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The closed future cone in inertial coordinates: description, finite sums, and the coordinate criterion for causal reachability

Statement

Let $(A, V, η, C^{+})$ be the oriented time-oriented affine Minkowski space (Oriented time-oriented affine Minkowski space (the spacetime primitives), Lorentz forms of inertia (3,1,0) on a 4-dimensional real vector space) and use the reachability objects $J^{+}$ , $I^{+}$ of Causal reachability, causal futures and pasts, causal diamonds, and reachability by future causal paths in affine Minkowski space. Let $(o; e_{0}, e_{1}, e_{2}, e_{3})$ be an inertial frame with chart $x$ (Inertial frames and inertial charts of the affine Minkowski model) and let $∣ v_{s} ∣ : = \sqrt{(v^{1})^{2} + (v^{2})^{2} + (v^{3})^{2}}$ be the Euclidean norm of the spatial part $v_{s} : = v^{1} e_{1} + v^{2} e_{2} + v^{3} e_{3}$ of a vector $v = v^{0} e_{0} + v_{s}$ (The Euclidean spatial structure and cross product of an inertial frame). Then:

(a) Cone description. $C^{+} = {v : v^{0} > ∣ v_{s} ∣}$ , $\overline{C^{+}} = {v : v^{0} \geq ∣ v_{s} ∣}$ , $C^{-} = {v : - v^{0} > ∣ v_{s} ∣}$ and $\overline{C^{-}} = {v : - v^{0} \geq ∣ v_{s} ∣}$ ; moreover every $v \in \overline{C^{+}}$ with $v \neq 0$ satisfies $- η (v, e_{0}) = v^{0} > 0$ and $∣ v_{s} ∣ \leq v^{0}$ , and $v$ is future null exactly when $∣ v_{s} ∣ = v^{0} > 0$ .

(b) Finite sums. If $v_{1}, \dots, v_{k}$ lie in $\overline{C^{+}}$ ( $k \geq 1$ ), then $v_{1} + \dots + v_{k} \in \overline{C^{+}}$ ; if in addition at least one $v_{i}$ lies in $C^{+}$ , then the sum lies in $C^{+}$ ; and if $v + w = 0$ with $v, w \in \overline{C^{+}}$ , then $v = w = 0$ .

(c) Coordinate criterion. For events $p, q \in A$ put $D^{a} : = x^{a} (q) - x^{a} (p)$ and $D_{s} : = (D^{1}, D^{2}, D^{3})$ . Then: $q \in J^{+} (p)$ if and only if $D^{0} \geq ∣ D_{s} ∣$ ; $q \in I^{+} (p)$ if and only if $D^{0} > ∣ D_{s} ∣$ ; $q \in J^{+} (p)$ with $q \neq p$ if and only if $D^{0} \geq ∣ D_{s} ∣$ and $D^{0} > 0$ ; and for $q \in J^{+} (p)$ with $q \neq p$ one has $0 < D^{0}$ and $∣ D_{s} ∣ \leq D^{0}$ . The criteria are stated for the chart of any inertial frame, because every inertial basis is $η$ -orthonormal with future first vector.

(d) Scope. The lemma is a statement about the affine model and the fixed time orientation; it attaches no physical meaning to $\leq$ , $J^{+}$ or $I^{+}$ and uses no choice principle.

Facts & Assumptions

Given: The oriented time-oriented affine Minkowski space $(A, V, η, C^{+})$ , an inertial frame $(o; e_{0}, e_{1}, e_{2}, e_{3})$ with chart $x$ , and for a vector $v = \sum_{a} v^{a} e_{a}$ the spatial vector $v_{s} = v^{1} e_{1} + v^{2} e_{2} + v^{3} e_{3} \in S$ with coordinate column $(v^{1}, v^{2}, v^{3})$ and norm $∣ v_{s} ∣ = \sqrt{(v^{1})^{2} + (v^{2})^{2} + (v^{3})^{2}}$ .

[L1]

In an $η$ -orthonormal ordered basis $(e_{0}, e_{1}, e_{2}, e_{3})$ with $v = v^{0} e_{0} + v^{1} e_{1} + v^{2} e_{2} + v^{3} e_{3}$ one has $η (v, v) = - (v^{0})^{2} + ∣ v_{s} ∣^{2}$ , and the timelike set $T = {v : η (v, v) < 0}$ is the disjoint union $T = T_{+} \cup T_{-}$ with $T_{+} = {v : v^{0} > ∣ v_{s} ∣}$ and $T_{-} = {v : v^{0} < - ∣ v_{s} ∣}$ , these being exactly the two connected components of $T$ ; moreover $cl (T_{+}) = {v : v^{0} \geq ∣ v_{s} ∣}$ , $cl (T_{-}) = {v : v^{0} \leq - ∣ v_{s} ∣}$ and $cl (T_{+}) \cap cl (T_{-}) = {0}$ (The timelike set has exactly two convex components, described in orthonormal coordinates).

