Up to now our considerations have been referred to a particular body of reference, which we have styled a “railway embankment.” We suppose a very long train travelling along the rails with the constant velocity v and in the direction indicated in Fig 1. People travelling in this train will with a vantage view the train as a rigid reference-body (co-ordinate system); they regard all events in reference to the train. Then every event which takes place along the line also takes place at a particular point of the train. Also the definition of simultaneity can be given relative to the train in exactly the same way as with respect to the embankment. As a natural consequence, however, the following question arises:
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Are two events (_e.g._ the two strokes of lightning _A_ and _B_) which are simultaneous _with reference to the railway embankment_ also simultaneous _relatively to the train?_ We shall show directly that the answer must be in the negative.
When we say that the lightning strokes _A_ and _B_ are simultaneous with respect to be embankment, we mean: the rays of light emitted at the places _A_ and _B_, where the lightning occurs, meet each other at the mid-point _M_ of the length _A_ → _B_ of the embankment. But the events _A_ and _B_ also correspond to positions _A_ and _B_ on the train. Let _M′_ be the mid-point of the distance _A_ → _B_ on the travelling train. Just when the flashes (as judged from the embankment) of lightning occur, this point _M′_ naturally coincides with the point _M_ but it moves towards the right in the diagram with the velocity v of the train. If an observer sitting in the position _M′_ in the train did not possess this velocity, then he would remain permanently at M, and the light rays emitted by the flashes of lightning _A_ and _B_ would reach him simultaneously, _i.e._ they would meet just where he is situated. Now in reality (considered with reference to the railway embankment) he is hastening towards the beam of light coming from _B_, whilst he is riding on ahead of the beam of light coming from _A_. Hence the observer will see the beam of light emitted from _B_ earlier than he will see that emitted from _A_. Observers who take the railway train as their reference-body must therefore come to the conclusion that the lightning flash _B_ took place earlier than the lightning flash _A_. We thus arrive at the important result:
Events which are simultaneous with reference to the embankment are not simultaneous with respect to the train, and _vice versa_ (relativity of simultaneity). Every reference-body (co-ordinate system) has its own particular time; unless we are told the reference-body to which the statement of time refers, there is no meaning in a statement of the time of an event.
Now before the advent of the theory of relativity it had always tacitly been assumed in physics that the statement of time had an absolute significance, _i.e._ that it is independent of the state of motion of the body of reference. But we have just seen that this assumption is incompatible with the most natural definition of simultaneity; if we discard this assumption, then the conflict between the law of the propagation of light _in vacuo_ and the principle of relativity (developed in Section VII) disappears.
We were led to that conflict by the considerations of Section VI, which are now no longer tenable. In that section we concluded that the man in the carriage, who traverses the distance _w per second_ relative to the carriage, traverses the same distance also with respect to the embankment _in each second_ of time. But, according to the foregoing considerations, the time required by a particular occurrence with respect to the carriage must not be considered equal to the duration of the same occurrence as judged from the embankment (as reference-body). Hence it cannot be contended that the man in walking travels the distance _w_ relative to the railway line in a time which is equal to one second as judged from the embankment.
Moreover, the considerations of Section VI are based on yet a second assumption, which, in the light of a strict consideration, appears to be arbitrary, although it was always tacitly made even before the introduction of the theory of relativity.
