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Energy Momentum

1 · Prerequisites

2 · Summary

This page develops the four-momentum, energy and momentum of relativistic particles on the affine Minkowski model, first as mathematics and then under the model's adopted physical premises. Observer energy, spatial momentum, invariant mass, rest energy and kinetic energy of a four-momentum fixes the observer energy $E_{n} (P) = - c η (P, n)$ , the spatial momentum $q_{n} (P) = h (P)$ , the invariant mass $m (P) = \sqrt{- η (P, P)} / c$ , the rest energy $m c^{2}$ and the kinetic energy $T_{n} (P) = E_{n} (P) - m c^{2}$ , together with the unique decomposition $P = (E_{n} (P) / c) n + q_{n} (P)$ . Positivity, mass shell and kinetic energy of the observer quantities of a four-momentum proves positivity of $E_{n}$ , the mass-shell identity $E_{n}^{2} = m^{2} c^{4} + c^{2} ∣ q_{n} ∣^{2}$ , the null relation $E_{n} = c ∣ q_{n} ∣$ and the velocity formulas $E_{n} = m c^{2} γ$ , $q_{n} = m γ w$ ; the mass $m (P)$ is a function of the scalar $η (P, P)$ alone.

Four-momentum components, energy-velocity relation and passive boost transformation in an inertial frame reads the components in an inertial frame, where $P = (E / c, p)$ with $E = m γ_{w} c^{2}$ and $p = m γ_{w} w$ , proves the passive-boost transformation law of the components and the frame independence of $E^{2} - c^{2} ∣ p ∣^{2}$ . The proper-time derivative and the four-force are introduced by Proper-time derivative of a vector field along a worldline and the four-force notation, and Four-force identities and the work-energy relation in an inertial frame proves $η (P, F) = 0$ , $F = m A$ , $F = γ (f \cdot w / c, f)$ and the work–energy relation $d E / d t = f \cdot w$ , with the variable-mass identity $η (P, F) = - m c^{2} d m / d τ$ as a separate clause.

For systems, Total four-momentum, invariant mass and center-of-momentum frame of a finite system defines the total four-momentum and the invariant mass $M$ of a future timelike total, and Finite sums of future causal vectors and additivity of observer energy and momentum shows that finite sums of nonzero future causal vectors are nonzero future causal, additive in the observer quantities and timelike as soon as one summand is timelike. Center-of-momentum frames of a future timelike four-momentum and their unique time vector constructs the center-of-momentum frame with $e_{0} = Q / (M c)$ , proves uniqueness up to origin choice and spatial rotations and shows that a nonzero future null total has no such frame. Collision data and four-momentum conservation fixes the collision-data vocabulary, and Invariant mass, center-of-momentum frame and necessary conditions for conserving collision data derives the necessary conditions: $M$ dominates each side's mass sum with a specific equality case, a null total forces parallel massless legs, and the two-leg projectile–target configuration obeys the displayed threshold inequalities.

The adopted physical premises are Adopted massive-particle momentum, observer measurement and isolated-collision conservation (constant positive mass with $P = m U$ , the measurement identification, isolated collision balance, and the equation of motion for a stipulated force) and Adopted momentum model for an ideal massless particle (ideal massless particles as nonzero future null momenta, with no four-velocity, proper time or rest frame). Energy, momentum, invariant mass and four-force of relativistic particles, and necessary collision thresholds assembles these into the conditional physical conclusions, including the necessary collision thresholds; it uses no empirical premise. Worked computations, including the $β = 3 / 5$ four-momentum, a boost to rest, the head-on collective mass, the null pair, the stationary-target threshold and the non-uniqueness of the final state, are on sr-energy-momentum-examples.

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

DefinitionDefinition: AI-adaptedProof: Not applicablejudge pass (gpt-6.1-sol)Open item page →

Observer energy, spatial momentum, invariant mass, rest energy and kinetic energy of a four-momentum

Definition

Throughout, $(A, V, η, C^{+})$ is the oriented time-oriented affine Minkowski model of Oriented time-oriented affine Minkowski space (the spacetime primitives), $c > 0$ , and $n \in V$ is future unit timelike, $η (n, n) = - 1$ (Lorentz forms of inertia (3,1,0) on a 4-dimensional real vector space, Time orientation, future and past timelike/null/causal vectors, and the closed cones). Write $n^{⊥} = {v \in V : η (n, v) = 0}$ for the rest space of $n$ , $h (v) : = v + η (n, v) n$ for the rest projection and $∣ w ∣ : = \sqrt{η (w, w)}$ for the norm on $n^{⊥}$ (Norms of four-velocity and four-acceleration, and the observer rest-space decomposition,(c)); recall from that clause that $h$ is linear with image $n^{⊥}$ and kernel $s p a n (n)$ , that $V = s p a n (n) \oplus n^{⊥}$ , and that the restriction of $η$ to $n^{⊥}$ is positive definite.

(a) Observer energy, spatial momentum and invariant mass. For every $P$ in the future closed cone $c l (C^{+})$ define

$E_{n} (P) : = - c η (P, n), q_{n} (P) : = h (P),$

and, when $P$ is nonzero future causal, define the invariant mass of $P$ by

$m (P) : = \frac{\sqrt{- η (P, P)}}{c} .$

Then $q_{n} (P) \in n^{⊥}$ , $η (n, P) = - E_{n} (P) / c$ , and

$P = \frac{E_{n} (P)}{c} n + q_{n} (P)$

is the unique decomposition of $P$ as a real multiple of $n$ plus a vector of $n^{⊥}$ : a vector $P$ has $P = λ n + w$ with $λ \in R$ and $w \in n^{⊥}$ if and only if $λ = E_{n} (P) / c$ and $w = q_{n} (P)$ . Equivalently, $E_{n} (P)$ and $q_{n} (P)$ are the unique pair consisting of a real number and a vector of $n^{⊥}$ with that property. The pair $(E_{n}, q_{n})$ is the restriction to $c l (C^{+})$ of the pair of linear maps $P \mapsto - c η (P, n)$ and $P \mapsto h (P)$ on $V$ . For future null $P$ the formula for $m (P)$ gives $m (P) = 0$ , because $η (P, P) = 0$ .

(b) Rest-mass parameter, rest energy and kinetic energy. Let $m > 0$ and let $P = m U$ with $U$ future timelike and $η (U, U) = - c^{2}$ , the four-velocity normalisation of Four-velocity and four-acceleration of a future timelike worldline. Then call $m$ the rest-mass parameter of $P$ as written, $m c^{2}$ its rest energy, and

$T_{n} (P) : = E_{n} (P) - m c^{2}$

its kinetic energy relative to the observer $n$ . Bilinearity gives $η (P, P) = m^{2} η (U, U) = - m^{2} c^{2}$ , so $m = m (P) > 0$ is determined by $P$ . Thus $T_{n} (P) = E_{n} (P) - m (P) c^{2}$ is a function of $(P, n)$ on the future timelike cone; recording $m$ supplies no independent parameter.

(c) No positivity, inequality or measurement claim. This definition claims no positivity of $E_{n}$ , no inequality or vanishing property of $T_{n}$ , and no physical measurement. The names observer energy, spatial momentum, invariant mass, rest energy and kinetic energy, and the units joule, kilogram metre per second and kilogram, belong to the unit conventions of SI units and coordinate conventions of the special-relativity framework and to the physical postulates of this framework; the mathematical content asserted here is only the explicit dependence of the five quantities on $P$ , $n$ and $m$ .

(d) What the objects depend on. $E_{n}$ and $q_{n}$ are functions of the pair $(P, n)$ : $q_{n} (P)$ is an element of the vector space $V$ , not a coordinate triple, and no basis of $V$ enters its definition, while $E_{n} (P) = - c η (P, n)$ uses the Lorentz form alone. $m (P)$ depends on the scalar $η (P, P)$ alone and hence on no observer and no basis; $T_{n} (P)$ uses the rest mass $m = m (P)$ fixed by that scalar. No choice principle is used.

LemmaStatement: AI-adaptedProof: AI-adaptedOpen item page →

Positivity, mass shell and kinetic energy of the observer quantities of a four-momentum

Statement

Let $(A, V, η, C^{+})$ be the oriented time-oriented affine Minkowski model (Oriented time-oriented affine Minkowski space (the spacetime primitives)), $c > 0$ , let $n \in V$ be future unit timelike, $η (n, n) = - 1$ , and let $E_{n}, q_{n}, m, T_{n}$ be the quantities of Observer energy, spatial momentum, invariant mass, rest energy and kinetic energy of a four-momentum.

(a) If $P$ is nonzero future causal, then $E_{n} (P) > 0$ , $q_{n} (P) \in n^{⊥}$ and

$η (q_{n} (P), q_{n} (P)) = \frac{E_{n} (P)^{2}}{c^{2}} + η (P, P) \geq 0.$

Consequently: (i) if $P$ is future null, then $η (P, P) = 0$ and $E_{n} (P) = c ∣ q_{n} (P) ∣ > 0$ ; (ii) if $P$ is future timelike with invariant mass $m (P)$ , then $η (P, P) = - m (P)^{2} c^{2}$ and the mass-shell identity

$E_{n} (P)^{2} = m (P)^{2} c^{4} + c^{2} ∣ q_{n} (P) ∣^{2}$

holds. The invariant mass $m (P)$ depends only on the scalar $η (P, P)$ and therefore on no observer $n$ and no basis, while $E_{n}$ and $q_{n}$ do depend on $n$ .

(b) Let $P = m U$ with $m > 0$ constant, let $U$ be future timelike with $η (U, U) = - c^{2}$ (the four-velocity normalisation of Four-velocity and four-acceleration of a future timelike worldline), and put $γ : = - η (U / c, n) \geq 1$ and $w : = c h (U / c) / γ \in n^{⊥}$ , the relative velocity of Norms of four-velocity and four-acceleration, and the observer rest-space decomposition,(d) applied to the future unit vector $u = U / c$ ; equivalently $w = h (U) / γ$ . Then

$E_{n} (P) = m c^{2} γ, T_{n} (P) = m c^{2} (γ - 1) \geq 0, q_{n} (P) = m γ w,$

and $m (P) = m$ . Moreover $T_{n} (P) = 0$ if and only if $γ = 1$ , that is $U = c n$ , equivalently $P = m c n$ (the particle is at rest relative to $n$ ). Finally $∣ q_{n} (P) ∣ = m γ ∣ w ∣ = m c \sqrt{γ^{2} - 1}$ .

Facts & Assumptions

Given: the affine Minkowski model $(A, V, η, C^{+})$ ; a real number $c > 0$ ; a future unit timelike $n \in V$ with $η (n, n) = - 1$ ; clause (a): a nonzero $P \in c l (C^{+})$ ; clause (b): a real $m > 0$ and a future timelike $U \in V$ with $η (U, U) = - c^{2}$ , with $P = m U$ .

[F1]

The definition of the observer quantities (Observer energy, spatial momentum, invariant mass, rest energy and kinetic energy of a four-momentum, Definition): for $P \in c l (C^{+})$ one has $E_{n} (P) = - c η (P, n)$ and $q_{n} (P) = h (P) = P + η (P, n) n \in n^{⊥}$ , and $η (n, P) = - E_{n} (P) / c$ ; for nonzero future causal $P$ the invariant mass is $m (P) = \sqrt{- η (P, P)} / c$ ; for $P = m U$ with $m > 0$ and $η (U, U) = - c^{2}$ the kinetic energy is $T_{n} (P) = E_{n} (P) - m c^{2}$ ; and $∣ w ∣ = \sqrt{η (w, w)}$ on $n^{⊥}$ .

[F2]

Norms of four-velocity and four-acceleration, and the observer rest-space decomposition, clauses (c) and (d): $h$ is linear with image $n^{⊥}$ and kernel $s p a n (n)$ , the restriction of $η$ to $n^{⊥}$ is positive definite, and $V = s p a n (n) \oplus n^{⊥}$ ; and for future unit timelike $n, u$ one has, with $γ : = - η (n, u)$ and $w : = c h (u) / γ$ , the relations $γ \geq 1$ , $w \in n^{⊥}$ , $u = γ (n + w / c)$ , $∣ w ∣^{2} = c^{2} (1 - γ^{- 2})$ , and $γ = 1$ if and only if $u = n$ .

[F3]

Lorentz forms of inertia (3,1,0) on a 4-dimensional real vector space: $η$ is a symmetric bilinear form on $V$ , so $η (v, w)$ is additive and homogeneous in each argument and $η (v, w) = η (w, v)$ ; in particular polarisation gives $η (v + w, v + w) = η (v, v) + 2 η (v, w) + η (w, w)$ and $η (λ v, μ w) = λ μ η (v, w)$ for real $λ, μ$ .

[F4]

Every Lorentz form admits an ordered basis with matrix diag(-1,1,1,1): there is an ordered basis $(e_{0}, e_{1}, e_{2}, e_{3})$ of $V$ with $η (e_{a}, e_{b}) = d i a g (- 1, 1, 1, 1)_{a b}$ .

[F5]

Existence of future-oriented inertial frames at every event with Inertial frames and inertial charts of the affine Minkowski model: in the oriented time-oriented affine Minkowski space there exists, for every event $o$ , an inertial frame with origin $o$ , and an inertial frame's basis is an $η$ -orthonormal ordered basis whose first vector lies in $C^{+}$ ; in particular such a basis of $V$ exists.

[F6]

The timelike set has exactly two convex components, described in orthonormal coordinates: in every $η$ -orthonormal ordered basis whose first vector is timelike, writing $v = v^{0} e_{0} + v_{s}$ with $v_{s} \in S : = s p a n (e_{1}, e_{2}, e_{3})$ and $∣ v_{s} ∣ : = \sqrt{(v^{1})^{2} + (v^{2})^{2} + (v^{3})^{2}}$ , one has $η (v, v) = - (v^{0})^{2} + ∣ v_{s} ∣^{2}$ , $C^{+} = {v : v^{0} > ∣ v_{s} ∣}$ (for the appropriate naming of the two components) and $c l (C^{+}) = {v : v^{0} \geq ∣ v_{s} ∣}$ .

