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Newtonian Bridge

1 · Prerequisites

2 · Summary

The controlled low-speed bridge of the special-relativity framework. A named inertial frame supplies comparison functionals directly from its spatial Euclidean structure; a uniform second-order Taylor remainder for $(1 - z)^{- 1 / 2}$ , with an explicit constant $K (z_{0})$ depending only on the speed bound, is transported through the model's observer identities to the measured energy, momentum and kinetic energy of a massive particle state. A separate family of affine Galilean limiting chart maps, together with bounded chart regions, has an explicit $O (C^{- 2})$ limit on each bounded region and a proved failure on the whole chart domain for nonzero boost velocity; at zero velocity both maps are the identity everywhere. The correspondence theorem then states the exact remainders and their uniform bounds in the bounded domain $∣ w ∣^{2} \leq z_{0} c^{2}$ ; a scope remark records that no speed-independent constant exists, that the ideal massless case is excluded rather than obtained as a limit, and that the comparison is conditional and not a framework identity. No measurement or empirical precision is asserted anywhere on this page.

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

DefinitionDefinition: AI-adaptedProof: Not applicableprecheck passOpen item page →

Newtonian comparison momentum and kinetic energy in a named inertial frame

Physical scope

Formulation definition inside the flat special-relativity framework: the Newtonian comparison momentum and kinetic energy are built from a named inertial frame, its spatial Euclidean structure, a positive mass parameter and a coordinate velocity, in the SI conventions of the framework. The definition asserts no approximation, correspondence or error bound, no Newtonian theory, no force or matter law, and no empirical content; the correspondence claim is pthm-sr-newtonian-p-e-correspondence.

Definition

Let $(A, V, η, C^{+})$ be the oriented time-oriented affine Minkowski model (Oriented time-oriented affine Minkowski space (the spacetime primitives)), let $c > 0$ with the SI conventions of SI units and coordinate conventions of the special-relativity framework, let $(o; e_{0}, e_{1}, e_{2}, e_{3})$ be an inertial frame with inertial chart $x = (x^{0}, x^{1}, x^{2}, x^{3})$ and spatial Euclidean structure $(S, δ, ∣ \cdot ∣)$ (Inertial frames and inertial charts of the affine Minkowski model, The Euclidean spatial structure and cross product of an inertial frame), and let $m > 0$ .

(a) For $w \in S$ define the Newtonian comparison momentum and the Newtonian comparison kinetic energy of the pair $(m, w)$ by $N_{p} (m, w) : = m w, N_{K} (m, w) : = \frac{1}{2} m ∣ w ∣^{2},$ where $m w$ is the scalar multiple of $w$ by the real number $m$ (Vector space over a field) and $∣ w ∣^{2} = δ (w, w)$ (Integer powers $a^{m}$ ). With the SI conventions these carry the units kilogram metre per second and joule respectively.

(b) Let $I \subseteq R$ be an interval (Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length) and let $y : I \to A$ have inertial coordinate path $t \mapsto (c t, r (t))$ ; writing $r (t) = r^{1} (t) e_{1} + r^{2} (t) e_{2} + r^{3} (t) e_{3}$ with real functions $r^{1}, r^{2}, r^{3}$ , at each parameter value $t$ where all three coordinate derivatives exist, define the coordinate velocity $w (t) \in S$ of the path by $w (t) : = (r^{1})^{'} (t) e_{1} + (r^{2})^{'} (t) e_{2} + (r^{3})^{'} (t) e_{3},$ the componentwise derivative of The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral. For a path differentiable at every parameter value these are functions on $I$ ; otherwise they are defined on its differentiability set. The comparison functionals along the path are $t \mapsto N_{p} (m, w (t))$ and $t \mapsto N_{K} (m, w (t))$ .

(c) The functionals $N_{p}$ and $N_{K}$ are comparison expressions built from a named inertial frame, the mass parameter $m$ and the path's coordinate velocity. They are not defined by, and do not presuppose, a separately developed Newtonian theory, and this definition asserts no approximation, correspondence, error bound or physical claim; those are the content of Specified low-speed momentum-energy correspondence between the relativistic measured quantities and the Newtonian comparison quantities.

Remarks

Well-definedness. Clause (a) applies the scalar multiple of a vector of the real vector space $V$ (restricted to $S \subseteq V$ ) and the square of the norm $∣ \cdot ∣$ supplied by the frame's spatial structure, both total on $S$ ; clause (b) uses the componentwise derivative of a function $I \to R^{3}$ and the unique expansion of a vector of $S$ in the ordered basis $(e_{1}, e_{2}, e_{3})$ of The Euclidean spatial structure and cross product of an inertial frame. No division occurs and no choice principle is used.
No correspondence is asserted here. The definition only names the two expressions that Specified low-speed momentum-energy correspondence between the relativistic measured quantities and the Newtonian comparison quantities compares with the model's adopted measured quantities, and the units are the ones fixed by SI units and coordinate conventions of the special-relativity framework. The derivative symbols in (b) are used only at points where the coordinate derivatives exist; no velocity is assigned to an arbitrary nondifferentiable path, and the physical correspondence consumers retain their stated differentiable future-timelike worldline hypotheses. The mass parameter $m > 0$ is part of the definition's data; the definition itself attaches no dynamics, force or measurement to it.

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The uniform second-order Taylor remainder of $z \mapsto (1 - z)^{- 1 / 2}$ on $[0, z_{0}]$

Statement

Let $z_{0} \in [0, 1)$ and define $f : (- \infty, 1) \to R$ by $f (z) : = (1 - z)^{- 1 / 2} = ((1 - z)^{1 / 2})^{- 1}$ , the reciprocal of the nonnegative square root, with rational powers as in Rational powers $a^{r}$ of a positive base; put $K (z_{0}) : = \frac{3}{8} (1 - z_{0})^{- 5 / 2}$ .

(a) $f$ is differentiable on $(- \infty, 1)$ with $f^{'} (z) = \frac{1}{2} (1 - z)^{- 3 / 2}$ , and twice differentiable there with $f^{''} (z) = \frac{3}{4} (1 - z)^{- 5 / 2}$ ; in particular $∣ f^{''} (z) ∣ \leq \frac{3}{4} (1 - z_{0})^{- 5 / 2}$ for every $z \in [0, z_{0}]$ .

(b) For every $z \in [0, z_{0}]$ there is a real $R (z)$ with $f (z) = 1 + \frac{z}{2} + R (z), ∣ R (z) ∣ \leq K (z_{0}) z^{2},$ so $∣ f (z) - 1 ∣ \leq \frac{z}{2} + K (z_{0}) z^{2}$ for $z \in [0, z_{0}]$ . Consequently, for every $η > 0$ , $0 \leq z \leq \min ⁣ {z_{0}, \frac{η}{\frac{1}{2} + K (z_{0}) z_{0}}} ⟹ ∣ f (z) - 1 ∣ \leq η;$ the display is the quantitative form of $f (z) \to 1$ as $z \to 0^{+}$ used by Uniform Galilean limit of the boost charts on bounded regions, and its failure on the whole chart domain.

The lemma is a statement of real analysis about the single function $f$ ; it attaches no physical meaning to $z$ and makes no claim about a Lorentz factor or a physical velocity.

Facts & Assumptions

Given: A real number $z_{0} \in [0, 1)$ ; the function $f : (- \infty, 1) \to R$ , $f (z) = (1 - z)^{- 1 / 2} = ((1 - z)^{1 / 2})^{- 1}$ , with rational powers as in Rational powers $a^{r}$ of a positive base; and the constant $K (z_{0}) = \frac{3}{8} (1 - z_{0})^{- 5 / 2}$ .

[L1]

For $a > 0$ and rational $r$ the rational power $a^{r}$ is defined and positive, and $a^{- r} = (a^{r})^{- 1}$ ; on rational exponents it agrees with the real power $\exp (r \log a)$ (Rational powers $a^{r}$ of a positive base, Laws of rational exponents, The exponential definition of real powers agrees with the existing rational powers). In particular $1^{- 1 / 2} = 1$ .

