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51
Relativity: The Special and General Theory
Albert Einstein: Relativity
Part II: The General Theory of Relativity
Euclidean and Non?Euclidean Continuum
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 square), 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 wander 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 temperature. We heat 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 position 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