# Relativity: The Special and General Theory

The Relativity of Simulatneity

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 :

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 M1 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 M1 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 M1 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 7) disappears.

We were led to that conflict by the considerations of Section 6, 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