[L2]

For real $α$ the function $x \mapsto x^{α}$ is differentiable on $(0, \infty)$ with derivative $α x^{α - 1}$ (Continuity and derivatives of positive-base real powers).

[L3]

The identity function $p_{1} (z) = z$ has derivative $1$ , the affine map $g (z) = 1 - z$ has derivative $- 1$ , and derivatives obey the sum, scalar-multiple and chain rules on their stated domains (For a natural $n \geq 1$ the function $x \mapsto x^{n}$ is differentiable everywhere with derivative $ι (n) x^{n - 1}$ ; for $n = 0$ it is the constant $1$ , with derivative $0$ ; for a natural $n \geq 1$ the function $x \mapsto x^{- n}$ is differentiable at every $x \neq 0$ with derivative $- ι (n) x^{- n - 1}$ ; consequently every polynomial function is differentiable at every real, with the derivative computed term by term, 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 $g (c) \neq 0$ , 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 $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ ).

[L4]

If $0 \leq z \leq z_{0}$ then $0 < 1 - z_{0} \leq 1 - z$ , and for positive bases $a \leq b$ and rational $r > 0$ one has $a^{r} \leq b^{r}$ (with equality exactly when $a = b$ ); reciprocals of positive elements reverse the order (Monotonicity of $r \mapsto a^{r}$ and of $a \mapsto a^{r}$ , Inverses of positives are positive, and reciprocation reverses order).

[L5]

Taylor's theorem with the Lagrange remainder: if $f$ has derivatives through order $n + 1$ on the closed interval between $a$ and $x$ with endpoint continuity, and $∣ f^{(n + 1)} (t) ∣ \leq M$ throughout that interval, then $∣ R_{n, a} f (x) ∣ \leq M ∣ x - a ∣^{n + 1} / (n + 1)!$ (A uniform derivative bound gives a uniform Taylor remainder bound).

[L6]

Absolute value: $∣ u + v ∣ \leq ∣ u ∣ + ∣ v ∣$ , $∣ u v ∣ = ∣ u ∣ ∣ v ∣$ , $∣ u ∣ = u$ for $u \geq 0$ , and $∣ u ∣ = 0$ if and only if $u = 0$ (Absolute value in an ordered field, Basic properties of the absolute value, The triangle inequality).

[L7]

The canonical natural numbers are cofinal in $R$ : for every real $b$ there is an integer $N > b$ (Every complete ordered field is Archimedean).

Proof

technique · direct

1.1givenL1L2

The composite form of $f$ . Define $g : (- \infty, 1) \to (0, \infty)$ by $g (z) : = 1 - z$ and $h : (0, \infty) \to (0, \infty)$ by $h (y) : = y^{- 1 / 2}$ . For every $z < 1$ the base $1 - z$ is positive, so $f = h \circ g$ ; and by [L1] the rational power $h (y) = y^{- 1 / 2}$ equals the real power $\exp (- \frac{1}{2} \log y)$ , so the derivative statement of [L2] applies to $h$ .

1.2L3

The inner derivative. The constant function $z \mapsto 1$ has derivative $0$ and the identity $p_{1}$ has $p_{1}^{'} \equiv 1$ by [L3]; the difference rule of 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 $g (c) \neq 0$ therefore gives $g^{'} (z) = - 1$ for every $z \in (- \infty, 1)$ , and each point of $(- \infty, 1)$ is a limit point of that interval.

2.1L2step 1.1

The outer derivative. By [L2] with $α = - \frac{1}{2}$ , the function $y \mapsto y^{- 1 / 2}$ is differentiable on $(0, \infty)$ with derivative $(y^{- 1 / 2})^{'} = - \frac{1}{2} y^{- 3 / 2}$ ; by step 1.1 this is $h^{'}$ .

3.1step 1.2step 2.1L3algebra

First derivative of $f$ . Fix $z \in (- \infty, 1)$ . Then $g (z) = 1 - z > 0$ is a limit point of $(0, \infty)$ , so the chain rule [L3] applied to $f = h \circ g$ at $z$ gives $f^{'} (z) = h^{'} (g (z)) g^{'} (z) = - \frac{1}{2} (1 - z)^{- 3 / 2} \cdot (- 1) = \frac{1}{2} (1 - z)^{- 3 / 2} .$

4.1step 3.1L1L2L3algebra

Second derivative of $f$ . The function $z \mapsto \frac{1}{2} (1 - z)^{- 3 / 2}$ is the scalar multiple by $\frac{1}{2}$ of the composite of $g$ with $y \mapsto y^{- 3 / 2}$ ; the rational power $y^{- 3 / 2}$ agrees with the real power by [L1], so [L2] with $α = - \frac{3}{2}$ gives it derivative $- \frac{3}{2} y^{- 5 / 2}$ on $(0, \infty)$ , and the scalar rule and the chain rule [L3] give $f^{''} (z) = \frac{1}{2} (- \frac{3}{2} (1 - z)^{- 5 / 2}) (- 1) = \frac{3}{4} (1 - z)^{- 5 / 2}$ for every $z \in (- \infty, 1)$ , which also covers the claim that $f$ is twice differentiable there.

5.1step 4.1L1L4L6algebra

Uniform bound on $[0, z_{0}]$ . Let $z \in [0, z_{0}]$ . By [L4], $0 < 1 - z_{0} \leq 1 - z$ , so $(1 - z_{0})^{5 / 2} \leq (1 - z)^{5 / 2}$ and, taking reciprocals of positive elements, $(1 - z)^{- 5 / 2} \leq (1 - z_{0})^{- 5 / 2}$ . Multiplying by $\frac{3}{4} > 0$ and using $f^{''} > 0$ by [L1] to read $∣ f^{''} ∣ = f^{''}$ ([L6]) gives $∣ f^{''} (z) ∣ = f^{''} (z) \leq \frac{3}{4} (1 - z_{0})^{- 5 / 2}$ for every $z \in [0, z_{0}]$ .

6.1step 3.1step 5.1L1L5algebra

The Taylor remainder. Fix $z \in [0, z_{0}]$ . Since $[0, z] \subseteq (- \infty, 1)$ , the function $f$ has derivatives through order $2$ on the closed interval between $0$ and $z$ , continuous at the endpoints; by step 5.1 the second derivative is bounded there by $M : = \frac{3}{4} (1 - z_{0})^{- 5 / 2}$ . Applying [L5] with $n = 1$ , $a = 0$ and this $z$ gives, for $R (z) : = R_{1, 0} f (z) = f (z) - f (0) - f^{'} (0) z$ , $∣ R (z) ∣ \leq \frac{M}{2!} z^{2} = \frac{3}{8} (1 - z_{0})^{- 5 / 2} z^{2} = K (z_{0}) z^{2} .$ Here $f (0) = 1^{- 1 / 2} = 1$ by [L1] and $f^{'} (0) = \frac{1}{2}$ by step 3.1, so the displayed expansion of $f$ is $f (z) = 1 + \frac{z}{2} + R (z)$ .

7.1step 6.1L1L6algebra

The quantitative limit. For $z \in [0, z_{0}]$ , steps 6.1 and 3.1 give $∣ f (z) - 1 ∣ = ∣ \frac{z}{2} + R (z) ∣ \leq \frac{z}{2} + K (z_{0}) z^{2} = z (\frac{1}{2} + K (z_{0}) z) \leq z (\frac{1}{2} + K (z_{0}) z_{0})$ by [L6] and $K (z_{0}) > 0$ , $z \leq z_{0}$ , which is the first displayed assertion of (b); if in addition $0 \leq z \leq η / (\frac{1}{2} + K (z_{0}) z_{0})$ for an $η > 0$ , then $∣ f (z) - 1 ∣ \leq η$ , which is the second displayed assertion; and the bound is not asserted for $z$ outside $[0, z_{0}]$ .

8.1step 6.1L1L6L7algebra∎

Failure of a single remainder constant on $[0, 1)$ . For every integer $N \geq 2$ put $z_{N} = 1 - N^{- 2} \in (0, 1)$ . By [L1], $f (z_{N}) = N$ , so the exact remainder $r_{N} : = f (z_{N}) - 1 - z_{N} / 2 = N - 1 - z_{N} / 2$ satisfies $r_{N} > N - 3 / 2$ . Given $B \geq 0$ , choose $N > \max {2, B + 3 / 2}$ by [L7]; then $∣ r_{N} ∣ > B \geq B z_{N}^{2}$ . Thus no finite $B \geq 0$ bounds $∣ f (z) - 1 - z / 2 ∣$ by $B z^{2}$ on all of $[0, 1)$ .

Remarks

The constants depend on $z_{0}$ alone, and the estimate is uniform over the whole interval $[0, z_{0}]$ ; as $z_{0} ↑ 1$ the constant $K (z_{0}) = \frac{3}{8} (1 - z_{0})^{- 5 / 2}$ grows without bound. The actual impossibility of a single remainder constant on $[0, 1)$ is proved by the family in step 8.1, which explains the bounded speed window in the consumers. No physical quantity and no physical restriction are introduced here: $z$ is an arbitrary real parameter, and the two in-run consumers named in the statement are the items that apply this estimate to the model's velocities.

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Uniform low-speed expansion of the observer energy, momentum and kinetic energy of a massive 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 with $η (n, n) = - 1$ , and let $P = m U$ with $m > 0$ and $U \in V$ future timelike, $η (U, U) = - c^{2}$ . Let $E_{n} (P)$ , $q_{n} (P)$ , $T_{n} (P)$ and $m (P)$ be the observer energy, spatial momentum, kinetic energy and invariant mass of Observer energy, spatial momentum, invariant mass, rest energy and kinetic energy of a four-momentum, put $h (v) : = v + η (n, v) n$ as in Norms of four-velocity and four-acceleration, and the observer rest-space decomposition,(c), and let $γ : = - η (U / c, n) \geq 1$ and $w : = h (U) / γ = c h (U / c) / γ \in n^{⊥}$ be the Lorentz factor and relative velocity of Norms of four-velocity and four-acceleration, and the observer rest-space decomposition,(d); then $u : = U / c$ satisfies $u = γ (n + w / c)$ , $∣ w ∣^{2} = c^{2} (1 - γ^{- 2})$ and $γ = (1 - ∣ w ∣^{2} / c^{2})^{- 1 / 2}$ , and Positivity, mass shell and kinetic energy of the observer quantities of a four-momentum,(b) gives $E_{n} (P) = m c^{2} γ$ , $q_{n} (P) = m γ w$ , $T_{n} (P) = m c^{2} (γ - 1)$ and $m (P) = m$ .

Let $z_{0} \in [0, 1)$ , suppose $z : = ∣ w ∣^{2} / c^{2} \leq z_{0}$ , put $K (z_{0}) : = \frac{3}{8} (1 - z_{0})^{- 5 / 2}$ , and let $f, R$ be as in The uniform second-order Taylor remainder of $z \mapsto (1 - z)^{- 1 / 2}$ on $[0, z_{0}]$ , so that $γ = f (z)$ and $γ = 1 + \frac{z}{2} + R (z)$ with $∣ R (z) ∣ \leq K (z_{0}) z^{2}$ . Then:

(a) $E_{n} (P) = m c^{2} + \frac{1}{2} m ∣ w ∣^{2} + R_{E}$ and $T_{n} (P) = \frac{1}{2} m ∣ w ∣^{2} + R_{E}$ for the common real number $R_{E} : = m c^{2} R (z)$ , and $∣ R_{E} ∣ \leq K (z_{0}) z m ∣ w ∣^{2} = K (z_{0}) \frac{m ∣ w ∣^{4}}{c^{2}}, ∣ T_{n} (P) - \frac{1}{2} m ∣ w ∣^{2} ∣ \leq 2 K (z_{0}) z \frac{1}{2} m ∣ w ∣^{2} .$

(b) $q_{n} (P) = m w + R_{q}$ with $R_{q} : = m (γ - 1) w$ , and $∣ R_{q} ∣ \leq m ∣ w ∣ (\frac{z}{2} + K (z_{0}) z^{2}) \leq (\frac{1}{2} + K (z_{0}) z_{0}) z m ∣ w ∣;$ moreover $∣ q_{n} (P) ∣ = m c \sqrt{γ^{2} - 1} = m γ ∣ w ∣$ and $m (P) = m$ .

(c) Every constant above depends only on $z_{0}$ , so the estimates are uniform over all future unit $n$ , all $m > 0$ and all $P = m U$ with $∣ w ∣^{2} \leq z_{0} c^{2}$ ; if $z = 0$ (equivalently $w = 0$ , $U = c n$ ) then $R_{E} = R_{q} = 0$ , $E_{n} (P) = m c^{2}$ , $q_{n} (P) = 0$ and $T_{n} (P) = 0$ . The lemma is a statement about the mathematical model only; it asserts no measured quantity, no approximation of a physical theory and no error estimate for any experiment.

Facts & Assumptions

Given: The affine Minkowski model $(A, V, η, C^{+})$ , $c > 0$ , a future unit $n$ , $P = m U$ with $m > 0$ and $U$ future timelike with $η (U, U) = - c^{2}$ , the projection $h (v) = v + η (n, v) n$ , the numbers $γ = - η (U / c, n)$ and $w = h (U) / γ$ , 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 observer quantities: $E_{n} (P) = - c η (P, n)$ , $q_{n} (P) = h (P) \in n^{⊥}$ , $m (P) = \sqrt{- η (P, P)} / c$ for nonzero future causal $P$ , and $T_{n} (P) = E_{n} (P) - m c^{2}$ after the constant $m > 0$ is fixed; $P = (E_{n} (P) / c) n + q_{n} (P)$ is the unique decomposition of $P$ along $n$ and $n^{⊥}$ (Observer energy, spatial momentum, invariant mass, rest energy and kinetic energy of a four-momentum).

[L2]

Projection and relative velocity: $h$ is linear with image $n^{⊥}$ , kernel $s p a n (n)$ , $h (n) = 0$ , and the restriction of $η$ to $n^{⊥}$ is positive definite with norm $∣ w ∣ = \sqrt{η (w, w)}$ ; for future unit $u$ one has $γ = - η (n, u) \geq 1$ , $w = c h (u) / γ \in n^{⊥}$ , $u = γ (n + w / c)$ , $∣ w ∣^{2} = c^{2} (1 - γ^{- 2})$ , $γ = (1 - ∣ w ∣^{2} / c^{2})^{- 1 / 2}$ , and $γ = 1$ if and only if $u = n$ (Norms of four-velocity and four-acceleration, and the observer rest-space decomposition,(c),(d)).

[L3]

Observer identities: for $P = m U$ with $m > 0$ constant and $U$ future timelike, $η (U, U) = - c^{2}$ , and $γ, w$ as above, one has $E_{n} (P) = m c^{2} γ$ , $T_{n} (P) = m c^{2} (γ - 1) \geq 0$ , $q_{n} (P) = m γ w$ , $m (P) = m$ and $∣ q_{n} (P) ∣ = m c \sqrt{γ^{2} - 1} = m γ ∣ w ∣$ (Positivity, mass shell and kinetic energy of the observer quantities of a four-momentum,(b)).

