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Invariant mass, center-of-momentum frame and necessary conditions for conserving collision data

Statement

Let $(A, V, η, C^{+})$ be the oriented time-oriented affine Minkowski model (Oriented time-oriented affine Minkowski space (the spacetime primitives)), $c > 0$ , and let $D = (p; (P_{1}, \dots, P_{N}); (P_{1}^{'}, \dots, P_{K}^{'}))$ be a mass-specified conserving collision datum of Collision data and four-momentum conservation, causal and on-shell, all of whose legs are nonzero; write $Q : = Q_{i n} = Q_{o u t}$ for the common side total.

(a) If $Q$ is future timelike with invariant mass $M$ , then $M$ is the common invariant mass of both sides, $Q$ has the center-of-momentum frame of Center-of-momentum frames of a future timelike four-momentum and their unique time vector with $e_{0} = Q / (M c)$ , and in that frame

$M c^{2} = \sum_{legs of a side} E_{e_{0}} (leg) for each side;$

each massive leg $ℓ = m U$ has $E_{e_{0}} (ℓ) = m c^{2} γ_{ℓ} \geq m c^{2}$ with $γ_{ℓ} : = - η (U / c, e_{0})$ and equality if and only if $U = c e_{0}$ , and each null leg has $E_{e_{0}} > 0$ . Hence for each side $S \in {i n, o u t}$ ,

$M \geq \sum_{massive legs ℓ \in S} m_{ℓ},$

with equality if and only if $S$ is massive-only and every massive leg of $S$ has four-velocity $c e_{0}$ (all massive particles of that side at rest in the center-of-momentum frame).

(b) If $Q$ is future null, then every leg is null, all legs are mutually parallel in one future null direction, $M = 0$ by the convention of Total four-momentum, invariant mass and center-of-momentum frame of a finite system,(b), and no center-of-momentum frame exists.

(c) Suppose the incoming side is two-leg, $P_{i} = m_{i} U_{i}$ with $m_{i} > 0$ , and put $n_{2} : = U_{2} / c$ (the future unit observer comoving with the second leg), $γ : = - η (U_{1} / c, n_{2}) \geq 1$ and $E : = E_{n_{2}} (P_{1}) = m_{1} c^{2} γ$ . Then

$M^{2} = m_{1}^{2} + m_{2}^{2} + \frac{2 m_{2} E}{c^{2}} .$

Consequently, if the outgoing side is mass-specified with massive masses $μ_{1}, \dots, μ_{L}$ , $L \geq 1$ (it may also contain null legs), the datum can conserve only if

$E \geq \frac{c^{2}}{2 m_{2}} [(\sum_{l = 1}^{L} μ_{l})^{2} - m_{1}^{2} - m_{2}^{2}], equivalently T : = E - m_{1} c^{2} \geq \frac{c^{2}}{2 m_{2}} [(\sum_{l = 1}^{L} μ_{l})^{2} - (m_{1} + m_{2})^{2}] .$

(d) Suppose the incoming side consists of exactly two future null legs $P_{1}, P_{2}$ . If there is an inertial frame with $P_{i} = (E_{i} / c) (e_{0} + s_{i})$ , $E_{i} > 0$ , $s_{1}, s_{2}$ unit spatial vectors with $s_{2} = - s_{1}$ , then $Q$ is future timelike and $M = 2 \sqrt{E_{1} E_{2}} / c^{2}$ . More generally, with $s_{1}, s_{2}$ unit spatial vectors in that frame and $σ : = s_{1} \cdot s_{2} \in [- 1, 1]$ (The Cauchy-Schwarz inequality for finite sums), if $θ : = \arccos σ \in [0, π]$ is their Euclidean angle, with $\arccos : [- 1, 1] \to [0, π]$ the inverse of the continuous strictly decreasing restriction of cosine in Principal inverse sine and inverse cosine. Here $\cos x = \sum_{j = 0}^{\infty} (- 1)^{j} x^{2 j} / (2 j)!$ is the function of Sine and cosine defined by their real power series, and $π$ is twice its smallest positive zero (Pi as twice the smallest positive zero of cosine). Thus $\cos θ = σ$ , $θ = 0 ⟺ σ = 1$ , and $θ = π ⟺ σ = - 1$ . Then $Q$ is future timelike for $θ \neq 0$ with

