Null vector

A nonzero spacetime vector $k$ is null when its Minkowski norm vanishes, $k^a{k}_a=0$. Null tangent vectors describe the local directions of light rays.

Special relativity describes inertial frames using Lorentz transformations and Minkowski spacetime.

In Minkowski spacetime with metric signature $(-,+,+,+)$, write two nonzero future-pointing null vectors as $A=a(1,\mathbf{n})$ and $B=b(1,\mathbf{m})$, where $a,b>0$ and $\mathbf{n},\mathbf{m}$ are unit vectors. Their Lorentzian inner product givesThus their sum is a future-pointing causal vector. It is a null vector exactly when the spatial directions coincide, and a timelike vector otherwise. The positivity of the time components is essential: adding past- and future-pointing null vectors can instead produce a spacelike vector.

The null condition for a nonzero vector $k$ in a Lorentzian metric is $g(k,k)=0$. Applied to the tangent of a curve it makes the curve a null curve. Nullity alone does not imply the geodesic equation; an additional zero or tangent-parallel covariant acceleration is required.