[L4]

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

[L5]

Model and powers: the model data are those of Oriented time-oriented affine Minkowski space (the spacetime primitives); $∣ w ∣^{2} = η (w, w) \geq 0$ on $n^{⊥}$ and $z = ∣ w ∣^{2} / c^{2} \geq 0$ ; rational powers such as $(1 - z)^{- 1 / 2}$ are those of Rational powers $a^{r}$ of a positive base and integer powers those of Integer powers $a^{m}$ .

[L6]

Scalar absolute values satisfy $∣ a b ∣ = ∣ a ∣ ∣ b ∣$ , $∣ a ∣ \geq 0$ , and $∣ a ∣ = a$ for $a \geq 0$ (Absolute value in an ordered field, Basic properties of the absolute value). The rest-space norm supplied by [L2] (Norms of four-velocity and four-acceleration, and the observer rest-space decomposition,(c)) satisfies $∣ λ v ∣ = ∣ λ ∣ ∣ v ∣$ for $v \in n^{⊥}$ .

Proof

technique · direct

1.1givenL2L4L5algebra

Relative velocity and the Taylor form of $γ$ . The map $h$ is linear by [L2], so $h (U) = c h (U / c)$ and $w = h (U) / γ = c h (U / c) / γ$ ; the vector $u : = U / c$ is future unit and $γ = - η (n, u) \geq 1$ , so [L2],(d) gives $w \in n^{⊥}$ , $u = γ (n + w / c)$ , $∣ w ∣^{2} = c^{2} (1 - γ^{- 2})$ and $γ = (1 - ∣ w ∣^{2} / c^{2})^{- 1 / 2} = (1 - z)^{- 1 / 2} = f (z)$ . By [L4] applied to $z \leq z_{0}$ we have $γ = 1 + \frac{z}{2} + R (z)$ with $∣ R (z) ∣ \leq K (z_{0}) z^{2}$ ; define $R_{E} : = m c^{2} R (z)$ .

1.2givenL3algebra

The observer identities. By [L3] at these data, $E_{n} (P) = m c^{2} γ$ , $T_{n} (P) = m c^{2} (γ - 1)$ , $q_{n} (P) = m γ w$ , $m (P) = m$ and $∣ q_{n} (P) ∣ = m c \sqrt{γ^{2} - 1} = m γ ∣ w ∣$ ; define $R_{q} : = q_{n} (P) - m w = m (γ - 1) w$ .

1.3givenL5L6algebra

Translating $z$ into norms. Since $z = ∣ w ∣^{2} / c^{2}$ , we have $m c^{2} z = m ∣ w ∣^{2}$ , and also $K (z_{0}) z^{2} m c^{2} = K (z_{0}) z m ∣ w ∣^{2} = K (z_{0}) m ∣ w ∣^{4} / c^{2}$ . Both expressions are nonnegative because $m > 0$ , $K (z_{0}) > 0$ and $∣ w ∣^{2} \geq 0$ .

2.1step 1.1step 1.2step 1.3L1algebra

Part (a), the expansions. By steps 1.1 and 1.2, $E_{n} (P) = m c^{2} (1 + \frac{z}{2} + R (z)) = m c^{2} + \frac{1}{2} m ∣ w ∣^{2} + R_{E}$ using $m c^{2} \frac{z}{2} = \frac{1}{2} m ∣ w ∣^{2}$ from step 1.3, and $T_{n} (P) = E_{n} (P) - m c^{2} = \frac{1}{2} m ∣ w ∣^{2} + R_{E}$ by the definition of $T_{n}$ in [L1].

2.2step 1.1step 1.3L4L6algebra

Part (a), the bounds. By step 1.1, $∣ R_{E} ∣ = m c^{2} ∣ R (z) ∣ \leq m c^{2} K (z_{0}) z^{2} = K (z_{0}) z m ∣ w ∣^{2} = K (z_{0}) m ∣ w ∣^{4} / c^{2}$ using step 1.3; the second displayed bound is $∣ T_{n} (P) - \frac{1}{2} m ∣ w ∣^{2} ∣ = ∣ R_{E} ∣ \leq K (z_{0}) z m ∣ w ∣^{2} = 2 K (z_{0}) z \cdot \frac{1}{2} m ∣ w ∣^{2}$ .

2.3step 1.1step 1.2L4L6algebra

Part (b). By step 1.2, $q_{n} (P) = m w + R_{q}$ with $R_{q} = m (γ - 1) w$ . Norm homogeneity [L6] gives $∣ R_{q} ∣ = m ∣ γ - 1 ∣ ∣ w ∣$ ; since $γ = f (z)$ , the bound on $∣ f (z) - 1 ∣$ of The uniform second-order Taylor remainder of $z \mapsto (1 - z)^{- 1 / 2}$ on $[0, z_{0}]$ ,(b) gives $∣ R_{q} ∣ \leq m ∣ w ∣ (\frac{z}{2} + K (z_{0}) z^{2}) \leq m ∣ w ∣ z (\frac{1}{2} + K (z_{0}) z_{0}),$ where the final inequality uses $z \leq z_{0}$ and $K (z_{0}) > 0$ . The norm identity $∣ q_{n} (P) ∣ = m c \sqrt{γ^{2} - 1} = m γ ∣ w ∣$ and $m (P) = m$ are those of step 1.2.

3.1step 2.1step 2.2step 2.3L1L2L4algebra

Uniformity and the case $z = 0$ . The remainders $R_{E} = m c^{2} R (z)$ and $R_{q} = m (γ - 1) w$ and the bounds of steps 2.2 and 2.3 depend on the state only through $z$ , $m$ and $∣ w ∣$ , and their constants depend on $z_{0}$ alone, so the estimates hold uniformly over all future unit $n$ , all $m > 0$ and all $P = m U$ with $∣ w ∣^{2} \leq z_{0} c^{2}$ . If $z = 0$ then $γ = (1 - z)^{- 1 / 2} = 1$ by step 1.1, so $u = n$ and $w = 0$ by [L2],(d), whence $R (0) = 0$ by [L4], $R_{E} = 0$ , $R_{q} = 0$ , and steps 1.2 and 2.1 give $E_{n} (P) = m c^{2}$ , $q_{n} (P) = 0$ , $T_{n} (P) = 0$ ; conversely $w = 0$ gives $z = 0$ , and $U = c n$ is the same case.

4.1step 3.1given∎

Conclusion. Steps 2.1 and 2.2 are clauses (a), step 2.3 is clause (b) including the norm identity and $m (P) = m$ , and step 3.1 is clause (c) with the uniformity statement and the $z = 0$ case; no step used more than the stated hypotheses $z \leq z_{0} < 1$ , $m > 0$ and the model's normalization, and the statement concerns the mathematical model only.

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Galilean limiting chart maps and bounded chart regions

Definition

Let $(A, V, η, C^{+})$ be the oriented time-oriented affine Minkowski model and let $c > 0$ be a fixed mathematical speed scale, with the coordinate convention $x^{0} = c t$ , let $(o; e_{0}, e_{1}, e_{2}, e_{3})$ be an inertial frame with inertial chart $x$ and spatial Euclidean structure $(S, δ, ∣ \cdot ∣)$ (Inertial frames and inertial charts of the affine Minkowski model, The Euclidean spatial structure and cross product of an inertial frame), and write the points of $R \times S$ as $(t, x)$ .

(a) Lorentz boost charts with a variable speed scale. For $v \in S$ and a real $C > ∣ v ∣$ put $γ_{C} (v) : = (1 - ∣ v ∣^{2} / C^{2})^{- 1 / 2}$ (Rational powers $a^{r}$ of a positive base) and define the affine map $Φ_{v, C} : R \times S \to R \times S$ by $Φ_{v, C} (t, x) : = (γ_{C} (v) (t - \frac{δ (v, x)}{C^{2}}), x + \frac{γ_{C} (v) - 1}{∣ v ∣^{2}} δ (v, x) v - γ_{C} (v) t v) (v \neq 0),$ with $Φ_{0, C} : = i d$ ; when $∣ v ∣ < c$ , at $C = c$ this is the coordinate expression, in the convention $x^{0} = c t$ , of the passive Lorentz boost $B (v / c)$ of The passive Lorentz boost with parameter beta: if an event has coordinates $(c t, x)$ in the unprimed inertial chart of the frame, then $Φ_{v, c} (t, x)$ are its coordinates $(x^{' 0} / c, x^{' 1}, x^{' 2}, x^{' 3})$ in the boosted inertial chart with parameter $v / c$ .