[L2]

$C^{+}$ is one component of $T$ , $C^{-} : = T ∖ C^{+}$ is the other; a vector is future timelike, future null, future causal exactly by membership in $C^{+}$ , or nonzero null in $cl (C^{+})$ , or the union of the two; past notions are the same with $C^{-}$ ; $\overline{C^{+}}$ and $\overline{C^{-}}$ are convex cones with $\overline{C^{+}} \cap \overline{C^{-}} = {0}$ , and a causal vector is future or past but not both (Time orientation, future and past timelike/null/causal vectors, and the closed cones).

[L3]

An inertial frame is a tuple $(o; e_{0}, e_{1}, e_{2}, e_{3})$ with $o \in A$ and $(e_{0}, e_{1}, e_{2}, e_{3})$ an $η$ -orthonormal basis of $V$ with $η (e_{a}, e_{b}) = diag (- 1, 1, 1, 1)_{a b}$ , whose first vector $e_{0}$ lies in $C^{+}$ and which is positively oriented; its chart is $x (r) = E^{- 1} (r - o)$ with $E (u) = \sum_{a} u^{a} e_{a}$ , and in particular $x (r) - x (s) = E^{- 1} (r - s)$ (Inertial frames and inertial charts of the affine Minkowski model).

[L4]

For a fixed inertial frame the spatial part $S = span (e_{1}, e_{2}, e_{3})$ carries the positive-definite form $δ = η ∣_{S}$ , $∣ w ∣ : = \sqrt{δ (w, w)}$ is a norm on $S$ , and for $w = w^{1} e_{1} + w^{2} e_{2} + w^{3} e_{3}$ one has $∣ w ∣ = \sqrt{(w^{1})^{2} + (w^{2})^{2} + (w^{3})^{2}}$ and $η (w, e_{0}) = 0$ ; the Euclidean scalar product on the coordinate space is the positive-definite inner product of The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ (The Euclidean spatial structure and cross product of an inertial frame).

[L5]

For the norm of [L4] the triangle inequality $∣ w + z ∣ \leq ∣ w ∣ + ∣ z ∣$ holds for all $w, z \in S$ (The inner-product norm is definite, homogeneous, and satisfies the triangle inequality).

[L6]

A vector $v$ is timelike when $η (v, v) < 0$ , null when $v \neq 0$ and $η (v, v) = 0$ , and spacelike when $η (v, v) > 0$ ; every nonzero vector has exactly one causal character (Timelike, null, spacelike and causal vectors of a Lorentz vector space).

[L7]

For events $p, q \in A$ : $q \in J^{+} (p)$ iff $q - p \in \overline{C^{+}}$ ; $q \in I^{+} (p)$ iff $q - p \in C^{+}$ ; $J^{+} (p) = p + \overline{C^{+}}$ (Causal reachability, causal futures and pasts, causal diamonds, and reachability by future causal paths in affine Minkowski space).

Proof

technique · direct

1.1givenL1L2L3

The component $C^{+}$ is one of $T_{+}$ , $T_{-}$ by [L2], and the coordinates of $e_{0}$ in the basis $(e_{0}, e_{1}, e_{2}, e_{3})$ are $(1, 0, 0, 0)$ , so $e_{0} \in T_{+}$ ; since $e_{0} \in C^{+}$ by [L3], it follows that $C^{+} = T_{+}$ and $C^{-} = T_{-}$ , that is $C^{+} = {v : v^{0} > ∣ v_{s} ∣}$ and $C^{-} = {v : - v^{0} > ∣ v_{s} ∣} = {v : v^{0} < - ∣ v_{s} ∣}$ . Taking closures with [L1] gives $\overline{C^{+}} = {v : v^{0} \geq ∣ v_{s} ∣}$ , $\overline{C^{-}} = {v : - v^{0} \geq ∣ v_{s} ∣}$ and $\overline{C^{+}} \cap \overline{C^{-}} = {0}$ ; also $C^{-} = - C^{+}$ , because $- (T_{+}) = T_{-}$ , whence $\overline{C^{-}} = - \overline{C^{+}}$ and the description $- v^{0} \geq ∣ v_{s} ∣$ agrees with $- v \in \overline{C^{+}}$ .

2.1step 1.1L1L3L4L6algebra

Let $v = v^{0} e_{0} + v_{s} \in \overline{C^{+}}$ . Then $v^{0} \geq ∣ v_{s} ∣ \geq 0$ , and $- η (v, e_{0}) = v^{0}$ because $η (v, e_{0}) = v^{0} η (e_{0}, e_{0}) + η (v_{s}, e_{0}) = - v^{0} + 0$ by the orthonormality and spatial orthogonality of [L3] and [L4]. If $v \neq 0$ then $v^{0} > 0$ : if $v^{0} = 0$ then $∣ v_{s} ∣ \leq 0$ , so $v_{s} = 0$ and $v = 0$ , a contradiction. Finally $v$ is future null exactly when $∣ v_{s} ∣ = v^{0} > 0$ : by [L6] nullity of $v$ is the conjunction $v \neq 0$ and $η (v, v) = - (v^{0})^{2} + ∣ v_{s} ∣^{2} = 0$ , which by [L1] means $∣ v_{s} ∣ = v^{0}$ , and by the preceding sentence this forces $v^{0} > 0$ .

2.2step 1.1L5algebra

Let $v, w \in \overline{C^{+}}$ . Then $(v + w)^{0} = v^{0} + w^{0} \geq ∣ v_{s} ∣ + ∣ w_{s} ∣ \geq ∣ v_{s} + w_{s} ∣$ by [L5] applied to the spatial parts, so $v + w \in \overline{C^{+}}$ by the description of step 1.1; if in addition $v \in C^{+}$ , that is $v^{0} > ∣ v_{s} ∣$ , the first inequality is strict and $(v + w)^{0} > ∣ v_{s} + w_{s} ∣$ , so $v + w \in C^{+}$ ; the same holds if $w \in C^{+}$ .

2.3step 1.1L2algebra

Let $v, w \in \overline{C^{+}}$ with $v + w = 0$ . Then $w = - v$ and $v \in \overline{C^{+}}$ , so $w \in - \overline{C^{+}} = \overline{C^{-}}$ by step 1.1; hence $w \in \overline{C^{+}} \cap \overline{C^{-}} = {0}$ by step 1.1, that is $w = 0$ and $v = - w = 0$ .

3.1step 2.2algebra

Let $k \geq 1$ and $v_{1}, \dots, v_{k} \in \overline{C^{+}}$ . For $k = 1$ the membership claim is the hypothesis and the strict claim is the hypothesis $v_{1} \in C^{+}$ ; for the passage from $k - 1$ to $k$ put $s^{'} : = v_{1} + \dots + v_{k - 1}$ and $s : = s^{'} + v_{k}$ , where $s^{'} \in \overline{C^{+}}$ by the induction hypothesis, so $s \in \overline{C^{+}}$ by step 2.2 applied to $s^{'}$ and $v_{k}$ . If some $v_{i}$ with $i < k$ lies in $C^{+}$ , the induction hypothesis gives $s^{'} \in C^{+}$ and the strict case of step 2.2 gives $s \in C^{+}$ ; if none of $v_{1}, \dots, v_{k - 1}$ is in $C^{+}$ but $v_{k}$ is, the strict case of step 2.2 applies directly to $s^{'}$ and $v_{k}$ . Induction on $k$ therefore proves both claims; finite sums are defined by recursion on $k$ , so no choice is used.

3.2step 1.1step 2.1L3L7algebra

Let $p, q \in A$ and $D : = x (q) - x (p)$ , so that $D$ is the coordinate column of $q - p$ by the linearity clause of [L3]. By [L7] and step 1.1, $q \in J^{+} (p)$ iff $q - p \in \overline{C^{+}}$ iff $D^{0} \geq ∣ D_{s} ∣$ , and $q \in I^{+} (p)$ iff $q - p \in C^{+}$ iff $D^{0} > ∣ D_{s} ∣$ . Moreover $D = 0$ iff $q - p = 0$ iff $q = p$ , because $E^{- 1}$ is injective; hence $q \in J^{+} (p)$ with $q \neq p$ iff $D^{0} \geq ∣ D_{s} ∣$ and $D \neq 0$ . For $D \in \overline{C^{+}}$ this is equivalent to $D^{0} \geq ∣ D_{s} ∣$ and $D^{0} > 0$ by step 2.1, which is the third criterion; the fourth is the same statement as read off from step 2.1.

4.1step 1.1step 2.1step 2.2step 2.3step 3.1step 3.2given∎

Clause (a) is steps 1.1 and 2.1, clause (b) is steps 2.2, 2.3 and 3.1, and clause (c) is step 3.2. Every inertial chart has the same description because each inertial basis is $η$ -orthonormal with first vector $e_{0} \in C^{+}$ ([L3]), so the criteria hold in the chart of any inertial frame. The objects $\leq$ , $J^{+}$ and $I^{+}$ are those of the affine model and no physical meaning, synchronization procedure or empirical claim is attached to them here; no choice principle is used, the only analytic input being the triangle inequality [L5] and the finite-dimensional algebra of the coordinates.

The closed future cone in inertial coordinates: description, finite sums, and the coordinate criterion for causal reachability

Statement

Facts & Assumptions

Proof

Depends on

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