Physics theoremStatement: AI-adaptedProof: AI-adaptedPipeline-generatedprecheck passverified 2026-10-15 (gpt-6-luna, gpt-6.1-sol, and delegated Codex review/repair agents; see completed run evidence)

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.

Mass shell inversion, observer work and variable mass

Physical scope

Conditional kinematic identities for a massive point particle of constant positive mass in one inertial chart of flat Minkowski spacetime, plus a separate variable-mass ansatz; no force law or variable-mass dynamics is adopted.

Statement

Let $I \subseteq R$ be a nonempty open interval, and let $x : I \to A$ be a $C^{2}$ future-timelike worldline in the flat Minkowski model of Particle spacetime, observers and worldlines. Use one inertial frame and parameterize the curve by that chart's time coordinate $t$ , so its coordinates are $(c t, x (t))$ (Inertial frames and inertial charts of the affine Minkowski model). Identify the observer rest space with $R^{3}$ by the frame's spatial orthonormal basis and use its Euclidean dot product and norm (The Euclidean spatial structure and cross product of an inertial frame). Write $v = d x / d t$ , $γ = (1 - ∣ v ∣^{2} / c^{2})^{- 1 / 2}$ , proper time $τ$ , and $U = d x / d τ$ . In the constant-mass sector of Adopted massive-particle momentum, observer measurement and isolated-collision conservation, let $m > 0$ be constant and $P = m U$ . In this chart write $P = (E / c, p)$ , where $E$ and $p$ are the observer energy and spatial momentum of Mechanical momentum and observer energy. Then $∣ v ∣ < c$ , $E = γ m c^{2} > 0$ , $p = γ m v$ , and

$E^{2} = m^{2} c^{4} + c^{2} ∣ p ∣^{2}, v = \frac{c^{2} p}{E} .$

Define the three-force by $F : = d p / d t$ and the four-force by $K : = d P / d τ$ in this flat inertial chart. In SI units $m$ is in kilograms, $c$ and $v$ in metres per second, $E$ in joules, $p$ in kilogram metres per second, and $F, K$ in newtons. Then

$\frac{d E}{d t} = F \cdot v, K = γ (\frac{F \cdot v}{c}, F) .$

Separately, let $I_{τ} \subseteq R$ be a nonempty open proper-time interval, let $m : I_{τ} \to (0, \infty)$ be $C^{1}$ , and let $U : I_{τ} \to V$ be a $C^{1}$ future timelike vector field with $η (U, U) = - c^{2}$ . Measure $m$ in kilograms and $d m / d τ$ in kilograms per second, and set $P (τ) : = m (τ) U (τ)$ . With $K : = d P / d τ$ , the kinematic ansatz satisfies

$η (P, K) = - m \frac{d m}{d τ} c^{2} .$

The last identity is only a differentiated mass-shell relation; it prescribes no evolution for $m$ . None of these conclusions assumes a force law.

Facts & Assumptions

Given: A $C^{2}$ future-timelike worldline on an open interval, one inertial chart with $x^{0} = c t$ , $c > 0$ , and either a constant mass $m > 0$ in the adopted massive sector or the separately stated $C^{1}$ fields $m (τ) > 0$ and $U (τ)$ with $P = m U$ and $η (U, U) = - c^{2}$ .

[A1]

The constant-mass sector of Adopted massive-particle momentum, observer measurement and isolated-collision conservation adopts $m > 0$ constant and $P = m U$ for massive particles. This theorem uses no force equation from that postulate.

[F1]

Mechanical momentum and observer energy defines $E_{n} (P) = - c η (P, n)$ and $q_{n} (P) = P + η (P, n) n$ ; in the inertial chart with time-axis observer $n = e_{0}$ , $P = (E / c, p)$ .

[F2]

Norms of four-velocity and four-acceleration, and the observer rest-space decomposition gives $∣ v ∣ < c$ , $U = γ (c, v)$ with $γ = (1 - ∣ v ∣^{2} / c^{2})^{- 1 / 2}$ , $η (U, U) = - c^{2}$ , and $d τ / d t = 1 / γ$ in the stated chart.

[F3]

Particle spacetime, observers and worldlines fixes the $C^{2}$ future-timelike curve domain, the distinction between an observer and a chart, and the SI convention $x^{0} = c t$ with $t$ in seconds and spatial coordinates in metres.

[F4]

Inertial frames and inertial charts of the affine Minkowski model and The Euclidean spatial structure and cross product of an inertial frame: the chosen chart has a future unit time-axis observer and an orthonormal Euclidean spatial basis; the coordinates of $q_{n} (P)$ are the spatial components of $P$ .

Proof

technique · direct differentiation and component decomposition

1.1A1F1F2F3F4algebra

By [A1] and [F2], $P = m U = (γ m c, γ m v)$ , and [F4] identifies these with the time and spatial components in the chosen frame. The definitions in [F1] give $E = γ m c^{2}$ and $p = γ m v$ . Hence $E^{2} - c^{2} ∣ p ∣^{2} = γ^{2} m^{2} c^{4} (1 - ∣ v ∣^{2} / c^{2}) = m^{2} c^{4}$ , and $γ m > 0$ gives $E > 0$ .

2.1step 1.1algebra

Dividing the spatial identity $p = γ m v$ by $E = γ m c^{2}$ yields $p / E = v / c^{2}$ , and therefore $v = c^{2} p / E$ . The division is valid because $E > 0$ by step 1.1.

3.1step 1.1step 2.1F4algebra

Differentiating the mass shell of step 1.1 with respect to $t$ gives $2 E d E / d t = 2 c^{2} p \cdot F$ . Substituting $p / E = v / c^{2}$ from step 2.1 yields $d E / d t = F \cdot v$ .

4.1F2step 3.1algebra

In inertial coordinates, $d P / d t = (c^{- 1} d E / d t, F)$ , and [F2] gives $d / d τ = (d t / d τ) d / d t = γ d / d t$ . Substituting the work identity of step 3.1 proves $K = γ (F \cdot v / c, F)$ . This is a kinematic identity, not a force law.

5.1F2F3algebra∎

For the separate variable-mass ansatz, bilinearity and $η (U, U) = - c^{2}$ give $η (P, P) = m (τ)^{2} η (U, U) = - m (τ)^{2} c^{2}$ . Since $m$ and $U$ are $C^{1}$ , $P$ is differentiable and $K = d P / d τ$ exists. Differentiating with respect to $τ$ gives $2 η (P, K) = - 2 m (d m / d τ) c^{2}$ , so $η (P, K) = - m (d m / d τ) c^{2}$ . No equation for $m$ was used.

Mass shell inversion, observer work and variable mass

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources