Invariant mass

Remarks

A sum of future-directed timelike or null vectors is future-directed timelike or null, so $-\eta(P, P) \ge 0$ and $M$ is a real number $\ge 0$. Since $\eta$ is preserved by Lorentz transformations, $M$ does not depend on the inertial frame. For one particle of mass $m$, $P = m u$ and $M = m$. Example: two photons with four-momenta $(E/c)(1, 1, 0, 0)$ and $(E/c)(1, -1, 0, 0)$ give $P = (2E/c, 0, 0, 0)$, so $M = 2E/c^2 > 0$ although each photon has mass $0$. Non-example: the sum $\sum_i \gamma_i m_i$ of the relativistic masses of the particles is not the invariant mass: it equals $E/c^2$, which depends on the frame. When $M > 0$ there is a frame where $\vec P = 0$, the centre-of-momentum frame, and there $M = E/c^2$.