In the laboratory frame a particle has four-momentum components

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

p^\mu = (750\,\mathrm{MeV}/c,\ 450\,\mathrm{MeV}/c,\ 0,\ 0)

. In a second frame, moving alongside the particle, the components are

p'^\mu = (600\,\mathrm{MeV}/c,\ 0,\ 0,\ 0)

. Someone argues that the rest mass can be found by adding the components instead of subtracting them, using

m^2c^2 = (p^0)^2 + (p^1)^2

in the laboratory frame, and obtains

mc^2 = 874.6\,\mathrm{MeV}

. ((a)) Find the particle's rest mass using the correct invariant product. ((b)) Show that the plus-sign rule above gives a different, inconsistent, answer when applied instead to the second frame's components, while the correct rule does not. ((c)) A second particle has components

k^\mu = (500\,\mathrm{MeV}/c,\ 180\,\mathrm{MeV}/c,\ 240\,\mathrm{MeV}/c,\ 0)

in the laboratory frame and

k'^\mu = (490\,\mathrm{MeV}/c,\ -150\,\mathrm{MeV}/c,\ 240\,\mathrm{MeV}/c,\ 0)

in the second frame. Find its rest mass, and compute the invariant product

p\cdot k

of the two particles' four-momenta in each frame.

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