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Relativity: The Special and General Theory

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" 1)), 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 identical construction ; the man at the
railway-carriage window is holding one of them, and the man on the footpath the other.
Each of the observers determines the position on his own reference-body occupied by the
stone at each tick of the clock he is holding in his hand. In this connection we have not
taken account of the inaccuracy involved by the finiteness of the velocity of propagation
of light. With this and with a second difficulty prevailing here we shall have to deal in
detail later.
Notes
1) That is, a curve along which the body moves.
 
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