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57
Relativity: The Special and General Theory
Albert Einstein: Relativity
Part II: The General Theory of Relativity
The Space?Time Continuum
of the General Theory
of Realtiivty
is Not a
Eculidean Continuum
In the first part of this book we were able to make use of space?time co?ordinates which allowed of
a simple and direct physical interpretation, and which, according to Section 26, can be regarded as
four?dimensional Cartesian co?ordinates. This was possible on the basis of the law of the
constancy of the velocity of tight. But according to
Section 21 the general theory 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 field 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
temperature, and with which we made acquaintance as an example of a two?dimensional
continuum. Just as it was there impossible to construct a Cartesian co?ordinate system from equal
rods, so here it is impossible to build up a system (reference?body) from rigid bodies and clocks,
which shall be of such a nature that measuring?rods and clocks, arranged rigidly with respect to
one another, shaIll indicate position 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, x1, x2, x3, 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 x1, x2, x3, 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 x1, x2, x3, 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 motion. If this point had
only a momentary existence without duration, then it would to described in space?time by a single
system of values x1, x2, x3, 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, x1, x2, x3, 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.