Time orientation, future and past timelike/null/causal vectors, and the closed cones

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 causal terminology as in Timelike, null, spacelike and causal vectors of a Lorentz vector space, and let

$T:=\{v\in V:\eta (v,v)<0\}$

be its timelike set. Components and closures refer to the topology obtained by using any ordered basis to identify $V$ with Euclidean ${\mathbb{R}}^4$; The timelike set has exactly two convex components, described in orthonormal coordinates ↗, step 5.1, proves that the resulting open sets are independent of the basis. By that lemma the set $T$ has exactly two connected components, and each of them is nonempty, open, convex and path-connected. A time orientation of $(V,\eta )$ is a choice of one of these two components. Fixing such a choice, write $C^{+}$ for the chosen component and

$C^{-}:=T\setminus C^{+}$

for the other component; thus $T=C^{+}\bigsqcup C^{-}$ is a disjoint union and $C^{+}$, $C^{-}$ are exactly the two components supplied by the lemma. Relative to this choice a vector $v\in V$ is

• future timelike when $v\in C^{+}$;
• past timelike when $v\in C^{-}$;
• future null when $v\mathrm{\ne }0$, $\eta (v,v)=0$ and $v\in \mathrm{cl}(C^{+})$;
• past null when $v\mathrm{\ne }0$, $\eta (v,v)=0$ and $v\in \mathrm{cl}(C^{-})$;
• future causal when it is future timelike or future null;
• past causal when it is past timelike or past null.

The sets $\mathrm{cl}(C^{+})$ and $\mathrm{cl}(C^{-})$ are the future closed cone and the past closed cone of the time-oriented space $(V,\eta ,C^{+})$. By The timelike set has exactly two convex components, described in orthonormal coordinates ↗ each of them is a convex cone, their intersection is $\{0\}$, and a vector is causal exactly when it is future causal or past causal, in which case exactly one of these two alternatives holds: no nonzero vector is both future causal and past causal. The timelike vectors are exactly the points of $T=C^{+}\cup C^{-}$, and $0\mathrm{\notin }T$ because $\eta (0,0)=0$; since a null vector is nonzero with $\eta (v,v)=0$, no vector is both timelike and null, so the classes above are mutually exclusive where the definitions say they are: no vector is both future and past causal, and no timelike vector is null.