Galilean limiting chart maps and bounded chart regions

Let $(A,V,\eta ,C^{+})$ be the oriented time-oriented affine Minkowski model and let $c>0$ be a fixed mathematical speed scale, with the coordinate convention $x^0=ct$, let $(o;e_0,e_1,e_2,e_3)$ be an inertial frame with inertial chart $x$ and spatial Euclidean structure $(S,\delta ,\mathrm{\mid }\cdot \mathrm{\mid })$ (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 $\mathbb{R}\times S$ as $(t,x)$.

(a) Lorentz boost charts with a variable speed scale. For $v\in S$ and a real $C>\mathrm{\mid }v\mathrm{\mid }$ put ${\gamma }_C(v):=(1-\mathrm{\mid }v{\mathrm{\mid }}^2\mathrm{/}{C}^2{)}^{-1\mathrm{/}2}$ (Rational powers $a^r$ of a positive base) and define the affine map ${\mathrm{\Phi }}_{v,C}:\mathbb{R}\times S\to \mathbb{R}\times S$ by ${\mathrm{\Phi }}_{v,C}(t,x):=({\gamma }_C(v)(t-\frac{\delta (v,x)}{C^2}),x+\frac{{\gamma }_C(v)-1}{\mathrm{\mid }v{\mathrm{\mid }}^2}\delta (v,x)v-{\gamma }_C(v)tv)(v\mathrm{\ne }0),$ with ${\mathrm{\Phi }}_{0,C}:=\mathrm{i}\mathrm{d}$; when $\mathrm{\mid }v\mathrm{\mid }<c$, at $C=c$ this is the coordinate expression, in the convention $x^0=ct$, of the passive Lorentz boost $B(v\mathrm{/}c)$ of The passive Lorentz boost with parameter beta: if an event has coordinates $(ct,x)$ in the unprimed inertial chart of the frame, then ${\mathrm{\Phi }}_{v,c}(t,x)$ are its coordinates $(x^{\mathrm{\prime }0}\mathrm{/}c,x^{\mathrm{\prime }1},x^{\mathrm{\prime }2},x^{\mathrm{\prime }3})$ in the boosted inertial chart with parameter $v\mathrm{/}c$.

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

(c) Bounded chart region. For reals $T,R>0$ put $D_{T,R}:=\{(t,x)\in \mathbb{R}\times S:\mathrm{\mid }t\mathrm{\mid }\le T,\mathrm{\mid }x\mathrm{\mid }\le R\}$, with $\mathrm{\mid }\cdot \mathrm{\mid }$ 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 $C\mathrm{\ne }c$ 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.

• Well-definedness. For $v\mathrm{\ne }0$ and $C>\mathrm{\mid }v\mathrm{\mid }$ the base $1-\mathrm{\mid }v{\mathrm{\mid }}^2\mathrm{/}{C}^2$ lies in $(0,1]$; it equals $1$ only at $v=0$, which the nonvanishing branch excludes, so ${\gamma }_C(v)$ is a positive rational power (Rational powers $a^r$ of a positive base) and the displayed formulas assign to each pair $(t,x)\in \mathbb{R}\times S$ a pair of elements of $\mathbb{R}\times S$, using the bilinear form $\delta$ and the vector-space operations of $S$. For $v=0$ the coefficient $({\gamma }_C(0)-1)\mathrm{/}\mathrm{\mid }0{\mathrm{\mid }}^2$ in the nonvanishing branch is the indeterminate form $0\mathrm{/}0$, 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 $C>\mathrm{\mid }v\mathrm{\mid }$, an open interval, and $C=c$ belongs to it precisely when $\mathrm{\mid }v\mathrm{\mid }<c$.

• The identification at $C=c$ is a substitution. Writing $\beta :=v\mathrm{/}c$ and $x^0:=ct$ in the definition of $B(\beta )$ in The passive Lorentz boost with parameter beta gives $x^{\mathrm{\prime }0}=\gamma (x^0-\beta \cdot x)$ and $x^{\mathrm{\prime }}=x+\frac{\gamma -1}{\mathrm{\mid }\beta {\mathrm{\mid }}^2}(\beta \cdot x)\beta -\gamma \beta x^0$ with $\gamma ={\gamma }_c(v)$; since $\beta \cdot x=\delta (v,x)\mathrm{/}c$, $\mathrm{\mid }\beta {\mathrm{\mid }}^2=\mathrm{\mid }v{\mathrm{\mid }}^2\mathrm{/}{c}^2$ and $\beta x^0=(v\mathrm{/}c)(ct)=vt$, this is exactly $x^{\mathrm{\prime }0}\mathrm{/}c={\gamma }_c(v)(t-\delta (v,x)\mathrm{/}{c}^2)$ and $x^{\mathrm{\prime }}=x+\frac{{\gamma }_c(v)-1}{\mathrm{\mid }v{\mathrm{\mid }}^2}\delta (v,x)v-{\gamma }_c(v)tv$, that is ${\mathrm{\Phi }}_{v,c}$. To verify that the new chart is inertial, let $B=B(v\mathrm{/}c)$ act on $V$ in the given frame and set $e_a^{\mathrm{\prime }}=B^{-1}{e}_a$, with the same origin $o$. By The passive boost is a proper orthochronous Lorentz map with inverse B(-beta),(i),(iii), $B^{-1}$ preserves $\eta$, the future cone and orientation, so $(o;e_0^{\mathrm{\prime }},\dots ,e_3^{\mathrm{\prime }})$ 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 $B(v\mathrm{/}c)$ is a linear map of $V$ acting on coordinates in the frame basis, while ${\mathrm{\Phi }}_{v,c}$ is the induced affine coordinate expression on the chart domain $\mathbb{R}\times S$; 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.