invariant mass

Two observers racing past a particle at different speeds will measure different energies and different momenta for it — there is no single 'true' value for those. Yet there is one number about the particle that every observer, no matter how fast they move, calculates to be exactly the same. That number is the invariant mass. It is the relativistic equivalent of a fingerprint: frame-changes shuffle energy and momentum around, but this particular combination of them stays fixed.

The recipe is to take the energy squared, subtract the momentum squared (times c-squared), and the result equals the mass squared (times c to the fourth). For a single particle this invariant mass is just its familiar rest mass — its energy when it is sitting still. The deeper power appears for a group of particles: you can add up the four-momenta of several particles and compute the invariant mass of the whole bunch. That combined invariant mass equals the mass of whatever single object could have produced them, even if no such object is present anymore.

This is the single most important measurement technique in collider physics. When a heavy particle decays, it vanishes before any detector sees it, but its decay products fly out and are measured. Compute the invariant mass of those products and you recover the mass of the parent. Plotting that number for millions of events makes the parent show up as a sharp bump at its mass — exactly how the Higgs boson revealed itself in 2012 as a peak near 125 GeV in the masses reconstructed from pairs of photons.