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)

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Mass shell, observer energy and inertial work

Physical scope

Conditional constant-mass mass-shell and observer-decomposition identities; coordinate work relation only in flat inertial charts; variable-mass formula is kinematic only.

Statement

For a massive body with constant $m > 0$ , $P = m U$ , $g (U, U) = - c^{2}$ , and the force law of Prescribed constant-mass force law,

$g (P, P) = - m^{2} c^{2} .$

For every observer $n$ at the event, the energy and spatial momentum defined in Mechanical momentum and observer energy satisfy

$E_{n} (P)^{2} = m^{2} c^{4} + c^{2} g (q_{n} (P), q_{n} (P)), E_{n} (P) > 0.$

For a nonzero future-null $P$ , $E_{n} (P) = c \sqrt{g (q_{n} (P), q_{n} (P))}$ . In a flat inertial frame $(o; e_{0}, e_{1}, e_{2}, e_{3})$ with $n = e_{0}$ and inertial chart $x^{0} = c t$ from Inertial frames and inertial charts of the affine Minkowski model, let $E = E_{n} (P)$ , let $p$ be the spatial coordinates of $q_{n} (P)$ in $(e_{1}, e_{2}, e_{3})$ , set $v = d x / d t$ , and define $F = d p / d t$ . Then $d E / d t = F \cdot v$ under the constant-mass force hypotheses. If $m (τ) > 0$ is $C^{1}$ , $U$ is $C^{1}$ with $g (U, U) = - c^{2}$ , and $P = m (τ) U$ , then the kinematic identity $g (U, D_{τ} P) = - c^{2} \dot{m}$ holds; it specifies no variable-mass law.

Facts & Assumptions

Given: A future unit timelike tangent $U$ , constant $m > 0$ on the massive branch, an observer $n$ , and the force assumptions of the Statement.

[F1]

The prescribed force is orthogonal to $U$ and the local IVP preserves $g (U, U) = - c^{2}$ (Prescribed constant-mass force law, Smooth shell-tangent force initial-value problem).

[F2]

For an observer, $q_{n} (P) = P + g (P, n) n$ and $g (q_{n} (P), q_{n} (P)) = g (P, P) + g (P, n)^{2}$ (Mechanical momentum and observer energy, The observer rest-space decomposition and the spatial projection).

[F3]

Pairing a future timelike vector with a future observer is strictly negative (Pairing of an observer with future causal vectors).

[F4]

In a flat inertial chart with constant mass, the four-force work identity gives $d E / d t = F \cdot v$ (Four-force identities and the work-energy relation in an inertial frame).

[F5]

Along a curve, $\frac{d}{d τ} g (V, W) = g (D_{τ} V, W) + g (V, D_{τ} W)$ (Covariant differentiation along curves: metric pairing, parallel transport and geodesic speed).

Proof

1.1givenF2algebra

From $P = m U$ and $g (U, U) = - c^{2}$ , $g (P, P) = m^{2} g (U, U) = - m^{2} c^{2}$ . Write $a = - g (P, n)$ and $q = q_{n} (P)$ ; then $P = a n + q$ , $g (q, n) = 0$ , and [F2] gives $g (q, q) = g (P, P) + g (P, n)^{2}$ . Since $E_{n} = c a$ , rearrangement yields $E_{n}^{2} = m^{2} c^{4} + c^{2} g (q, q)$ .

1.2F2F3algebra

The vectors $P$ and $n$ are future timelike, so [F3] gives $g (P, n) < 0$ and therefore $E_{n} = - c g (P, n) > 0$ . For null $P$ , since $g (P, P) = 0$ , [F2] yields $g (q, q) = g (P, n)^{2}$ , and positivity selects $E_{n} = c \sqrt{g (q, q)}$ .

2.1givenF1F4F5algebra∎

Under the flat inertial constant-mass hypotheses, [F1] identifies the prescribed force with the derivative of the mechanical momentum, and [F4] supplies the coordinate work relation. If $P = m (τ) U$ , differentiating $g (U, U) = - c^{2}$ using metric compatibility gives $g (U, D_{τ} U) = 0$ ; hence $g (U, D_{τ} P) = \dot{m} g (U, U) + m g (U, D_{τ} U) = - c^{2} \dot{m}$ . This last identity imposes no dynamics for $m$ .

Mass shell, observer energy and inertial work

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources