Causal reachability, causal futures and pasts, causal diamonds, and reachability by future causal paths in affine Minkowski space

Throughout, $(A,V,\eta ,C^{+})$ is an oriented time-oriented affine Minkowski space (Oriented time-oriented affine Minkowski space (the spacetime primitives)): $A$ is a nonempty affine space over the $4$-dimensional real vector space $V$ with displacement vectors $q-p$ and translations $v\mapsto p+v$ (Affine spaces over a real vector space and displacement vectors), and $C^{+}$ is one of the two connected components of the timelike set $T=\{v\in V:\eta (v,v)<0\}$, chosen as the time orientation, with $C^{-}:=T\setminus C^{+}$ the other component (Time orientation, future and past timelike/null/causal vectors, and the closed cones). Write $\overline{C^{+}}:=\mathrm{cl}(C^{+})$ and $\overline{C^{-}}:=\mathrm{cl}(C^{-})$ for the future and past closed cones. A vector $v\in V$ is timelike, null, spacelike or causal according to the sign of $\eta (v,v)$ as in Timelike, null, spacelike and causal vectors of a Lorentz vector space; in particular the zero vector is none of timelike, null or spacelike and hence is not causal, and the causal vectors are exactly the nonzero elements of $\overline{C^{+}}\cup \overline{C^{-}}$.

The closed cones and their boundaries. By Time orientation, future and past timelike/null/causal vectors, and the closed cones the sets $\overline{C^{+}}$ and $\overline{C^{-}}$ are convex cones with $\overline{C^{+}}\cap \overline{C^{-}}=\{0\}$. By The timelike set has exactly two convex components, described in orthonormal coordinates, applied in an $\eta$-orthonormal basis whose first vector lies in $C^{+}$, the closure of the component $\{v:v^0>\mathrm{\mid }{v}_s\mathrm{\mid }\}$ of the timelike set is $\{v:v^0\ge \mathrm{\mid }{v}_s\mathrm{\mid }\}$ with boundary points exactly $0$ and the nonzero null vectors of $\overline{C^{+}}$. Writing $N^{+}$ and $N^{-}$ for the sets of nonzero future and past null vectors (Time orientation, future and past timelike/null/causal vectors, and the closed cones), this gives

$\overline{C^{+}}=C^{+}\cup N^{+}\cup \{0\},\overline{C^{-}}=C^{-}\cup N^{-}\cup \{0\}\mathrm{.}$

The objects fixed by this definition.

(a) Causal and chronological reachability. For events $p,q\in A$ set $p\le q$ if and only if $q-p\in \overline{C^{+}}$, and $p<q$ if and only if $q-p\in C^{+}$. The first relation is read "$q$ is causally reachable from $p$" and the second "$q$ is strictly (chronologically) reachable from $p$"; both are relations on the set of events, and neither is asserted here to have any property beyond being a relation.

(b) Futures and pasts. For an event $p\in A$ set

$J^{+}(p):=\{q\in A:p\le q\}=p+\overline{C^{+}},I^{+}(p):=\{q\in A:p<q\}=p+C^{+},$

$J^{-}(p):=\{q\in A:q\le p\}=p+\overline{C^{-}},I^{-}(p):=\{q\in A:q<p\}=p+C^{-}\mathrm{.}$

The displayed equalities hold because $q\in J^{+}(p)$ says exactly that the displacement $q-p$ lies in $\overline{C^{+}}$, that is $q=p+v$ for some $v\in \overline{C^{+}}$; here $p+v$ is the affine translation of Affine spaces over a real vector space and displacement vectors. Since $0\in \overline{C^{+}}\cap \overline{C^{-}}$ one has $p\in J^{+}(p)\cap J^{-}(p)$, and since $0$ is not timelike one has $p\mathrm{\notin }{I}^{+}(p)$ and $p\mathrm{\notin }{I}^{-}(p)$. The sets $J^{\pm }(p)$ and $I^{\pm }(p)$ are called the causal and chronological future and past of $p$.

(c) Causal diamonds. For events $p,q\in A$ with $p\le q$, the causal diamond of the ordered pair $(p,q)$ is the set

$J^{+}(p)\cap J^{-}(q)=\{r\in A:p\le r\le q\}\mathrm{.}$

The diamond is defined only under the hypothesis $p\le q$; some of its properties are computed in the examples page of this pair.

(d) Reachability by future causal paths. A piecewise $C^1$ future causal path is a piecewise $C^1$ curve $x:[a,b]\to A$ on a compact interval with $a<b$ (Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length, Curves, regular causal curves and future timelike worldlines in affine Minkowski space) all of whose tangents are future causal: at every interior point of every open smooth piece the tangent $x^{\mathrm{\prime }}(s)$ exists and is future causal, and at every partition point of $[a,b]$ and at the endpoints $a$ and $b$ both one-sided difference-quotient tangents that exist there exist and are future causal vectors in the sense of Timelike, null, spacelike and causal vectors of a Lorentz vector space and Time orientation, future and past timelike/null/causal vectors, and the closed cones. For events $p,q\in A$, say that $q$ is reachable from $p$ by a future causal path when $p=q$, or when $p\mathrm{\ne }q$ and there are real numbers $a<b$ and a piecewise $C^1$ future causal path $x:[a,b]\to A$ with $x(a)=p$ and $x(b)=q$. Write $R^{+}(p)$ for the set of events reachable from $p$ in this sense; thus $p\in R^{+}(p)$ by the first clause.

(e) Conventions and scope. All objects above are defined from the affine structure of $A$, the form $\eta$ and the time orientation alone: no chart, no choice of origin, no spatial orientation of $V$ and no unit convention enters their definition. This item asserts no property of these objects beyond their being well-defined sets and relations: the reflexivity of $\le$ is part of (b), while the transitivity of $\le$, the identification $R^{+}(p)=J^{+}(p)$, the antisymmetry "$p\le q$ and $q\le p$ imply $p=q$", the monotonicity of inertial coordinate time along future causal paths and the invariance of these objects under proper orthochronous affine Lorentz maps are proved later on this page and are not used here. No choice principle is used: all objects are subsets or finite data built from a finite-dimensional real vector space.