Physics theoremStatement: AI-adaptedProof: AI-adaptedprecheck pass

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Specified low-speed momentum-energy correspondence between the relativistic measured quantities and the Newtonian comparison quantities

Physical scope

Flat special-relativity massive classical particles (constant mass m>0, P=mU) with the measured energy, spatial momentum and kinetic energy identified as E_n, q_n, T_n by post-sr-massive-particle-dynamics, in inertial frames and in the bounded low-speed domain |w|^2 <= z_0 c^2 with z_0<1, in the SI conventions. The error bounds are uniform on that domain and depend on z_0; nothing is asserted uniformly as |w| -> c. Excludes massless particles, variable mass, forces, interactions and interaction mechanisms, accelerated observers, 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 (massive-particle momentum, the identification of measured quantities and isolated collision balance) and the SI conventions of SI units and coordinate conventions of the special-relativity framework. Let $(A, V, η, C^{+})$ be the affine Minkowski model, $c > 0$ , let $(o; e_{0}, e_{1}, e_{2}, e_{3})$ be an inertial frame with stationary observer $n : = e_{0}$ , spatial Euclidean structure $(S, δ, ∣ \cdot ∣)$ and inertial chart $x$ (Inertial frames and inertial charts of the affine Minkowski model, The Euclidean spatial structure and cross product of an inertial frame), and let a massive particle have constant mass $m > 0$ and four-momentum $P = m U$ with $U$ future timelike, $η (U, U) = - c^{2}$ . Let $y$ be a worldline of the particle whose inertial coordinate path is $t \mapsto (c t, r (t))$ and whose coordinate velocity $w (t) = r^{'} (t) \in S$ of Newtonian comparison momentum and kinetic energy in a named inertial frame,(b) equals a fixed $w$ at the instant considered; then in this frame $P = m γ_{w} (c, w)$ with $γ_{w} = (1 - ∣ w ∣^{2} / c^{2})^{- 1 / 2}$ (Four-momentum components, energy-velocity relation and passive boost transformation in an inertial frame,(b), Norms of four-velocity and four-acceleration, and the observer rest-space decomposition,(e)).

Fix $z_{0} \in [0, 1)$ and suppose $z : = ∣ w ∣^{2} / c^{2} \leq z_{0}$ ; put $K (z_{0}) : = \frac{3}{8} (1 - z_{0})^{- 5 / 2}$ , and let $N_{p} (m, w) = m w$ and $N_{K} (m, w) = \frac{1}{2} m ∣ w ∣^{2}$ be the Newtonian comparison quantities of Newtonian comparison momentum and kinetic energy in a named inertial frame. With $E_{n} (P)$ , $q_{n} (P)$ and $T_{n} (P)$ the measured energy, spatial momentum and kinetic energy identified in clause (ii) of the adopted postulate, $m (P)$ the invariant mass of Observer energy, spatial momentum, invariant mass, rest energy and kinetic energy of a four-momentum and $R_{E}, R_{q}$ the remainders of Uniform low-speed expansion of the observer energy, momentum and kinetic energy of a massive four-momentum:

(a) [Energy] $E_{n} (P) = m c^{2} + N_{K} (m, w) + R_{E}$ and $E_{n} (P) - T_{n} (P) = m c^{2}$ ; $∣ R_{E} ∣ \leq K (z_{0}) z m ∣ w ∣^{2}$ . So the measured energy of the moving particle exceeds its rest energy by the Newtonian kinetic energy up to the displayed uniform bound of order $m ∣ w ∣^{4} / c^{2}$ .

(b) [Kinetic energy] $T_{n} (P) = N_{K} (m, w) + R_{E}$ , hence $∣ T_{n} (P) - N_{K} (m, w) ∣ \leq 2 K (z_{0}) z N_{K} (m, w)$ : for $w \neq 0$ the relative deviation from the Newtonian kinetic energy is at most $2 K (z_{0}) z$ .

(c) [Momentum] $q_{n} (P) = N_{p} (m, w) + R_{q}$ , and $∣ R_{q} ∣ \leq (\frac{1}{2} + K (z_{0}) z_{0}) z ∣ N_{p} (m, w) ∣$ ; for $w \neq 0$ the relative deviation from the Newtonian momentum is at most $(\frac{1}{2} + K (z_{0}) z_{0}) z$ .

