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One-null decay and massive recoil

Physical scope

Hypothetical two-product decay in the classical massive-plus-null test-particle sectors, conditional on finite isolated momentum balance; no decay probability, quantum photon-energy relation, or empirical claim is asserted.

Statement

Hypothetically, a parent of rest mass $M > 0$ is initially at rest and undergoes an isolated event into one massive daughter of rest mass $m$ and one nonzero future-null leg, where $0 < m < M$ ; there are no other legs or external transfers. Under the balance postulate, in the parent rest frame the daughter and null momenta are opposite with common magnitude

$k = \frac{(M^{2} - m^{2}) c}{2 M} > 0.$

Their energies and the daughter speed are

$E_{n u l l} = \frac{(M^{2} - m^{2}) c^{2}}{2 M}, E_{d} = \frac{(M^{2} + m^{2}) c^{2}}{2 M}, ∣ v_{d} ∣ = \frac{(M^{2} - m^{2}) c}{M^{2} + m^{2}} < c .$

The null-leg energy is strictly less than the full rest-energy difference $(M - m) c^{2}$ ; the remainder is the daughter's kinetic energy $(M - m)^{2} c^{2} / (2 M)$ . This is a conditional classical kinematic construction, not an assertion about a decay channel in nature.

Facts & Assumptions

Given: A parent at rest, masses $M > m > 0$ , a hypothetical daughter and one nonzero null output, no omitted legs, and the adopted isolated-balance postulate.

[A1]

Isolated event four-momentum balance: a complete isolated event has equal incoming and outgoing four-momentum totals.

[F1]

Finite event collisions and observer quantities: a null leg is a nonzero future-null vector; a massive leg at the event $p \in A$ has positive recorded rest mass $a$ and $P = a U$ with $U$ future timelike and $η (U, U) = - c^{2}$ . Observer energy and spatial momentum are the time and spatial components $P = (E / c, p)$ in a named inertial chart.

[F2]

Complete two-body rest-frame kinematics: in a timelike total rest frame, the two-body energy and momentum formulas hold; a massless output requires strict energy above the zero-momentum endpoint.

[F3]

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 $P = a U$ , its coordinate velocity obeys $v = c^{2} p / E$ , $E > 0$ , and $E^{2} = a^{2} c^{4} + c^{2} ∣ p ∣^{2}$ .

Proof

technique · specialize the two-body formulas, then use the inverse momentum relation

1.1A1F1F2F3constructalgebra

For either massive leg $P = a U$ of [F1], write $U = (U^{0}, U)$ in the parent rest chart, with $U^{0} > 0$ by future orientation. The auxiliary affine curve $y : R \to A$ , $y (τ) = p + τ U$ , is smooth and future timelike with proper-time parameter $τ$ since $η (y^{'}, y^{'}) = - c^{2}$ . Its chart time is $t = t_{p} + τ U^{0} / c$ , where $t_{p}$ is the event's time coordinate, so it has a smooth time parametrization on the open interval $R$ . Assign constant mass $a$ and momentum $a d y / d τ = P$ . This meets every hypothesis of [F3], giving the shell and $v_{P} = c U / U^{0} = c^{2} p / E$ . This velocity depends only on the leg's normalized tangent and agrees with any trajectory's coordinate velocity at that tangent; no smooth continuation through the decay is assumed. For the rest parent, $U = 0$ and normalization gives $U^{0} = c$ , hence its total is $(M c, 0)$ with $W = M c^{2}$ . Balance [A1] therefore makes this rest frame the final total's center-of-momentum frame. Apply [F2] with outgoing masses $a = m$ and $b = 0$ . Since $M > m$ , the strict condition $W > m c^{2}$ holds, so the null output is nonzero. The formula for $k$ gives $k = \sqrt{[W^{2} - m^{2} c^{4}]^{2}} / (2 W c) = (M^{2} - m^{2}) c / (2 M)$ ; the energy formulas give the stated $E_{d}$ and $E_{n u l l}$ and opposite spatial momenta.

2.1step 1.1algebra∎

The daughter velocity identity established in step 1.1 gives $∣ v_{d} ∣ = c^{2} k / E_{d} = (M^{2} - m^{2}) c / (M^{2} + m^{2}) < c$ , since $0 < M^{2} - m^{2} < M^{2} + m^{2}$ . Also $E_{d} - m c^{2} = (M - m)^{2} c^{2} / (2 M)$ , so $E_{n u l l} + (E_{d} - m c^{2}) = (M - m) c^{2}$ ; for $0 < m < M$ , this daughter kinetic energy, defined as energy minus rest energy, is positive and $E_{n u l l} < (M - m) c^{2}$ .

One-null decay and massive recoil

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources