Two particles, moving at relativistic speeds in the x direction, are observed to have energies E1 and E2, and momenta p1 and p2 in frame A. Frame B moves at relativistic speed v relative to frame A, also in the x direction. The particle energies and momenta in frame B are E1′ , E2′ , p′1 and p′2.

How do I proof the following relationship?

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closed as off-topic by AccidentalFourierTransform, John Rennie, user36790, Jon Custer, Kyle Kanos Dec 26 '16 at 19:08

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  • $\begingroup$ You need some extra conditions here. How are the 4-vectors on each side related? $\endgroup$ – probably_someone Dec 25 '16 at 9:32
  • $\begingroup$ @probably_someone I've added some more information to the question $\endgroup$ – foxielmao Dec 25 '16 at 9:35
  • $\begingroup$ So does i run from 1 to 2 here? $\endgroup$ – probably_someone Dec 25 '16 at 9:36
  • $\begingroup$ @probably_someone yup, i run from 1 to 2 here $\endgroup$ – foxielmao Dec 25 '16 at 9:37

We know that Lorentz transformations preserve the length of the 4-momentum vector for each particle. In Minkowski space, this length is $E^2-p^2c^2$ up to sign convention, which is irrelevant here. Since this quantity is frame-independent (frame changes are Lorentz transformations) for each particle, the sum must also be frame-independent.


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