[F7]

The Euclidean spatial structure and cross product of an inertial frame: for an inertial frame, $S = s p a n (e_{1}, e_{2}, e_{3})$ equals $e_{0}^{⊥}$ , the restriction $δ$ of $η$ to $S$ is positive definite, and $∣ w ∣ = \sqrt{δ (w, w)}$ is a norm on $S$ ; in coordinates $δ (v_{s}, w_{s}) = v_{s} \cdot w_{s} = \sum_{i = 1}^{3} v^{i} w^{i}$ .

[F8]

The Cauchy-Schwarz inequality for finite sums: for real numbers $a_{i}, b_{i}$ , $∣ \sum_{i = 1}^{3} a_{i} b_{i} ∣ \leq (\sum_{i = 1}^{3} a_{i}^{2})^{1 / 2} (\sum_{i = 1}^{3} b_{i}^{2})^{1 / 2}$ .

[F9]

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 vector $v$ is causal exactly when $v \neq 0$ and $η (v, v) \leq 0$ (it is timelike when $η (v, v) < 0$ and null when $v \neq 0$ and $η (v, v) = 0$ ), and $C^{+}$ is a component of the timelike set with closed cone $c l (C^{+})$ .

[F10]

Rational powers $a^{r}$ of a positive base and Laws of rational exponents: for $a, b > 0$ and rationals $r, s$ the power $a^{r}$ is the positive real number defined there, with $(a b)^{r} = a^{r} b^{r}$ , $(a^{r})^{s} = a^{r s}$ and $a^{1 / 2}$ the unique nonnegative square root of $a$ ; in particular $\sqrt{λ^{2}} = λ$ for $λ > 0$ and $γ \sqrt{1 - γ^{- 2}} = \sqrt{γ^{2} - 1}$ for $γ \geq 1$ .

Proof

technique · fix orthonormal coordinates, prove the algebraic identity and positivity there, then specialise to $P=mU$ using the relative-velocity decomposition

1.1F1F2F3

By [F1], $q_{n} (P) = P + η (P, n) n \in n^{⊥}$ , $η (P, n) = - E_{n} (P) / c$ , $m (P)^{2} = - η (P, P) / c^{2}$ for nonzero future causal $P$ , and $T_{n} (P) = E_{n} (P) - m c^{2}$ in the setting of (b). Expanding with [F3] and $η (n, n) = - 1$ gives $η (q_{n} (P), q_{n} (P)) = η (P, P) + 2 η (P, n)^{2} + η (P, n)^{2} η (n, n) = η (P, P) + η (P, n)^{2} = η (P, P) + E_{n} (P)^{2} / c^{2}$ . The formula $m (P)^{2} = - η (P, P) / c^{2}$ involves only the scalar $η (P, P)$ , so $m (P)$ depends on no observer and no basis, whereas $E_{n}$ and $q_{n}$ are built from $η (P, n)$ and $h$ and depend on $n$ .

1.2F3F4F5F6F7F8

Choose by [F4] and [F5] an $η$ -orthonormal ordered basis $(e_{0}, e_{1}, e_{2}, e_{3})$ with $e_{0} \in C^{+}$ ; write $v = v^{0} e_{0} + v_{s}$ with $v_{s} \in S = s p a n (e_{1}, e_{2}, e_{3})$ and $∣ v_{s} ∣$ as in [F6]. Bilinearity [F3] extends the quadratic form of [F6] to $η (v, w) = - v^{0} w^{0} + δ (v_{s}, w_{s})$ with $δ (v_{s}, w_{s}) = v_{s} \cdot w_{s}$ the Euclidean inner product of [F7]. By [F6], $C^{+} = {v : v^{0} > ∣ v_{s} ∣}$ and $c l (C^{+}) = {v : v^{0} \geq ∣ v_{s} ∣}$ ; hence $n^{0} > ∣ n_{s} ∣ \geq 0$ and $P^{0} \geq ∣ P_{s} ∣ \geq 0$ , while $η (n, n) = - 1$ gives $(n^{0})^{2} = 1 + ∣ n_{s} ∣^{2}$ . By [F8], $∣ P_{s} \cdot n_{s} ∣ \leq ∣ P_{s} ∣ ∣ n_{s} ∣$ .

1.3F2F3F7

In the setting of (b), put $u : = U / c$ . Since $η (U, U) = - c^{2}$ , bilinearity [F3] gives $η (u, u) = c^{- 2} η (U, U) = - 1$ , so $u$ is future unit timelike, and [F2] clause (d) applies: with $γ : = - η (n, u) \geq 1$ and $w : = c h (u) / γ$ one has $w \in n^{⊥}$ , $u = γ (n + w / c)$ , $∣ w ∣^{2} = c^{2} (1 - γ^{- 2})$ , and $γ = 1$ if and only if $u = n$ ; equivalently $U = c n$ , and $P = m U = m c n$ when $γ = 1$ .

2.1step 1.1step 1.2F8

Positivity of $E_{n}$ : using [step 1.2], $c^{- 1} E_{n} (P) = - η (P, n) = P^{0} n^{0} - P_{s} \cdot n_{s}$ , so $c^{- 1} E_{n} (P) \geq P^{0} n^{0} - ∣ P_{s} ∣ ∣ n_{s} ∣ \geq ∣ P_{s} ∣ n^{0} - ∣ P_{s} ∣ ∣ n_{s} ∣ = ∣ P_{s} ∣ (n^{0} - ∣ n_{s} ∣) \geq 0$ , where the first inequality uses [F8], the second uses $P^{0} \geq ∣ P_{s} ∣$ and $n^{0} > 0$ , and the last uses $n^{0} > ∣ n_{s} ∣$ of [step 1.2]; if $∣ P_{s} ∣ > 0$ the last term is strictly positive, while if $∣ P_{s} ∣ = 0$ then $P \neq 0$ forces $P^{0} > 0$ and $c^{- 1} E_{n} (P) = P^{0} n^{0} > 0$ , so $E_{n} (P) > 0$ in both cases.

2.2step 1.1F1F2

Since $q_{n} (P) \in n^{⊥}$ by step 1.1 and the restriction of $η$ to $n^{⊥}$ is positive definite by [F2], $η (q_{n} (P), q_{n} (P)) = ∣ q_{n} (P) ∣^{2} \geq 0$ . Combining this with the identity of step 1.1 gives the displayed equality and inequality of (a).

2.3step 1.1step 1.3F2F3

The energy and momentum of $P = m U$ : by [step 1.1] and [step 1.3], $E_{n} (P) = - c η (m U, n) = - m c^{2} η (U / c, n) = m c^{2} γ$ and $q_{n} (P) = h (m U) = m h (U) = m c h (u) = m γ w$ , the last equality because $h (u) = h (γ (n + w / c)) = γ h (w / c) = γ w / c$ by linearity of $h$ and $h (n) = 0$ in [F2].

3.1step 2.1step 2.2F1F9

The null case: if $P$ is future null then $η (P, P) = 0$ by [F9], so [step 2.2] gives $η (q_{n} (P), q_{n} (P)) = E_{n} (P)^{2} / c^{2} > 0$ by [step 2.1], and since $q_{n} (P) \in n^{⊥}$ the norm of [F1] gives $∣ q_{n} (P) ∣ = \sqrt{η (q_{n} (P), q_{n} (P))} = E_{n} (P) / c$ , that is $E_{n} (P) = c ∣ q_{n} (P) ∣ > 0$ .

3.2step 1.1step 2.2F1

The timelike case and the mass shell: if $P$ is future timelike, then $η (P, P) = - m (P)^{2} c^{2}$ by the definition of $m (P)$ in [F1]; substituting this into [step 2.2] and multiplying by $c^{2}$ gives $E_{n} (P)^{2} = c^{2} η (q_{n} (P), q_{n} (P)) + m (P)^{2} c^{4} = c^{2} ∣ q_{n} (P) ∣^{2} + m (P)^{2} c^{4}$ , using $∣ q_{n} (P) ∣^{2} = η (q_{n} (P), q_{n} (P))$ for $q_{n} (P) \in n^{⊥}$ .

3.3step 1.3step 2.3F1

The kinetic energy: by [step 2.3] and $T_{n} (P) = E_{n} (P) - m c^{2}$ of [F1], $T_{n} (P) = m c^{2} γ - m c^{2} = m c^{2} (γ - 1) \geq 0$ because $γ \geq 1$ in [step 1.3]; moreover $T_{n} (P) = 0$ if and only if $γ = 1$ , which by [step 1.3] is equivalent to $u = n$ , that is to $U = c n$ , that is to $P = m U = m c n$ , and conversely $P = m c n$ gives $γ = 1$ and $T_{n} (P) = 0$ .

3.4step 1.3step 2.3F10

The momentum magnitude: by [step 2.3] and [step 1.3], $∣ q_{n} (P) ∣ = m γ ∣ w ∣ = m γ c \sqrt{1 - γ^{- 2}}$ , and since $γ \geq 1$ , [F10] gives $γ \sqrt{1 - γ^{- 2}} = \sqrt{γ^{2} (1 - γ^{- 2})} = \sqrt{γ^{2} - 1}$ , so $∣ q_{n} (P) ∣ = m c \sqrt{γ^{2} - 1}$ .

4.1step 1.1step 2.1step 2.3step 3.1step 3.2step 3.3step 3.4F3∎

Collect the conclusions and evaluate the invariant mass of $P = m U$ : by [step 1.1], $m (P)^{2} = - η (P, P) / c^{2}$ , and for $P = m U$ with $η (U, U) = - c^{2}$ bilinearity [F3] gives $η (P, P) = m^{2} η (U, U) = - m^{2} c^{2}$ , so $m (P)^{2} = m^{2}$ and, since both $m (P)$ and $m$ are nonnegative and $c > 0$ , $m (P) = m$ ; together with [step 2.1] and [step 3.1] this proves clause (a) including (i), with [step 3.2] proving clause (a)(ii), and with [step 2.3], [step 3.3] and [step 3.4] it proves clause (b), the vanishing statement of (b) included by [step 3.3].

LemmaStatement: AI-adaptedProof: AI-adaptedOpen item page →

Four-momentum components, energy-velocity relation and passive boost transformation in an inertial frame

Statement

(a) For every $P \in c l (C^{+})$ the coordinate components of $P$ in the frame are $(P^{0}, P^{1}, P^{2}, P^{3}) = (E_{n} (P) / c, q^{1}, q^{2}, q^{3})$ , where $q_{n} (P) = \sum_{i = 1}^{3} q^{i} e_{i}$ ; if $P$ is future timelike then $E^{2} = m (P)^{2} c^{4} + c^{2} \sum_{i = 1}^{3} (q^{i})^{2}$ with $E = E_{n} (P)$ .

(b) Let $x : I \to A$ be a $C^{1}$ future timelike worldline (Curves, regular causal curves and future timelike worldlines in affine Minkowski space) whose chart coordinate path is $t \mapsto (c t, r (t))$ , with coordinate velocity $w : = r^{'}$ and $γ_{w} : = (1 - ∣ w ∣^{2} / c^{2})^{- 1 / 2}$ (The Euclidean spatial structure and cross product of an inertial frame, Rational powers $a^{r}$ of a positive base); then $∣ w ∣ < c$ , the four-velocity is $U = γ_{w} (c, w)$ by Norms of four-velocity and four-acceleration, and the observer rest-space decomposition,(e), and for $P = m U$ with $m > 0$ constant the components are $P = m γ_{w} (c, w)$ , that is $E_{n} (P) = m γ_{w} c^{2}$ and $q_{n} (P) = \sum_{i = 1}^{3} m γ_{w} w^{i} e_{i}$ .

(c) Let $B (β)$ be the passive boost of The passive Lorentz boost with parameter beta with parameter $β = (v / c) s$ , where $v \in R$ , $∣ v ∣ < c$ , and $s \in R^{3}$ is a unit spatial coordinate vector ( $s \cdot s = 1$ ), and let $P^{'}$ denote the components of the same vector $P$ after the coordinate change $x^{'} = B (β) x$ , with $B (β)$ here denoting its frame matrix. Put $γ_{v} : = (1 - v^{2} / c^{2})^{- 1 / 2}$ , $p : = (P^{1}, P^{2}, P^{3})$ , $p_{∥} : = s \cdot p$ , $p_{∥} : = p_{∥} s$ and $p_{⊥} : = p - p_{∥} s$ , with the same projections defining the primed components. Then

$P^{' 0} = γ_{v} (P^{0} - (v / c) p_{∥}), p_{∥}^{'} = γ_{v} (p_{∥} - (v / c) P^{0} s), p_{⊥}^{'} = p_{⊥},$

equivalently $E^{'} = γ_{v} (E - v p_{∥})$ , $p_{∥}^{'} = γ_{v} (p_{∥} - v E / c^{2})$ , $p_{⊥}^{'} = p_{⊥}$ . The quadratic combination is invariant, $E^{' 2} - c^{2} ∣ p^{'} ∣^{2} = E^{2} - c^{2} ∣ p ∣^{2} = m (P)^{2} c^{4}$ for future timelike $P$ , while for future null $P$ one has $E = c ∣ p ∣$ and $E^{'} = c ∣ p^{'} ∣$ in both coordinate systems; in particular the invariant mass is frame-independent.

Facts & Assumptions

Given: the affine Minkowski model $(A, V, η, C^{+})$ ; $c > 0$ ; an inertial frame $(o; e_{0}, e_{1}, e_{2}, e_{3})$ with $n = e_{0}$ ; $P \in c l (C^{+})$ for (a); a $C^{1}$ future timelike worldline $x$ with chart path $t \mapsto (c t, r (t))$ for (b); a passive boost $B (β)$ with $β = (v / c) s$ , $∣ v ∣ < c$ , $∣ s ∣ = 1$ , for (c).

[F1]

(Observer energy, spatial momentum, invariant mass, rest energy and kinetic energy of a four-momentum, Definition): for $P \in c l (C^{+})$ , $P = (E_{n} (P) / c) n + q_{n} (P)$ is the unique decomposition with $q_{n} (P) \in n^{⊥}$ ; for nonzero future causal $P$ , $m (P) = \sqrt{- η (P, P)} / c$ ; and $q_{n} (P)$ is an element of $V$ , not a coordinate triple.

