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With metric signature $(+,-,-,-)$, four-momentum is $p^{\mu }=(E/c,\mathbf{p})$; for a particle of rest mass $m$ it satisfies $p^{\mu }{p}_{\mu }=m^2{c}^2$. For a massive particle $p^{\mu }=m{u}^{\mu }$, where $u^{\mu }$ is its four-velocity. The total four-momentum is the sum of the individual four-momenta. If that sum is timelike, its center-of-momentum frame exists and the total energy in that frame isThis follows because the spatial momentum is zero in that frame and the Minkowski norm is invariant under a Lorentz boost. A null total momentum, such as parallel photons, would not have such a rest frame.

For the photon striking an electron initially at rest, the initial invariant gives $W^2=m_e^2{c}^4+2m_e{c}^2{E}_{\gamma }$. The three final particles have total rest energy $3m_e{c}^2$. Their center-of-momentum energy is at least this value, with equality when all have zero relative momentum. Energy-momentum conservation therefore requiresThe photon-induced triplet pair-production threshold is consequently four electron rest-energy units. This is attainable as a kinematic threshold: at equality the center-of-momentum speed in the laboratory is $4c/5$, and boosting the three resting final particles gives each energy $5m_e{c}^2/3$ and momentum $4m_{e}c/3$. Their totals are the initial $5m_e{c}^2$ and $4m_{e}c$. Recoil of the original electron is why twice the rest energy is insufficient.