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%%% Primary Title: Gaussian Co-Ordinates
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\begin{document}
\subsection{Gaussian Co-Ordinates}
From \htmladdnormallink{Relativity: The Special and General Theory}{http://planetphysics.us/encyclopedia/SpecialTheoryOfRelativity.html} by \htmladdnormallink{Albert Einstein}{http://planetphysics.us/encyclopedia/AlbertEinstein.html}
\begin{figure}[bthp]
\centering
\caption{}
\label{fig:4}
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According to Gauss, this combined analytical and geometrical mode of
handling the problem can be arrived at in the following way. We
imagine a \htmladdnormallink{system}{http://planetphysics.us/encyclopedia/SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence.html} of arbitrary curves (see Fig. \ref{fig:4}) drawn on the surface
of the table. These we designate as $u$-curves, and we indicate each of
them by means of a number. The curves $u=1$, $u=2$ and $u=3$ are drawn in
the \htmladdnormallink{diagram}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html}. Between the curves $u=1$ and $u=2$ we must imagine an
infinitely large number to be drawn, all of which correspond to real
numbers lying between 1 and 2. fig. 04 We have then a system of
u-curves, and this ``infinitely dense" system covers the whole surface
of the table. These u-curves must not intersect each other, and
through each point of the surface one and only one curve must pass.
Thus a perfectly definite value of u belongs to every point on the
surface of the marble slab. In like manner we imagine a system of
v-curves drawn on the surface. These satisfy the same conditions as
the u-curves, they are provided with numbers in a corresponding
manner, and they may likewise be of arbitrary shape. It follows that a
value of u and a value of v belong to every point on the surface of
the table. We call these two numbers the co-ordinates of the surface
of the table (Gaussian co-ordinates). For example, the point $P$ in the
diagram has the Gaussian co-ordinates $u=3$, $v=1$. Two neighbouring
points $P$ and $P_1$ on the surface then correspond to the co-ordinates
\begin{eqnarray*}
P: & u ~~,~~v \\
P': & u + du , v + dv
\end{eqnarray*}
where $du$ and $dv$ signify very small numbers. In a similar manner we may
indicate the distance (line-interval) between $P$ and $P_1$, as measured
with a little rod, by means of the very small number $ds$. Then
according to Gauss we have
$$ds_2 = g_{11}du^2 + 2g_{12}dudv = g_{22}dv^2$$
\noindent where $g_{11}, g_{12}, g_{22}$, are magnitudes which depend in a perfectly
definite way on $u$ and $v$. The magnitudes $g_{11}$, $g_{12}$ and $g_{22}$,
determine the behaviour of the rods relative to the $u$-curves and
$v$-curves, and thus also relative to the surface of the table. For the
case in which the points of the surface considered form a Euclidean
continuum with reference to the measuring-rods, but only in this case,
it is possible to draw the $u$-curves and $v$-curves and to attach numbers
to them, in such a manner, that we simply have:
$$ds^2 = du^2 + dv^2$$
Under these conditions, the $u$-curves and $v$-curves are straight lines
in the sense of Euclidean geometry, and they are perpendicular to each
other. Here the Gaussian coordinates are simply Cartesian ones. It is
clear that Gauss co-ordinates are nothing more than an association of
two sets of numbers with the points of the surface considered, of such
a nature that numerical values differing very slightly from each other
are associated with neighbouring points ``in space."
So far, these considerations hold for a continuum of two dimensions.
But the Gaussian method can be applied also to a continuum of three,
four or more dimensions. If, for instance, a continuum of four
dimensions be supposed available, we may represent it in the following
way. With every point of the continuum, we associate arbitrarily four
numbers, $x_1, x_2, x_3, x_4$, which are known as ``co-ordinates."
Adjacent points correspond to adjacent values of the coordinates. If a
distance $ds$ is associated with the adjacent points $P$ and $P_1$, this
distance being measurable and well defined from a physical point of
view, then the following formula holds:
$$ds^2 = g_{11}dx_1^2 + 2g_{12}dx_1dx_2 . . . . g_{44}dx_4^2$$
\noindent where the magnitudes g[11], etc., have values which vary with the
position in the continuum. Only when the continuum is a Euclidean one
is it possible to associate the co-ordinates $x_1 \ldots x_4$. with the
points of the continuum so that we have simply
$$ds2 = dx_1^2 + dx_2^2 + dx_3^2 + dx_4^2$$
In this case relations hold in the four-dimensional continuum which
are analogous to those holding in our three-dimensional measurements.
However, the Gauss treatment for $ds^2$ which we have given above is not
always possible. It is only possible when sufficiently small regions
of the continuum under consideration may be regarded as Euclidean
continua. For example, this obviously holds in the case of the marble
slab of the table and local variation of temperature. The temperature
is practically constant for a small part of the slab, and thus the
geometrical behaviour of the rods is almost as it ought to be
according to the rules of Euclidean geometry. Hence the imperfections
of the construction of squares in the previous section do not show
themselves clearly until this construction is extended over a
considerable portion of the surface of the table.
We can sum this up as follows: Gauss invented a method for the
mathematical treatment of continua in general, in which
``size-relations''`(``distances'' between neighbouring points) are
defined. To every point of a continuum are assigned as many numbers
(Gaussian coordinates) as the continuum has dimensions. This is done
in such a way, that only one meaning can be attached to the
assignment, and that numbers (Gaussian coordinates) which differ by an
indefinitely small amount are assigned to adjacent points. The
Gaussian coordinate system is a logical generalisation of the
Cartesian co-ordinate system. It is also applicable to non-Euclidean
continua, but only when, with respect to the defined ``size'' or
``distance,'' small parts of the continuum under consideration behave
more nearly like a Euclidean system, the smaller the part of the
continuum under our notice.
\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.
\end{document}
