But
according to Section 21 the \htmladdnormallink{general theory}{http://planetphysics.us/encyclopedia/GeneralTheory.html} 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 \htmladdnormallink{field}{http://planetphysics.us/encyclopedia/CosmologicalConstant2.html} 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 \htmladdnormallink{temperature}{http://planetphysics.us/encyclopedia/BoltzmannConstant.html}, and with which we made acquaintance
as an example of a \htmladdnormallink{two-dimensional}{http://planetphysics.us/encyclopedia/CoriolisEffect.html} continuum. Just as it was there
impossible to construct a Cartesian co-ordinate \htmladdnormallink{system}{http://planetphysics.us/encyclopedia/SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence.html} from equal
rods, so here it is impossible to build up a system (reference-body)
from \htmladdnormallink{rigid bodies}{http://planetphysics.us/encyclopedia/CenterOfGravity.html} and clocks, which shall be of such a nature that
measuring-rods and clocks, arranged rigidly with respect to one
another, shaIll indicate \htmladdnormallink{position}{http://planetphysics.us/encyclopedia/Position.html} 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, $x_1, x_2, x_3,
x_4$ (co-ordinates), which have not the least direct physical
significance, but only serve the purpose of numbering the points of
the continuum in a definite but arbitrary manner. This arrangement
does not even need to be of such a kind that we must regard $x_1,
x_2, x_3$, as ``space" co-ordinates and $x_4$, 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 $x_1, x_2, x_3, x_4$, 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 \htmladdnormallink{motion}{http://planetphysics.us/encyclopedia/CosmologicalConstant.html}. If this point had
only a momentary existence without duration, then it would to
described in space-time by a single system of values $x_1, x_2, x_3,
x_4$. Thus its permanent existence must be characterised by an
infinitely large number of such systems of values, the co-ordinate
values of which are so close together as to give continuity;
corresponding to the material point, we thus have a (uni-dimensional)
line in the four-dimensional continuum. In the same way, any such
lines in our continuum correspond to many points in motion. The only
statements having regard to these points which can claim a physical
existence are in reality the statements about their encounters. In our
mathematical treatment, such an encounter is expressed in the fact
that the two lines which represent the motions of the points in
question have a particular system of co-ordinate values, $x_1, x_2,
x_3, x_4$, in common. After mature consideration the reader will
doubtless admit that in reality such encounters constitute the only
actual evidence of a time-space nature with which we meet in physical
statements.

When we were describing the motion of a material point relative to a
body of reference, we stated nothing more than the encounters of this
point with particular points of the reference-body. We can also
determine the corresponding values of the time by the observation of
encounters of the body with clocks, in conjunction with the
observation of the encounter of the hands of clocks with particular
points on the dials. It is just the same in the case of
space-measurements by means of measuring-rods, as a litttle
consideration will show.

The following statements hold generally: Every physical description
resolves itself into a number of statements, each of which refers to
the space-time coincidence of two events A and B. In terms of \htmladdnormallink{Gaussian Co-Ordinates}{http://planetphysics.us/encyclopedia/GaussianCoOrdinates.html}, every such statement is expressed by the agreement of
their four co-ordinates $x_1, x_2, x_3, x_4$. 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.

\subsection{References}
This article is derived from the Einstein Reference Archive (marxists.org) 1999, 2002. \htmladdnormallink{Einstein Reference Archive}{http://www.marxists.org/reference/archive/einstein/index.htm} which is under the FDL copyright.

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