A smooth time-dependent translation chart need not be an inertial frame

[The chart is a bijection] The map $G(t,x)=(t,x+t^{2}c)$ is a bijection of $\mathbb{R}\times {\mathbb{R}}^3$ with inverse $(t,y)\mapsto (t,y-t^{2}c)$, since $G(t,y-t^{2}c)=(t,y)$ and $G=(G^{-1}{)}^{-1}$ is smooth as well; hence $G$ is a legitimate chart of the coordinate space in the smooth structure of the arena.

[The shear is not Galilean] Suppose $G=g_{(b,a,v,R)}$. Comparing time coordinates gives $t=t+b$ for every $t$, so $b=0$. Then $x+t^{2}c=Rx+vt+a$ for all $(t,x)\in \mathbb{R}\times {\mathbb{R}}^3$. At $t=0$ this gives $x=Rx+a$ for all $x$; taking $x=0$ gives $a=0$, and $(R-I_3)x=0$ for all $x$ gives $R=I_3$ by comparing the action on $e_1,e_2,e_3$. The terms $t^{2}c$ and $vt$ both have dimension $\mathrm{\ell }$. Evaluating at $t=T_0$ yields $v=T_{0}c$; at $t=2T_0$ the same identity would then give $4T_0^{2}c=2T_{0}v=2T_0^{2}c$, contradicting $T_0>0$ and $c\mathrm{\ne }0$. Thus no parameter quadruple represents $G$, so by the set definition in [L1] it is not Galilean.

[A zero-acceleration curve becomes accelerating] Let $\gamma :\mathbb{R}\to \mathbb{R}\times {\mathbb{R}}^3$ be $\gamma (s):=(s,us)$, where $s$ is a time coordinate. In the standard chart its coordinate curve is $x(s)=us$ with $x^{\mathrm{\prime }\mathrm{\prime }}=0$; in the chart $G$ it is $x^{\mathrm{\prime }}(s)=us+s^{2}c$, whose second derivative is $2c\mathrm{\ne }0$. The terms $us$ and $s^{2}c$ both have dimension $\mathrm{\ell }$, and their first and second derivatives have dimensions $\mathrm{\ell }{\tau }^{-1}$ and $\mathrm{\ell }{\tau }^{-2}$. Hence zero coordinate acceleration is not preserved by this chart, although it is preserved by every Galilean change of frame by clause 3 of Transformation between adapted charts and its kinematic consequences.

Step 1.1 makes $G$ a smooth chart. Every change between the inertial frames in the Given paragraph belongs to the Galilean class, whereas step 1.2 proves that $G$ does not; hence this chart is not an inertial frame. Independently, step 1.3 shows that it fails preservation of zero coordinate acceleration, which clause 3 of Transformation between adapted charts and its kinematic consequences supplies as a necessary property of inertial-frame changes, without asserting sufficiency. Thus the refuted statement fails: not every smooth time-dependent spatial translation is an inertial frame of the model, and the inertial frames form a proper subclass of the smooth charts of the arena.