$M = \frac{\sqrt{2 E_{1} E_{2} (1 - \cos θ)}}{c^{2}},$

while for $θ = 0$ the total $Q$ is future null, $M = 0$ and no center-of-momentum frame exists. In particular, a conserving on-shell collision datum whose incoming side is a single null leg and whose outgoing side contains at least one massive leg does not exist.

(e) All clauses are conditional implications from the hypotheses of a conserving, causal, on-shell datum. They assert no interaction, no existence of a conservation process for prescribed legs, no cross section, amplitude or probability, and no empirical fact.

Facts & Assumptions

Given: the affine Minkowski model $(A, V, η, C^{+})$ ; $c > 0$ ; a mass-specified conserving causal on-shell collision datum with nonzero legs; the additional hypotheses of clauses (a)–(d) as stated.

[F1]

(Collision data and four-momentum conservation, Definition): a datum conserves when $Q_{i n} = Q_{o u t}$ ; a mass-specified datum is on-shell, with recorded mass $m > 0$ on each massive leg and $0$ on each null leg; a causal datum has every leg in $c l (C^{+})$ ; a massive leg $P = m U$ has $U$ future timelike with $η (U, U) = - c^{2}$ and recorded mass $m = m (P) = \sqrt{- η (P, P)} / c$ ; one compares the sides only through their totals.

[F2]

(Finite sums of future causal vectors and additivity of observer energy and momentum, Statement): if $P_{1}, \dots, P_{N}$ , $N \geq 1$ , are nonzero future causal vectors then $\sum_{i} P_{i}$ is nonzero future causal, and future timelike if some $P_{i}$ is future timelike; for $v, w \in c l (C^{+})$ one has $- η (v, w) \geq 0$ , with equality if and only if $v = 0$ , or $w = 0$ , or both are nonzero future null with $w = λ v$ , $λ > 0$ ; and $E_{n}$ and $q_{n}$ are additive over finite sums of elements of $c l (C^{+})$ , with $E_{n} (\sum_{i} P_{i}) > 0$ when the $P_{i}$ are nonzero future causal.

[F3]

(Center-of-momentum frames of a future timelike four-momentum and their unique time vector, Statement): for future timelike $Q$ with invariant mass $M$ , an inertial frame with $e_{0} = Q / (M c)$ exists and in it $q_{e_{0}} (Q) = 0$ , $E_{e_{0}} (Q) = M c^{2}$ ; conversely $q_{e_{0}} (Q) = 0$ forces $e_{0} = Q / (M c)$ ; and a nonzero future null vector has no center-of-momentum frame.

[F4]

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

[F5]

(Total four-momentum, invariant mass and center-of-momentum frame of a finite system, Definition): the invariant mass of a future timelike total $Q$ is $M = \sqrt{- η (Q, Q)} / c$ ; if the total is future null then $M = 0$ by convention; the side totals are the finite sums of the legs.

[F6]

(Observer energy, spatial momentum, invariant mass, rest energy and kinetic energy of a four-momentum, Definition): $E_{n} (P) = - c η (P, n)$ and $q_{n} (P) = h (P)$ for $P \in c l (C^{+})$ .

[F7]

(Lorentz forms of inertia (3,1,0) on a 4-dimensional real vector space, Definition): $η$ is symmetric and bilinear, so $η (\sum_{i} P_{i}, \sum_{j} P_{j}) = \sum_{i, j} η (P_{i}, P_{j})$ for finite families.

[F8]

(Inertial frames and inertial charts of the affine Minkowski model and The Euclidean spatial structure and cross product of an inertial frame): in an inertial frame, $S = s p a n (e_{1}, e_{2}, e_{3}) = e_{0}^{⊥}$ with positive definite restriction $δ$ , a unit spatial vector is $s \in S$ with $δ (s, s) = 1$ , and $δ (s_{1}, s_{2}) = s_{1} \cdot s_{2}$ ; a vector with components $(v^{0}, v_{s})$ satisfies $η (v, v) = - (v^{0})^{2} + ∣ v_{s} ∣^{2}$ .

[F9]

(The Cauchy-Schwarz inequality for finite sums, Statement): for real $a_{i}, b_{i}$ , $∣ \sum_{i = 1}^{3} a_{i} b_{i} ∣ \leq (\sum_{i} a_{i}^{2})^{1 / 2} (\sum_{i} b_{i}^{2})^{1 / 2}$ ; hence $∣ s_{1} \cdot s_{2} ∣ \leq 1$ for unit spatial vectors.

[F10]

(Timelike, null, spacelike and causal vectors of a Lorentz vector space and Time orientation, future and past timelike/null/causal vectors, and the closed cones): a nonzero vector has exactly one causal character; it is timelike when $η (v, v) < 0$ , null when $η (v, v) = 0$ and $v \neq 0$ , and future null when moreover $v \in c l (C^{+})$ ; the closed cones meet only at $0$ .

[F11]

(Rational powers $a^{r}$ of a positive base and Laws of rational exponents): for $a \geq 0$ the number $\sqrt{a}$ is the unique nonnegative square root of $a$ , and for $a, b \geq 0$ one has $\sqrt{a b} = \sqrt{a} \sqrt{b}$ ; in particular $\sqrt{4 E_{1} E_{2}} = 2 \sqrt{E_{1} E_{2}}$ for $E_{1}, E_{2} > 0$ .

[F12]

Principal inverse sine and inverse cosine: cosine restricts to a continuous strictly decreasing bijection $[0, π] \to [- 1, 1]$ , with values $1$ and $- 1$ at the endpoints; $\arccos$ is its inverse. Hence for $θ = \arccos σ$ , $θ \neq 0 ⟺ σ < 1$ and $θ = π ⟺ σ = - 1$ . For unit vectors, $∣ s_{1} - s_{2} ∣^{2} = 2 - 2 σ$ , so $σ = 1 ⟺ s_{1} = s_{2}$ by positive definiteness in [F8].

Proof

technique · read both sides as the same future causal total, use the center-of-momentum frame where the total is timelike, and compute the pairwise Lorentz products for the two-leg and null-leg clauses

1.1F1F2

The datum conserves, so $Q_{i n} = Q_{o u t} = Q$ by [F1], and every side is a finite nonempty list of nonzero future causal vectors; hence by [F2] $Q$ is nonzero future causal, $E_{n}$ and $q_{n}$ are additive over each side, and if some leg of a side is future timelike then that side's total $Q$ is future timelike.

1.2F3F4F6

Assume $Q$ future timelike with invariant mass $M$ ; by [F3] the frame with $e_{0} = Q / (M c)$ exists and satisfies $q_{e_{0}} (Q) = 0$ and $E_{e_{0}} (Q) = M c^{2}$ ; by [F4], a massive leg $ℓ = m U$ has $E_{e_{0}} (ℓ) = m c^{2} γ_{ℓ}$ with $γ_{ℓ} = - η (U / c, e_{0}) \geq 1$ and $γ_{ℓ} = 1$ if and only if $U = c e_{0}$ , and a null leg satisfies $E_{e_{0}} (ℓ) = c ∣ q_{e_{0}} (ℓ) ∣ > 0$ .

1.3F2F7

For $v, w \in c l (C^{+})$ : by [F2] $- η (v, w) \geq 0$ , with equality exactly in the three listed cases; consequently, if $Q = \sum_{i} P_{i}$ with all $P_{i}$ nonzero future causal, [F7] gives $- η (Q, Q) = \sum_{i, j} - η (P_{i}, P_{j})$ , a sum of nonnegative terms each of which vanishes only when the corresponding pair is zero or parallel future null.

