Oriented time-oriented affine Minkowski space (the spacetime primitives)

An oriented time-oriented affine Minkowski space is a quadruple $(A,V,\eta ,C^{+})$ with the following data. In this notation $V$ includes its specified orientation; forgetting that orientation gives only the underlying real vector space, not the full datum $V$ of the model.

1. A nonempty affine space $A$ over a $4$-dimensional real vector space $V$, with action $(p,v)\mapsto p+v$ and displacement $q-p\in V$ (Affine spaces over a real vector space and displacement vectors). In particular $V$ is a real vector space of dimension $4$.
2. A Lorentz form $\eta$ on $V$, i.e. a nondegenerate symmetric bilinear form of inertia $(3,1,0)$ (Lorentz forms of inertia (3,1,0) on a 4-dimensional real vector space). Its associated quadratic form is $q(v)=\eta (v,v)$.
3. An orientation of $V$: a positive ray of the determinant line ${\mathrm{\Lambda }}^{4}V$, fixed once and for all (Determinant-line orientations of finite-dimensional real vector spaces).
4. A time orientation $C^{+}$ of $(V,\eta )$: one of the two connected components of the timelike set $T=\{v\in V:\eta (v,v)<0\}$, the other being $C^{-}:=T\setminus C^{+}$ (Time orientation, future and past timelike/null/causal vectors, and the closed cones).

The following terminology is attached to such a quadruple.

• The elements $p$ of $A$ are the events; for events $p,q$ the vector $q-p\in V$ is the displacement from $p$ to $q$.
• The interval of $p,q\in A$ is the real number $I(p,q):=\eta (q-p,q-p)=q(q-p)$.
• Events $p,q$ are timelike, null, spacelike or causally related when the displacement $q-p$ is timelike, null, spacelike or causal in the sense of Timelike, null, spacelike and causal vectors of a Lorentz vector space; they are future-related when $q-p$ is future causal and past-related when $q-p$ is past causal in the sense of Time orientation, future and past timelike/null/causal vectors, and the closed cones.

No point of $A$ is distinguished. For every $v\in V$ the translation ${\tau }_v:A\to A$, ${\tau }_v(p):=p+v$, is a bijection, since it is inverse to ${\tau }_{-v}$: ${\tau }_{-v}({\tau }_v(p))=(p+v)+(-v)=p+(v-v)=p$ and symmetrically, by the affine action axioms (Affine spaces over a real vector space and displacement vectors). Displacements are translation invariant, ${\tau }_v(q)-{\tau }_v(p)=q-p$, so every ${\tau }_v$ preserves the interval, the causal classification of every ordered pair, and the sets $C^{+}$, $C^{-}$ (which are subsets of $V$, independent of the point of $A$); it changes only the labelling of points. The mathematical structure of the model is exactly the quadruple $(A,V,\eta ,C^{+})$; no derived physical quantity (proper time, four-velocity, momentum, energy, charge or any field) is part of this definition, and all later structure of the framework (frames, charts, tensor fields, worldlines) is built on this model.

Well-definedness. The affine-space definition supplies the action and the unique displacements; the Lorentz-form definition supplies $\eta$ and its inertia; the determinant-line definition supplies orientations; and the two components of $T$ between which the time-orientation definition chooses exist by the components lemma recorded in the well-definedness clause of Time orientation, future and past timelike/null/causal vectors, and the closed cones, which is also what makes the names future and past causal in item 4 well defined.