(b) Galilean boost. For $v \in S$ define the affine map $G_{v} : R \times S \to R \times S$ by $G_{v} (t, x) : = (t, x - t v)$ .

(c) Bounded chart region. For reals $T, R > 0$ put $D_{T, R} : = {(t, x) \in R \times S : ∣ t ∣ \leq T, ∣ x ∣ \leq R}$ , with $∣ \cdot ∣$ the norm of the named frame.

(d) Scope of the family. The parameter $C$ is a variable speed scale inside a family of coordinate expressions; it is not a change of the model's adopted constant $c$ , and the members with $C \neq c$ are auxiliary mathematical maps used only to state the limit of Uniform Galilean limit of the boost charts on bounded regions, and its failure on the whole chart domain. Nothing here asserts a physical limit of the framework, and no limiting trajectory, force or interaction is defined.

Remarks

Well-definedness. For $v \neq 0$ and $C > ∣ v ∣$ the base $1 - ∣ v ∣^{2} / C^{2}$ lies in $(0, 1]$ ; it equals $1$ only at $v = 0$ , which the nonvanishing branch excludes, so $γ_{C} (v)$ is a positive rational power (Rational powers $a^{r}$ of a positive base) and the displayed formulas assign to each pair $(t, x) \in R \times S$ a pair of elements of $R \times S$ , using the bilinear form $δ$ and the vector-space operations of $S$ . For $v = 0$ the coefficient $(γ_{C} (0) - 1) / ∣ 0 ∣^{2}$ in the nonvanishing branch is the indeterminate form $0 / 0$ , so the map is declared separately to be the identity; this is the same convention as in The passive Lorentz boost with parameter beta. The family is total in the parameter: for every $v$ the admissible speeds are exactly $C > ∣ v ∣$ , an open interval, and $C = c$ belongs to it precisely when $∣ v ∣ < c$ .
The identification at $C = c$ is a substitution. Writing $β : = v / c$ and $x^{0} : = c t$ in the definition of $B (β)$ in The passive Lorentz boost with parameter beta gives $x^{' 0} = γ (x^{0} - β \cdot x)$ and $x^{'} = x + \frac{γ - 1}{∣ β ∣^{2}} (β \cdot x) β - γ β x^{0}$ with $γ = γ_{c} (v)$ ; since $β \cdot x = δ (v, x) / c$ , $∣ β ∣^{2} = ∣ v ∣^{2} / c^{2}$ and $β x^{0} = (v / c) (c t) = v t$ , this is exactly $x^{' 0} / c = γ_{c} (v) (t - δ (v, x) / c^{2})$ and $x^{'} = x + \frac{γ_{c} (v) - 1}{∣ v ∣^{2}} δ (v, x) v - γ_{c} (v) t v$ , that is $Φ_{v, c}$ . To verify that the new chart is inertial, let $B = B (v / c)$ act on $V$ in the given frame and set $e_{a}^{'} = B^{- 1} e_{a}$ , with the same origin $o$ . By The passive boost is a proper orthochronous Lorentz map with inverse B(-beta),(i),(iii), $B^{- 1}$ preserves $η$ , the future cone and orientation, so $(o; e_{0}^{'}, \dots, e_{3}^{'})$ is an inertial frame; its coordinate map is $E^{- 1} B (p - o)$ , since its frame isomorphism is $B^{- 1} E$ . Thus its coordinates are the displayed passive-boost coordinates. The boost $B (v / c)$ is a linear map of $V$ acting on coordinates in the frame basis, while $Φ_{v, c}$ is the induced affine coordinate expression on the chart domain $R \times S$ ; those two descriptions agree by the displayed computation, which is why the definition records the identification rather than asserting a new map.
No uniform limit and no physical interpretation are asserted here. The definition proves no limit of the family and states no convergence of trajectories; the bounded-region rate and the failure off bounded regions are proved in Uniform Galilean limit of the boost charts on bounded regions, and its failure on the whole chart domain. The variable $C$ is an auxiliary parameter of a mathematical family, not a change of the model's adopted constant $c$ ; the model's Lorentz form, time orientation and inertial frames are those of Oriented time-oriented affine Minkowski space (the spacetime primitives), and no choice principle is used.

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Uniform Galilean limit of the boost charts on bounded regions, and its failure on the whole chart domain

Statement

Let $v \in S$ , $T, R > 0$ , and let $(Φ_{v, C})_{C > ∣ v ∣}$ , $G_{v}$ and $D_{T, R}$ be as in Galilean limiting chart maps and bounded chart regions. For $C > ∣ v ∣$ write $γ_{C} : = (1 - ∣ v ∣^{2} / C^{2})^{- 1 / 2}$ and $z : = ∣ v ∣^{2} / C^{2}$ , and for $(t, x) \in R \times S$ write $Φ_{v, C} (t, x) = (t^{'}, x^{'})$ .

(a) Exact differences. For every $C > ∣ v ∣$ and all $(t, x)$ , $t^{'} - t = (γ_{C} - 1) t - γ_{C} \frac{δ (v, x)}{C^{2}}, x^{'} - (x - t v) = (γ_{C} - 1) (x_{∥} - t v),$ where $x_{∥} : = δ (v, x) v / ∣ v ∣^{2}$ for $v \neq 0$ ; for $v = 0$ one has $Φ_{0, C} = i d$ and both differences vanish.

(b) Uniform bounds on bounded regions. For every $v \neq 0$ , every $C > ∣ v ∣$ and every $(t, x) \in D_{T, R}$ , $∣ t^{'} - t ∣ \leq (γ_{C} - 1) T + γ_{C} \frac{∣ v ∣ R}{C^{2}}, ∣ x^{'} - (x - t v) ∣ \leq (γ_{C} - 1) (R + ∣ v ∣ T) .$

(c) Quantitative rate and uniform convergence to the Galilean chart map. Fix $z_{0} \in (0, 1)$ and set $K_{0} = \frac{3}{8} (1 - z_{0})^{- 5 / 2}$ , $A_{0} = \frac{1}{2} + K_{0} z_{0}$ and $Γ_{0} = (1 - z_{0})^{- 1 / 2}$ . Whenever $v \neq 0$ and $C \geq ∣ v ∣ / \sqrt{z_{0}}$ , the following hold uniformly on $D_{T, R}$ : $∣ t^{'} - t ∣ \leq \frac{A_{0} ∣ v ∣^{2} T + Γ_{0} ∣ v ∣ R}{C^{2}}, ∣ x^{'} - (x - t v) ∣ \leq \frac{A_{0} ∣ v ∣^{2} (R + ∣ v ∣ T)}{C^{2}} .$ Thus the rate is $O (C^{- 2})$ with explicit constants for fixed $v, T, R, z_{0}$ . For every $ε > 0$ there is $C_{ε} > ∣ v ∣$ such that for every $C \geq C_{ε}$ and every $(t, x) \in D_{T, R}$ , $\max {∣ t^{'} - t ∣, ∣ x^{'} - (x - t v) ∣} \leq ε .$ For $v \neq 0$ , put $H = T + R + ∣ v ∣ T$ , choose any $z_{1}$ with $0 < z_{1} \leq \min {z_{0}, (2 A_{0})^{- 1}, ε / (2 A_{0} H)}$ (for example the displayed minimum), and set $C_{ε} = \max {∣ v ∣ / \sqrt{z_{1}}, \sqrt{3 ∣ v ∣ R / ε}} .$ This is sufficient, and since $z_{1} < 1$ it gives $C_{ε} > ∣ v ∣$ . For $v = 0$ both errors vanish identically and any $C_{ε} > 0$ suffices. Hence $\sup_{D_{T, R}} \max {∣ t^{'} - t ∣, ∣ x^{'} - (x - t v) ∣} \to 0$ as $C \to \infty$ with $v, T, R$ fixed, the convergence is uniform on $D_{T, R}$ , and for each fixed $(t, x)$ the image $Φ_{v, C} (t, x)$ converges to the Galilean value $G_{v} (t, x) = (t, x - t v)$ .