(d) [Momentum-to-velocity inversion] $w = q_{n} (P) / (m γ_{w})$ exactly, and $∣ w - q_{n} (P) / m ∣ = (γ_{w} - 1) ∣ w ∣ \leq (\frac{1}{2} + K (z_{0}) z_{0}) z ∣ w ∣$ ; at $z = 0$ one has $w = 0$ , $q_{n} (P) = 0$ , $E_{n} (P) = m c^{2}$ and $T_{n} (P) = 0$ , and $m (P) = m$ always.

(e) [Scope and uniformity] The constants $K (z_{0})$ and $\frac{1}{2} + K (z_{0}) z_{0}$ depend on $z_{0}$ alone, so the estimates hold uniformly over all masses $m > 0$ , all states, instants and inertial frames with $∣ w ∣^{2} \leq z_{0} c^{2}$ . The statement is a conditional correspondence between the model's adopted measured quantities and the comparison functionals of a named frame: it identifies no framework with another, asserts no convergence of trajectories, forces or interactions and no limit in which the constant $c$ varies, gives no bound uniform in $∣ w ∣ < c$ (see Limits of the Newtonian bridge: speed-dependent error constants, the excluded massless case, and bounded chart regions) and makes no empirical claim.

Facts & Assumptions

Given: The affine Minkowski model with a fixed $c > 0$ and the SI conventions $x^{0} = c t$ ; an inertial frame with stationary observer $n = e_{0}$ , spatial structure $(S, δ, ∣ \cdot ∣)$ and chart $x$ ; a massive particle state $P = m U$ with constant $m > 0$ , $U$ future timelike and $η (U, U) = - c^{2}$ , realized as the momentum of a $C^{1}$ worldline with future timelike tangent (Curves, regular causal curves and future timelike worldlines in affine Minkowski space) whose coordinate velocity at the instant considered is a fixed $w \in S$ ; a bound $z_{0} \in [0, 1)$ with $z = ∣ w ∣^{2} / c^{2} \leq z_{0}$ ; and $K (z_{0}) = \frac{3}{8} (1 - z_{0})^{- 5 / 2}$ .

[L1]

The adopted postulate: $P = m U$ with constant $m > 0$ and $η (U, U) = - c^{2}$ (clause (i)); the model identifies the measured energy, spatial momentum, kinetic energy and rest energy of the particle with $E_{n} (P)$ , $q_{n} (P)$ , $T_{n} (P) = E_{n} (P) - m c^{2}$ and $m c^{2}$ (clause (ii)); units as in SI units and coordinate conventions of the special-relativity framework (Adopted massive-particle momentum, observer measurement and isolated-collision conservation).

[L2]

The comparison functionals: for $w \in S$ of the named frame, $N_{p} (m, w) = m w$ and $N_{K} (m, w) = \frac{1}{2} m ∣ w ∣^{2}$ ; the coordinate velocity of a chart path $t \mapsto (c t, r (t))$ is $w (t) = r^{'} (t)$ where it exists (Newtonian comparison momentum and kinetic energy in a named inertial frame).

[L3]

The SI conventions fix $x^{0} = c t$ in every inertial chart and the units metre, kilogram, second, joule and kilogram metre per second for the quantities used here (SI units and coordinate conventions of the special-relativity framework).

[L4]

The observer quantities: $E_{n} (P) = - c η (P, n)$ , $q_{n} (P) = h (P)$ , $m (P) = \sqrt{- η (P, P)} / c$ , and $T_{n} (P) = E_{n} (P) - m c^{2}$ after the constant $m > 0$ is fixed (Observer energy, spatial momentum, invariant mass, rest energy and kinetic energy of a four-momentum).

[L5]

The observer identities: for $P = m U$ with $m > 0$ constant, $U$ future timelike with $η (U, U) = - c^{2}$ , relative velocity $w$ and Lorentz factor $γ$ of Norms of four-velocity and four-acceleration, and the observer rest-space decomposition,(d): $E_{n} (P) = m c^{2} γ$ , $T_{n} (P) = m c^{2} (γ - 1)$ , $q_{n} (P) = m γ w$ and $m (P) = m$ (Positivity, mass shell and kinetic energy of the observer quantities of a four-momentum,(b)).

[L6]

The frame's stationary observer and the chart coordinate velocity: for $n = e_{0}$ the relative velocity of clause (d) is the chart coordinate velocity $w (t) = r^{'} (t)$ of the worldline, with $∣ w ∣ < c$ and $γ_{w} = (1 - ∣ w ∣^{2} / c^{2})^{- 1 / 2}$ (Norms of four-velocity and four-acceleration, and the observer rest-space decomposition,(e)).

