Page:BatemanTime.djvu/6

In order that Galileo's method of comparing times at different points of space may be suitable for a sober world, it seems necessary to suppose that a body cannot move with a velocity greater than that of light, and it may be of interest to remark that this view is supported by modern electrical theories.

Now let us suppose that a second system of observers, $B_{1},B_{2},\dots B_{n}$, find that their observations are in agreement, and so can regard themselves as a standard system. It may happen that according to their measurements the first system of observers $A_{1},A_{2},\dots A_{n}$ are in motion, and then it is easy to see that the specifications of position and time as made by the A's and the B's will not agree.

internally by a given sphere associated with A, or if we make $t_{2}=0$ it is clear that there is only one sphere of the series which passes through a given point in space, provided the radii of the spheres associated with B are all positive; the equivalent supposition in the other case is that $t_{1}>t_{2}$.

If B is moving with a velocity greater than that of light, two of the spheres associated with it may intersect, and so two of them may be touched internally by the same sphere belonging to A, and then B is able to see more than one picture of the same event. See Figs. 2 and 3.



Again, if A is moving with a velocity less than that of light the spheres associated with A lie within one another or surround one another in succession, and it is clear that there is only one sphere of the series which touches internally a given sphere associated with B. Hence B cannot see two different states of A at the same time. If A is moving with a velocity greater than that of light two of its spheres may intersect, and then it is quite possible for B to see two or more different positions of A at a given time.

If B and A are moving with velocities less than that of light, and we