(d) Failure on the whole chart domain. For every $v \neq 0$ and every $C > ∣ v ∣$ , $\sup_{R \times S} ∣ t^{'} - t ∣ = + \infty$ and $\sup_{R \times S} ∣ x^{'} - (x - t v) ∣ = + \infty$ ; so the boundedness of $T, R$ in (c) is essential and the limit is not uniform on the whole chart domain.

Facts & Assumptions

Given: A vector $v \in S$ , reals $T, R > 0$ , the family $(Φ_{v, C})_{C > ∣ v ∣}$ , the Galilean map $G_{v} (t, x) = (t, x - t v)$ , the region $D_{T, R} = {(t, x) : ∣ t ∣ \leq T, ∣ x ∣ \leq R}$ , and for $C > ∣ v ∣$ the numbers $γ_{C} = (1 - ∣ v ∣^{2} / C^{2})^{- 1 / 2}$ and $z = ∣ v ∣^{2} / C^{2}$ (Galilean limiting chart maps and bounded chart regions).

[L1]

The definitions of $Φ_{v, C}$ , $G_{v}$ , $D_{T, R}$ and $γ_{C}$ are those of Galilean limiting chart maps and bounded chart regions: for $v \neq 0$ , $Φ_{v, C} (t, x) = (γ_{C} (t - δ (v, x) / C^{2}), x + ((γ_{C} - 1) / ∣ v ∣^{2}) δ (v, x) v - γ_{C} t v)$ , and $Φ_{0, C} = i d$ .

[L2]

Uniform Taylor remainder: for $z_{0} \in (0, 1)$ , $K_{0} = \frac{3}{8} (1 - z_{0})^{- 5 / 2}$ and $0 \leq z \leq z_{0}$ , one has $(1 - z)^{- 1 / 2} = 1 + \frac{z}{2} + R (z)$ with $∣ R (z) ∣ \leq K_{0} z^{2}$ , hence $(γ - 1) \leq (\frac{1}{2} + K_{0} z_{0}) z$ for $γ = (1 - z)^{- 1 / 2}$ (The uniform second-order Taylor remainder of $z \mapsto (1 - z)^{- 1 / 2}$ on $[0, z_{0}]$ ).

[L3]

Cauchy–Schwarz in the frame's orthonormal basis: writing $v = \sum_{i} v^{i} e_{i}$ and $x = \sum_{i} x^{i} e_{i}$ in the $δ$ -orthonormal basis of The Euclidean spatial structure and cross product of an inertial frame, one has $δ (v, x) = \sum_{i} v^{i} x^{i}$ , $∣ v ∣^{2} = \sum_{i} (v^{i})^{2}$ , $∣ x ∣^{2} = \sum_{i} (x^{i})^{2}$ and $∣ δ (v, x) ∣ \leq ∣ v ∣ ∣ x ∣$ (The Cauchy-Schwarz inequality for finite sums).

[L4]

Norm and absolute-value calculus: $∣ λ u ∣ = ∣ λ ∣ ∣ u ∣$ and $∣ u + w ∣ \leq ∣ u ∣ + ∣ w ∣$ on $S$ (The inner-product norm is definite, homogeneous, and satisfies the triangle inequality); $λ, μ \in R$ satisfy $∣ λ + μ ∣ \leq ∣ λ ∣ + ∣ μ ∣$ , $∣ λ μ ∣ = ∣ λ ∣ ∣ μ ∣$ , $∣ w ∣ = w$ for $w \geq 0$ (The triangle inequality, Absolute value in an ordered field, Basic properties of the absolute value).

[L5]

Rational-power order: for $0 < a \leq b$ and rational $r > 0$ , $a^{r} \leq b^{r}$ ; for $0 < a < b$ , $0 < b^{- 1} < a^{- 1}$ ; for $0 < a < 1$ and rational $r < 0$ , $a^{r} > 1$ , and $1^{r} = 1$ (Monotonicity of $r \mapsto a^{r}$ and of $a \mapsto a^{r}$ , Inverses of positives are positive, and reciprocation reverses order, Laws of rational exponents, Rational powers $a^{r}$ of a positive base).

[L6]

The norm $∣ w ∣ = \sqrt{δ (w, w)}$ of The Euclidean spatial structure and cross product of an inertial frame is the nonnegative square root of Square roots exist: a unique $\sqrt{a} \geq 0$ with $(\sqrt{a})^{2} = a$ ; the positives are ${x^{2} : x \neq 0}$ , so $∣ w ∣^{2} = δ (w, w)$ and $∣ w ∣ \geq 0$ .

Proof

technique · direct

1.1L1algebra

The exact time difference. For $v \neq 0$ and $C > ∣ v ∣$ , subtraction in $R$ using [L1] gives $t^{'} - t = γ_{C} (t - \frac{δ (v, x)}{C^{2}}) - t = (γ_{C} - 1) t - γ_{C} \frac{δ (v, x)}{C^{2}},$ for every $(t, x) \in R \times S$ .

1.2L1algebra

The exact spatial difference. In the same case, subtraction in the vector space $S$ using [L1] gives $x^{'} - (x - t v) = x + \frac{γ_{C} - 1}{∣ v ∣^{2}} δ (v, x) v - γ_{C} t v - x + t v = (γ_{C} - 1) (\frac{δ (v, x)}{∣ v ∣^{2}} v - t v) = (γ_{C} - 1) (x_{∥} - t v),$ with $x_{∥} = δ (v, x) v / ∣ v ∣^{2}$ ; if $v = 0$ then $Φ_{0, C} = i d$ and $t^{'} - t = 0$ , $x^{'} - (x - t v) = x - x = 0$ . This proves (a).

1.3L2algebra

The Taylor estimate for $γ_{C}$ . Fix $z_{0} \in (0, 1)$ , let $0 \leq z \leq z_{0}$ and put $γ = (1 - z)^{- 1 / 2}$ ; by [L2] $γ = 1 + \frac{z}{2} + R (z)$ with $∣ R (z) ∣ \leq K_{0} z^{2}$ , so $γ - 1 = \frac{z}{2} + R (z) \leq \frac{z}{2} + K_{0} z^{2} \leq (\frac{1}{2} + K_{0} z_{0}) z = A_{0} z,$ because $K_{0} \geq 0$ and $z \leq z_{0}$ .

1.4L5

The reciprocal bound for $γ_{C}$ . With the same data, $0 < 1 - z_{0} \leq 1 - z \leq 1$ gives $(1 - z_{0})^{1 / 2} \leq (1 - z)^{1 / 2}$ by [L5]; if $1 - z_{0} < 1 - z$ the strict reciprocal rule of [L5] gives $(1 - z)^{- 1 / 2} < (1 - z_{0})^{- 1 / 2}$ , and if $1 - z_{0} = 1 - z$ the two quantities are equal, so in all cases $γ = (1 - z)^{- 1 / 2} \leq (1 - z_{0})^{- 1 / 2} = Γ_{0}$ .

1.5L1L5algebra

The domain inclusion. For $v \neq 0$ and $C \geq ∣ v ∣ / \sqrt{z_{0}}$ one has $z = ∣ v ∣^{2} / C^{2} \leq ∣ v ∣^{2} / (∣ v ∣^{2} / z_{0}) = z_{0}$ ; hence steps 1.3 and 1.4 apply with this $z$ .

