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54
Relativity: The Special and General Theory
1
4
1
2
3
4
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 samply
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, x1, x2, x3, 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 P¹, this distance being measurable and well defined from
a physical point of view, then the following formula holds:
ds² = g11dx
2
+ 2g12dx1dx2 . . . . g
44
dx
2
,
where the magnitudes g11, 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 x1 . . x
4
.
with the
points of the continuum so that we have simply
ds² = dx
2
+ dx
2
+ dx
2
+ dx
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 ds² 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.
Next: The Space?Time Continuum of the Speical Theory of Relativity Considered as a Euclidean
Continuum
Relativity: The Special and General Theory