Timelike, null, spacelike and causal vectors of a Lorentz vector space

Let $(V,\eta )$ be a Lorentz vector space of signature $(-,+,+,+)$ (Lorentz forms of inertia (3,1,0) on a 4-dimensional real vector space), with associated quadratic form $q(v)=\eta (v,v)$. A vector $v\in V$ is

• timelike when $q(v)<0$;
• null, or lightlike, when $v\mathrm{\ne }0$ and $q(v)=0$;
• spacelike when $q(v)>0$;
• causal when it is timelike or null.

The zero vector $0\in V$ satisfies $q(0)=0$ and is consequently none of timelike, null or spacelike: the null case explicitly excludes it, and the timelike and spacelike cases exclude it by $q(0)=0$ together with the strict inequalities. The causal character of a nonzero vector $v$ is the sign of the real number $q(v)$, and the three names are attached to the three signs as above. When no confusion arises, the pairing $\eta$ itself is written $⟨v,w⟩:=\eta (v,w)$.

Each nonzero vector has exactly one causal character. For a nonzero $v\in V$ the real number $q(v)$ satisfies exactly one of $q(v)<0$, $q(v)=0$, $q(v)>0$ by trichotomy of the order on $\mathbb{R}$, so $v$ is timelike, null or spacelike, and no two of the three alternatives can hold at once. The causal vectors are therefore exactly the nonzero vectors that are timelike or null, and the zero vector is causal in no sense introduced here.