§ 1. Definition of Synchronism.
Let us have a co-ordinate system, in which the Newtonian equations hold.
For distinguishing this system from another which will be introduced
hereafter, we shall always call it “the stationary system.”
If a material point be at rest in this system, then its position in this
system can be found out by a measuring rod, and can be expressed by the
methods of Euclidean Geometry, or in Cartesian co-ordinates.
If we wish to describe the motion of a material point, the values of its
coordinates must be expressed as functions of time. It is always to be
borne in mind that _such a mathematical definition has a physical sense,
only when we have a clear notion of what is meant by time. We have to
take into consideration the fact that those of our conceptions, in which
time plays a part, are always conceptions of synchronism._ For example,
we say that a train arrives here at 7 o’clock; this means that the exact
pointing of the little hand of my watch to 7, and the arrival of the
train are synchronous events.
It may appear that all difficulties connected with the definition of
time can be removed when in place of time, we substitute the position of
the little hand of my watch. Such a definition is in fact sufficient,
when it is required to define time exclusively for the place at which
the clock is stationed. But the definition is not sufficient when it is
required to connect by time events taking place at different
stations,—or what amounts to the same thing,—to estimate by means of
time (zeitlich werten) the occurrence of events, which take place at
stations distant from the clock.
Now with regard to this attempt;—the time-estimation of events, we can
satisfy ourselves in the following manner. Suppose an observer—who is
stationed at the origin of coordinates with the clock—associates a ray
of light which comes to him through space, and gives testimony to the
event of which the time is to be estimated,—with the corresponding
position of the hands of the clock. But such an association has this
defect,—it depends on the position of the observer provided with the
clock, as we know by experience. We can attain to a more practicable
result by the following treatment.
If an observer be stationed at A with a clock, he can estimate the time
of events occurring in the immediate neighbourhood of A, by looking for
the position of the hands of the clock, which are synchronous with the
event. If an observer be stationed at B with a clock,—we should add that
the clock is of the same nature as the one at A,—he can estimate the
time of events occurring about B. But without further premises, it is
not possible to compare, as far as time is concerned, the events at B
with the events at A. We have hitherto an A-time, and a B-time, but no
time common to A and B. This last time (_i.e._, common time) can be
defined, if we establish by definition that the time which light
requires in travelling from A to B is equivalent to the time which light
requires in travelling from B to A. For example, a ray of light proceeds
from A at A-time t_{A} towards B, arrives and is reflected from B at
B-time t_{B}, and returns to A at A-time t′_{A}. According to the
definition, both clocks are synchronous, if
t_{B} - t_{A} = t′_{A} - t_{B}.
We assume that this definition of synchronism is possible without
involving any inconsistency, for any number of points, therefore the
following relations hold:—
1. If the clock at B be synchronous with the clock at A, then the clock
at A is synchronous with the clock at B.
2. If the clock at A as well as the clock at B are both synchronous with
the clock at C, then the clocks at A and B are synchronous.
Thus with the help of certain physical experiences, we have established
what we understand when we speak of clocks at rest at different
stations, and synchronous with one another; and thereby we have arrived
at a definition of synchronism and time.