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Classification of future and past causal vectors by a timelike field

Statement

Let $(M, g, N)$ be a time-oriented smooth Lorentzian four-manifold and let $p \in M$ (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model). Write $N_{p}^{⊥} = {v \in T_{p} M : g_{p} (v, N_{p}) = 0}$ (eta-orthogonality and orthogonal complements in a Lorentz vector space) and let $T_{p}$ be the timelike set, with components $C_{p}^{+}$ (the one containing $N_{p}$ ) and $C_{p}^{-}$ , and with future and past causality as fixed in Time orientation, future and past timelike/null/causal vectors, and the closed cones and Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model. Then:

$N_{p}^{⊥}$ is a $3$ -dimensional linear subspace of $T_{p} M$ , and $g_{p}$ is positive definite on it; equivalently, every nonzero $v \in T_{p} M$ with $g_{p} (v, N_{p}) = 0$ is spacelike.
$g_{p} (v, N_{p}) \neq 0$ for every nonzero causal $v \in T_{p} M$ .
$C_{p}^{+} = {v \in T_{p} M : g_{p} (v, v) < 0 and g_{p} (v, N_{p}) < 0}$ , and $C_{p}^{-} = {v \in T_{p} M : g_{p} (v, v) < 0 and g_{p} (v, N_{p}) > 0}$ .
A nonzero $v \in T_{p} M$ is causal and future-directed if and only if $v \in cl (C_{p}^{+})$ , and causal and past-directed if and only if $v \in cl (C_{p}^{-})$ . Consequently $cl (C_{p}^{+}) ∖ {0}$ is exactly the set of future-directed causal vectors and $cl (C_{p}^{-}) ∖ {0}$ exactly the set of past-directed causal vectors, $cl (C_{p}^{+}) \cap cl (C_{p}^{-}) = {0}$ , and every nonzero causal vector is exactly one of future-directed and past-directed.

Facts & Assumptions

Given: A time-oriented smooth Lorentzian four-manifold $(M, g, N)$ and a point $p \in M$ .

[L1]

In an $η$ -orthonormal ordered basis $(e_{0}, e_{1}, e_{2}, e_{3})$ of a Lorentz vector space, coordinate writing $v = \sum_{a} v^{a} e_{a}$ gives $η (v, w) = - v^{0} w^{0} + v^{1} w^{1} + v^{2} w^{2} + v^{3} w^{3}$ ; the timelike set is $T = T_{+} \cup T_{-}$ with $T_{+} = {v : v^{0} > ∣ v_{v e c} ∣}$ and $T_{-} = {v : v^{0} < - ∣ v_{v e c} ∣}$ , these are exactly the two connected components and each is convex, their closures are $cl (T_{+}) = {v : v^{0} \geq ∣ v_{v e c} ∣}$ and $cl (T_{-}) = {v : v^{0} \leq - ∣ v_{v e c} ∣}$ , $cl (T_{+}) \cap cl (T_{-}) = {0}$ , the nonzero causal vectors are exactly the nonzero points of $cl (T_{+}) \cup cl (T_{-})$ , and the topology, closures and components do not depend on the orthonormal basis used (The timelike set has exactly two convex components, described in orthonormal coordinates).

[L2]

Every Lorentz form on a $4$ -dimensional real vector space admits an ordered basis $(e_{0}, e_{1}, e_{2}, e_{3})$ with $g_{p} (e_{a}, e_{b}) = diag (- 1, 1, 1, 1)_{a b}$ (Every Lorentz form admits an ordered basis with matrix diag(-1,1,1,1)).

[L3]

A vector is timelike when $g_{p} (v, v) < 0$ , null when $v \neq 0$ and $g_{p} (v, v) = 0$ , spacelike when $g_{p} (v, v) > 0$ , and causal when it is timelike or null; each nonzero vector has exactly one causal character (Timelike, null, spacelike and causal vectors of a Lorentz vector space).

[L4]

For a Lorentz vector space, $C^{+}$ is the chosen component of the timelike set and future null vectors are the nonzero null points of $cl (C^{+})$ ; future causal means future timelike or future null (Time orientation, future and past timelike/null/causal vectors, and the closed cones).

[L5]

In the time-oriented manifold $(M, g, N)$ , a nonzero causal $v \in T_{p} M$ is future-directed when $g_{p} (v, N_{p}) < 0$ and past-directed when $g_{p} (v, N_{p}) > 0$ (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model).

[L6]

Coordinates of a vector with respect to an ordered basis are unique (A finite list $v : n \to V$ is an ordered basis if and only if every $x \in V$ equals $\sum_{i < n} λ_{i} v_{i}$ for exactly one $λ : n \to F$ ; those scalars are the coordinates of $x$ in that ordered basis), and the coordinate derivations of a smooth chart form a basis of the tangent space, so $T_{p} M$ is $4$ -dimensional (Coordinate derivations form a basis of the tangent space).

[L7]

The Euclidean inner product of $x, y \in R^{3}$ is $⟨ x, y ⟩ = \sum_{i = 1}^{3} x_{i} y_{i}$ with norm $∥ x ∥_{2}$ (The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ), and $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥_{2} ∥ y ∥_{2}$ (Cauchy–Schwarz: $∣ ⟨ u, v ⟩ ∣ \leq ∥ u ∥ ∥ v ∥$ , with equality exactly for linearly dependent vectors).

Proof

technique · direct

1.1L1L2L6given

By [L2] choose a $g_{p}$ -orthonormal ordered basis $(e_{0}, e_{1}, e_{2}, e_{3})$ of $T_{p} M$ , and write $N_{p} = \sum_{a} n^{a} e_{a}$ and $v = \sum_{a} v^{a} e_{a}$ for the unique coordinates of [L6]; then $g_{p} (v, w) = - v^{0} w^{0} + v^{1} w^{1} + v^{2} w^{2} + v^{3} w^{3}$ by [L1]. Replacing $e_{0}$ by $- e_{0}$ if necessary, which leaves the basis orthonormal and replaces $n^{0}$ by $- n^{0}$ , we may and do assume $n^{0} > 0$ . Since $N_{p}$ is timelike, $- (n^{0})^{2} + ∣ n_{v e c} ∣^{2} < 0$ , so $n^{0} > ∣ n_{v e c} ∣ \geq 0$ .

2.1step 1.1L7algebra

In the coordinates of step 1.1, $v_{v e c} \cdot n_{v e c}$ is the Euclidean inner product of the spatial coordinate triples, so [L7] gives $∣ v_{v e c} \cdot n_{v e c} ∣ \leq ∣ v_{v e c} ∣ ∣ n_{v e c} ∣$ , and $g_{p} (v, N_{p}) = - v^{0} n^{0} + v_{v e c} \cdot n_{v e c}$ .

2.2step 1.1L1

In the basis of step 1.1, [L1] gives $T_{p} = T_{+} \cup T_{-}$ with $T_{+} = {v : v^{0} > ∣ v_{v e c} ∣}$ and $T_{-} = {v : v^{0} < - ∣ v_{v e c} ∣}$ the two connected components, and $N_{p} \in T_{+}$ because $n^{0} > ∣ n_{v e c} ∣$ by step 1.1; hence $C_{p}^{+} = T_{+}$ and $C_{p}^{-} = T_{-}$ .

3.1step 2.1step 2.2L1algebra

For $v \in T_{+}$ we have $v^{0} > 0$ , and step 2.1 gives $g_{p} (v, N_{p}) = - v^{0} n^{0} + v_{v e c} \cdot n_{v e c} \leq - v^{0} n^{0} + ∣ v_{v e c} ∣ ∣ n_{v e c} ∣ < - v^{0} n^{0} + v^{0} n^{0} = 0$ . For $v \in T_{-}$ we have $v^{0} < 0$ and $∣ v_{v e c} ∣ < ∣ v^{0} ∣ = - v^{0}$ , so $g_{p} (v, N_{p}) = - v^{0} n^{0} + v_{v e c} \cdot n_{v e c} \geq - v^{0} n^{0} - ∣ v_{v e c} ∣ ∣ n_{v e c} ∣ > - v^{0} n^{0} - ∣ v^{0} ∣ n^{0} = - v^{0} n^{0} + v^{0} n^{0} = 0$ . Together with step 2.2 this shows that on $C_{p}^{+} = T_{+}$ the function $g_{p} (\cdot, N_{p})$ is strictly negative and on $C_{p}^{-} = T_{-}$ strictly positive; since every timelike vector lies in exactly one of $T_{+}, T_{-}$ , claim 3 follows.

3.2step 2.1L1L3algebra

Suppose $v \neq 0$ is timelike with $g_{p} (v, N_{p}) = 0$ . Then $∣ v^{0} ∣ > ∣ v_{v e c} ∣$ by [L3] and [L1], so step 2.1 gives $∣ v^{0} ∣ n^{0} = ∣ v_{v e c} \cdot n_{v e c} ∣ \leq ∣ v_{v e c} ∣ ∣ n_{v e c} ∣ < ∣ v^{0} ∣ n^{0}$ , a contradiction.

3.3step 2.1L1L3algebra

Suppose $v \neq 0$ is null with $g_{p} (v, N_{p}) = 0$ . Then $∣ v^{0} ∣ = ∣ v_{v e c} ∣$ by [L3] and [L1] and $∣ v_{v e c} ∣ \neq 0$ because $v \neq 0$ , so step 2.1 gives $∣ v^{0} ∣ n^{0} = ∣ v_{v e c} ∣ n^{0} = ∣ v_{v e c} \cdot n_{v e c} ∣ \leq ∣ v_{v e c} ∣ ∣ n_{v e c} ∣ < ∣ v_{v e c} ∣ n^{0}$ , again a contradiction.

3.4step 2.1L1L3L5algebra

The past-directed statements follow by the same computation with $T_{-}$ : if $v \in cl (T_{-}) ∖ {0}$ then $v^{0} \leq - ∣ v_{v e c} ∣ \leq 0$ and $v \neq 0$ , so $v$ is causal by [L1] and, since $- v^{0} \geq ∣ v_{v e c} ∣ \geq 0$ gives $- ∣ v_{v e c} ∣ ∣ n_{v e c} ∣ \geq v^{0} ∣ n_{v e c} ∣$ , $g_{p} (v, N_{p}) \geq - v^{0} n^{0} - ∣ v_{v e c} ∣ ∣ n_{v e c} ∣ \geq - v^{0} n^{0} + v^{0} ∣ n_{v e c} ∣ = - v^{0} (n^{0} - ∣ n_{v e c} ∣) \geq 0$ , with equality only if $v^{0} = 0$ , which would force $v = 0$ ; hence $g_{p} (v, N_{p}) > 0$ . Conversely a nonzero timelike $v$ with $g_{p} (v, N_{p}) > 0$ lies in $C_{p}^{-}$ by claim 3, and a nonzero null $v$ with $g_{p} (v, N_{p}) > 0$ has $v^{0} = - ∣ v_{v e c} ∣$ by the estimate of step 2.1 with the sign of $g_{p} (v, N_{p})$ reversed, hence in this case too $v \in cl (C_{p}^{-})$ . Thus $cl (C_{p}^{-}) ∖ {0}$ is exactly the set of past-directed causal vectors.

4.1step 3.2step 3.3L1L3L6algebra

By [L3] every nonzero $v$ with $g_{p} (v, N_{p}) = 0$ is timelike, null or spacelike, and steps 3.2 and 3.3 exclude the first two alternatives; hence such a $v$ is spacelike and $g_{p} (v, v) > 0$ . Thus $g_{p}$ is positive definite on the set $N_{p}^{⊥}$ , which is a linear subspace by bilinearity of $g_{p}$ and equals the kernel of the nonzero linear functional $g_{p} (\cdot, N_{p})$ (nonzero because $g_{p} (N_{p}, N_{p}) < 0$ ), hence has dimension $4 - 1 = 3$ in the $4$ -dimensional space $T_{p} M$ of [L6]. This proves claim 1, and steps 3.2 and 3.3 together prove claim 2.

4.2step 3.1L1L3L4algebra

By [L4] the future timelike vectors are exactly the points of $C_{p}^{+}$ , which by claim 3 are exactly the timelike $v$ with $g_{p} (v, N_{p}) < 0$ . For a null $v$ with $g_{p} (v, N_{p}) < 0$ : if $v^{0} < ∣ v_{v e c} ∣$ then, since $∣ v^{0} ∣ = ∣ v_{v e c} ∣$ and $v_{v e c} \neq 0$ by [L3] and [L1], necessarily $v^{0} = - ∣ v_{v e c} ∣$ , so $g_{p} (v, N_{p}) = ∣ v_{v e c} ∣ n^{0} + v_{v e c} \cdot n_{v e c} \geq ∣ v_{v e c} ∣ n^{0} - ∣ v_{v e c} ∣ ∣ n_{v e c} ∣ = ∣ v_{v e c} ∣ (n^{0} - ∣ n_{v e c} ∣) > 0$ , a contradiction; hence $v^{0} \geq ∣ v_{v e c} ∣$ , that is, $v \in cl (T_{+}) = cl (C_{p}^{+})$ by [L1]. Conversely, if $v \in cl (C_{p}^{+}) ∖ {0} = cl (T_{+}) ∖ {0}$ , then $v^{0} \geq ∣ v_{v e c} ∣ \geq 0$ and $v \neq 0$ , so by [L1] $v$ is causal and $g_{p} (v, N_{p}) = - v^{0} n^{0} + v_{v e c} \cdot n_{v e c} \leq - v^{0} n^{0} + ∣ v_{v e c} ∣ ∣ n_{v e c} ∣ \leq - v^{0} n^{0} + v^{0} ∣ n_{v e c} ∣ = - v^{0} (n^{0} - ∣ n_{v e c} ∣) \leq 0$ , with equality only if $v^{0} = 0$ , which would force $∣ v_{v e c} ∣ \leq v^{0} = 0$ and hence $v = 0$ ; so $g_{p} (v, N_{p}) < 0$ and $v$ is a nonzero causal vector. This proves the future-directed half of claim 4, including that $cl (C_{p}^{+}) ∖ {0}$ consists exactly of the future-directed causal vectors.

5.1step 3.4step 4.1step 4.2L1L5∎

Claims 1-3 and steps 3.4, 4.2 show that a nonzero $v \in T_{p} M$ is causal and future-directed exactly when $v \in cl (C_{p}^{+})$ and causal and past-directed exactly when $v \in cl (C_{p}^{-})$ ; these two sets meet only at $0$ because $cl (T_{+}) \cap cl (T_{-}) = {0}$ by [L1]. By claims 1 and 2 every nonzero causal vector has $g_{p} (v, N_{p}) \neq 0$ , so it is exactly one of future-directed and past-directed by the definition [L5]; this completes claim 4.

Classification of future and past causal vectors by a timelike field

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources