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In metric signature $(+---)$, a future-pointing timelike four-vector has positive time component larger than its spatial magnitude. Two such vectors have strictly positive Minkowski metric scalar product, since $P\cdot Q=P^0{Q}^0-\mathbf{P}\cdot \mathbf{Q}\ge P^0{Q}^0-|\mathbf{P}||\mathbf{Q}|>0$. In the opposite signature the scalar-product sign reverses, while the causal condition is the same. The future timelike cone is closed under addition, which rules out a null photon momentum equalling a sum of two massive future four-momenta.

The Lorentz group consists of real matrices preserving the Minkowski metric: ${\Lambda }^T\eta \Lambda =\eta$.

A Lorentz transformation preserves the Minkowski metric. A boost of speed $u$ along the $x$-axis uses $\beta =u/c$ and

The mass shell is the locus of four-momenta satisfying the relativistic dispersion relation for a fixed mass $m$. With mostly-minus Minkowski metric, $p^2=m^2$; with mostly-plus Minkowski metric, $p^2=-m^2$. The positive-energy sheet represents particle four-momenta. For $m=0$ this is the light cone. The condition of being on shell refers more generally to satisfying the field equations.

Starting from the Minkowski metric in oblate spheroidal coordinates, set $u=t-r$, $R=r$, and choose an azimuth $\phi$ satisfyingThenso the coordinate vector ${\partial }_R$ is null.