TheoremStatement: AI-adaptedProof: AI-generatedprecheck pass

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Relative velocity decomposition of a future unit timelike vector

Statement

Let $(M, g, N)$ be a time-oriented smooth Lorentzian four-manifold, let $p \in M$ , let $n$ be an observer at $p$ , let $c > 0$ be a real parameter, and let $u \in T_{p} M$ be future unit timelike, $g_{p} (u, u) = - 1$ (Observers, rest spaces and the spatial projection at an event). Put

$γ : = - g_{p} (n, u), w : = \frac{c h_{n} (u)}{γ},$

where $h_{n}$ is the spatial projection. Then:

$γ \geq 1$ , with $γ = 1$ if and only if $u = n$ ; in particular $w$ is well defined.
$w \in n^{⊥}$ , $g_{p} (w, w) = c^{2} (1 - γ^{- 2}) < c^{2}$ , and

$u = γ (n + \frac{w}{c}) .$
Uniqueness. If $u = λ (n + v / c)$ with $λ \in R$ and $v \in n^{⊥}$ , then $λ = γ$ and $v = w$ .
Converse. Conversely, if $λ \geq 1$ and $v \in n^{⊥}$ satisfy $g_{p} (v, v) = c^{2} (1 - λ^{- 2})$ , then $u^{'} : = λ (n + v / c)$ is future unit timelike, $- g_{p} (n, u^{'}) = λ$ and $h_{n} (u^{'}) = λ v / c$ ; the maps $u \mapsto (γ, w)$ and $(λ, v) \mapsto u^{'}$ are mutually inverse bijections between the future unit timelike vectors and the pairs $(λ, v)$ with $λ \geq 1$ , $v \in n^{⊥}$ and $g_{p} (v, v) = c^{2} (1 - λ^{- 2})$ .

Facts & Assumptions

Given: A time-oriented smooth Lorentzian four-manifold $(M, g, N)$ , a point $p$ , an observer $n$ at $p$ , a real $c > 0$ and a future unit timelike $u \in T_{p} M$ .

[L1]

An observer is a vector $n$ with $g_{p} (n, n) = - 1$ and $g_{p} (n, N_{p}) < 0$ ; the spatial projection is $h_{n} (v) = v + g_{p} (n, v) n$ , and a future unit timelike vector is a future-directed vector $u$ with $g_{p} (u, u) = - 1$ (Observers, rest spaces and the spatial projection at an event, Timelike, null, spacelike and causal vectors of a Lorentz vector space, Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model).

[L2]

$h_{n}$ is linear, $\ker h_{n} = R n$ , $im h_{n} = n^{⊥}$ , $T_{p} M = R n \oplus n^{⊥}$ with $h_{n} (v) = v$ for $v \in n^{⊥}$ , the projection identity $g_{p} (h (v), h (v^{'})) = g_{p} (v, v^{'}) + g_{p} (v, n) g_{p} (n, v^{'})$ holds, and $g_{p}$ is positive definite on $n^{⊥}$ (The observer rest-space decomposition and the spatial projection).

[L3]

If $z$ is nonzero future causal then $- g_{p} (n, z) > 0$ ; if $z$ is unit timelike then $z$ is future-directed exactly when $g_{p} (n, z) < 0$ (Pairing of an observer with future causal vectors).

Proof

technique · direct

1.1L2L3given

The vector $u$ is nonzero and future causal, so [L3] gives $γ = - g_{p} (n, u) > 0$ ; thus $w = c h_{n} (u) / γ$ is well defined and belongs to $n^{⊥}$ because $h_{n} (u)$ does by [L2].

1.2L1L2L3algebra

Conversely let $λ \geq 1$ and $v \in n^{⊥}$ with $g_{p} (v, v) = c^{2} (1 - λ^{- 2})$ , and set $u^{'} : = λ (n + v / c)$ . Then $g_{p} (u^{'}, u^{'}) = λ^{2} (g_{p} (n, n) + 2 g_{p} (n, v) / c + g_{p} (v, v) / c^{2}) = λ^{2} (- 1 + 0 + 1 - λ^{- 2}) = - 1$ , so $u^{'}$ is unit timelike, and $g_{p} (n, u^{'}) = λ g_{p} (n, n) = - λ < 0$ , so $u^{'}$ is future-directed by [L3]. Moreover $- g_{p} (n, u^{'}) = λ$ and $h_{n} (u^{'}) = λ h_{n} (v) / c = λ v / c$ by [L2].

2.1step 1.1L1L2algebra

By the projection identity of [L2] applied to $v = v^{'} = u$ , and using $g_{p} (u, u) = - 1$ , we get $g_{p} (h_{n} (u), h_{n} (u)) = γ^{2} - 1$ ; the left-hand side is $\geq 0$ because it is the square of the $g_{p}$ -norm of $h_{n} (u) \in n^{⊥}$ and $g_{p}$ is positive definite on $n^{⊥}$ , with equality if and only if $h_{n} (u) = 0$ . Since $γ > 0$ by step 1.1, $γ^{2} \geq 1$ gives $γ \geq 1$ , and $γ = 1$ holds exactly when $h_{n} (u) = 0$ , i.e. $u \in \ker h_{n} = R n$ by [L2], which for future unit timelike $u$ means $u = n$ . This proves claim 1.

2.2step 1.1L2algebra

By the decomposition $u = - g_{p} (u, n) n + h_{n} (u)$ of [L2] and $γ = - g_{p} (n, u)$ , we have $u = γ n + h_{n} (u) = γ (n + h_{n} (u) / γ) = γ (n + w / c)$ , which is the formula of claim 2.

2.3step 1.1L2algebra

Suppose $u = λ (n + v / c)$ with $λ \in R$ and $v \in n^{⊥}$ . Pairing with $n$ gives $g_{p} (u, n) = λ (g_{p} (n, n) + g_{p} (n, v) / c) = - λ$ , hence $λ = - g_{p} (n, u) = γ$ . Applying $h_{n}$ and using linearity, $h_{n} (n) = 0$ and $h_{n} (v) = v$ [L2] gives $h_{n} (u) = λ h_{n} (v) / c = λ v / c$ , hence $v = c h_{n} (u) / λ = c h_{n} (u) / γ = w$ . This proves claim 3.

3.1step 1.1step 2.1algebra

From $w = c h_{n} (u) / γ$ and step 2.1, $g_{p} (w, w) = c^{2} γ^{- 2} g_{p} (h_{n} (u), h_{n} (u)) = c^{2} γ^{- 2} (γ^{2} - 1) = c^{2} (1 - γ^{- 2}) < c^{2}$ , because $γ$ is a finite real number with $γ \geq 1$ by step 2.1, so $γ^{- 2} > 0$ and $c^{2} > 0$ . This strict bound also holds when $u = n$ , in which case $γ = 1$ and $w = 0$ . This completes claim 2.

4.1step 3.1step 2.3step 1.2∎

Steps 1.1-2.2 prove claims 1 and 2; step 2.3 proves claim 3; and step 1.2 together with steps 1.1-2.2 shows that the two assignments are mutually inverse on the stated sets, completing claim 4.

Relative velocity decomposition of a future unit timelike vector

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources