ExampleConstruction: AI-adaptedVerification: 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.

Observer energy, spatial momentum and mass shell of a boosted test body

Example

Keep the setting of The relative-velocity decomposition of a boost, checked in an explicit tetrad: the flat metric $g$ on $R^{4}$ , the tetrad $(E_{0}, E_{1}, E_{2}, E_{3})$ , the observer $n = E_{0}$ , the parameters $β \in (- 1, 1)$ , $γ = (1 - β^{2})^{- 1 / 2}$ , $c > 0$ , and the future unit timelike vector $u = γ (E_{0} + β E_{1})$ with $g (u, u) = - 1$ and $- g (n, u) = γ$ . Let $m > 0$ and set

$P : = m c u .$

Then $P$ is future timelike, its rest mass is $m (P) = m$ , and relative to the observer $n$ its energy and spatial momentum are

$E_{n} (P) = m c^{2} γ, q_{n} (P) = m c γ β E_{1},$

so that $g (q_{n} (P), q_{n} (P)) = m^{2} c^{2} γ^{2} β^{2}$ and the mass shell $E_{n} (P)^{2} = m^{2} c^{4} + c^{2} g (q_{n} (P), q_{n} (P))$ holds. This is the standard boosted test-body kinematics in the abstract observer split.

Facts & Assumptions

Given: The flat metric $g$ and tetrad $(E_{A})$ of the boosted-frame example, the observer $n = E_{0}$ , parameters $β, γ, c$ , the vector $u = γ (E_{0} + β E_{1})$ , a mass $m > 0$ , and the vector $P = m c u$ .

[L1]

$u$ is future unit timelike, $- g (n, u) = γ$ , $h_{n} (u) = γ β E_{1}$ , and $g (E_{1}, E_{1}) = 1$ with $g (n, E_{1}) = 0$ (The relative-velocity decomposition of a boost, checked in an explicit tetrad).

[L2]

For $P \in T_{p} M$ the observer energy and spatial momentum are $E_{n} (P) = - c g (P, n)$ and $q_{n} (P) = h_{n} (P)$ ; for $g (P, P) \leq 0$ the rest mass is $m (P) = \sqrt{- g (P, P)} / c \geq 0$ , and if $m (P) > 0$ then $U (P) = P / m (P)$ satisfies $g (U (P), U (P)) = - c^{2}$ (Observer-relative energy, spatial momentum and rest mass of a tangent vector).

[L3]

If $P$ is future timelike with rest mass $m$ , then with $u = P / (m c)$ and $γ = - g (n, u)$ one has $E_{n} (P) = m c^{2} γ$ , $q_{n} (P) = m c h_{n} (u)$ , $g (q_{n} (P), q_{n} (P)) = m^{2} c^{2} (γ^{2} - 1)$ and $E_{n} (P)^{2} = m^{2} c^{4} + c^{2} g (q_{n} (P), q_{n} (P))$ (Positivity, relative-velocity identities and the mass shell of observer-relative energy and momentum).

[L4]

For $a > 0$ the nonnegative square root satisfies $(\sqrt{a})^{2} = a$ and $\sqrt{a} > 0$ (Rational powers $a^{r}$ of a positive base).

Verification

technique · direct

1.1L1L2L4algebra

Since $g (u, u) = - 1$ by [L1], we get $g (P, P) = m^{2} c^{2} g (u, u) = - m^{2} c^{2}$ , so $- g (P, P) = m^{2} c^{2} > 0$ ; by [L4] and [L2] the rest mass is $m (P) = \sqrt{m^{2} c^{2}} / c = m$ . In particular $P = m c u$ is future timelike and $P / (m (P) c) = u$ .

2.1step 1.1L1L2algebra

By [L2] and [L1], $E_{n} (P) = - c g (m c u, n) = - m c^{2} g (u, n) = m c^{2} γ$ because $g (u, n) = - g (n, u) = - γ$ ; and $q_{n} (P) = h_{n} (P) = m c h_{n} (u) = m c γ β E_{1}$ .

3.1step 2.1L1L3algebra∎

Using $g (q_{n} (P), q_{n} (P)) = m^{2} c^{2} γ^{2} β^{2}$ and $γ^{2} (1 - β^{2}) = 1$ , we compute $E_{n} (P)^{2} - c^{2} g (q_{n} (P), q_{n} (P)) = m^{2} c^{4} γ^{2} (1 - β^{2}) = m^{2} c^{4}$ , which is the mass shell relation of [L3]; equivalently $E_{n} (P)^{2} = m^{2} c^{4} + c^{2} g (q_{n} (P), q_{n} (P))$ . The energy is positive because $m, c, γ > 0$ .

Remarks

The example computes the same numbers as the elementary boost formula in an inertial frame, but only through the observer split of a single tetrad: the energy $m c^{2} γ$ and momentum $m c γ β$ are vector and scalar data in $T_{p} M$ and its dual, not coordinate components. No measurement or empirical claim is made; the physical reading of $E_{n}$ and $q_{n}$ belongs to the framework's units and postulates.

Observer energy, spatial momentum and mass shell of a boosted test body

Example

Facts & Assumptions

Verification

Remarks

Depends on

Dependency tree · 2 levels

Sources