[L7]

Frame components of the four-momentum: for $P = m U$ with $m > 0$ constant the components in the frame are $P = m γ_{w} (c, w)$ , that is $E_{n} (P) = m γ_{w} c^{2}$ and $q_{n} (P) = m γ_{w} w$ (Four-momentum components, energy-velocity relation and passive boost transformation in an inertial frame,(b)).

[L8]

The uniform low-speed expansion of the observer quantities: with $γ = γ_{w}$ and $z = ∣ w ∣^{2} / c^{2} \leq z_{0}$ , $E_{n} (P) = m c^{2} + \frac{1}{2} m ∣ w ∣^{2} + R_{E}$ , $T_{n} (P) = \frac{1}{2} m ∣ w ∣^{2} + R_{E}$ and $q_{n} (P) = m w + R_{q}$ for $R_{E} = m c^{2} R (z)$ , $R_{q} = m (γ_{w} - 1) w$ , with $∣ R_{E} ∣ \leq K (z_{0}) z m ∣ w ∣^{2} = K (z_{0}) m ∣ w ∣^{4} / c^{2}$ , $∣ T_{n} (P) - \frac{1}{2} m ∣ w ∣^{2} ∣ \leq 2 K (z_{0}) z \frac{1}{2} m ∣ w ∣^{2}$ and $∣ R_{q} ∣ \leq (\frac{1}{2} + K (z_{0}) z_{0}) z m ∣ w ∣$ ; the constants depend only on $z_{0}$ , and at $z = 0$ the remainders vanish (Uniform low-speed expansion of the observer energy, momentum and kinetic energy of a massive four-momentum,(a)-(c)).

[L9]

The Taylor remainder: $γ_{w} = 1 + \frac{z}{2} + R (z)$ with $∣ R (z) ∣ \leq K (z_{0}) z^{2}$ for $0 \leq z \leq z_{0}$ (The uniform second-order Taylor remainder of $z \mapsto (1 - z)^{- 1 / 2}$ on $[0, z_{0}]$ ,(b)).

[L10]

The model and frame structure: $(A, V, η, C^{+})$ is the oriented time-oriented affine Minkowski model and the frame is an inertial frame with chart $x$ and spatial structure $(S, δ, ∣ \cdot ∣)$ (Oriented time-oriented affine Minkowski space (the spacetime primitives), Inertial frames and inertial charts of the affine Minkowski model, The Euclidean spatial structure and cross product of an inertial frame).

Proof

technique · direct

1.1givenL6L10algebra

Identification of the state's data in the frame. By [L6], for the frame's stationary observer $n = e_{0}$ the relative velocity of [L5] coincides with the chart coordinate velocity $w (t) = r^{'} (t)$ and $γ = γ_{w} = (1 - ∣ w ∣^{2} / c^{2})^{- 1 / 2}$ ; by the hypothesis of the statement the value at the instant considered is the fixed $w$ , so the state $P = m U$ of the postulate has relative velocity $w$ and Lorentz factor $γ_{w}$ in this frame, with $∣ w ∣ < c$ .

1.2L2L3

The comparison quantities. By [L2], $N_{p} (m, w) = m w$ and $N_{K} (m, w) = \frac{1}{2} m ∣ w ∣^{2}$ ; the momenta $N_{p}, q_{n}$ have unit kilogram metre per second, the energies $N_{K}, E_{n}, T_{n}$ have unit joule, and $m$ has unit kilogram by [L3].

2.1step 1.1L1L5L7algebra

The measured quantities at the state. By [L5] applied with $γ = γ_{w}$ , $E_{n} (P) = m c^{2} γ_{w}$ , $T_{n} (P) = m c^{2} (γ_{w} - 1)$ , $q_{n} (P) = m γ_{w} w$ and $m (P) = m$ ; by [L7] the frame components are $P = m γ_{w} (c, w)$ , i.e. $E_{n} (P) = m γ_{w} c^{2}$ and $q_{n} (P) = m γ_{w} w$ , consistently. By [L1], $E_{n} (P)$ , $q_{n} (P)$ and $T_{n} (P)$ are the model's measured energy, spatial momentum and kinetic energy; its rest energy is $m c^{2}$ .

