Page:BatemanTime.djvu/7

For instance, if two observers $A_{r},A_{s}$ pass an observer B at different times, their distance apart as measured by B is zero, while measured from A's point of view it is not.

The relation between the two sets of measurements may be obtained by taking into account the fact that the analytical conditions that an observer P should be able at time $t_{1}$ to observe an event which happened at another point Q at time $t_{2}$, ought to be of the same form in the two systems of coordinates.

If $\left(x_{1},y_{1},z_{1}\right),$  $\left(x_{2},y_{2},z_{2}\right)$ are the coordinates of P and Q, we have, in the first place, the necessary conditions

$\left(x_{1}-x_{2}\right)^{2}+\left(y_{1}-y_{2}\right)^{2}+\left(z_{1}-z_{2}\right)^{2}=c^{2}\left(t_{1}-t_{2}\right)^{2}\ t_{1}>t_{2}.$

These conditions, combined with the kinematical character of the motion of the B's relative to the A's, are

establish a correspondence between the spheres connected with the different positions of B and A by associating together two spheres which touch internally, then since the two series of spheres are such that any sphere of a series surrounds all the smaller ones of the series, it is clear that as the radius of a B sphere increases the radius of the corresponding A sphere also increases. Fig. 4. This shows that B observes the events happening to A in the



correct order. This is not necessarily the case if one or both of the observers are moving with a velocity greater than that of light; in fact under certain conditions it would be possible for one observer to witness the other's experiences in the reverse order. For instance in figure 3 if the four crosses denote successive positions of B, it appears that the first and last positions are seen by A at one instant and the two intermediate positions at another instant. The two earlier positions of B are thus seen in the reverse order, and the two later positions in the correct order.