CounterexampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedjudge pass (gpt-6.1-sol)

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AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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Kinetic energy depends on the observer even though the invariant mass does not

Statement refuted

The kinetic energy $T_{n} (P) = E_{n} (P) - m c^{2}$ of a massive particle is a property of the particle alone, independent of the observer $n$ .

Facts & Assumptions

Given: the affine Minkowski model $(A, V, η, C^{+})$ ; $c > 0$ ; $m > 0$ ; $w = (3 c / 5, 0, 0)$ ; $U = γ (c, w)$ with $γ = 5 / 4$ ; $P = m U$ ; the observers $n_{1} = e_{0}$ of the displayed inertial frame and $n_{2} = U / c$ .

[F1]

(Four-momentum and kinetic energy at speed three-fifths c, Example): with $w = (3 c / 5, 0, 0)$ and $γ = 5 / 4$ the four-momentum $P = m U$ has $E_{e_{0}} (P) = 5 m c^{2} / 4$ , spatial momentum $(3 m c / 4, 0, 0)$ and kinetic energy $T_{e_{0}} (P) = m c^{2} / 4$ .

[F2]

(A passive boost that brings a moving particle to rest, Example): in the frame obtained by the passive boost with parameter $3 / 5$ along $e_{1}$ one has $E^{'} = m c^{2}$ , $p^{'} = 0$ , the mass shell $E^{' 2} - c^{2} ∣ p^{'} ∣^{2} = m^{2} c^{4}$ , and that frame is the particle's rest frame, its four-velocity being $c e_{0}^{'}$ .

[F3]

(Positivity, mass shell and kinetic energy of the observer quantities of a four-momentum, Statement (b)): for $P = m U$ with $m > 0$ , $U$ future timelike with $η (U, U) = - c^{2}$ , and $n$ future unit, $T_{n} (P) = m c^{2} (γ - 1)$ with $γ = - η (U / c, n) \geq 1$ , and $T_{n} (P) = 0$ if and only if $U = c n$ .

[F4]

(Observer energy, spatial momentum, invariant mass, rest energy and kinetic energy of a four-momentum, Definition): $T_{n} (P) = E_{n} (P) - m c^{2}$ and $E_{n} (P) = - c η (P, n)$ ; the invariant mass $m (P)$ depends on the scalar $η (P, P)$ alone and hence on no observer.

[F5]

(Four-velocity and four-acceleration of a future timelike worldline, clause (c)): a point observer of the model at an event is a future unit timelike vector $n$ with $η (n, n) = - 1$ , and the four-velocity of the observer's own worldline is $c n$ ; the four-velocity $U$ of the particle satisfies $η (U, U) = - c^{2}$ (Norms of four-velocity and four-acceleration, and the observer rest-space decomposition,(a)).

[F6]

(Lorentz forms of inertia (3,1,0) on a 4-dimensional real vector space, Definition): $η$ is bilinear, so $η (n_{2}, n_{2}) = c^{- 2} η (U, U) = - 1$ for $n_{2} = U / c$ .

Counterexample

1.1F5F6

Two observers: $n_{1} = e_{0}$ is future unit by definition of an inertial frame, and $n_{2} = U / c$ satisfies $η (n_{2}, n_{2}) = c^{- 2} η (U, U) = - 1$ by [F5] and [F6] and is future timelike; both are point observers of the particle in the sense of [F5], and $n_{1} \neq n_{2}$ because $U = γ (c, w) \neq c e_{0}$ has nonzero spatial part.

1.2F1F3

Kinetic energies: in the frame of $n_{1} = e_{0}$ , [F1] gives $T_{n_{1}} (P) = m c^{2} / 4 \neq 0$ ; for $n_{2} = U / c$ one has $U = c n_{2}$ , so [F3] gives $γ = - η (n_{2}, n_{2}) = 1$ and $T_{n_{2}} (P) = m c^{2} (γ - 1) = 0$ .

1.3F2F4

Energies and mass: $E_{n_{1}} (P) = 5 m c^{2} / 4$ by [F1] and $E_{n_{2}} (P) = m c^{2}$ by [F2]; the invariant mass is $m (P) = m$ in both cases, depending on the scalar $η (P, P)$ alone by [F4].

2.1step 1.2step 1.3

The same particle and the same vector $P$ therefore have kinetic energy $m c^{2} / 4$ for the observer $n_{1}$ and kinetic energy $0$ for the observer $n_{2}$ , while $P$ and the invariant mass $m$ are unchanged; hence the displayed statement is false: $T_{n} (P)$ depends on the observer $n$ , and so does $E_{n} (P)$ ( $5 m c^{2} / 4$ versus $m c^{2}$ ), whereas the invariant mass does not.

3.1step 1.1step 1.2step 1.3step 2.1∎

The refutation is complete: two future unit observers of the same massive particle give different values of $T_{n} (P)$ and of $E_{n} (P)$ , so neither quantity is a property of the particle alone, while the invariant mass $m (P) = m$ is the same in both frames.

Kinetic energy depends on the observer even though the invariant mass does not

Statement refuted

Facts & Assumptions

Counterexample

Depends on

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