Thought experimentStatement: 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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Ideal two-particle fusion and internal mass

Physical scope

Hypothetical flat-spacetime event of two equal-mass classical particles forming one product with no other leg, conditional on the isolated-balance postulate; no actual reaction, interaction mechanism, probability, or empirical observation is asserted.

Statement

In a hypothetical isolated event, let two massive particles of equal rest mass $m > 0$ have opposite velocities $v$ and $- v$ in one inertial laboratory, with $∣ v ∣ < c$ and $γ = (1 - ∣ v ∣^{2} / c^{2})^{- 1 / 2}$ . Suppose, as a stipulated possible channel, that they form one massive product at rest in that laboratory and that there are no other incoming or outgoing legs. Conditional on the balance postulate, the product rest mass is

$M = 2 m γ,$

so its excess rest mass over the two incoming rest masses is $2 m (γ - 1)$ , and the corresponding rest-energy increase is $2 m (γ - 1) c^{2}$ . This equals the incoming kinetic energy in the specified laboratory. When $v = 0$ , the excess is zero. The item does not assert that such a perfectly complete fusion channel occurs in nature.

Facts & Assumptions

Given: Two hypothetical equal-mass incoming legs, a single hypothetical massive output leg at rest, no omitted legs or external transfer, and the adopted model balance.

[A1]

Isolated event four-momentum balance: for this complete isolated event, the incoming and outgoing four-momentum sums are equal.

[F1]

Finite event collisions and observer quantities: each massive leg at the event $p \in A$ has positive recorded rest mass $a$ and momentum $P = a U$ , where $U$ is future timelike with $η (U, U) = - c^{2}$ . The laboratory is one inertial chart with a future orthonormal time axis and Euclidean spatial coordinates.

[F2]

Mass shell inversion, observer work and variable mass: for a $C^{2}$ future-timelike worldline on a nonempty open interval, parameterized by one inertial chart's time, with constant mass $a > 0$ and $P = a U$ , the coordinate velocity $v$ satisfies $E = γ a c^{2}$ and $p = γ a v$ , where $γ = (1 - ∣ v ∣^{2} / c^{2})^{- 1 / 2}$ .

Proof

technique · compute the incoming total and use the stipulated balance

1.1F1F2constructalgebra

For any massive leg $P = a U$ of [F1], write $U = (U^{0}, U)$ in the laboratory, with $U^{0} > 0$ by future orientation. The auxiliary curve $y : R \to A$ , $y (τ) = p + τ U$ , is smooth and future timelike, and $τ$ is proper time since $η (y^{'}, y^{'}) = - c^{2}$ . Its laboratory time is $t = t_{p} + τ U^{0} / c$ , where $t_{p}$ is the event's time coordinate, so it admits a smooth time parametrization on the open interval $R$ with velocity $v_{P} = c U / U^{0}$ . Assign constant mass $a$ and momentum $a d y / d τ = P$ ; [F2] therefore applies. This velocity depends only on the leg's normalized tangent and agrees with the coordinate velocity of any massive trajectory with that tangent; the auxiliary curve assumes no smooth continuation of the collision itself. Applying these identities to the specified incoming velocities gives $P_{1} = (γ m c, γ m v)$ and $P_{2} = (γ m c, - γ m v)$ . Thus $Q_{i n} = P_{1} + P_{2} = (2 γ m c, 0)$ , which is future timelike and has invariant mass $\sqrt{- η (Q_{i n}, Q_{i n})} / c = 2 m γ$ .

2.1A1step 1.1algebra∎

The output is one massive leg at rest, so the pointwise identities established in step 1.1 give $P_{o u t} = (M c, 0)$ . Balance [A1] gives $P_{o u t} = Q_{i n}$ and therefore $M = 2 m γ$ . Subtracting the two incoming rest masses gives $M - 2 m = 2 m (γ - 1)$ ; multiplying by $c^{2}$ gives the rest-energy increase. The incoming total energy is $2 γ m c^{2}$ and its rest-energy baseline is $2 m c^{2}$ , so the same difference is the incoming kinetic energy, defined as energy minus rest energy. Since $γ = 1$ exactly at $v = 0$ , the zero-speed boundary gives zero excess.

Ideal two-particle fusion and internal mass

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources