§ 1. Definition of Simultaneity.
Let us have a co-ordinate system, in which the Newtonian equations hold. For verbally distinguishing this system from another which will be introduced hereafter, and for clarification of the idea, we shall call it the "stationary system."
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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 simultaneous with the event. If a clock be stationed also at point B in space, — we should add that "the clock is exactly of the same nature as the one at A", — then the chronological evaluations of the events occurring in the immediate vicinity of B, is possible for an observer located in B. But without further premises, it is not possible to chronologically compare the events at B with the events at A. We have hitherto only an "A-time", and a "B-time", but no "time" common to A and B. This last time (i.e., common time) can now be defined, however, if we establish by definition that the "time" which light requires in traveling from A to B is equivalent to the "time" which light requires in traveling 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:
- If the clock at B be synchronous with the clock at A, then the clock at A is synchronous with the clock at B.
- If the clock at A be synchronous with the clock at B as well as with the clock at C, then also the clocks at B and C are synchronous.
Thus with the help of certain (imagined) physical experiences, we have established what we understand when we speak of clocks at rest at different places, and synchronous with one another; and thereby we have arrived at a definition of "synchronism" and "time". The "time" of an event is the simultaneous indication of a stationary clock located at the place of the event, which is synchronous with a certain stationary clock for all time determinations.
In accordance with experience we shall assume that the magnitude
$$\frac {2{\overline {AB}}}{t'_{A}-t_{A}}=V,$$
is a universal constant (the speed of light in empty space).
We have defined time essentially with a clock at rest in a stationary system. On account of its affiliation to the stationary system, we call the time defined in this way "the time of the stationary system."