2.1step 1.1L3L4L5algebra

The time bound. Let $v \neq 0$ , $C > ∣ v ∣$ and $(t, x) \in D_{T, R}$ ; by [L5], $(1 - z)^{- 1 / 2} > 1$ for $0 < z < 1$ and $γ_{C} = 1$ for $z = 0$ , so $γ_{C} - 1 \geq 0$ and $γ_{C} \geq 1$ . Taking absolute values in step 1.1 and using [L4], $∣ t^{'} - t ∣ \leq (γ_{C} - 1) ∣ t ∣ + γ_{C} ∣ δ (v, x) ∣ / C^{2}$ , and [L3] plus $∣ t ∣ \leq T$ , $∣ x ∣ \leq R$ give $∣ t^{'} - t ∣ \leq (γ_{C} - 1) T + γ_{C} ∣ v ∣ R / C^{2}$ .

2.2step 1.2L3L4algebra

The spatial bound. In the same case, step 1.2 and the homogeneity and triangle inequality of [L4] give $∣ x^{'} - (x - t v) ∣ = (γ_{C} - 1) ∣ x_{∥} - t v ∣ \leq (γ_{C} - 1) (∣ x_{∥} ∣ + ∣ t ∣ ∣ v ∣)$ ; by [L3] $∣ x_{∥} ∣ = ∣ δ (v, x) ∣ / ∣ v ∣ \leq ∣ x ∣$ , so on $D_{T, R}$ we get $∣ x^{'} - (x - t v) ∣ \leq (γ_{C} - 1) (∣ x ∣ + ∣ v ∣ T) \leq (γ_{C} - 1) (R + ∣ v ∣ T)$ . This proves (b).

2.3step 1.2L1

The case $v = 0$ . Here $G_{0} (t, x) = (t, x)$ and $Φ_{0, C} = i d$ for every $C > 0$ , so $∣ t^{'} - t ∣ = 0$ and $∣ x^{'} - (x - 0 \cdot v) ∣ = ∣ x^{'} - x ∣ = 0$ for every $(t, x)$ ; hence every $ε > 0$ is met already with any $C_{ε} > 0$ .

3.1step 2.1step 2.2step 1.3step 1.4step 1.5algebra

The explicit $O (C^{- 2})$ bounds. Let $v \neq 0$ , $C \geq ∣ v ∣ / \sqrt{z_{0}}$ and $(t, x) \in D_{T, R}$ . By step 2.1 and steps 1.3, 1.5, $∣ t^{'} - t ∣ \leq (γ_{C} - 1) T + γ_{C} \frac{∣ v ∣ R}{C^{2}} \leq A_{0} z T + Γ_{0} \frac{∣ v ∣ R}{C^{2}} = \frac{A_{0} ∣ v ∣^{2} T + Γ_{0} ∣ v ∣ R}{C^{2}},$ and by step 2.2 and steps 1.3, 1.5, $∣ x^{'} - (x - t v) ∣ \leq (γ_{C} - 1) (R + ∣ v ∣ T) \leq A_{0} z (R + ∣ v ∣ T) = \frac{A_{0} ∣ v ∣^{2} (R + ∣ v ∣ T)}{C^{2}},$ since $z = ∣ v ∣^{2} / C^{2}$ .

3.2step 2.1step 2.2step 1.3step 1.5L5algebra

The $ε$ -threshold for $v \neq 0$ . Fix $ε > 0$ , put $H = T + R + ∣ v ∣ T > 0$ , and choose $z_{1}$ with $0 < z_{1} \leq \min {z_{0}, (2 A_{0})^{- 1}, ε / (2 A_{0} H)}$ ; the minimum is positive because $z_{0}, A_{0}, ε, H$ are positive. Let $C \geq \max {∣ v ∣ / \sqrt{z_{1}}, \sqrt{3 ∣ v ∣ R / ε}}$ . Then $z \leq z_{1} \leq z_{0}$ , so by step 1.3, $γ_{C} - 1 \leq A_{0} z_{1} \leq \frac{1}{2}$ , whence $γ_{C} \leq \frac{3}{2}$ , and $A_{0} z_{1} T \leq \frac{ε}{2 H} T \leq \frac{ε}{2}$ while $A_{0} z_{1} (R + ∣ v ∣ T) \leq \frac{ε}{2 H} (R + ∣ v ∣ T) \leq \frac{ε}{2}$ . Thus step 2.1 gives $∣ t^{'} - t ∣ \leq A_{0} z_{1} T + \frac{3}{2} \frac{∣ v ∣ R}{C^{2}} \leq \frac{ε}{2} + \frac{ε}{2} = ε$ and step 2.2 gives $∣ x^{'} - (x - t v) ∣ \leq A_{0} z_{1} (R + ∣ v ∣ T) \leq \frac{ε}{2} \leq ε$ on $D_{T, R}$ . Since $z_{1} \leq z_{0} < 1$ and $v \neq 0$ , the number $C_{ε} = \max {∣ v ∣ / \sqrt{z_{1}}, \sqrt{3 ∣ v ∣ R / ε}}$ satisfies $C_{ε} \geq ∣ v ∣ / \sqrt{z_{1}} > ∣ v ∣$ .

4.1step 3.2step 2.3L1algebra

Uniform and pointwise convergence. By steps 3.2 and 2.3, for every fixed $v, T, R$ and every $ε > 0$ there is $C_{ε} > ∣ v ∣$ with $\max {∣ t^{'} - t ∣, ∣ x^{'} - (x - t v) ∣} \leq ε$ for all $(t, x) \in D_{T, R}$ and all $C \geq C_{ε}$ ; taking the supremum over $D_{T, R}$ gives $\sup_{D_{T, R}} \max {∣ t^{'} - t ∣, ∣ x^{'} - (x - t v) ∣} \leq ε$ , so the supremum tends to $0$ as $C \to \infty$ . Since $Φ_{v, C} (t, x) - (t, x - t v)$ has both components bounded by that maximum, $Φ_{v, C} (t, x) \to G_{v} (t, x)$ for each fixed $(t, x)$ , which lies in $D_{T, R}$ for $T, R$ large; this proves (c).

5.1step 1.1step 1.2L1L6algebra∎

Failure on the whole chart domain. Let $v \neq 0$ and $C > ∣ v ∣$ , so $z > 0$ , $γ_{C} > 1$ and $x_{∥}$ is defined. For $λ > 0$ take $(t, x) = (0, λ v / ∣ v ∣)$ ; then $x_{∥} = λ v / ∣ v ∣$ and step 1.2 gives $x^{'} - (x - 0 \cdot v) = (γ_{C} - 1) λ v / ∣ v ∣$ , of norm $(γ_{C} - 1) λ$ , which is unbounded as $λ \to \infty$ ; hence $\sup_{R \times S} ∣ x^{'} - (x - t v) ∣ = + \infty$ . For $μ > 0$ take $(t, x) = (0, - μ v)$ ; then $δ (v, x) = - μ ∣ v ∣^{2}$ and step 1.1 gives $t^{'} - t = γ_{C} μ ∣ v ∣^{2} / C^{2}$ , which is unbounded as $μ \to \infty$ ; hence $\sup_{R \times S} ∣ t^{'} - t ∣ = + \infty$ . This proves (d), so the boundedness of $T, R$ is essential and no uniform limit holds on the whole chart domain.

Remarks

The statement and proof concern the auxiliary coordinate family $Φ_{v, C}$ of Galilean limiting chart maps and bounded chart regions alone: $C$ is varied with $v, T, R$ fixed, the model's adopted scale $c$ is unchanged, and no trajectory, force, interaction or physical limiting process is defined. The constants $A_{0}, Γ_{0}, K_{0}$ depend on the chosen $z_{0}$ (equivalently on the admissible ratio $∣ v ∣ / C$ ), so the rate is uniform only on the declared bounded region and speed window; clause (d) exhibits failure on the whole chart domain. This does not imply failure on every unbounded subset: on ${(0, x) : δ (v, x) = 0}$ both exact differences in (a) vanish, even though that subset is unbounded. The quantitative threshold of clause (c) is sharp enough for the worked evaluation of Worked uniform Galilean chart bound on a bounded region ↗.

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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.

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Limits of the Newtonian bridge: speed-dependent error constants, the excluded massless case, and bounded chart regions

Physical scope

Scope remark for the flat special-relativity massive-particle model and the same bounded low-speed domain as pthm-sr-newtonian-p-e-correspondence: it records the failure of any speed-independent error constant as |w| -> c, the exclusion of ideal massless particles, and the Galilean chart limit on bounded regions and its whole-domain failure for nonzero boost velocity (the zero-velocity maps coincide everywhere). It asserts no new physical law, no dynamics, and no empirical statement.

Limits of the low-speed bridge

Scope remark on the limits of the low-speed bridge. Each quantitative assertion is an immediate consequence of the cited displayed identities; none of them is used as a premise elsewhere on this page, and the remark asserts no new physical law, dynamics or empirical content.

(i) No speed-independent error constant

Let $m > 0$ . For every $B \geq 0$ there are a future unit $n \in V$ , a vector $U$ with $η (U, U) = - c^{2}$ and a state $P = m U$ whose relative velocity $w$ of Norms of four-velocity and four-acceleration, and the observer rest-space decomposition,(d) satisfies $∣ w ∣ < c$ and $∣ T_{n} (P) - \frac{1}{2} m ∣ w ∣^{2} ∣ > B$ , with $T_{n}$ as in Observer energy, spatial momentum, invariant mass, rest energy and kinetic energy of a four-momentum. Explicitly, fix a future unit $n$ and a unit $e \in n^{⊥}$ and, for every integer $γ \geq 2$ , take the state with Lorentz factor $γ$ , namely $w = c \sqrt{1 - γ^{- 2}} e$ , $U = c γ (n + w / c)$ and $P = m U$ . Bilinearity gives $η (U, U) = - c^{2}$ . The continuous segment $s \mapsto n + s w / c$ , $0 \leq s \leq 1$ , is timelike because its squared norm is $- 1 + s^{2} ∣ w ∣^{2} / c^{2} < 0$ ; it starts at $n \in C^{+}$ and its endpoint lies in that same component (Time orientation, future and past timelike/null/causal vectors, and the closed cones, Every path-connected space is connected, and every path component lies inside a component,(3)). Positive scaling gives $U \in C^{+}$ , while $- η (n, U / c) = γ$ and the rest projection gives $h (U) / γ = w$ . Hence these states have the claimed relative velocity and Lorentz factor; Positivity, mass shell and kinetic energy of the observer quantities of a four-momentum,(b) gives $T_{n} (P) = m c^{2} (γ - 1)$ while $\frac{1}{2} m ∣ w ∣^{2} = \frac{1}{2} m c^{2} (1 - γ^{- 2})$ , so $T_{n} (P) - \frac{1}{2} m ∣ w ∣^{2} = m c^{2} (γ - \frac{3}{2} + \frac{1}{2 γ^{2}}) \to \infty (γ \to \infty),$ and the relative deviation from the Newtonian value equals $\frac{(2 γ + 1) (γ - 1)}{γ + 1} \to \infty$ ; the Archimedean property of $R$ (Every complete ordered field is Archimedean) turns either display into the explicit violation of a prescribed $B$ at a sufficiently large integer $γ$ . Consequently the constants $K (z_{0})$ and $\frac{1}{2} + K (z_{0}) z_{0}$ of Specified low-speed momentum-energy correspondence between the relativistic measured quantities and the Newtonian comparison quantities necessarily depend on the speed bound $z_{0}$ , no bound of that form is uniform on the whole domain $∣ w ∣ < c$ , and $E_{n} (P) = m c^{2} γ$ and $∣ q_{n} (P) ∣ = m c \sqrt{γ^{2} - 1}$ are unbounded as $∣ w ∣ \to c$ although $\frac{1}{2} m ∣ w ∣^{2} \leq \frac{1}{2} m c^{2}$ there. The mechanism is explicit: the remainder is $R_{E} = m c^{2} R (z)$ with $∣ R (z) ∣ \leq K (z_{0}) z^{2}$ , and $K (z_{0}) = \frac{3}{8} (1 - z_{0})^{- 5 / 2}$ diverges as $z_{0} ↑ 1$ (Uniform low-speed expansion of the observer energy, momentum and kinetic energy of a massive four-momentum).

(ii) The massless case is excluded, not a limit

An ideal massless particle of Adopted momentum model for an ideal massless particle carries a nonzero future null four-momentum $P$ and, by Positivity, mass shell and kinetic energy of the observer quantities of a four-momentum,(a)(i), satisfies $E_{n} (P) = c ∣ q_{n} (P) ∣ > 0$ exactly; it has no four-velocity, proper time or relative velocity, and no rest mass, rest energy or kinetic energy is assigned to it. The hypotheses $m > 0$ and $z \leq z_{0} < 1$ of Specified low-speed momentum-energy correspondence between the relativistic measured quantities and the Newtonian comparison quantities exclude it, and no substitution $w$ with $∣ w ∣ = c$ is available because $γ_{w} = (1 - ∣ w ∣^{2} / c^{2})^{- 1 / 2}$ is undefined there. The bridge therefore concerns massive particles only, and it neither derives nor constrains the ideal massless case or light.

(iii) Bounded chart regions, not a limit of frameworks

For fixed $v \in S$ , the uniform chart statement Uniform Galilean limit of the boost charts on bounded regions, and its failure on the whole chart domain holds on each bounded region $D_{T, R}$ as $C \to \infty$ (its clause (c)). For $v \neq 0$ it fails on the whole chart domain $R \times S$ (its clause (d)); for $v = 0$ , $Φ_{0, C} = G_{0} = i d$ for every $C > 0$ , so convergence is uniform on the whole chart domain. The nonzero-velocity failure does not assert failure on every unbounded subset; the correspondence is a conditional statement about the adopted model in a bounded low-speed domain, and it is neither an equivalence between the relativistic and Newtonian frameworks, nor a physical process in which the constant $c$ varies, nor empirical evidence for either framework. The Newtonian comparison quantities Newtonian comparison momentum and kinetic energy in a named inertial frame are comparison expressions built from a named frame and the mass parameter, not the derived quantities of a separate theory; the correspondence theorem's clause (e) records the same scope restriction on the pair.

5 · Examples, experiments and counterexamples

None yet.

Newtonian Bridge

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

Newtonian comparison momentum and kinetic energy in a named inertial frame

Definition

Remarks

The uniform second-order Taylor remainder of z↦(1−z)−1/2z↦(1−z)−1/2 on [0,z0][0,z0​]

Statement

Facts & Assumptions

Proof

Remarks

Uniform low-speed expansion of the observer energy, momentum and kinetic energy of a massive four-momentum

Statement

Facts & Assumptions

Proof

Galilean limiting chart maps and bounded chart regions

Definition

Remarks

Uniform Galilean limit of the boost charts on bounded regions, and its failure on the whole chart domain

Statement

Facts & Assumptions

Proof

Remarks

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

Statement

Facts & Assumptions

Proof

Limits of the Newtonian bridge: speed-dependent error constants, the excluded massless case, and bounded chart regions

Limits of the low-speed bridge

(i) No speed-independent error constant

(ii) The massless case is excluded, not a limit

(iii) Bounded chart regions, not a limit of frameworks

5 · Examples, experiments and counterexamples

Sources

The uniform second-order Taylor remainder of $z \mapsto (1 - z)^{- 1 / 2}$ on $[0, z_{0}]$