(a) Suppose that the total three-momentum P = of an isolated system is conserved in all inertial frames. Show that if this is true (which it is), then the fourth component P4 of the total four-momentum P = (P, P 4) has to be conserved as well.

(b) Using the zero-component theorem of Problem 15.35, you can prove the following stronger result very quickly: If any one component of the total four-momentum P is conserved in all frames, then all four components are conserved.

Problem 15.35

Prove the following useful result, called the zero-component theorem: Let q be a four-vector, and suppose that one component of q is found to be zero in all inertial frames. (For example, q4= 0 in all frames.) Then all four components of q are zero in all frames.