Albert Einstein: Relativity

Part II: The General Theory of Relativity

Hitherto I have purposely refrained from speaking about the physical interpretation of space- and time-data in the case of the general theory of relativity. As a consequence, I am guilty of a certain slovenliness of treatment, which, as we know from the special theory of relativity, is far from being unimportant and pardonable. It is now high time that we remedy this defect; but I would mention at the outset, that this matter lays no small claims on the patience and on the power of abstraction of the reader.

We start off again from quite special cases, which we
have frequently used before. Let us consider a space time
domain in which no gravitational field exists relative to a
reference-body K whose state of motion has been suitably
chosen. K is then a Galileian reference-body as regards the
domain considered, and the results of the special theory of
relativity hold relative to K. Let us supposse the same
domain referred to a second body of reference K^{1}, which is
rotating uniformly with respect to K. In order to fix our
ideas, we shall imagine K^{1} to be in the form of a plane
circular disc, which rotates uniformly in its own plane
about its centre. An observer who is sitting eccentrically
on the
disc K^{1} is sensible of a force which acts outwards in a radial
direction, and which would be interpreted as an effect of
inertia (centrifugal force) by an observer who was at rest
with respect to the original reference-body K. But the
observer on the disc may regard his disc as a reference-body
which is " at rest " ; on the basis of the general principle of
relativity he is justified in doing this. The force acting on
himself, and in fact on all other bodies which are at rest
relative to the disc, he regards as the effect of a
gravitational field. Nevertheless, the space-distribution of
this gravitational field is of a kind that would not be
possible on Newton's theory of gravitation.^{1)} But since the
observer believes in the general theory of relativity, this
does not disturb him; he is quite in the right when he
believes that a general law of gravitation can be formulated-
a law which not only explains the motion of the stars
correctly, but also the field of force experienced by himself.

The observer performs experiments on his circular disc
with clocks and measuring-rods. In doing so, it is his
intention to arrive at exact definitions for the signification
of time- and space-data with reference to the circular disc
K^{1}, these definitions being based on his observations. What
will be his experience in this enterprise ?

To start with, he places one of two identically
constructed clocks at the centre of the circular disc, and the
other on the edge of the disc, so that they are at rest
relative to it. We now ask ourselves whether both clocks
go at the same rate from the standpoint of the
non-rotating Galileian reference-body K. As judged from this
body, the clock at the centre of the disc has no velocity,
whereas the clock at the edge of the disc is in motion
relative to K in consequence of the rotation. According to a
result obtained in Section 12, it follows that the latter
clock goes at a rate permanently slower than that of the
clock at the centre of the circular disc, *i.e.* as observed from
K. It is obvious that the same effect would be noted by an
observer whom we will imagine sitting alongside his clock
at the centre of the circular disc. Thus on our circular disc,
or, to make the case more general, in every gravitational
field, a clock will go more quickly or less quickly,
according to the position in which the clock is situated (at
rest). For this reason it is not possible to obtain a
reasonable definition of time with the aid of clocks which
are arranged at rest with respect to the body of reference. A
similar difficulty presents itself when we attempt to apply
our earlier definition of simultaneity in such a case, but I do
not wish to go any farther into this question.

Moreover, at this stage the definition of the space co-ordinates also presents insurmountable difficulties. If the
observer applies his standard measuring-rod (a rod which is
short as compared with the radius of the disc) tangentially
to the edge of the disc, then, as judged from the Galileian
system, the length of this rod will be less than I, since,
according to Section 12, moving bodies suffer a shortening
in the direction of the motion. On the other hand, the
measaring-rod will not experience a shortening in length, as
judged from K, if it is applied to the disc in the direction of
the radius. If, then, the observer first measures the
circumference of the disc with his measuring-rod and then
the diameter of the
disc, on dividing the one by the other, he will not obtain
as quotient the familiar number π = 3.14 . . ., but a larger
number,^{2)} whereas of course, for a disc which is at rest with
respect to K, this operation would yield π exactly. This
proves that the propositions of Euclidean geometry cannot
hold exactly on the rotating disc, nor in general in a
gravitational field, at least if we attribute the length I to
the rod in all positions and in every orientation. Hence the
idea of a straight line also loses its meaning. We are
therefore not in a position to define exactly the co-ordinates
x, y, z relative to the disc by means of the method used in
discussing the special theory, and as long as the co-
ordinates and times of events have not been defined, we
cannot assign an exact meaning to the natural laws in
which these occur.

Thus all our previous conclusions based on general relativity would appear to be called in question. In reality we must make a subtle detour in order to be able to apply the postulate of general relativity exactly. I shall prepare the reader for this in the following paragraphs.

Next: Euclidean and Non-Euclidean Continuum

^{1)}
The field disappears at the centre of the disc and increases proportionally to the distance from the centre as we proceed outwards.

^{2)}
Throughout this consideration we have to use the Galileian
(non-rotating) system K as reference-body, since we may
only assume the validity of the results of the special theory of
relativity relative to K (relative to K^{1} a gravitational field
prevails).