Energy–momentum relation

For a particle of rest mass $m$,

For a massive free particle with invariant rest mass $m$, speed $v<c$, and Lorentz factor $\gamma =(1-v^2/c^2{)}^{-1/2}$, its total energy is $E=\gamma m{c}^2$. It includes the rest energy $m{c}^2$ and the relativistic kinetic energy $(\gamma -1)m{c}^2$. Here $c$ is the speed of light. At fixed total energy, a larger speed requires a smaller rest mass.

For an isolated decay of a parent of rest mass $M$ into products of rest masses $m_i$, the difference between total final and initial kinetic energies is $(M-{\sum }_i{m}_i)c^2$ in every inertial frame. Conservation of total energy and inclusion of all products are essential.

Rest mass is the invariant mass $m$ in the relativistic energy-momentum relation. It equals the energy measured in the particle's rest frame divided by $c^2$.

Adding two null photon four-momenta gives the displayed invariant mass, with opening angle $\theta$ measured in the same frame as the energies. A nonzero mass of the pair does not give either photon a rest mass. At fixed energies it vanishes for parallel photons and is greatest for opposite directions. A reconstructed mass peak must be assessed with background, resolution and identification controls before assigning a parent particle.

Logarithmic differentiation of the diphoton mass gives $dM/M=(d{E}_1/E_1+d{E}_2/E_2)/2$ at fixed angle. If $s_i$ are fractional energy standard deviations and $\rho$ their correlation coefficient, covariance propagation gives the displayed formula. A common multiplicative calibration error transfers directly to the reconstructed mass; independent errors combine differently. Angular errors, if present, add the corresponding derivative and covariance terms.

For sequential on-shell two-body decays $C\to B+t$ and $B\to A+s$, with massless $s,t$, evaluate their energies in the B rest frame. Their pair mass is $2E_s{E}_t(1-\cos \theta )$, so its squared invariant mass ranges from zero to the displayed endpoint. The mass ordering is $m_C>m_B>m_A>0$. spin correlations can alter the distribution inside these limits, but cannot move the kinematic endpoints while the same on-shell decay chain applies.

For the sequential decay defining the cascade-decay invariant-mass endpoint, momentum conservation gives the displayed eventwise identity. Thus $m_{At}^2$ has minimum $m_A^2{m}_C^2/m_B^2$ and maximum $m_A^2+m_C^2-m_B^2$, attained at the opposite relative angles from the $s,t$ pair endpoints. The final A particle is massive, so its lower pair-mass bound need not vanish.

In an isolated relativistic collision, total four-momentum is conserved; its time component states conservation of total relativistic energy.

This is the relativistic form of conservation of energy, expressed through the conserved four-momentum.

For $\mathbf{E}=(0,0,E)$, $\mathbf{B}=0$, and initial momentum $(p_0,0,0)$,With the initial position at the origin,