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%%% Primary Title: Euclidean and Non-Euclidean Continuum
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\begin{document}
\subsection{Euclidean and Non-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}
The surface of a marble table is spread out in front of me. I can get
from any one point on this table to any other point by passing
continuously from one point to a ``neighbouring'' one, and repeating
this process a (large) number of times, or, in other words, by going
from point to point without executing ``jumps." I am sure the reader
will appreciate with sufficient clearness what I mean here by
``neighbouring'' and by ``jumps'' (if he is not too pedantic). We
express this property of the surface 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 lengths being 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 these little rods on the marble slab so that they constitute a
quadrilateral figure (a \htmladdnormallink{square}{http://planetphysics.us/encyclopedia/PiecewiseLinear.html}), the diagonals of which are equally
long. To ensure the equality of the diagonals, we make use of a little
testing-rod. To this square we add similar ones, each of which has one
rod in common with the first. We proceed in like manner with each of
these squares until finally the whole marble slab is laid out with
squares. The arrangement is such, that each side of a square belongs
to two squares and each corner to four squares.
It is a veritable wonder that we can carry out this business without
getting into the greatest difficulties. We only need to think of the
following. If at any moment three squares meet at a corner, then two
sides of the fourth square are already laid, and, as a consequence,
the arrangement of the remaining two sides of the square is already
completely determined. But I am now no longer able to adjust the
quadrilateral so that its diagonals may be equal. If they are equal of
their own accord, then this is an especial favour of the marble slab
and of the little rods, about which I can only be thankfully
surprised. We must experience many such surprises if the construction
is to be successful.
If everything has really gone smoothly, then I say that the points of
the marble slab constitute a Euclidean continuum with respect to the
little rod, which has been used as a ``distance'' (line-interval). By
choosing one corner of a square as ``origin" I can characterise every
other corner of a square with reference to this origin by means of two
numbers. I only need state how many rods I must pass over when,
starting from the origin, I proceed towards the ``right'' and then
``upwards," in order to arrive at the corner of the square under
consideration. These two numbers are then the ``Cartesian co-ordinates"
of this corner with reference to the ``Cartesian co-ordinate system"
which is determined by the arrangement of little rods.
By making use of the following modification of this abstract
experiment, we recognise that there must also be cases in which the
experiment would be unsuccessful. We shall suppose that the rods
``expand'' by in amount proportional to the increase of \htmladdnormallink{temperature}{http://planetphysics.us/encyclopedia/BoltzmannConstant.html}. We
\htmladdnormallink{heat}{http://planetphysics.us/encyclopedia/Heat.html} the central part of the marble slab, but not the periphery, in
which case two of our little rods can still be brought into
coincidence at every \htmladdnormallink{position}{http://planetphysics.us/encyclopedia/Position.html} on the table. But our construction of
squares must necessarily come into disorder during the heating,
because the little rods on the central region of the table expand,
whereas those on the outer part do not.
With reference to our little rods---defined as unit lengths---the
marble slab is no longer a Euclidean continuum, and we are also no
longer in the position of defining Cartesian co-ordinates directly
with their aid, since the above construction can no longer be carried
out. But since there are other things which are not influenced in a
similar manner to the little rods (or perhaps not at all) by the
temperature of the table, it is possible quite naturally to maintain
the point of view that the marble slab is a ``Euclidean continuum."
This can be done in a satisfactory manner by making a more subtle
stipulation about the measurement or the comparison of lengths.
But if rods of every kind ({\it i.e.} of every material) were to behave in
the same way as regards the influence of temperature when they are on
the variably heated marble slab, and if we had no other means of
detecting the effect of temperature than the geometrical behaviour of
our rods in experiments analogous to the one described above, then our
best plan would be to assign the distance one to two points on the
slab, provided that the ends of one of our rods could be made to
coincide with these two points; for how else should we define the
distance without our proceeding being in the highest measure grossly
arbitrary? The method of Cartesian coordinates must then be
discarded, and replaced by another which does not assume the validity
of Euclidean geometry for \htmladdnormallink{rigid bodies}{http://planetphysics.us/encyclopedia/CenterOfGravity.html}.\footnotemark\ The reader will notice
that the situation depicted here corresponds to the one brought about
by the general postitlate of relativity (\htmladdnormallink{section}{http://planetphysics.us/encyclopedia/IsomorphicObjectsUnderAnIsomorphism.html} 23).
\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.
\footnotetext{Mathematicians have been confronted with our problem in the
following form. If we are given a surface ({\it e.g.} an ellipsoid) in
Euclidean three-dimensional space, then there exists for this surface
a \htmladdnormallink{two-dimensional}{http://planetphysics.us/encyclopedia/CoriolisEffect.html} geometry, just as much as for a plane surface. Gauss
undertook the task of treating this two-dimensional geometry from
first principles, without making use of the fact that the surface
belongs to a Euclidean continuum of three dimensions. If we imagine
constructions to be made with rigid rods in the surface (similar to
that above with the marble slab), we should find that different laws
hold for these from those resulting on the basis of Euclidean plane
geometry. The surface is not a Euclidean continuum with respect to the
rods, and we cannot define Cartesian co-ordinates in the surface.
Gauss indicated the principles according to which we can treat the
geometrical relationships in the surface, and thus pointed out the way
to the method of Riemman of treating multi-dimensional, non-Euclidean
continuum. Thus it is that mathematicians long ago solved the formal
problems to which we are led by the general postulate of relativity.}
\end{document}
