Relativity - The Special and General Theory | Page 4

rigidly attached to a
rigid body. Referred to a system of co-ordinates, the scene of any event will be
determined (for the main part) by the specification of the lengths of the three
perpendiculars or co-ordinates (x, y, z) which can be dropped from the scene of the event
to those three plane surfaces. The lengths of these three perpendiculars can be determined
by a series of manipulations with rigid measuring-rods performed according to the rules

and methods laid down by Euclidean geometry.
In practice, the rigid surfaces which constitute the system of co-ordinates are generally
not available ; furthermore, the magnitudes of the co-ordinates are not actually
determined by constructions with rigid rods, but by indirect means. If the results of
physics and astronomy are to maintain their clearness, the physical meaning of
specifications of position must always be sought in accordance with the above
considerations. ***
We thus obtain the following result: Every description of events in space involves the use
of a rigid body to which such events have to be referred. The resulting relationship takes
for granted that the laws of Euclidean geometry hold for "distances;" the "distance" being
represented physically by means of the convention of two marks on a rigid body.
Notes
* Here we have assumed that there is nothing left over i.e. that the measurement gives a
whole number. This difficulty is got over by the use of divided measuring-rods, the
introduction of which does not demand any fundamentally new method.
**A Einstein used "Potsdamer Platz, Berlin" in the original text. In the authorised
translation this was supplemented with "Tranfalgar Square, London". We have changed
this to "Times Square, New York", as this is the most well known/identifiable location to
English speakers in the present day. [Note by the janitor.]
**B It is not necessary here to investigate further the significance of the expression
"coincidence in space." This conception is sufficiently obvious to ensure that differences
of opinion are scarcely likely to arise as to its applicability in practice.
*** A refinement and modification of these views does not become necessary until we
come to deal with the general theory of relativity, treated in the second part of this book.

SPACE AND TIME IN CLASSICAL MECHANICS
The purpose of mechanics is to describe how bodies change their position in space with
"time." I should load my conscience with grave sins against the sacred spirit of lucidity
were I to formulate the aims of mechanics in this way, without serious reflection and
detailed explanations. Let us proceed to disclose these sins.
It is not clear what is to be understood here by "position" and "space." I stand at the
window of a railway carriage which is travelling uniformly, and drop a stone on the
embankment, without throwing it. Then, disregarding the influence of the air resistance, I
see the stone descend in a straight line. A pedestrian who observes the misdeed from the
footpath notices that the stone falls to earth in a parabolic curve. I now ask: Do the
"positions" traversed by the stone lie "in reality" on a straight line or on a parabola?
Moreover, what is meant here by motion "in space" ? From the considerations of the

previous section the answer is self-evident. In the first place we entirely shun the vague
word "space," of which, we must honestly acknowledge, we cannot form the slightest
conception, and we replace it by "motion relative to a practically rigid body of reference."
The positions relative to the body of reference (railway carriage or embankment) have
already been defined in detail in the preceding section. If instead of " body of reference "
we insert " system of co-ordinates," which is a useful idea for mathematical description,
we are in a position to say : The stone traverses a straight line relative to a system of
co-ordinates rigidly attached to the carriage, but relative to a system of co-ordinates
rigidly attached to the ground (embankment) it describes a parabola. With the aid of this
example it is clearly seen that there is no such thing as an independently existing
trajectory (lit. "path-curve"*), but only a trajectory relative to a particular body of
reference.
In order to have a complete description of the motion, we must specify how the body
alters its position with time ; i.e. for every point on the trajectory it must be stated at what
time the body is situated there. These data must be supplemented by such a definition of
time that, in virtue of this definition, these time-values can be regarded essentially as
magnitudes (results of measurements) capable of observation. If we take our stand on the
ground of classical mechanics, we can satisfy this requirement for our illustration in the
following manner. We imagine two clocks of