[F2]

(Positivity, mass shell and kinetic energy of the observer quantities of a four-momentum, Statement): if $P$ is nonzero future causal, then $E_{n} (P) > 0$ and $η (q_{n} (P), q_{n} (P)) = E_{n} (P)^{2} / c^{2} + η (P, P)$ ; if $P$ is future null, then $E_{n} (P) = c ∣ q_{n} (P) ∣$ ; if $P$ is future timelike with invariant mass $m (P)$ , then $E_{n} (P)^{2} = m (P)^{2} c^{4} + c^{2} ∣ q_{n} (P) ∣^{2}$ ; and $m (P)$ depends only on the scalar $η (P, P)$ .

[F3]

(Inertial frames and inertial charts of the affine Minkowski model, Definition): the inertial chart $x$ sends $p$ to the coordinates relative to $(o; e_{0}, \dots, e_{3})$ , so a vector of $V$ has components $P^{a}$ with $P = \sum_{a = 0}^{3} P^{a} e_{a}$ .

[F4]

(Norms of four-velocity and four-acceleration, and the observer rest-space decomposition, clause (e)): for a $C^{1}$ future timelike worldline with chart path $t \mapsto (c t, r (t))$ , coordinate velocity $w = r^{'}$ and $γ_{w} = (1 - ∣ w ∣^{2} / c^{2})^{- 1 / 2}$ , one has $∣ w (t) ∣ < c$ and $U = γ_{w} (c, w)$ .

[F5]

(The Euclidean spatial structure and cross product of an inertial frame, Definition): $S = s p a n (e_{1}, e_{2}, e_{3}) = e_{0}^{⊥}$ , the restriction $δ$ of $η$ to $S$ is positive definite, $∣ w ∣ = \sqrt{δ (w, w)}$ is a norm, and with $w = \sum w^{i} e_{i}$ one has $δ (w, z) = w \cdot z = \sum_{i} w^{i} z^{i}$ and $∣ w ∣^{2} = \sum_{i} (w^{i})^{2}$ ; the cross product is not used here.

[F6]

(Rational powers $a^{r}$ of a positive base and Laws of rational exponents): for $0 \leq q < c^{2}$ the expression $(1 - q / c^{2})^{- 1 / 2}$ is a positive real number whose square is $(1 - q / c^{2})^{- 1}$ ; with $q = ∣ w ∣^{2}$ this gives $γ_{w}^{2} (1 - ∣ w ∣^{2} / c^{2}) = 1$ and $γ_{w} > 0$ for $∣ w ∣ < c$ .

[F7]

(The passive Lorentz boost with parameter beta, Definition): $B (β)$ is the linear map with matrix $B^{0}_{0} = γ$ , $B^{0}_{i} = B^{i}_{0} = - γ β_{i}$ and $B^{i}_{j} = δ_{j}^{i} + ((γ - 1) / ∣ β ∣^{2}) β_{i} β_{j}$ for $β \neq 0$ , and $B (0) = I$ , in the basis $(e_{0}, e_{1}, e_{2}, e_{3})$ , where $γ = (1 - ∣ β ∣^{2})^{- 1 / 2}$ .

[F8]

(Transformation of tensor, covector, derivative and current components under affine Lorentz maps, clauses (i) and (vi)): under the coordinate change $x^{'} = Λ x + b$ the components of a vector transform as $P^{' a} = Λ^{a}_{b} P^{b}$ ; for $Λ = B (β)$ this gives $P^{' 0} = γ (P^{0} - β \cdot p)$ and the displayed spatial formula.

[F9]

(The passive boost is a proper orthochronous Lorentz map with inverse B(-beta), clauses (i) and (iii)): $B (β)$ preserves $η$ , is invertible, and maps $C^{+}$ onto $C^{+}$ .

[F10]

(Lorentz forms of inertia (3,1,0) on a 4-dimensional real vector space, Definition): $η$ is bilinear, so for a vector $P = \sum_{a} P^{a} e_{a}$ in an $η$ -orthonormal basis, $η (P, P) = - (P^{0})^{2} + ∣ p ∣^{2}$ where $∣ p ∣^{2} = \sum_{i} (P^{i})^{2}$ by [F5].

Proof

technique · read the decomposition of the observer definition in the frame basis, then transform components with the boost matrix and cancel the quadratic combination algebraically

1.1F1F3F5

Since $n = e_{0}$ , the decomposition of [F1] reads $P = (E_{n} (P) / c) e_{0} + q_{n} (P)$ ; as $q_{n} (P) \in n^{⊥} = S = s p a n (e_{1}, e_{2}, e_{3})$ by [F5], write $q_{n} (P) = \sum_{i} q^{i} e_{i}$ , so the coordinate components of $P$ in the basis $(e_{0}, e_{1}, e_{2}, e_{3})$ are $(P^{0}, P^{1}, P^{2}, P^{3}) = (E_{n} (P) / c, q^{1}, q^{2}, q^{3})$ by [F3].

1.2F4F5F6

Under the hypotheses of (b), [F4] gives $∣ w ∣ < c$ and $U = γ_{w} (c, w)$ , so the chart components of $U$ are $U^{0} = γ_{w} c$ and $U^{i} = γ_{w} w^{i}$ ; since $w^{i}$ are the components of $w \in S$ , the norm identity $∣ w ∣^{2} = \sum_{i} (w^{i})^{2}$ of [F5] holds and $γ_{w} > 0$ with $γ_{w}^{2} (1 - ∣ w ∣^{2} / c^{2}) = 1$ by [F6].

1.3F7F8

Under the hypotheses of (c), write $p_{∥} : = s \cdot p$ and $p_{⊥} : = p - p_{∥} s$ , so that $p = p_{∥} s + p_{⊥}$ with $s \cdot p_{⊥} = 0$ ; for $v \neq 0$ the boost $B (β)$ with $β = (v / c) s$ has the matrix of [F7] with $∣ β ∣ = ∣ v ∣ / c$ , $β \cdot p = (v / c) p_{∥}$ and $((γ_{v} - 1) / ∣ β ∣^{2}) (β \cdot p) β = (γ_{v} - 1) p_{∥} s$ , and [F8] gives $P^{' a} = B (β)^{a}_{b} P^{b}$ ; hence for $v \neq 0$ with $∣ v ∣ < c$ , $p^{'} = p + (γ_{v} - 1) p_{∥} s - γ_{v} (v / c) s P^{0} = p_{⊥} + γ_{v} (p_{∥} - (v / c) P^{0}) s$ and $P^{' 0} = γ_{v} (P^{0} - (v / c) p_{∥})$ . For $v = 0$ , the definition gives $B (0) = I$ and $γ_{v} = 1$ , so the same two final formulas reduce to $p' = p$ and $P'^{0} = P^{0}$ without division by $∣ β ∣^{2}$ .

2.1F2F5step 1.1

From [step 1.1], $∣ q_{n} (P) ∣^{2} = \sum_{i} (q^{i})^{2}$ by [F5] and $E_{n} (P) = c P^{0}$ , so for future timelike $P$ the mass-shell identity of [F2] becomes $E^{2} = m (P)^{2} c^{4} + c^{2} \sum_{i} (q^{i})^{2}$ ; this is the second claim of (a).

2.2step 1.1step 1.2

For $P = m U$ with $m > 0$ constant, the components of $P$ are $m$ times those of $U$ , so by [step 1.2] $P^{0} = m γ_{w} c$ and $P^{i} = m γ_{w} w^{i}$ ; [step 1.1] identifies $E_{n} (P) = c P^{0} = m γ_{w} c^{2}$ and $q_{n} (P) = \sum_{i} m γ_{w} w^{i} e_{i}$ , which is (b).

2.3F5step 1.1step 1.3

Specialise [step 1.3] and use $E = c P^{0}$ , $E^{'} = c P^{' 0}$ of [step 1.1]: reading the components, $E^{'} = γ_{v} (E - v p_{∥})$ and $s \cdot p^{'} = γ_{v} (p_{∥} - v E / c^{2})$ because $(v / c) P^{0} = v E / c^{2}$ , while $p_{⊥}^{'} = p_{⊥}$ because $s \cdot p_{⊥} = 0$ ; this is the component law of (c).

3.1F2F6F10step 2.3

Invariance of the quadratic combination: with the formulas of [step 2.3], $γ_{v}^{2} [(E - v p_{∥})^{2} - c^{2} (p_{∥} - v E / c^{2})^{2}] = γ_{v}^{2} (1 - v^{2} / c^{2}) [E^{2} - c^{2} p_{∥}^{2}] = E^{2} - c^{2} p_{∥}^{2}$ by $γ_{v}^{2} (1 - v^{2} / c^{2}) = 1$ from [F6], so $E^{' 2} - c^{2} ∣ p^{'} ∣^{2} = E^{2} - c^{2} p_{∥}^{2} - c^{2} ∣ p_{⊥} ∣^{2} = E^{2} - c^{2} ∣ p ∣^{2}$ ; by [F10] and [F2] this equals $- c^{2} η (P, P) = m (P)^{2} c^{4}$ for future timelike $P$ (and $0$ for future null $P$ , since $η (P, P) = 0$ ).

4.1F2F8F9step 3.1

If $P$ is future null, [F2] gives $E = c ∣ p ∣$ in the unprimed frame. The primed column is $[P]^{'} = B (β) [P]$ , the coordinates of the same vector in the primed inertial frame; by [F9] it is nonzero and lies in the future null coordinate cone. Applying [F2] in that frame gives $E^{'} > 0$ and $E^{'} = c ∣ p^{'} ∣$ . The invariant mass depends only on $η (P, P)$ , and step 3.1 shows that both frames compute the same quadratic combination. This completes (c).

5.1step 1.1step 2.1step 2.2step 2.3step 3.1step 4.1∎

Clause (a) is [step 1.1] with [step 2.1], clause (b) is [step 2.2], and clause (c) is [step 2.3] with [step 3.1] and [step 4.1]; hence all assertions of the lemma hold.

DefinitionDefinition: AI-adaptedProof: Not applicablejudge pass (gpt-6.1-sol)Open item page →

Proper-time derivative of a vector field along a worldline and the four-force notation

Definition

Throughout, $c > 0$ and $(A, V, η, C^{+})$ is the oriented time-oriented affine Minkowski model of Oriented time-oriented affine Minkowski space (the spacetime primitives), with displacement space $V$ ; $I \subseteq R$ is an interval with at least two elements (Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length) and $x : I \to A$ is a $C^{1}$ future timelike worldline (Curves, regular causal curves and future timelike worldlines in affine Minkowski space) with proper-time function $τ : I \to J$ and inverse $ρ : J \to I$ (Proper time along a future timelike worldline, The proper-time function is a strictly increasing C^1 parametrization with C^1 inverse). By that lemma $τ$ is $C^{1}$ with

$τ^{'} (s) = \frac{1}{c} \sqrt{- η (x^{'} (s), x^{'} (s))} > 0 (s \in I),$

$τ$ is strictly increasing, $J = τ [I]$ is an interval with at least two elements, and $ρ : J \to I$ is $C^{1}$ with $ρ^{'} (τ (s)) = 1 / τ^{'} (s) > 0$ for every $s \in I$ .

(a) The proper-time derivative. Let $P : I \to V$ be a $C^{1}$ map, its derivative $P^{'}$ being the derivative of a $V$ -valued function of one real variable, read componentwise in any ordered basis of $V$ (The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral). The proper-time derivative of $P$ is the map $\dot{P} : I \to V$ defined by

$\dot{P} (s) : = \frac{P^{'} (s)}{τ^{'} (s)} .$

This is well defined because $τ^{'} (s) > 0$ for every $s \in I$ , so the scalar division by $τ^{'} (s)$ is a legitimate vector-space operation on $V$ . The definition does not depend on the basis of $V$ in which $P^{'}$ is read: a change of basis acts on the coordinates of $P^{'} (s)$ , and hence on those of $\dot{P} (s)$ , by the same constant invertible linear map, which commutes with multiplication by the scalar $1 / τ^{'} (s)$ . Since $P^{'}$ and $1 / τ^{'}$ are continuous, $\dot{P}$ is continuous; equivalently, the proper-time derivative of a $C^{1}$ field is a continuous field.

The proper-time reading. Since $τ$ is a strictly increasing bijection $I \to J$ with inverse $ρ$ , one has $ρ (τ (s)) = s$ and therefore $(P \circ ρ) \circ τ = P$ on $I$ . Fix $s \in I$ ; the components of $P$ in a basis of $V$ are differentiable at $s$ and $ρ$ is differentiable at $τ (s)$ , so the scalar chain rule (The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ ) applied to each component of $P \circ ρ$ gives

$P^{'} (s) = (P \circ ρ)^{'} (τ (s)) τ^{'} (s), equivalently \dot{P} (s) = (P \circ ρ)^{'} (τ (s)) .$

At an endpoint of $I$ this is the one-sided reading of the derivative, as in The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral. Thus $\dot{P}$ is exactly the derivative with respect to the proper-time parameter of the field $P$ read along the parametrisation $s \mapsto τ (s)$ ; for every $u \in J$ the chain rule on $J$ gives $(P \circ ρ)^{'} (u) = P^{'} (ρ (u)) ρ^{'} (u) = \dot{P} (ρ (u))$ .

(b) The four-force notation. Let $m > 0$ be constant and suppose the four-velocity $U : I \to V$ of $x$ is $C^{1}$ (Four-velocity and four-acceleration of a future timelike worldline); by that definition $U$ is always continuous, and $U$ is $C^{1}$ whenever the worldline $x$ is $C^{2}$ . Then $P : = m U$ is a $C^{1}$ map $I \to V$ and $F : = \dot{P}$ is called the four-force on the particle. In an inertial frame the components of $F$ are the derivatives of the components of the four-momentum with respect to the proper-time parameter: writing $P^{a} (u)$ for the components of $P$ as a function of the proper-time parameter $u = τ (s)$ , one has $F^{a} (s) = \frac{d}{d u} P^{a} (u) ∣_{u = τ (s)}$ .

