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A conserving two-into-one collision datum and the non-uniqueness of its final state

Example

In the setting of Two equal masses in head-on motion: system mass exceeds the sum of the masses, the datum with incoming side $(P_{1}, P_{2})$ and the single outgoing vector $Q = (5 m c / 2, 0, 0, 0) = (5 m / 2) c e_{0}$ is an on-shell conserving collision datum: $Q$ is massive with mass $5 m / 2$ and four-velocity $c e_{0}$ , and $P_{1} + P_{2} = Q$ . The outgoing rest mass $5 m / 2$ exceeds the incoming sum $2 m$ by $m / 2$ , and the incoming total kinetic energy $m c^{2} / 2 = (M - 2 m) c^{2}$ has become rest energy. The final state is not unique: the same total $Q$ is also the sum of two outgoing massive legs at rest with masses $m$ and $3 m / 2$ (four-velocities $c e_{0}$ ), so four-momentum conservation fixes only the total, not the number or masses of the outgoing particles. The datum is a mathematical configuration and no interaction is asserted.

Facts & Assumptions

Given: the affine Minkowski model $(A, V, η, C^{+})$ ; $c > 0$ ; $m > 0$ ; the inertial frame and the incoming four-momenta $P_{1}, P_{2}$ of the equal-mass example.

[F1]

(Two equal masses in head-on motion: system mass exceeds the sum of the masses, Example): in the displayed frame $P_{1}, P_{2}$ are the four-momenta of two massive particles of mass $m$ with components $(5 m c / 4, \pm 3 m c / 4, 0, 0)$ , their total is $Q = P_{1} + P_{2} = (5 m c / 2, 0, 0, 0)$ , $Q$ is future timelike with invariant mass $M = 5 m / 2$ , the displayed frame is a center-of-momentum frame, each particle has kinetic energy $m c^{2} / 4$ there, and $(M - 2 m) c^{2} = m c^{2} / 2$ is the total kinetic energy.

[F2]

(Collision data and four-momentum conservation, Definition): a collision datum records an event and the incoming and outgoing legs with side totals formed as finite sums, and conserves when the totals are equal; a leg $P$ is massive with mass $m^{'} > 0$ if $P = m^{'} U^{'}$ for some future timelike $U^{'}$ with $η (U^{'}, U^{'}) = - c^{2}$ ; a mass-specified datum is on-shell with recorded masses, and a causal datum has all legs in $c l (C^{+})$ .

[F3]

(Lorentz forms of inertia (3,1,0) on a 4-dimensional real vector space and Inertial frames and inertial charts of the affine Minkowski model): the frame basis is $η$ -orthonormal with $η (e_{0}, e_{0}) = - 1$ , components of sums and scalar multiples are computed componentwise, and for $λ > 0$ the vector $λ (c e_{0})$ has components $(λ c, 0, 0, 0)$ .

[F4]

(Invariant mass, center-of-momentum frame and necessary conditions for conserving collision data, Statement (a)): for a mass-specified conserving causal on-shell datum with future timelike total $Q$ of invariant mass $M$ , each side satisfies $M \geq \sum m_{ℓ}$ over its massive legs, with equality if and only if the side is massive-only and every massive leg is at rest in the center-of-momentum frame.

[F5]

(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 $U^{'} = c n$ ; in particular a massive leg with four-velocity $c e_{0}$ is at rest in the frame of $e_{0}$ .

Verification

1.1F1F2F3

The outgoing leg: by [F3], $Q = (5 m c / 2, 0, 0, 0) = (5 m / 2) (c e_{0})$ with $η (c e_{0}, c e_{0}) = c^{2} η (e_{0}, e_{0}) = - c^{2}$ , and $5 m / 2 > 0$ ; hence by the massive-leg clause of [F2] the vector $Q$ is a massive leg with mass $5 m / 2$ and four-velocity $c e_{0}$ , future timelike, and its recorded mass equals the invariant mass $5 m / 2$ computed in [F1].

2.1F1F2step 1.1

The datum: $(p; (P_{1}, P_{2}); (Q))$ with any event $p$ is mass-specified (recorded masses $m, m, 5 m / 2$ ), on-shell (all three legs massive by [F1] and [step 1.1]), causal (all legs future timelike), and conserving because $P_{1} + P_{2} = Q$ by [F1] and [F2]; its common total is $Q$ with invariant mass $M = 5 m / 2$ .

3.1F1F4F5step 2.1

The mass inequality and the rest-energy interpretation: by [F4] applied to the incoming side, $M = 5 m / 2 \geq m_{1} + m_{2} = 2 m$ , with equality only if both incoming massive legs are at rest in the center-of-momentum frame; by [F1] each has kinetic energy $m c^{2} / 4 \neq 0$ there, so [F5] gives $U_{i} \neq c e_{0}$ and equality fails, $5 m / 2 > 2 m$ . The excess $(M - 2 m) c^{2} = m c^{2} / 2$ equals the incoming total kinetic energy by [F1], and in the datum it is the rest energy of the single outgoing leg beyond the incoming rest energies.

4.1F1F2F3step 3.1

Non-uniqueness of the final state: define the alternative outgoing side $(Q_{1}^{'}, Q_{2}^{'})$ with $Q_{1}^{'} : = m (c e_{0}) = (m c, 0, 0, 0)$ and $Q_{2}^{'} : = (3 m / 2) (c e_{0}) = (3 m c / 2, 0, 0, 0)$ . By [F2] and [F3] both are massive legs with four-velocities $c e_{0}$ and masses $m$ and $3 m / 2$ , so they are at rest in the center-of-momentum frame, and $Q_{1}^{'} + Q_{2}^{'} = (5 m c / 2, 0, 0, 0) = Q$ ; hence $(p; (P_{1}, P_{2}); (Q_{1}^{'}, Q_{2}^{'}))$ is a second conserving mass-specified causal on-shell datum with the same incoming side and the same total but a different number and masses of outgoing legs.

5.1step 3.1step 4.1∎

Summary: the two-into-one datum with outgoing leg $Q$ conserves and has outgoing rest mass $5 m / 2 > 2 m$ , the mass excess being $m / 2$ , whose rest energy $(M - 2 m) c^{2} = m c^{2} / 2$ equals the incoming total kinetic energy; and the same total $Q$ is realised by the alternative outgoing pair of masses $m$ and $3 m / 2$ at rest, so four-momentum conservation alone fixes only the total, not the number or masses of the outgoing particles.

Depends on

Dependency tree · 2 levels

Sources

B. Crowell, Special Relativity, revision 2020-12-10 (author-published undergraduate textbook, 246 PDF pages); signature (+---) converted to (-+++) and c = 1 formulas converted to x^0 = c t; Section 4.3.2 p. 90 (all the centre-of-mass energy is available for producing new particles, and it is just enough to create a single particle of mass M)