and
on the embankment.
The relativity of simultaneity states that every rigid body of reference (co-ordinate system) has its own time: a time-datum only has meaning when the body of reference is specified, or we may say that simultaneity is dependent on the state of motion of the body of reference.
Similar reasoning applies in the case of the distance between two points on a rigid body. The length of a rod is defined as the distance, measured by (say) a metre-rule, between the two points which are occupied simultaneously by the two ends. Since simultaneity, as we have just seen, is relative, the distance between two points, since they depend on a simultaneous reading of two events, is also relative, and length only has a meaning if the body of reference is likewise specified: any change of motion entails a corresponding change of length: we cannot detect the change since our measures alter in the same ratio. Length is thus a relative conception, and only reveals a relation between the observer and an object: the "actual" length of a body in the sense we usually understand it does not exist: there is no meaning in the term. The length of a body measured parallel to its direction of motion will always yield a greater result when judged from a system attached to it than from any other system. These few remarks may suffice to indicate the relativity of distance.
In classical mechanics it had always been assumed that the time which elapsed between the happening of two events, and also the distance between two points of a rigid body were independent of the state of motion of the body of reference: these hypotheses must, as a result of the relativity of simultaneity and distance, be rejected. We may now ask whether a mathematical relation between the place and time of occurrence of various events is possible, such that every ray of light travels with the same constant velocity
whichever rigid body of reference be chosen, e.g. such that the rays measured by an observer either in the train or on the embankment travel with the same apparent velocity.
In other words, if we assume the constancy of propagation of light in vacuo for two systems,