the electrical impulse to the projectors is sent. Let us assume further that the mark 5 on his moving film is exactly at the focal point of the camera as C is passed. The electrical impulses travelling from C to A and B would travel the distance AC =BC, produce the flashes A and B which again would travel with finite velocity in all directions. During the interval these impulses and lightwaves are travelling, our observer is moving from A toward B, and spot 5 on his moving film is no more at the focus of the camera. Obviously he will meet the light-wave from B first, at C, let us say, when mark 6 on his film is at the focus (Fig. 3). After another short interval when he reaches C" and mark 7 on his film is at the focus, the light-wave from A overtakes him (Fig. 4).
So we see that what was 'simultaneous' (by definition) and produced one impression on the moving film of the stationary observer, was not 'simultaneous', (again by definition), for the moving observer, as his film registers two pictures.
As both observers use similar instruments and one set of definitions, obviously both are entitled to claim that their records on the film are conclusive. So the first can claim that the flashes were 'simultaneous', the second can claim that they were not 'simultaneous'. The reverse is equally true. If the moving observer had one picture, and claimed 'simultaneity', the stationary observer would have two pictures, and deny 'simultaneity'.
But when two observers are equally justified in making two opposing claims where, by their very meanings, there is only one possible, we must conclude that the claim itself is meaningless. We see that 'absolute simultaneity' is a fiction and impossible to ascertain, as it would depend on some impossible 'absolute motion', or 'infinite velocity' of propagation of signals.
The analytical form of showing the impossibility of 'absolute simultaneity' is very simple, and follows directly from the Lorentz-Einstein transformation.
Let us imagine two observers, one in an S system of co-ordinates (x, y, z, t) and another in an S' system of co-ordinatesmoving relatively with
the velocity v.
Let us assume two events happening in the unprimed system at the point (*i> yii 2i) at the 'time' h, and the other at the pointat the 'time' h.
According to the Lorentz-Einstein transformation the 'times' at which the two events occur relatively to the primed system are given by the formulae:
, where as usual
If we assume that in our unprimed system S the two events were 'simultaneous', which means that they 'occurred at the same time', h would be equal to h, that is, h=h, or h ^=0. Let us find the difference between the two primed 'times' in the moving system S', and see if this difference is zero, which would mean that the primed 'times' are equal.
Returning to our formulae which give us the values for the primed system 'times', we express their difference as
. (1) But we assumed h h = 0; therefore
This last formula shows clearly thatcannot be zero; or in other words,
cannot be equal tounless