1.4F1F6F7

For the two-leg incoming side of (c), $P_{i} = m_{i} U_{i}$ and $n_{2} = U_{2} / c$ is future unit; by [F7] and $η (U_{i}, U_{i}) = - c^{2}$ one has $η (P_{1}, P_{1}) = - m_{1}^{2} c^{2}$ , $η (P_{2}, P_{2}) = - m_{2}^{2} c^{2}$ , and $η (P_{1}, P_{2}) = m_{2} c η (P_{1}, n_{2}) = - m_{2} E$ because $E = E_{n_{2}} (P_{1}) = - c η (P_{1}, n_{2})$ by [F6]; hence $- η (Q, Q) = m_{1}^{2} c^{2} + m_{2}^{2} c^{2} + 2 m_{2} E$ .

1.5F7F8F9F10

For the null pair of (d), in the displayed inertial frame the vectors $P_{i} = (E_{i} / c) (e_{0} + s_{i})$ satisfy $η (P_{i}, P_{i}) = (E_{i} / c)^{2} (- 1 + δ (s_{i}, s_{i})) = 0$ and are nonzero future null (their components in the frame are $(E_{i} / c) (1, s_{i})$ with $∣ s_{i} ∣ = 1$ ); by [F7] and [F8], $η (P_{1}, P_{2}) = (E_{1} E_{2} / c^{2}) (- 1 + σ)$ with $σ = s_{1} \cdot s_{2} = δ (s_{1}, s_{2}) \in [- 1, 1]$ by [F9], so $- η (Q, Q) = 2 E_{1} E_{2} (1 - σ) / c^{2}$ .

2.1F3F5step 1.1step 1.2

Under the hypothesis of (a), additivity of $E_{e_{0}}$ over each side [step 1.1] and $E_{e_{0}} (Q) = M c^{2}$ [step 1.2] give $M c^{2} = \sum_{legs of a side} E_{e_{0}} (leg)$ for each of the two sides, since both sides have total $Q$ ; and $M = M (Q)$ is the invariant mass of the common total, hence the common invariant mass of both sides.

2.2F3F5F10step 1.1step 1.3

Under the hypothesis of (b), $Q$ is future null and nonzero: every leg is null, because if some leg were future timelike then $Q$ would be future timelike by [step 1.1] and a nonzero vector has exactly one causal character by [F10]. Pairwise, [step 1.3] applied to the incoming list, whose total is $Q$ , gives $0 = - η (Q, Q) = \sum_{i, j} - η (P_{i}, P_{j})$ over the incoming legs as a sum of nonnegative terms, so each term vanishes and for $i \neq j$ the nonzero future null legs $P_{i}, P_{j}$ are parallel with $P_{j} = λ P_{i}$ , $λ > 0$ ; fixing the first incoming leg makes every incoming leg a positive multiple of it, and the same argument applied to the outgoing list makes every outgoing leg a positive multiple of the first outgoing leg $P_{1}^{'}$ . Since $Q = λ P_{1}$ and $Q = λ^{'} P_{1}^{'}$ with $λ, λ^{'} > 0$ , the two directions coincide and all legs lie in one future null direction. Then $M = 0$ by the convention of [F5], and no center-of-momentum frame exists by [F3] applied to the nonzero future null total.

2.3F4F5step 1.1step 1.4

Under the hypothesis of (c), $M$ is defined: $- η (Q, Q) = m_{1}^{2} c^{2} + m_{2}^{2} c^{2} + 2 m_{2} E > 0$ by [step 1.4] because $m_{1}, m_{2}, c > 0$ and $E = m_{1} c^{2} γ \geq m_{1} c^{2} > 0$ by [F4], so $Q$ is future timelike; hence [F5] and [step 1.4] give $M^{2} = - η (Q, Q) / c^{2} = m_{1}^{2} + m_{2}^{2} + 2 m_{2} E / c^{2}$ .

2.4F2F5F11F12step 1.5

Under the hypothesis of (d), if $σ < 1$ then $- η (Q, Q) = 2 E_{1} E_{2} (1 - σ) / c^{2} > 0$ by [step 1.5], while $Q = P_{1} + P_{2}$ is nonzero future causal by [F2], so $Q$ is future timelike and [F5] gives $M^{2} = - η (Q, Q) / c^{2} = 2 E_{1} E_{2} (1 - σ) / c^{4}$ , hence $M = \sqrt{2 E_{1} E_{2} (1 - σ)} / c^{2}$ by [F11] with $σ = \cos θ$ ; for the antiparallel case $s_{2} = - s_{1}$ one has $θ = π$ and $\cos θ = - 1$ , giving $M = \sqrt{4 E_{1} E_{2}} / c^{2} = 2 \sqrt{E_{1} E_{2}} / c^{2}$ by [F11].

2.5F3F5F10F12step 1.1step 1.5

Under the hypothesis of (d) with $σ = 1$ , i.e. $θ = 0$ : by [step 1.5] $η (Q, Q) = 0$ , and $Q = P_{1} + P_{2}$ is nonzero future causal by [step 1.1], hence future null by [F10]; therefore $M = 0$ by [F5], and no center-of-momentum frame exists by [F3].

2.6F10step 1.1

A conserving on-shell datum with a single null incoming leg and an outgoing side containing a massive leg does not exist: its incoming total $Q$ would be that single future null leg, while its outgoing side would contain a future timelike leg, so $Q_{o u t} = Q$ would be future timelike by [step 1.1]; a vector cannot be both future null and future timelike by [F10], and conservation would force the two totals to be equal.

3.1step 1.2step 2.1

Under the hypothesis of (a), split each side into its massive and null legs: by [step 1.2] each massive leg contributes $m_{ℓ} c^{2} γ_{ℓ} \geq m_{ℓ} c^{2}$ with $γ_{ℓ} = 1$ exactly when $U_{ℓ} = c e_{0}$ , and each null leg contributes $E_{e_{0}} > 0$ ; substituting into [step 2.1] gives $M = \sum_{massive} m_{ℓ} + \sum_{massive} m_{ℓ} (γ_{ℓ} - 1) + \sum_{null} E_{e_{0}} / c^{2} \geq \sum_{massive} m_{ℓ}$ , with equality if and only if no null leg occurs and every massive leg has $γ_{ℓ} = 1$ , that is $U_{ℓ} = c e_{0}$ .

4.1step 2.3step 3.1

Under the hypothesis of (c), if the datum conserves and the outgoing side is mass-specified with massive masses $μ_{1}, \dots, μ_{L}$ and possibly null legs, then [step 3.1] applied to that side gives $M \geq \sum_{l} μ_{l}$ ; combining with [step 2.3], $(\sum_{l} μ_{l})^{2} \leq M^{2} = m_{1}^{2} + m_{2}^{2} + 2 m_{2} E / c^{2}$ , so $E \geq \frac{c^{2}}{2 m_{2}} [(\sum_{l} μ_{l})^{2} - m_{1}^{2} - m_{2}^{2}]$ by rearrangement; subtracting $m_{1} c^{2}$ from both sides and expanding $(\sum_{l} μ_{l})^{2} - (m_{1} + m_{2})^{2} = (\sum_{l} μ_{l})^{2} - m_{1}^{2} - m_{2}^{2} - 2 m_{1} m_{2}$ gives the equivalent bound on $T = E - m_{1} c^{2}$ .

5.1step 2.1step 2.2step 2.3step 2.4step 2.5step 2.6step 3.1step 4.1∎

Clause (a) is [step 2.1] with [step 3.1], clause (b) is [step 2.2], clause (c) is [step 2.3] with [step 4.1], and clause (d) is [step 2.4], [step 2.5] and [step 2.6]; clause (e) records the conditional scope of these implications and asserts nothing further.

Invariant mass, center-of-momentum frame and necessary conditions for conserving collision data

Statement

Facts & Assumptions

Proof

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