2.2step 1.1L8algebra

The uniform expansion at the state. Since $z = ∣ w ∣^{2} / c^{2} \leq z_{0}$ , [L8] gives $E_{n} (P) = m c^{2} + \frac{1}{2} m ∣ w ∣^{2} + R_{E}, T_{n} (P) = \frac{1}{2} m ∣ w ∣^{2} + R_{E}, q_{n} (P) = m w + R_{q},$ with $R_{E} = m c^{2} R (z)$ , $R_{q} = m (γ_{w} - 1) w$ and the bounds $∣ R_{E} ∣ \leq K (z_{0}) z m ∣ w ∣^{2}$ , $∣ T_{n} (P) - \frac{1}{2} m ∣ w ∣^{2} ∣ \leq 2 K (z_{0}) z \frac{1}{2} m ∣ w ∣^{2}$ and $∣ R_{q} ∣ \leq (\frac{1}{2} + K (z_{0}) z_{0}) z m ∣ w ∣$ ; if $z = 0$ then $w = 0$ , $R_{E} = R_{q} = 0$ , $E_{n} (P) = m c^{2}$ , $q_{n} (P) = 0$ and $T_{n} (P) = 0$ .

3.1step 2.1step 2.2step 1.2L4algebra

Clause (a). By steps 1.1 and 2.2, $E_{n} (P) = m c^{2} + N_{K} (m, w) + R_{E}$ with $∣ R_{E} ∣ \leq K (z_{0}) z m ∣ w ∣^{2} = K (z_{0}) m ∣ w ∣^{4} / c^{2}$ , and $E_{n} (P) - T_{n} (P) = m c^{2}$ by the definition of $T_{n}$ in [L4]; this is the displayed energy correspondence and its uniform bound.

3.2step 2.2step 1.2algebra

Clause (b). By step 2.2, $T_{n} (P) = N_{K} (m, w) + R_{E}$ with $∣ R_{E} ∣ \leq K (z_{0}) z m ∣ w ∣^{2} = 2 K (z_{0}) z N_{K} (m, w)$ ; for $w \neq 0$ the comparison value $N_{K} (m, w) > 0$ and division gives the relative bound $2 K (z_{0}) z$ , while at $w = 0$ both sides vanish by step 2.2.

3.3step 2.2step 1.2algebra

Clause (c). By steps 2.2 and 1.2, $q_{n} (P) = N_{p} (m, w) + R_{q}$ with $∣ R_{q} ∣ \leq (\frac{1}{2} + K (z_{0}) z_{0}) z m ∣ w ∣ = (\frac{1}{2} + K (z_{0}) z_{0}) z ∣ N_{p} (m, w) ∣;$ for $w \neq 0$ , $∣ N_{p} (m, w) ∣ = m ∣ w ∣ > 0$ and division gives the relative bound $(\frac{1}{2} + K (z_{0}) z_{0}) z$ .

3.4step 2.1step 2.2L9algebra

Clause (d). From $q_{n} (P) = m γ_{w} w$ of step 2.1, $w = q_{n} (P) / (m γ_{w})$ exactly, and $w - q_{n} (P) / m = (1 - γ_{w}) w$ , so $∣ w - q_{n} (P) / m ∣ = (γ_{w} - 1) ∣ w ∣$ ; by $γ_{w} - 1 = \frac{z}{2} + R (z)$ with $∣ R (z) ∣ \leq K (z_{0}) z^{2}$ from [L9] and $z \leq z_{0}$ , this is at most $(\frac{1}{2} + K (z_{0}) z_{0}) z ∣ w ∣$ . At $z = 0$ the state is $w = 0$ , and step 2.2 gives $q_{n} (P) = 0$ , $E_{n} (P) = m c^{2}$ , $T_{n} (P) = 0$ , while $m (P) = m$ by step 2.1 always.

4.1step 3.1step 3.2step 3.3step 3.4L1L8algebra

Clause (e). Every constant used above is $K (z_{0})$ or $\frac{1}{2} + K (z_{0}) z_{0}$ , functions of $z_{0}$ alone, while $m > 0$ , $n$ and the state enter only through their displayed occurrences; hence the estimates hold uniformly over all masses and all states, instants and inertial frames satisfying $∣ w ∣^{2} \leq z_{0} c^{2}$ , as stated in [L8],(c). The proof used the postulate only through clauses (i) and (ii), so the conclusion is a conditional correspondence between the model's measured quantities and the comparison functionals of the named frame; it asserts no identification of frameworks, no trajectory or force convergence and no variation of $c$ , and no empirical claim.

5.1step 3.1step 3.2step 3.3step 3.4step 4.1∎

Conclusion. Steps 3.1 to 3.4 are clauses (a) to (d) with their stated bounds and the $w = 0$ case, and step 4.1 is clause (e); all hypotheses were used as stated, and nothing beyond the adopted postulate, the SI conventions and the model's mathematics is assumed.

Specified low-speed momentum-energy correspondence between the relativistic measured quantities and the Newtonian comparison quantities

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources