Question

15.52** (a) Suppose that the total three-momentum P = Σ 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) 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.

Answer 1

a) 3-momentum part of a momentum four-vector is a sum of 3-momentum for each particle. We can denote this vector of dimension 3 by p. Corresponding four-vector p is constructed as: Now lets have to frames. Frame S with unprimed coordinates and frame S' with primed coordinates. We know that 3-momentum is conserved in every squared is also invariant when transforming from S to S', so we can write: Subtracting this and using invariance of inner product of four-vectors and invariance of 3-vectors (given in the problem) 12 PI-

Ok. now we are not done, we have to add that γ > 0, lul > 0→P4,Pi>0: P4 = p4 b) Theorem states if one component of the p four-vector is 0 in all frames than p is null-vector in all frames. Using this we will prove, if one component is conserved, than all components are conserved in Lorentz transformations p-p'=(0. A, B, C) =q Here we defined without loss of generality that first component of p is conserved while S to S, transformation. This mean that p-p' = q vector has a 0 first component. Lets apply some A Lorentz transformation to last equation: Here we know from the theorem that we again have a 0 component, and also that all components are 0 so q must be a null-vector Thus: Since both p and p' are from arbitrary frames, entire four-vector p is conserved in all frames

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