A gamma ray is emitted with energy as measured in the laboratory frame

PHYSICS · Multi-part · Fermi University Physics · Relativity & Gravitation · Relativistic dynamics and fields in flat spacetime · Relativistic dynamics and fields in flat spacetime

Particle Physics, Ch Scales, units and the four-momentum

Problem

A gamma ray is emitted with energy 500 MeV500\,\mathrm{MeV} as measured in the laboratory frame. A second observer, moving along with the source, measures the same photon to have energy 250 MeV250\,\mathrm{MeV} in her frame. Someone argues: "The invariant product of the photon's four-momentum with itself is p⋅p=(E/c)2p\cdot p = (E/c)^2, since the photon carries no rest mass to worry about. In the laboratory frame this is (500 MeV)2/c2=2.5×105 MeV2/c2(500\,\mathrm{MeV})^2/c^2 = 2.5\times10^5\,\mathrm{MeV^2}/c^2, and in the moving observer's frame it is (250 MeV)2/c2=6.25×104 MeV2/c2(250\,\mathrm{MeV})^2/c^2 = 6.25\times10^4\,\mathrm{MeV^2}/c^2. Two different numbers, so the invariant product is not actually invariant for a massless particle." ((a)) Find the magnitude of the photon's momentum in each frame. ((b)) Identify the error in the argument, and compute the correct value of p⋅pp\cdot p in each frame. ((c)) State, in one sentence, why no Lorentz factor γ\gamma can be assigned to this photon.

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