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

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 $B\ge 0$ there are a future unit $n\in V$, a vector $U$ with $\eta (U,U)=-c^2$ and a state $P=mU$ whose relative velocity $w$ of Norms of four-velocity and four-acceleration, and the observer rest-space decomposition,(d) satisfies $\mathrm{\mid }w\mathrm{\mid }<c$ and $\mid T_n(P)-\frac{1}{2}m\mathrm{\mid }w{\mathrm{\mid }}^2\mid >B$, 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 $e\in n^{\perp }$ and, for every integer $\gamma \ge 2$, take the state with Lorentz factor $\gamma$, namely $w=c\sqrt{1-{\gamma }^{-2}}e$, $U=c\gamma (n+w\mathrm{/}c)$ and $P=mU$. Bilinearity gives $\eta (U,U)=-c^2$. The continuous segment $s\mapsto n+sw\mathrm{/}c$, $0\le s\le 1$, is timelike because its squared norm is $-1+s^2\mathrm{\mid }w{\mathrm{\mid }}^2\mathrm{/}{c}^2<0$; 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 $-\eta (n,U\mathrm{/}c)=\gamma$ and the rest projection gives $h(U)\mathrm{/}\gamma =w$. 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 $T_n(P)=m{c}^2(\gamma -1)$ while $\frac{1}{2}m\mathrm{\mid }w{\mathrm{\mid }}^2=\frac{1}{2}m{c}^2(1-{\gamma }^{-2})$, so $T_n(P)-\frac{1}{2}m\mathrm{\mid }w{\mathrm{\mid }}^2=m{c}^2(\gamma -\frac{3}{2}+\frac{1}{2{\gamma }^2})\to \mathrm{\infty }(\gamma \to \mathrm{\infty }),$ and the relative deviation from the Newtonian value equals $\frac{(2\gamma +1)(\gamma -1)}{\gamma +1}\to \mathrm{\infty }$; the Archimedean property of $\mathbb{R}$ (Every complete ordered field is Archimedean) turns either display into the explicit violation of a prescribed $B$ at a sufficiently large integer $\gamma$. 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 $\mathrm{\mid }w\mathrm{\mid }<c$, and $E_n(P)=m{c}^2\gamma$ and $\mathrm{\mid }{q}_n(P)\mathrm{\mid }=mc\sqrt{{\gamma }^2-1}$ are unbounded as $\mathrm{\mid }w\mathrm{\mid }\to c$ although $\frac{1}{2}m\mathrm{\mid }w{\mathrm{\mid }}^2\le \frac{1}{2}m{c}^2$ there. The mechanism is explicit: the remainder is $R_E=m{c}^{2}R(z)$ with $\mathrm{\mid }R(z)\mathrm{\mid }\le K(z_0)z^2$, and $K(z_0)=\frac{3}{8}(1-z_0{)}^{-5\mathrm{/}2}$ diverges as $z_0\uparrow 1$ (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 $E_n(P)=c\mathrm{\mid }{q}_n(P)\mathrm{\mid }>0$ 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 $z\le z_0<1$ of Specified low-speed momentum-energy correspondence between the relativistic measured quantities and the Newtonian comparison quantities exclude it, and no substitution $w$ with $\mathrm{\mid }w\mathrm{\mid }=c$ is available because ${\gamma }_w=(1-\mathrm{\mid }w{\mathrm{\mid }}^2\mathrm{/}{c}^2{)}^{-1\mathrm{/}2}$ 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 $C\to \mathrm{\infty }$ (its clause (c)). For $v\mathrm{\ne }0$ it fails on the whole chart domain $\mathbb{R}\times S$ (its clause (d)); for $v=0$, ${\mathrm{\Phi }}_{0,C}=G_0=\mathrm{i}\mathrm{d}$ 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.