(c) Scope. This item introduces a derivative and a name only. It asserts no force law, no field equation, no equation of motion and no existence or uniqueness of solutions of any equation of motion; in particular the hypothesis that $U$ be $C^{1}$ is a regularity hypothesis on the given worldline, not a dynamical statement. The identities satisfied by $F$ for a stipulated four-momentum field are proved in Four-force identities and the work-energy relation in an inertial frame, and a physical force law requires the separate model premise Adopted massive-particle momentum, observer measurement and isolated-collision conservation. The names four-momentum and four-force, and the units in which the components of $P$ and $F$ are measured, belong to the framework's coordinate and unit conventions (SI units and coordinate conventions of the special-relativity framework); the mathematical content asserted here is the derivative construction (a) and the notation (b) alone. No choice principle is used.

LemmaStatement: AI-adaptedProof: AI-adaptedOpen item page →

Four-force identities and the work-energy relation in an inertial frame

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 $(o; e_{0}, e_{1}, e_{2}, e_{3})$ be an inertial frame with $n : = e_{0}$ (Inertial frames and inertial charts of the affine Minkowski model). Let $x$ be a $C^{2}$ future timelike worldline whose chart coordinate path is $t \mapsto (c t, r (t))$ , with coordinate velocity $w : = r^{'}$ , coordinate acceleration $a : = w^{'}$ and $γ : = (1 - ∣ w ∣^{2} / c^{2})^{- 1 / 2}$ (The Euclidean spatial structure and cross product of an inertial frame); let $m > 0$ be constant, $P : = m U$ with $U = γ (c, w)$ the four-velocity (Four-velocity and four-acceleration of a future timelike worldline, Norms of four-velocity and four-acceleration, and the observer rest-space decomposition,(e)), $E : = E_{n} (P)$ , $p : = m γ w : I \to R^{3}$ , so that $q_{n} (P) = \sum_{i = 1}^{3} p^{i} e_{i}$ , $F : = \dot{P}$ the four-force of Proper-time derivative of a vector field along a worldline and the four-force notation and $f : = \frac{d}{d t} p$ the coordinate three-force.

(a) $F = m A$ , where $A$ is the four-acceleration, and $η (P, F) = 0$ ; equivalently $η (U, F) = 0$ and the mass shell $η (P, P) = - m^{2} c^{2}$ has vanishing proper-time derivative.

(b) $\frac{d E}{d t} = f \cdot w$ , the Euclidean inner product of the coordinate vectors.

(c) $F = γ (f \cdot w / c, f)$ , that is $F^{0} = γ f \cdot w / c$ and $F^{i} = γ f^{i}$ for $i = 1, 2, 3$ .

(d) If instead $m : I \to (0, \infty)$ is $C^{1}$ and $P = m U$ with this variable parameter, define $F : = \dot{P}$ by the general derivative clause (a) of Proper-time derivative of a vector field along a worldline and the four-force notation. Then $η (P, F) = - m c^{2} \frac{d m}{d τ}$ , the derivative being taken with respect to the proper-time parameter.

Facts & Assumptions

Given: the affine Minkowski model $(A, V, η, C^{+})$ ; $c > 0$ ; an inertial frame with $n = e_{0}$ ; a $C^{2}$ future timelike worldline $x$ with chart path $t \mapsto (c t, r (t))$ ; a constant $m > 0$ with $P = m U$ ; for (d) a $C^{1}$ function $m : I \to (0, \infty)$ with $P = m U$ .

[F1]

(Proper-time derivative of a vector field along a worldline and the four-force notation, Definition): for a $C^{1}$ field $P : I \to V$ along $x$ the proper-time derivative is $\dot{P} (s) = P^{'} (s) / τ^{'} (s)$ ; when $P = m U$ with $m > 0$ constant and $U$ of class $C^{1}$ , $F : = \dot{P}$ is the four-force.

[F2]

(Four-velocity and four-acceleration of a future timelike worldline, Definition): $U = x^{'} / τ^{'}$ and, for $C^{2}$ worldlines, $A = U^{'} / τ^{'}$ ; $U$ and $A$ are continuous and $U$ is future timelike.

[F3]

(Norms of four-velocity and four-acceleration, and the observer rest-space decomposition, clauses (a), (b) and (e)): $η (U, U) = - c^{2}$ ; if $x$ is $C^{2}$ then $η (U, A) = 0$ ; and in the chart, $∣ w ∣ < c$ , $U = γ_{w} (c, w)$ and $τ^{'} = c^{- 1} \sqrt{c^{2} - ∣ w ∣^{2}} = 1 / γ_{w}$ , so $d t / d τ = γ_{w}$ .

[F4]

(The proper-time function is a strictly increasing C^1 parametrization with C^1 inverse, Statement): $τ^{'} > 0$ , $τ$ is $C^{1}$ and strictly increasing, and the proper-time parametrisation is related to the chart parametrisation by the chain rule; for a $C^{1}$ field $Q$ along $x$ , $\dot{Q} = γ_{w} d Q / d t$ in the chart parametrisation of [F3].

[F5]

(Inertial frames and inertial charts of the affine Minkowski model and The Euclidean spatial structure and cross product of an inertial frame): in the frame the components of a vector are read in $(e_{0}, e_{1}, e_{2}, e_{3})$ , the spatial part lies in $S = e_{0}^{⊥}$ with positive definite restriction, and $s \cdot t$ is the Euclidean inner product there, with $∣ w ∣^{2} = w \cdot w$ .

[F6]

(Four-momentum components, energy-velocity relation and passive boost transformation in an inertial frame, clause (b)): $E_{n} (P) = m γ_{w} c^{2}$ and $q_{n} (P) = \sum_{i} m γ_{w} w^{i} e_{i}$ .

[F7]

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

[F8]

(Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ , Continuity and derivatives of positive-base real powers, The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ , The exponential definition of real powers agrees with the existing rational powers and Rational powers $a^{r}$ of a positive base): sums, scalar multiples and products of real differentiable functions and the positive power $t \mapsto (1 - t / c^{2})^{- 1 / 2}$ are differentiable with the usual rules; with $t = ∣ w ∣^{2}$ and $w$ of class $C^{1}$ this gives $γ_{w}^{'} = γ_{w}^{3} (w \cdot a) / c^{2}$ and $γ_{w}^{2} (1 - ∣ w ∣^{2} / c^{2}) = 1$ .

[F9]

(The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral, Definition): a $V$ -valued or $R^{3}$ -valued function of one real variable is differentiable exactly when all its components are, and then the derivative is componentwise; the coordinate vector $p$ of [F6] is differentiable when $w$ and $γ_{w}$ are.

[F10]

(Lorentz forms of inertia (3,1,0) on a 4-dimensional real vector space, Definition): $η$ is symmetric and bilinear, so for differentiable $P$ the product rule gives $\frac{d}{d τ} η (P, P) = 2 η (P, \dot{P})$ .

Proof

technique · differentiate the explicit frame components $E=m\gamma_wc^2$, $\mathbf p=m\gamma_w\mathbf w$ in chart time, then translate to proper time with $dt/d\tau=\gamma_w$

1.1F1F2F3F4F8

By [F3] and [F4], in this chart parametrisation $τ^{'} = 1 / γ_{w}$ , hence $d τ / d t = 1 / γ_{w}$ and $d t / d τ = γ_{w}$ ; therefore for every $C^{1}$ field $Q$ along $x$ , $\dot{Q} = γ_{w} d Q / d t$ , and in particular $A = U^{'} / τ^{'} = γ_{w} d U / d t$ and $F = \dot{P} = γ_{w} d P / d t$ by [F1] and [F2]. Since $x$ is $C^{2}$ , $w$ is $C^{1}$ and $γ_{w}$ is $C^{1}$ by [F8], so $U = γ_{w} (c, w)$ is $C^{1}$ and $F$ exists as a continuous field.

1.2F3F10

By [F3], $η (U, U) = - c^{2}$ and $η (U, A) = 0$ ; since $P = m U$ with $m$ constant, [F10] gives $η (P, P) = m^{2} η (U, U) = - m^{2} c^{2}$ , so the mass shell of the statement holds.

1.3F3F6F7F8F9

By [F3], $∣ w ∣ < c$ and $γ_{w}^{2} (1 - ∣ w ∣^{2} / c^{2}) = 1$ by [F8], while [F6] and [F7] give $E = m γ_{w} c^{2}$ and the coordinate vector $p = m γ_{w} w$ ; differentiating in $t$ by [F9] and [F8], $\frac{d E}{d t} = m c^{2} γ_{w}^{'} = m γ_{w}^{3} (w \cdot a)$ and $f = \frac{d p}{d t} = m γ_{w}^{'} w + m γ_{w} a$ , with $a$ the coordinate vector of $a$ .

2.1F9F10step 1.1step 1.2

Clause (a): since $m$ is constant, $F = γ_{w} \frac{d P}{d t} = γ_{w} m \frac{d U}{d t} = m A$ by [step 1.1] and [F9]; hence $η (P, F) = m^{2} η (U, A) = 0$ by [step 1.2], equivalently $η (U, F) = 0$ after dividing by $m > 0$ ; and [F10] with $η (P, P) = - m^{2} c^{2}$ of [step 1.2] gives $\frac{d}{d τ} η (P, P) = 2 η (P, \dot{P}) = 2 η (P, F) = 0$ , so the mass shell has vanishing proper-time derivative.

2.2F5F8step 1.3

Clause (b): from [step 1.3], $f \cdot w = m γ_{w}^{3} (w \cdot a) \frac{∣ w ∣^{2}}{c^{2}} + m γ_{w} (a \cdot w) = m γ_{w} (w \cdot a) (γ_{w}^{2} \frac{∣ w ∣^{2}}{c^{2}} + 1)$ ; since $γ_{w}^{2} (1 - ∣ w ∣^{2} / c^{2}) = 1$ gives $γ_{w}^{2} ∣ w ∣^{2} / c^{2} = γ_{w}^{2} - 1$ by [F8], this is $m γ_{w} (w \cdot a) γ_{w}^{2} = m γ_{w}^{3} (w \cdot a) = \frac{d E}{d t}$ , which is (b).

2.3F1F8F10step 1.2

Clause (d): let $m : I \to (0, \infty)$ be $C^{1}$ and $P = m U$ , so that $P$ is $C^{1}$ and $F = \dot{P}$ exists by [F1]; along the worldline $η (P, P) = m^{2} η (U, U) = - m (t)^{2} c^{2}$ by [step 1.2], and differentiating with respect to proper time using [F10] and the product and chain rules [F8] gives $2 η (P, F) = \frac{d}{d τ} η (P, P) = - 2 m c^{2} \frac{d m}{d τ}$ , hence $η (P, F) = - m c^{2} \frac{d m}{d τ}$ .

3.1F8F9step 1.1step 2.2

Clause (c): by [step 1.1], $F = γ_{w} \frac{d P}{d t}$ with $P = m γ_{w} (c, w)$ , so $\frac{d P}{d t} = (m c γ_{w}^{'}, m γ_{w}^{'} w + m γ_{w} a)$ ; substituting $γ_{w}^{'} = γ_{w}^{3} (w \cdot a) / c^{2}$ from [F8] gives $F^{0} = γ_{w} m c γ_{w}^{'} = γ_{w} \frac{m γ_{w}^{3} (w \cdot a)}{c} = γ_{w} \frac{f \cdot w}{c}$ and $F^{i} = γ_{w} (m γ_{w}^{'} w^{i} + m γ_{w} a^{i}) = γ_{w} f^{i}$ , where the first equality uses [step 2.2]; this is (c).

4.1step 2.1step 2.2step 2.3step 3.1∎

Collecting: clause (a) is [step 2.1], clause (b) is [step 2.2], clause (c) is [step 3.1], clause (d) is [step 2.3]; no force law, no equation of motion and no measurement is used, the identities being conditional on the stated four-momentum field.

DefinitionDefinition: AI-adaptedProof: Not applicablejudge pass (gpt-6.1-sol)Open item page →

Total four-momentum, invariant mass and center-of-momentum frame of a finite system

Definition

Throughout, $(A, V, η, C^{+})$ is the oriented time-oriented affine Minkowski model of Oriented time-oriented affine Minkowski space (the spacetime primitives) and $c > 0$ .

(a) Total four-momentum. For a finite list $(P_{1}, \dots, P_{N})$ , $N \geq 0$ , of vectors of $V$ , define the total four-momentum $P_{t o t} : = \sum_{i = 1}^{N} P_{i}$ as the finite sum of A finite sum in a commutative monoid indexed by an arbitrary finite set in the additive group $(V, +)$ of the vector space $V$ ; the empty sum is $0$ . The sum uses the vectors $P_{i}$ alone: no event, no position and no simultaneity datum enters it, and the order of the list does not affect $P_{t o t}$ because addition in $V$ is commutative.

(b) Invariant mass of the total. If $P_{t o t}$ is future timelike (Timelike, null, spacelike and causal vectors of a Lorentz vector space, Time orientation, future and past timelike/null/causal vectors, and the closed cones), define the invariant mass of the system and its rest energy by

$M : = M (P_{t o t}) : = \frac{\sqrt{- η (P_{t o t}, P_{t o t})}}{c} > 0$

and its rest energy is $M c^{2}$ .

The square root is a real number because $η (P_{t o t}, P_{t o t}) < 0$ for a timelike vector, and it is strictly positive, so $M > 0$ (Rational powers $a^{r}$ of a positive base). If $P_{t o t}$ is future null, set $M : = 0$ and say that the system has no rest frame. For $P_{t o t}$ equal to zero, spacelike or past causal, this convention defines no invariant mass.

(c) Center-of-momentum frame. A center-of-momentum frame of a nonzero $P_{t o t}$ is an inertial frame $(o; e_{0}, e_{1}, e_{2}, e_{3})$ (Inertial frames and inertial charts of the affine Minkowski model) whose first vector satisfies

$h_{e_{0}} (P_{t o t}) = 0, h_{e_{0}} (v) : = v + η (v, e_{0}) e_{0} (v \in V),$

where the projection is defined on all of $V$ ; for a future causal total this is $q_{e_{0}} (P_{t o t}) = 0$ , meaning that its total spatial momentum vanishes for the observer stationary in the frame (Observer energy, spatial momentum, invariant mass, rest energy and kinetic energy of a four-momentum). The condition is equivalent to $P_{t o t} = - η (P_{t o t}, e_{0}) e_{0}$ , i.e. to $P_{t o t} \in s p a n (e_{0})$ ; if $P_{t o t}$ is future timelike this holds exactly when $P_{t o t} = λ e_{0}$ with $λ > 0$ , and then $E_{e_{0}} (P_{t o t}) = M c^{2}$ for $M$ as in (b).

(d) Comparison with the single-vector convention. For a single future timelike vector $P$ , the invariant mass $M$ of (b) coincides with $m (P)$ of Observer energy, spatial momentum, invariant mass, rest energy and kinetic energy of a four-momentum; in particular if $P = m U$ with $m > 0$ and $η (U, U) = - c^{2}$ (the four-velocity normalisation of Four-velocity and four-acceleration of a future timelike worldline), then $M = m$ . Both quantities are functions of the scalar $η (P, P)$ alone.

(e) Scope. This definition asserts no existence of a center-of-momentum frame (that is Center-of-momentum frames of a future timelike four-momentum and their unique time vector), no physical conservation law and no dynamics; only the total vector carries the invariant mass, and the individual summands need not be causal for the sum to be defined. No choice principle is used.

LemmaStatement: AI-adaptedProof: AI-adaptedOpen item page →

Finite sums of future causal vectors and additivity of observer energy and momentum

Statement

Let $(A, V, η, C^{+})$ be the oriented time-oriented affine Minkowski model (Oriented time-oriented affine Minkowski space (the spacetime primitives)) and $c > 0$ .

(a) If $v, w \in c l (C^{+})$ , then $v + w \in c l (C^{+})$ ; if in addition $v \in C^{+}$ or $w \in C^{+}$ , then $v + w \in C^{+}$ . Moreover $- η (v, w) \geq 0$ for $v, w \in c l (C^{+})$ , with equality if and only if $v = 0$ , or $w = 0$ , or both $v$ and $w$ are nonzero future null and $w = λ v$ for some $λ > 0$ .

(b) If $P_{1}, \dots, P_{N}$ with $N \geq 1$ are nonzero future causal vectors, then $P_{t o t} = \sum_{i = 1}^{N} P_{i}$ (Total four-momentum, invariant mass and center-of-momentum frame of a finite system,(a)) is a nonzero future causal vector; if at least one $P_{i}$ is future timelike, then $P_{t o t}$ is future timelike. The converse implication does not hold: two future null vectors that are not parallel have a future timelike sum.

(c) For every future unit $n$ (Lorentz forms of inertia (3,1,0) on a 4-dimensional real vector space) and all $P_{1}, \dots, P_{N} \in c l (C^{+})$ ,

$E_{n} (\sum_{i = 1}^{N} P_{i}) = \sum_{i = 1}^{N} E_{n} (P_{i}), q_{n} (\sum_{i = 1}^{N} P_{i}) = \sum_{i = 1}^{N} q_{n} (P_{i}),$

where $E_{n}, q_{n}$ are the observer quantities of Observer energy, spatial momentum, invariant mass, rest energy and kinetic energy of a four-momentum; if the $P_{i}$ are nonzero future causal, then $E_{n} (P_{t o t}) > 0$ .

Facts & Assumptions

Given: the affine Minkowski model $(A, V, η, C^{+})$ ; $c > 0$ ; vectors $v, w \in c l (C^{+})$ for (a); nonzero future causal $P_{1}, \dots, P_{N} \in c l (C^{+})$ , $N \geq 1$ , for (b) and (c); a future unit $n$ for (c).

[F1]

(The timelike set has exactly two convex components, described in orthonormal coordinates, Statement): in an $η$ -orthonormal ordered basis whose first vector is timelike, writing $v = v^{0} e_{0} + v_{s}$ with $v_{s} \in S = s p a n (e_{1}, e_{2}, e_{3})$ and $∣ v_{s} ∣ = \sqrt{(v^{1})^{2} + (v^{2})^{2} + (v^{3})^{2}}$ , one has $η (v, v) = - (v^{0})^{2} + ∣ v_{s} ∣^{2}$ , and the timelike set is the disjoint union of the convex cones $T_{+} = {v : v^{0} > ∣ v_{s} ∣}$ and $T_{-} = {v : v^{0} < - ∣ v_{s} ∣}$ ; the time orientation names $C^{+}$ as one of $T_{\pm}$ , so $c l (C^{+}) = {v : v^{0} \geq ∣ v_{s} ∣}$ for the appropriate naming, and the two closed cones meet only at $0$ .

[F2]

(Time orientation, future and past timelike/null/causal vectors, and the closed cones, Statement): the closed cones $c l (C^{+})$ and $c l (C^{-})$ are convex cones meeting only at $0$ , a causal vector is future or past but not both, and a nonzero vector has exactly one causal character.

[F3]

(Timelike, null, spacelike and causal vectors of a Lorentz vector space, Definition): $v$ is timelike when $η (v, v) < 0$ , null when $v \neq 0$ and $η (v, v) = 0$ , spacelike when $η (v, v) > 0$ , and causal when it is timelike or null.

[F4]

(Lorentz forms of inertia (3,1,0) on a 4-dimensional real vector space, Definition): $η$ is symmetric and bilinear, so $η (v + w, v + w) = η (v, v) + 2 η (v, w) + η (w, w)$ and $η (λ v, w) = λ η (v, w)$ for real $λ$ .

[F5]

(The Cauchy-Schwarz inequality for finite sums, Statement): for real $a_{i}, b_{i}$ , $∣ \sum_{i = 1}^{3} a_{i} b_{i} ∣ \leq (\sum_{i = 1}^{3} a_{i}^{2})^{1 / 2} (\sum_{i = 1}^{3} b_{i}^{2})^{1 / 2}$ , with $(\sum_{i = 1}^{3} a_{i} b_{i})^{2} = (\sum_{i = 1}^{3} a_{i}^{2}) (\sum_{i = 1}^{3} b_{i}^{2})$ if and only if the two lists are proportional, that is there is a pair $(λ, μ) \neq (0, 0)$ of reals with $λ a_{i} = μ b_{i}$ for every $i$ ; consequently $∣ v_{s} + w_{s} ∣ \leq ∣ v_{s} ∣ + ∣ w_{s} ∣$ for the Euclidean norm on $S$ .

[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 $q_{n} (P) = h (P)$ , the restrictions to $c l (C^{+})$ of a linear map and of $h$ , for $P \in c l (C^{+})$ .

[F7]

(Norms of four-velocity and four-acceleration, and the observer rest-space decomposition, clause (c)): $h$ is linear with image $n^{⊥}$ and kernel $s p a n (n)$ .

[F8]

(Positivity, mass shell and kinetic energy of the observer quantities of a four-momentum, Statement (a)): if $P$ is nonzero future causal and $n$ is future unit, then $E_{n} (P) > 0$ .

[F9]

(A finite sum in a commutative monoid indexed by an arbitrary finite set) and (The principle of mathematical induction): the finite sum $\sum_{i = 1}^{N} P_{i}$ of the abelian group $(V, +)$ is defined by recursion, so $\sum_{i = 1}^{N} P_{i} = \sum_{i = 1}^{N - 1} P_{i} + P_{N}$ , and a statement about all $N \geq 1$ follows from the base case $N = 1$ and the induction step.

[F10]

(Every Lorentz form admits an ordered basis with matrix diag(-1,1,1,1) and Existence of future-oriented inertial frames at every event): there is an $η$ -orthonormal ordered basis $(e_{0}, e_{1}, e_{2}, e_{3})$ of $V$ with $e_{0} \in C^{+}$ .

Proof

technique · fix an orthonormal basis with future first vector, prove the two pointwise inequalities by Cauchy-Schwarz, then close (b) by induction and (c) by linearity

1.1F1F4F5F10

Choose by [F10] an $η$ -orthonormal basis with $e_{0} \in C^{+}$ , so that $C^{+}$ is the set $T_{+}$ of [F1] and $c l (C^{+}) = {v : v^{0} \geq ∣ v_{s} ∣}$ . For $v, w \in c l (C^{+})$ one has $v^{0} \geq ∣ v_{s} ∣ \geq 0$ and $w^{0} \geq ∣ w_{s} ∣ \geq 0$ , hence $v^{0} w^{0} \geq ∣ v_{s} ∣ ∣ w_{s} ∣$ and, by [F5], $v_{s} \cdot w_{s} \leq ∣ v_{s} ∣ ∣ w_{s} ∣$ ; therefore $- η (v, w) = v^{0} w^{0} - v_{s} \cdot w_{s} \geq ∣ v_{s} ∣ ∣ w_{s} ∣ - v_{s} \cdot w_{s} \geq 0$ by [F4]. If $v = 0$ or $w = 0$ equality is clear. Conversely, suppose $v, w \neq 0$ and $- η (v, w) = 0$ ; then both inequalities above are equalities, so $v^{0} w^{0} = ∣ v_{s} ∣ ∣ w_{s} ∣$ and $v_{s} \cdot w_{s} = ∣ v_{s} ∣ ∣ w_{s} ∣$ . Equality in Cauchy-Schwarz [F5] makes $v_{s}, w_{s}$ collinear with a nonnegative factor, and since $v^{0} w^{0} = ∣ v_{s} ∣ ∣ w_{s} ∣$ with $v^{0} \geq ∣ v_{s} ∣$ , $w^{0} \geq ∣ w_{s} ∣$ and $v^{0}, w^{0} > 0$ , one gets $v^{0} = ∣ v_{s} ∣$ and $w^{0} = ∣ w_{s} ∣$ ; hence $v$ and $w$ are future null, $w_{s} = μ v_{s}$ with $μ = ∣ w_{s} ∣ / ∣ v_{s} ∣ > 0$ , and $w = μ v$ . Thus equality holds exactly in the three cases listed in (a).

1.2F1F5F10

Under the same choice of basis, for $v, w \in c l (C^{+})$ the first coordinate of $v + w$ satisfies $(v + w)^{0} = v^{0} + w^{0} \geq ∣ v_{s} ∣ + ∣ w_{s} ∣ \geq ∣ (v + w)_{s} ∣$ by [F5], so $v + w \in c l (C^{+})$ ; if $v \in C^{+}$ then $v^{0} > ∣ v_{s} ∣$ and the first inequality is strict, giving $(v + w)^{0} > ∣ (v + w)_{s} ∣$ , that is $v + w \in C^{+}$ , and symmetrically for $w \in C^{+}$ .

1.3F9

By [F9] the finite sums of (b) and (c) are defined recursively with $\sum_{i = 1}^{N} P_{i} = \sum_{i = 1}^{N - 1} P_{i} + P_{N}$ for $N \geq 2$ , so a claim for all $N \geq 1$ may be proved by induction on $N$ with base case $N = 1$ and the recursive step.

1.4F2F3base

Base case of the induction for (b): for $N = 1$ one has $P_{t o t} = P_{1} \neq 0$ by hypothesis, $P_{1}$ is future causal, and $P_{t o t}$ is future timelike if $P_{1}$ is; the assertion of (b) therefore holds for $N = 1$ .

2.1F2F9step 1.2ih

Inductive step for (b): assume as induction hypothesis that for every list of $N - 1 \geq 1$ nonzero future causal vectors the sum is nonzero future causal, and future timelike if at least one entry is; given nonzero future causal $P_{1}, \dots, P_{N}$ , put $S : = \sum_{i = 1}^{N - 1} P_{i}$ with $N \geq 2$ , so $S \neq 0$ is future causal and future timelike if some $P_{i}$ , $i < N$ , is timelike. By [step 1.2] applied to $S$ and $P_{N}$ , $P_{t o t} = S + P_{N} \in c l (C^{+})$ , and $P_{t o t} \in C^{+}$ if $S \in C^{+}$ or $P_{N} \in C^{+}$ . If $P_{t o t}$ were $0$ , then $S = - P_{N} \in - c l (C^{+})$ while $S \in c l (C^{+})$ , so [F2] gives $S = 0$ , contradicting $S \neq 0$ ; hence $P_{t o t} \neq 0$ . This proves the assertion for lists of length $N$ , and [F9]'s induction principle closes the proof of (b) except its final sentence.

2.2F1F3F4step 1.1step 1.2

For any two nonparallel future null vectors $P_{1}, P_{2}$ , step 1.1 gives $η (P_{1}, P_{2}) < 0$ . Hence $η (P_{1} + P_{2}, P_{1} + P_{2}) = 2 η (P_{1}, P_{2}) < 0$ by [F4], and their sum is future by step 1.2, so it is future timelike. Such pairs exist: in the basis of step 1.1 take $P_{1} = e_{0} + e_{1}$ and $P_{2} = e_{0} + e_{2}$ . Their coordinates show that they are future null and nonparallel by [F1] and [F3], while $η (P_{1} + P_{2}, P_{1} + P_{2}) = - 2$ . This proves the final sentence of (b) and witnesses the failure of the converse.

3.1F6F7F8step 1.3step 2.1

Prove (c): by [F6] and [F7], $E_{n}$ and $q_{n}$ agree on $c l (C^{+})$ with linear maps into $R$ and $n^{⊥}$ , so additivity over the recursive sums [step 1.3] gives $E_{n} (\sum_{i = 1}^{N} P_{i}) = \sum_{i = 1}^{N} E_{n} (P_{i})$ and $q_{n} (\sum_{i = 1}^{N} P_{i}) = \sum_{i = 1}^{N} q_{n} (P_{i})$ for all $P_{i} \in c l (C^{+})$ , again by induction on $N$ ; if the $P_{i}$ are nonzero future causal, then [step 2.1] makes $P_{t o t}$ nonzero future causal and [F8] gives $E_{n} (P_{t o t}) > 0$ .

4.1step 1.1step 1.2step 1.4step 2.1step 2.2step 3.1discharge-induction: step 2.1∎

Clause (a) is [step 1.1] together with [step 1.2], clause (b) is the induction of [step 1.4] and [step 2.1] together with the counterexample of [step 2.2], and clause (c) is [step 3.1]; hence all the assertions of the lemma hold, the induction hypothesis of [step 2.1] having been discharged by its proof of the successor case.

LemmaStatement: AI-adaptedProof: AI-adaptedOpen item page →

Center-of-momentum frames of a future timelike four-momentum and their unique time vector

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 $Q \in V$ be future timelike with invariant mass $M : = \sqrt{- η (Q, Q)} / c$ (Total four-momentum, invariant mass and center-of-momentum frame of a finite system,(b)); put $n_{Q} : = Q / (M c)$ , so that $η (n_{Q}, n_{Q}) = - 1$ , $n_{Q} \in C^{+}$ and $Q = M c n_{Q}$ .

(a) There exists an inertial frame $(o; e_{0}, e_{1}, e_{2}, e_{3})$ with $e_{0} = n_{Q}$ ; in every such frame $q_{e_{0}} (Q) = 0$ , $E_{e_{0}} (Q) = M c^{2}$ and the components of $Q$ are $(M c, 0, 0, 0)$ . Such a frame is a center-of-momentum frame of $Q$ in the sense of Total four-momentum, invariant mass and center-of-momentum frame of a finite system,(c).

(b) Conversely, if an inertial frame satisfies $q_{e_{0}} (Q) = 0$ , then $e_{0} = n_{Q}$ . Hence the time vector of a center-of-momentum frame of $Q$ is unique, and two center-of-momentum frames of $Q$ differ only by the choice of origin and by a positively oriented $η$ -orthonormal change of the spatial basis $(e_{1}, e_{2}, e_{3})$ of $n_{Q}^{⊥}$ ; in particular the center-of-momentum frame at a fixed origin is unique up to spatial rotations.

(c) If $Q$ is a nonzero future null vector, then no inertial frame satisfies $q_{e_{0}} (Q) = 0$ : a nonzero future null vector has no center-of-momentum frame.

Facts & Assumptions

Given: the affine Minkowski model $(A, V, η, C^{+})$ ; $c > 0$ ; a future timelike $Q \in V$ with invariant mass $M = \sqrt{- η (Q, Q)} / c$ and $n_{Q} : = Q / (M c)$ ; for (c) a nonzero future null $Q$ .

[F1]

(Total four-momentum, invariant mass and center-of-momentum frame of a finite system, Definition): for future timelike $P$ the invariant mass is $M (P) = \sqrt{- η (P, P)} / c > 0$ ; a center-of-momentum frame of a nonzero $P \in V$ is an inertial frame with $h_{e_{0}} (P) = 0$ , where $h_{e_{0}} (v) = v + η (v, e_{0}) e_{0}$ for every $v \in V$ . This is equivalent to $P \in s p a n (e_{0})$ and to $P = - η (P, e_{0}) e_{0}$ ; for future causal $P$ , as in every use below, $h_{e_{0}} (P) = q_{e_{0}} (P)$ .

[F2]

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

[F3]

(Positivity, mass shell and kinetic energy of the observer quantities of a four-momentum, Statement (a)): if $P$ is nonzero future causal and $n$ future unit, then $E_{n} (P) > 0$ ; if $P$ is future timelike with invariant mass $m (P)$ , then $m (P)^{2} = - η (P, P) / c^{2}$ .

[F4]

(Norms of four-velocity and four-acceleration, and the observer rest-space decomposition, clause (c)): for future unit $n$ , the map $h$ is linear with image $n^{⊥}$ and kernel $s p a n (n)$ , $V = s p a n (n) \oplus n^{⊥}$ , and the restriction of $η$ to $n^{⊥}$ is positive definite; hence $η$ restricted to $n^{⊥}$ is an inner product.

[F5]

(Every finite-dimensional real or complex inner product space has an orthonormal basis, Statement): every finite-dimensional real inner product space has an orthonormal basis.

[F6]

(Inertial frames and inertial charts of the affine Minkowski model, Definition): an inertial frame is a tuple $(o; e_{0}, e_{1}, e_{2}, e_{3})$ with $o \in A$ and $(e_{0}, e_{1}, e_{2}, e_{3})$ an $η$ -orthonormal ordered basis of $V$ that is future and positively oriented; its inertial chart sends $p$ to the coordinates relative to $(o; e_{0}, \dots, e_{3})$ .

[F7]

(The oriented volume form epsilon and the permutation symbols of an inertial frame, Definition): the oriented volume form $ϵ (v_{0}, \dots, v_{3})$ is the determinant of the coordinate matrix in an inertial frame, an alternating $4$ -linear form; replacing one basis vector by its negative changes the determinant's sign.

[F8]

(Existence of future-oriented inertial frames at every event, Statement): in every oriented time-oriented affine Minkowski space and for every event $o$ there exists an inertial frame with origin $o$ ; explicitly, from an $η$ -orthonormal basis an orientation may be corrected by negating one spatial vector without destroying $η$ -orthonormality or the future first vector.

[F9]

(The timelike set has exactly two convex components, described in orthonormal coordinates, Statement): in an $η$ -orthonormal basis whose first vector is timelike, the timelike set is the disjoint union of $T_{+} = {v : v^{0} > ∣ v_{s} ∣}$ and $T_{-} = {v : v^{0} < - ∣ v_{s} ∣}$ , which are its two connected components, are convex and are closed under multiplication by positive reals; $c l (T_{+}) = {v : v^{0} \geq ∣ v_{s} ∣}$ , $c l (T_{-}) = {v : v^{0} \leq - ∣ v_{s} ∣}$ , and the two closed cones meet only at $0$ ; the time orientation names $C^{+}$ as one of $T_{\pm}$ .

[F10]

(Lorentz forms of inertia (3,1,0) on a 4-dimensional real vector space, Definition): $η$ is a symmetric bilinear form: $η (λ v, μ w) = λ μ η (v, w)$ for real $λ, μ$ .

[F11]

(Timelike, null, spacelike and causal vectors of a Lorentz vector space, Definition): a vector $v$ is timelike when $η (v, v) < 0$ , null when $v \neq 0$ and $η (v, v) = 0$ , and causal when it is timelike or null.

Proof

technique · build the frame along $n_Q$ by an explicit orthonormal basis of $n_Q^\perp$ and an orientation correction, then compare any center-of-momentum time vector with $n_Q$ algebraically

1.1F2F10

Since $Q$ is future timelike, $- η (Q, Q) > 0$ and hence $M > 0$ and $n_{Q} = Q / (M c)$ is defined; by [F10] and $η (Q, Q) = - M^{2} c^{2}$ , $η (n_{Q}, n_{Q}) = (M^{2} c^{2})^{- 1} η (Q, Q) = - 1$ and $Q = M c n_{Q}$ with $M c > 0$ , so $Q$ is a positive multiple of the unit vector $n_{Q}$ . Moreover $η (Q, n_{Q}) = M c η (n_{Q}, n_{Q}) = - M c$ , whence $h (Q) = Q + η (Q, n_{Q}) n_{Q} = M c n_{Q} - M c n_{Q} = 0$ and $E_{n_{Q}} (Q) = - c η (Q, n_{Q}) = M c^{2}$ by [F2].

1.2F9F10

By [F9] the future component $C^{+}$ of the timelike set is one of the two convex cones $T_{\pm}$ , each closed under multiplication by positive reals; since $Q \in C^{+}$ and $n_{Q} = (M c)^{- 1} Q$ with $(M c)^{- 1} > 0$ , [F10] gives $n_{Q} \in C^{+}$ .

1.3F4F5F10

By [F4] the restriction $δ$ of $η$ to $n_{Q}^{⊥}$ is positive definite, so by [F5] there is a $δ$ -orthonormal basis $(e_{1}, e_{2}, e_{3})$ of $n_{Q}^{⊥}$ ; setting $e_{0} : = n_{Q}$ , the ordered basis $(e_{0}, e_{1}, e_{2}, e_{3})$ is $η$ -orthonormal, because $η (e_{0}, e_{0}) = η (n_{Q}, n_{Q}) = - 1$ , $η (e_{0}, e_{i}) = 0$ by $e_{i} \in n_{Q}^{⊥}$ , and $η (e_{i}, e_{j}) = δ (e_{i}, e_{j}) = δ_{i j}$ for $i, j \in {1, 2, 3}$ .

1.4F1F2F3F10

For the converse, let $(o; e_{0}, e_{1}, e_{2}, e_{3})$ be an inertial frame with $q_{e_{0}} (Q) = 0$ . By [F1] and [F2], $Q = - η (Q, e_{0}) e_{0}$ and $η (Q, e_{0}) = - E_{e_{0}} (Q) / c$ , and [F3] gives $E_{e_{0}} (Q) > 0$ because $Q$ is nonzero future causal, so $λ : = - η (Q, e_{0}) > 0$ and $Q = λ e_{0}$ . Then $η (Q, Q) = λ^{2} η (e_{0}, e_{0}) = - λ^{2}$ by [F10] and $η (e_{0}, e_{0}) = - 1$ , while $η (Q, Q) = - M^{2} c^{2}$ by definition of $M$ , so $λ^{2} = M^{2} c^{2}$ and $λ = M c$ because $λ > 0$ and $M c > 0$ . Substituting $Q = λ e_{0} = M c e_{0}$ into $Q = M c n_{Q}$ and dividing by $M c \neq 0$ gives $e_{0} = n_{Q}$ .

1.5F10F11F1

If $Q$ is a nonzero future null vector and an inertial frame satisfied $q_{e_{0}} (Q) = 0$ , then [F1] gives $Q = - η (Q, e_{0}) e_{0} = λ e_{0}$ with $λ \neq 0$ because $Q \neq 0$ ; then $η (Q, Q) = λ^{2} η (e_{0}, e_{0}) = - λ^{2} < 0$ by [F10], contradicting $η (Q, Q) = 0$ for a null vector in [F11]. Hence no inertial frame satisfies $q_{e_{0}} (Q) = 0$ , which is (c).

2.1F6F7F8step 1.2step 1.3

The ordered basis of [step 1.3] has some orientation class; if that class is negative, replace $e_{1}$ by $- e_{1}$ , which leaves $η$ -orthonormality and the first vector unchanged and reverses the sign of the determinant of the coordinate matrix, hence changes the orientation class, by [F7] and [F8]. The resulting $(e_{0}, e_{1}, e_{2}, e_{3})$ is $η$ -orthonormal, future (its first vector is $e_{0} = n_{Q} \in C^{+}$ by [step 1.2]) and positively oriented, so it is the basis of an inertial frame by [F6].

2.2step 1.3step 1.4

Two inertial frames whose first vectors both equal $n_{Q}$ differ by their choice of origin $o$ , which is arbitrary, and by their spatial triples $(e_{1}, e_{2}, e_{3})$ and $(f_{1}, f_{2}, f_{3})$ ; both triples are $δ$ -orthonormal and positively oriented bases of the same oriented Euclidean space $n_{Q}^{⊥}$ , so the change-of-basis map between them preserves $δ$ and the induced orientation, that is it is a spatial rotation. Together with [step 1.4] this gives the uniqueness statement of (b): the time vector is forced to be $n_{Q}$ .

3.1F1step 1.1step 2.1

In the frame of [step 2.1] the first vector is $e_{0} = n_{Q}$ , so $q_{e_{0}} (Q) = q_{n_{Q}} (Q) = 0$ and $E_{e_{0}} (Q) = M c^{2}$ by [step 1.1]; by [F1] this frame is a center-of-momentum frame of $Q$ , and the components of $Q = M c n_{Q} = M c e_{0}$ are $(M c, 0, 0, 0)$ . This proves (a).

3.2step 1.4step 2.1step 2.2

Applying the spatial-rotation freedom of [step 2.2] to the frames produced by [step 2.1], two center-of-momentum frames of $Q$ differ only by origin choice and a spatial rotation, and by [step 1.4] no other time vector is possible; this completes (b).

4.1step 1.5step 3.1step 3.2∎

By [step 3.1] clause (a) holds, by [step 3.2] clause (b) holds, and by [step 1.5] clause (c) holds; hence an inertial frame with $e_{0} = n_{Q}$ exists, its time vector is unique up to the stated rotations, and a nonzero future null vector has no center-of-momentum frame.

DefinitionDefinition: AI-adaptedProof: Not applicableOpen item page →

Collision data and four-momentum conservation

Definition

Throughout, $(A, V, η, C^{+})$ is the oriented time-oriented affine Minkowski model of Oriented time-oriented affine Minkowski space (the spacetime primitives) and $c > 0$ .

(a) Collision datum and conservation. A collision datum is a triple $(p; (P_{1}, \dots, P_{N}); (P_{1}^{'}, \dots, P_{K}^{'}))$ consisting of an event $p \in A$ , a finite nonempty list $(P_{1}, \dots, P_{N})$ , $N \geq 1$ , of vectors of $V$ , called the incoming legs, and a finite nonempty list $(P_{1}^{'}, \dots, P_{K}^{'})$ , $K \geq 1$ , of vectors of $V$ , called the outgoing legs. Its side totals are the finite sums of A finite sum in a commutative monoid indexed by an arbitrary finite set in the additive group $(V, +)$ , formed as in Total four-momentum, invariant mass and center-of-momentum frame of a finite system:

$Q_{i n} : = \sum_{i = 1}^{N} P_{i}, Q_{o u t} : = \sum_{j = 1}^{K} P_{j}^{'} .$

The datum conserves when $Q_{i n} = Q_{o u t}$ . The two sides are compared only through these totals: if two data have the same incoming legs and the same outgoing legs, their conservation behaviour is the same, and for a datum with a single leg on each side conservation is exactly $P_{1} = P_{1}^{'}$ .

(b) Legs, mass data and mass sums. A leg $P$ is massive with mass $m > 0$ if there is a future timelike $U \in V$ with $η (U, U) = - c^{2}$ (the four-velocity normalisation of Four-velocity and four-acceleration of a future timelike worldline) and $P = m U$ ; a leg is null if it is a nonzero future null vector (Timelike, null, spacelike and causal vectors of a Lorentz vector space, Time orientation, future and past timelike/null/causal vectors, and the closed cones). For a massive leg both $U$ and $m$ are determined by $P$ : $η (P, P) = - m^{2} c^{2}$ forces $m = \sqrt{- η (P, P)} / c$ , the invariant mass $m (P)$ of Observer energy, spatial momentum, invariant mass, rest energy and kinetic energy of a four-momentum, and then $U = P / m$ ; so the recorded mass of a massive leg is necessarily its invariant mass. A datum is on-shell if every leg is massive or null; a mass-specified datum is an on-shell datum together with the recorded mass $m > 0$ of each massive leg and the recorded value $0$ for each null leg. The mass sum of a side of a mass-specified datum is the sum of the recorded positive masses of its massive legs, and the side is massive-only if all its legs are massive.

(c) Special classes of data. A datum is causal if every leg lies in $c l (C^{+})$ ; two-leg if $N = K = 2$ ; and equal-mass if, in addition to being mass-specified and two-leg, the recorded masses agree pair by pair, that is the incoming masses $(m_{1}, m_{2})$ equal the outgoing masses $(m_{1}^{'}, m_{2}^{'})$ under the identification of the pairing.

(d) Scope. The definition is purely mathematical: it neither asserts that any datum describes a physical process, nor that a conserving datum exists for a prescribed list of legs, nor any law of nature. Position, simultaneity, mechanism and interaction strength are not part of a datum; only the event, the two finite lists and, for mass-specified data, their recorded masses are. In particular conservation here is an equation between two vectors of $V$ and is not the adopted conservation law of a physical model, which is a separate postulate; the existence question for prescribed masses is answered only in special cases by Invariant mass, center-of-momentum frame and necessary conditions for conserving collision data. No choice principle is used.

LemmaStatement: AI-adaptedProof: AI-adaptedOpen item page →

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.

PostulateStatement: AI-adaptedProof: Not applicableOpen item page →

Adopted massive-particle momentum, observer measurement and isolated-collision conservation

Physical scope

Classical massive point particles with positive constant invariant mass in flat Minkowski spacetime, with ideal clocks, ideal light signals and SI conventions. The model identifies E_n, q_n, T_n with observer measurements and adopts four-momentum balance for isolated collisions, including ideal massless legs as defined by the separate massless postulate. The single-particle equation applies only to massive C2 worldlines. Excludes internal structure, spin, radiation reaction, variable-mass dynamics, gravity, continuum matter, quantum and statistical effects, and empirical reports; a force law and interaction mechanism are separate premises.

Postulate

Adopt as physical assumptions of the relativistic particle model, on top of the flat spacetime model of Oriented time-oriented affine Minkowski space (the spacetime primitives) and the SI unit conventions of SI units and coordinate conventions of the special-relativity framework, together with the framework's idealisations of point particles, ideal clocks and ideal light signals:

(i) Mass and momentum. Every massive classical particle of the model carries a positive constant invariant mass parameter $m > 0$ and has four-momentum $P = m U$ , where $U$ is the four-velocity of its future timelike worldline (Curves, regular causal curves and future timelike worldlines in affine Minkowski space, Four-velocity and four-acceleration of a future timelike worldline, Norms of four-velocity and four-acceleration, and the observer rest-space decomposition); $P$ is a vector of $V$ whose SI unit is kilogram metre per second. The formula is adopted only for $m > 0$ ; the case $m = 0$ is not defined by it and is the separate premise Adopted momentum model for an ideal massless particle. In particular $η (P, P) = - m^{2} c^{2}$ is a consequence of the four-velocity normalisation, not a separate assumption.

(ii) Observer measurement identification. Relative to a future unit observer $n$ , the model identifies the measured energy of the particle with $E_{n} (P)$ , its measured spatial momentum with $q_{n} (P)$ , its kinetic energy with $T_{n} (P) = E_{n} (P) - m c^{2}$ and its rest energy with $m c^{2}$ , where $E_{n}, q_{n}, T_{n}$ are the quantities of Observer energy, spatial momentum, invariant mass, rest energy and kinetic energy of a four-momentum; the units are the joule, kilogram metre per second and kilogram of SI units and coordinate conventions of the special-relativity framework. This clause is a formulation convention identifying mathematical quantities with the quantities the model calls measured; it is not a measurement report and no experiment is claimed.

(iii) Isolated collision balance. An isolated collision is represented by a mass-specified on-shell collision datum at an event, with finite nonempty incoming and outgoing lists of the massive particles of (i) and/or the ideal massless particles of Adopted momentum model for an ideal massless particle, and with no external four-momentum transfer or omitted leg. The interaction is idealized as confined to that event. The sum of the incoming four-momenta equals the sum of the outgoing four-momenta, in the sense of the collision data of Collision data and four-momentum conservation: the model adopts $Q_{i n} = Q_{o u t}$ for such an isolated datum. No particular collision is asserted to occur, and no mechanism, probability, cross section or amplitude is supplied by this clause.

(iv) Equation of motion for a stipulated force. Put $H_{c}^{+} : = {U \in C^{+} : η (U, U) = - c^{2}}$ . A stipulated force law is a given map $F : D \to V$ on a given subset $D \subseteq A \times H_{c}^{+}$ , with force components in $k g m / s^{2}$ . No regularity away from the specified worldline is assumed. Along a given $C^{2}$ massive worldline $x : I \to A$ with $(x (s), U (s)) \in D$ for all $s \in I$ , the model's particle equation is $m d U / d τ = F (x, U)$ , with $d U / d τ$ the proper-time derivative of Proper-time derivative of a vector field along a worldline and the four-force notation, so that the stipulated $F$ is the four-force of the motion. A particular force law, and the existence, uniqueness and domain of solutions of the resulting equation, are not adopted here and belong to a separate model.

Empirical support for these assumptions is a nonlogical relation of the framework, not a proof; the idealisations of point particles, ideal clocks, ideal light signals and flat spacetime are inherited and travel to every consumer of this premise.

PostulateStatement: AI-adaptedProof: Not applicablejudge pass (gpt-6.1-sol)Open item page →

Adopted momentum model for an ideal massless particle

Physical scope

Ideal classical massless point particles in flat Minkowski spacetime, modelled by a nonzero future null four-momentum whose energy and momentum are identified with the observer quantities, and whose scale is fixed by a measured energy. Excludes four-velocities, proper time and rest frames for the particle, any massive limit m -> 0, the identification with light or classical waves, quantum properties, and every interaction, decay, production or empirical claim.

Postulate

Adopt as a separate model premise for ideal massless particles, independent of Adopted massive-particle momentum, observer measurement and isolated-collision conservation whose formula $P = m U$ uses $m > 0$ and does not define a massless case:

(i) The object. An ideal massless particle of the model is described by a nonzero future null vector $P \in V$ (Timelike, null, spacelike and causal vectors of a Lorentz vector space, Time orientation, future and past timelike/null/causal vectors, and the closed cones) whose SI unit is kilogram metre per second (SI units and coordinate conventions of the special-relativity framework). Such a $P$ is not of the form $m U$ for any $m > 0$ and any future timelike $U$ with $η (U, U) = - c^{2}$ : a positive multiple of a timelike vector is timelike, and the zero vector is neither timelike, null nor spacelike, so the massive expression $m U$ with this normalized $U$ degenerates to $0$ at $m = 0$ and does not describe a nonzero null momentum. The particle therefore has no four-velocity, no proper-time parametrisation and no rest frame. Its scale is fixed by a measured energy in an observer's frame.

(ii) Observer identification. Relative to a future unit observer $n$ , the model identifies the measured energy and spatial momentum of the particle with the quantities $E_{n} (P) = - c η (P, n)$ and $q_{n} (P) = h (P)$ of Observer energy, spatial momentum, invariant mass, rest energy and kinetic energy of a four-momentum. Necessarily $E_{n} (P) = c ∣ q_{n} (P) ∣ > 0$ , by Positivity, mass shell and kinetic energy of the observer quantities of a four-momentum,(a)(i), and no rest mass, rest energy or kinetic energy is assigned to the particle — the definition of $T_{n}$ requires a rest-mass parameter $m$ and no such parameter is introduced here. This clause is a formulation convention identifying mathematical quantities with the quantities the model calls measured; it is not a measurement report and no experiment is claimed.

(iii) No identification with light or quantum objects. The ideal massless particle is not identified here with light, with a classical wave or with any quantum object: no frequency, wavelength, quantisation, polarisation or Planck constant enters this premise, and the identification of light with massless particles requires the wave model and its own postulates on a separate page.

(iv) No process. This premise asserts no interaction of massless particles and no decay or production process; such processes require a stipulated dynamics. Empirical support for the assumption is a nonlogical relation of the framework, not a proof, and the idealisations inherited from the flat spacetime model and the ideal-point description travel to every consumer of this premise.

Physics theoremStatement: AI-adaptedProof: AI-adaptedOpen item page →

Energy, momentum, invariant mass and four-force of relativistic particles, and necessary collision thresholds

Physical scope

Massive particles with constant positive mass and ideal massless particles described by future null four-momenta in flat Minkowski spacetime with ideal clocks and SI conventions; measured energy, momentum and kinetic energy are the model quantities E_n, q_n, T_n; isolated collisions obey the adopted four-momentum balance at one event, and the stated thresholds are necessary kinematic conditions only. The variable-mass clause is a mathematical derivative identity, not an adopted variable-mass dynamics. Excludes internal structure, spin, radiation reaction, force laws and interaction mechanisms (a separate premise), continuum matter, gravity, quantum and statistical descriptions, and every empirical claim, measurement value or precision statement.

Statement

Adopt Adopted massive-particle momentum, observer measurement and isolated-collision conservation and Adopted momentum model for an ideal massless particle. Then the following conditional physical claims hold in the flat classical model, with the units of SI units and coordinate conventions of the special-relativity framework.

(a) Massive particle. Let a massive particle have constant mass $m > 0$ , four-momentum $P = m U$ and a future unit observer $n$ . Its measured energy $E_{n} (P) = - c η (P, n)$ , spatial momentum $q_{n} (P) = h (P)$ and kinetic energy $T_{n} (P) = E_{n} (P) - m c^{2}$ satisfy $E_{n} (P) > 0$ , $T_{n} (P) \geq 0$ and the mass-shell identity $E_{n} (P)^{2} = m^{2} c^{4} + c^{2} ∣ q_{n} (P) ∣^{2}$ ; $T_{n} (P) = 0$ if and only if the particle is at rest in the observer's frame ( $U = c n$ ). In an inertial frame whose stationary observer is $n = e_{0}$ , the components are $P = (E / c, p)$ with $E = E_{e_{0}} (P) = m γ_{w} c^{2}$ and $p = m γ_{w} w \in R^{3}$ the coordinates of $q_{e_{0}} (P) = \sum_{i = 1}^{3} p^{i} e_{i}$ for the coordinate velocity $w$ , and under a passive boost the components transform as the components of the vector $P$ ; the combination $E^{2} - c^{2} ∣ p ∣^{2} = m^{2} c^{4}$ is frame-independent.

(b) Four-force. Along a $C^{2}$ worldline with four-acceleration $A$ , the four-force $F = \dot{P}$ satisfies $η (P, F) = 0$ and $F = m A$ ; in an inertial frame $F = γ (f \cdot w / c, f)$ and $d E / d t = f \cdot w$ , where $f = d p / d t$ is the coordinate three-force. The constancy of $m$ is used; if instead $m$ is a $C^{1}$ function $m : I \to (0, \infty)$ along the same worldline, define $P : = m U$ and $F : = \dot{P}$ by the general derivative construction. As a mathematical identity, $η (P, F) = - m c^{2} d m / d τ$ , and the constant-mass identities are not transferred to variable-mass systems.

(c) System mass. For a finite nonempty system of particles all of whose four-momenta are nonzero future causal, let $Q : = \sum_{i} P_{i}$ . If $Q$ is future timelike, its invariant mass $M = \sqrt{- η (Q, Q)} / c$ satisfies: in the center-of-momentum frame with $e_{0} = Q / (M c)$ one has $M c^{2} = \sum_{i} E_{e_{0}} (P_{i})$ ; every massive constituent has $E_{e_{0}} (P_{i}) \geq m_{i} c^{2}$ , with equality exactly when it is at rest in that frame, and every massless constituent has $E_{e_{0}} (P_{i}) > 0$ . Hence $M \geq \sum_{i} m_{i}$ over the massive constituents, with equality if and only if the system is massive-only and all its constituents are at rest in the center-of-momentum frame. If $Q$ is future null, then $M = 0$ , all constituents are massless with parallel momenta, and no center-of-momentum frame exists.

(d) Massless particle. An ideal massless particle has $E_{n} (P) = c ∣ q_{n} (P) ∣ > 0$ , has no rest frame and is not $m U$ for any $m > 0$ ; it is not obtained from the massive formula by setting $m = 0$ .

(e) Necessary collision thresholds. Suppose that, under the adopted isolated-collision balance, an isolated collision occurs whose incoming legs are massive particles of constant masses $m_{1}, \dots, m_{N}$ and whose outgoing legs are massive particles of masses $μ_{1}, \dots, μ_{K}$ and possibly ideal massless particles. Let $M$ be the invariant mass of the common total four-momentum $Q$ . Then the necessary inequalities $M \geq \sum_{i} m_{i}$ and $M \geq \sum_{j} μ_{j}$ hold; if the incoming side is a projectile of mass $m_{1}$ and a target of mass $m_{2}$ at rest in the observer frame of the target, with projectile energy $E$ and kinetic energy $T = E - m_{1} c^{2}$ in that frame, then

$E \geq \frac{c^{2}}{2 m_{2}} [(\sum_{j = 1}^{K} μ_{j})^{2} - m_{1}^{2} - m_{2}^{2}], T \geq \frac{c^{2}}{2 m_{2}} [(\sum_{j = 1}^{K} μ_{j})^{2} - (m_{1} + m_{2})^{2}],$

Separately, under the same adopted balance, a single incoming massless particle cannot produce an outgoing state containing a massive particle. These are kinematic necessity statements under the adopted conservation: they do not assert that any collision occurs, do not fix probabilities, amplitudes or cross sections, and are not empirical findings.

(f) Scope. All clauses are conditional on the two postulates and on the stated idealisations; the conclusions carry the same idealisations. No measurement, precision, statistical or empirical claim is made.

Facts & Assumptions

Given: the adopted premises Adopted massive-particle momentum, observer measurement and isolated-collision conservation and Adopted momentum model for an ideal massless particle; the flat Minkowski model $(A, V, η, C^{+})$ with $c > 0$ ; for (a) a massive particle with constant $m > 0$ and future unit observer $n$ ; for (b) a $C^{2}$ worldline; for (c) a finite nonempty system of particles with nonzero future causal four-momenta; for (e) an isolated collision with massive incoming and outgoing legs as stated.

[F1]

(Adopted massive-particle momentum, observer measurement and isolated-collision conservation, Postulate (i) and (ii)): every massive classical particle carries a positive constant mass parameter $m > 0$ and has four-momentum $P = m U$ ; relative to a future unit observer $n$ the model identifies the measured energy with $E_{n} (P)$ , the measured spatial momentum with $q_{n} (P)$ , the kinetic energy with $T_{n} (P) = E_{n} (P) - m c^{2}$ and the rest energy with $m c^{2}$ ; the SI unit of $P$ is kilogram metre per second.

[F2]

(Adopted momentum model for an ideal massless particle, Postulate (i) and (ii)): an ideal massless particle is described by a nonzero future null vector $P$ , with measured energy and spatial momentum identified with $E_{n} (P)$ and $q_{n} (P)$ ; it is not of the form $m U$ for any $m > 0$ and future timelike $U$ with $η (U, U) = - c^{2}$ , and it has no four-velocity, no proper-time parametrisation and no rest frame.

[F3]

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

[F4]

(Four-momentum components, energy-velocity relation and passive boost transformation in an inertial frame, Statement (a)–(c)): in an inertial frame with $n = e_{0}$ the components are $(P^{0}, P^{1}, P^{2}, P^{3}) = (E_{n} (P) / c, q^{1}, q^{2}, q^{3})$ and the mass-shell identity holds in components; for $P = m U$ with chart coordinate velocity $w$ one has $E = m γ_{w} c^{2}$ , $p = m γ_{w} w$ ; under a passive boost the components transform as the components of $P$ , the combination $E^{2} - c^{2} ∣ p ∣^{2}$ is invariant and equals $m (P)^{2} c^{4}$ for future timelike $P$ and $0$ for future null $P$ .

[F5]

(Four-force identities and the work-energy relation in an inertial frame, Statement (a)–(d)): with $F = \dot{P}$ the four-force of Proper-time derivative of a vector field along a worldline and the four-force notation along a $C^{2}$ worldline, $F = m A$ , $η (P, F) = 0$ , $F = γ (f \cdot w / c, f)$ , $d E / d t = f \cdot w$ for constant $m$ , and $η (P, F) = - m c^{2} d m / d τ$ for a $C^{1}$ variable mass parameter.

[F6]

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

[F7]

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

[F8]

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

[F9]

(Collision data and four-momentum conservation, Definition): a collision datum consists of an event and two finite nonempty lists of legs with side totals; it conserves when the totals are equal; a mass-specified datum is on-shell with recorded masses; a causal datum has all legs in $c l (C^{+})$ .

[F10]

(Invariant mass, center-of-momentum frame and necessary conditions for conserving collision data, Statement (a), (c) and (d)): for a mass-specified conserving causal on-shell datum with nonzero legs and future timelike total $Q$ of invariant mass $M$ , each side satisfies $M \geq \sum m_{ℓ}$ over its massive legs with equality exactly when the side is massive-only and all its massive legs are at rest in the center-of-momentum frame; for the two-leg incoming side $P_{i} = m_{i} U_{i}$ with $n_{2} = U_{2} / c$ , $γ = - η (U_{1} / c, n_{2})$ and $E = E_{n_{2}} (P_{1}) = m_{1} c^{2} γ$ , one has $M^{2} = m_{1}^{2} + m_{2}^{2} + 2 m_{2} E / c^{2}$ , giving the stated thresholds for an outgoing massive-mass sum $\sum_{j} μ_{j}$ ; and a conserving datum with a single null incoming leg and an outgoing massive leg does not exist.

[F11]

(SI units and coordinate conventions of the special-relativity framework, Definition): in the framework's conventions the four-momentum has the unit kilogram metre per second and the observer energy the unit joule.

[F12]

(Timelike, null, spacelike and causal vectors of a Lorentz vector space and Lorentz forms of inertia (3,1,0) on a 4-dimensional real vector space): a nonzero vector is timelike when $η (v, v) < 0$ and null when $η (v, v) = 0$ ; $η$ is bilinear, so $η (m U, m U) = m^{2} η (U, U)$ and $0 \cdot U = 0$ .

Proof

technique · quote the two adopted premises for the physical identification, then apply the page's mathematical suppliers clause by clause; the collision thresholds are the necessary conditions of the collision lemma under the adopted balance

1.1F1F2F11

The adopted premises [F1] and [F2] provide the physical identification: for a massive particle the model's measured energy, spatial momentum, kinetic energy and rest energy are $E_{n} (P), q_{n} (P), T_{n} (P), m c^{2}$ with $P = m U$ and $m > 0$ constant, and for an ideal massless particle they are $E_{n} (P), q_{n} (P)$ with $P$ a nonzero future null vector, no rest mass, rest energy or kinetic energy being assigned; [F11] fixes the units.

1.2F3F4

For $P = m U$ with $m > 0$ , $U$ future timelike, $η (U, U) = - c^{2}$ and $n$ future unit, [F3] gives $E_{n} (P) = m c^{2} γ$ with $γ = - η (U / c, n) \geq 1$ , $T_{n} (P) = m c^{2} (γ - 1) \geq 0$ , $T_{n} (P) = 0$ if and only if $U = c n$ , the mass-shell identity $E_{n} (P)^{2} = m^{2} c^{4} + c^{2} ∣ q_{n} (P) ∣^{2}$ and $m (P) = m$ ; [F4] adds the frame components $(E / c, p)$ with $E = m γ_{w} c^{2}$ , $p = m γ_{w} w$ , the passive-boost transformation law and the invariance $E^{2} - c^{2} ∣ p ∣^{2} = m^{2} c^{4}$ .

1.3F2F3F12

For every nonzero future null $P$ and future unit $n$ , [F3] gives $E_{n} (P) = c ∣ q_{n} (P) ∣ > 0$ ; [F2] gives that such a $P$ has no four-velocity, proper time or rest frame; and [F12] shows that $m = 0$ in the massive formula yields $0 \cdot U = 0 \neq P$ , while $m > 0$ yields $η (m U, m U) = - m^{2} c^{2} < 0$ , so $m U$ is timelike and cannot equal the null $P$ : no $m \geq 0$ produces $P$ from $P = m U$ .

1.4F5

Along a $C^{2}$ worldline with four-force $F = \dot{P}$ , [F5] gives the identities $η (P, F) = 0$ , $F = m A$ for constant $m$ , the frame form $F = γ (f \cdot w / c, f)$ , $d E / d t = f \cdot w$ , and the variable-mass identity $η (P, F) = - m c^{2} d m / d τ$ .

1.5F6F7F8

For nonzero future causal $P_{1}, \dots, P_{N}$ , [F7] makes $Q = \sum_{i} P_{i}$ nonzero future causal, future timelike if some $P_{i}$ is future timelike, additive in $E_{n}$ and $q_{n}$ with $E_{n} (Q) > 0$ , and gives the equality characterization of $- η (v, w) = 0$ for $v, w \in c l (C^{+})$ ; [F6] and [F8] give the invariant-mass convention, the existence of the center-of-momentum frame with $e_{0} = Q / (M c)$ and $E_{e_{0}} (Q) = M c^{2}$ when $Q$ is future timelike, and the nonexistence of such a frame when $Q$ is nonzero future null.

1.6F1F2F7F9F10

Postulate (iii) of Adopted massive-particle momentum, observer measurement and isolated-collision conservation adopts balance also for isolated data containing the ideal massless legs of [F2]. Thus the collision data of (e) form a mass-specified, conserving, causal, on-shell datum with nonzero legs in the sense of [F9]. For massive incoming legs, [F7] makes the total timelike and [F10] supplies the side mass inequalities and the two-leg incoming invariant-mass identity. If the outgoing massive list is empty, its mass sum is 0 and the side bound follows directly from M>0; the same identity and algebraic rearrangement still give the displayed energy thresholds. Separately, if the incoming list consists of a single null leg, its total is null, whereas any outgoing massive leg would make the outgoing total timelike by [F7], contradicting balance.

2.1step 1.1step 1.2

Clause (a): [step 1.1] identifies the measured quantities with $E_{n}, q_{n}, T_{n}, m c^{2}$ and [step 1.2] supplies $E_{n} (P) > 0$ , $T_{n} (P) \geq 0$ , the mass-shell identity, the rest criterion $T_{n} (P) = 0 ⟺ U = c n$ , the frame components, the boost transformation law and the frame independence of $E^{2} - c^{2} ∣ p ∣^{2} = m^{2} c^{4}$ .

2.2step 1.1step 1.4

Clause (b): [step 1.1] makes $\dot{P}$ the model's four-force and $E_{n} (P)$ its measured energy, and [step 1.4] gives $η (P, F) = 0$ , $F = m A$ , $F = γ (f \cdot w / c, f)$ and $d E / d t = f \cdot w$ for constant $m$ , together with the variable-mass identity and the warning that the constant-mass identities are not transferred to variable-mass systems.

2.3step 1.2step 1.3step 1.5

Clause (c), timelike total: assume $Q$ future timelike with invariant mass $M$ ; by [step 1.5] the center-of-momentum frame with $e_{0} = Q / (M c)$ exists and $E_{e_{0}} (Q) = M c^{2}$ , and additivity gives $M c^{2} = E_{e_{0}} (Q) = \sum_{i} E_{e_{0}} (P_{i})$ . Each massive constituent satisfies $E_{e_{0}} (P_{i}) = m_{i} c^{2} γ_{i} \geq m_{i} c^{2}$ with equality exactly when $U_{i} = c e_{0}$ by [step 1.2], and each massless constituent satisfies $E_{e_{0}} (P_{i}) = c ∣ q_{e_{0}} (P_{i}) ∣ > 0$ by [step 1.3]; hence $M = \sum_{massive} m_{i} + \sum_{massive} m_{i} (γ_{i} - 1) + \sum_{massless} E_{e_{0}} (P_{i}) / c^{2} \geq \sum_{massive} m_{i}$ , with equality if and only if the system is massive-only and every massive constituent has $γ_{i} = 1$ , that is is at rest in the center-of-momentum frame.

2.4step 1.5

Clause (c), null total: if $Q$ is future null, then [step 1.5] gives $M = 0$ by the convention and the nonexistence of a center-of-momentum frame; moreover $0 = - η (Q, Q) = \sum_{i, j} - η (P_{i}, P_{j})$ with each term nonnegative by [step 1.5], so each term vanishes and the equality characterization forces every $P_{i}$ to be nonzero future null and parallel to any other, that is all constituents are massless with parallel momenta.

2.5step 1.1step 1.3

Clause (d): for an ideal massless particle [step 1.1] identifies the measured energy and momentum with $E_{n}$ and $q_{n}$ , and [step 1.3] gives $E_{n} (P) = c ∣ q_{n} (P) ∣ > 0$ ; the same step gives that $P$ has no four-velocity, proper time or rest frame, that $P \neq m U$ for every $m > 0$ , and that no $m \geq 0$ produces $P$ from the massive formula.

2.6step 1.5step 1.6

Clause (e): the collision data of [step 1.6] form a conserving mass-specified causal on-shell datum; its incoming side contains the massive legs of masses $m_{1}, \dots, m_{N}$ , so $Q$ is future timelike by [step 1.5] and the invariant mass $M$ is defined; [step 1.6] then gives $M \geq \sum_{i} m_{i}$ and, applied to the outgoing side with massive-mass sum $\sum_{j} μ_{j}$ , $M \geq \sum_{j} μ_{j}$ ; in the two-leg projectile-target case with $n_{2} = U_{2} / c$ , [step 1.6] gives $M^{2} = m_{1}^{2} + m_{2}^{2} + 2 m_{2} E / c^{2}$ , and combining this with $M \geq \sum_{j} μ_{j}$ yields the two displayed threshold inequalities by rearrangement, while the one-null-leg impossibility is the separate case in [step 1.6]. No interaction, probability, amplitude or cross section is asserted.

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

Collecting: clause (a) is [step 2.1], clause (b) is [step 2.2], clause (c) is [step 2.3] together with [step 2.4], clause (d) is [step 2.5] and clause (e) is [step 2.6]; clause (f) records that every conclusion is conditional on the two adopted postulates and the stated idealisations, with no measurement or empirical claim, and no empirical premise is used in this deduction.

5 · Examples, experiments and counterexamples

None yet.

Energy Momentum

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

Observer energy, spatial momentum, invariant mass, rest energy and kinetic energy of a four-momentum

Definition

Positivity, mass shell and kinetic energy of the observer quantities of a four-momentum

Statement

Facts & Assumptions

Proof

Four-momentum components, energy-velocity relation and passive boost transformation in an inertial frame

Statement

Facts & Assumptions

Proof

Proper-time derivative of a vector field along a worldline and the four-force notation

Definition

Four-force identities and the work-energy relation in an inertial frame

Statement

Facts & Assumptions

Proof

Total four-momentum, invariant mass and center-of-momentum frame of a finite system

Definition

Finite sums of future causal vectors and additivity of observer energy and momentum

Statement

Facts & Assumptions

Proof

Center-of-momentum frames of a future timelike four-momentum and their unique time vector

Statement

Facts & Assumptions

Proof

Collision data and four-momentum conservation

Definition

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

Statement

Facts & Assumptions

Proof

Adopted massive-particle momentum, observer measurement and isolated-collision conservation

Postulate

Adopted momentum model for an ideal massless particle

Postulate

Energy, momentum, invariant mass and four-force of relativistic particles, and necessary collision thresholds

Statement

Facts & Assumptions

Proof

5 · Examples, experiments and counterexamples

Sources