Page:Eddington A. Space Time and Gravitation. 1920.djvu/160

144 observers. The rule for determining momentum ought to be such that both observers make the same construction, independent of their partitions, so that both arrive by the two routes at the same point $$C$$. Then it will not matter if, through their different measures of time, one observer measures momentum by horizontal progress and the other by oblique progress; both will agree that the momentum has not been altered by the collision. To describe such a construction, we must use the interval which is alike for both observers; make the interval-length of $$OB$$ equal to 2 units, and that of $$BC$$ equal to 3 units, disregarding the mesh-system altogether. Then both observers will make the same diagram and arrive at the same point $$C$$ (different from $$C$$ or $$C^\prime$$ in the previous diagrams). Then if momentum is conserved for one observer, it will be conserved for the other.

This involves a modified definition of momentum. Momentum must now be the mass multiplied by the change of position $$\partial{x}$$ per lapse of interval $$\partial{s}$$, instead of per lapse of time $$\partial{t}$$. Thus and the mass $$m$$ still preserves its character as an invariant number associated with the particle.

Whether the momentum as now defined is actually conserved or not, is a matter for experiment, or for theoretical deduction from the law of gravitation. The point is that with the original definition general conservation is impossible, because if it held good for one observer it could not hold for another. The new definition makes general conservation possible. Actually this form of the momentum is the one deduced from the law of gravitation through the identities already described. With regard to experimental confirmation it is sufficient at present to state that in all ordinary cases the interval and the time are so nearly equal that such experimental foundation as existed for the law of conservation of the old momentum is just as applicable to the new momentum.

Thus in the theory of relativity momentum appears as an