Inertial frames, stationary observers and the adopted Galilean transformations between frames

Adopt as physical assumptions of the framework the following statements about frames, in the spacetime structure of Adopted Newtonian spacetime structure: absolute time, Euclidean spatial slices and their smooth structure.

(i) Rotations. A rotation is a matrix $R\in SO(3)$, where

$SO(3)=\{R\in {\mathbb{R}}^{3\times 3}:R^{\mathsf{T}}R=I_3,\det R=1\}$

in the sense of The orthogonal group $O(3)$ and the rotation group $SO(3)$. A rotation acts on spatial column vectors by matrix multiplication; its entries are dimensionless under the radian convention of Dimensions and SI unit conventions of the non-relativistic classical mechanics framework. Rotations are part of the formulation of spatial changes of basis; the definition of $SO(3)$ itself asserts nothing about physical space.

(ii) Inertial frames. An inertial frame is the global adapted affine chart of the spacetime data determined by

$o\in A,u\in V\text{ with }\tau (u)=1,(e_1,e_2,e_3)\text{ an ordered, }g\text{-orthonormal, positively oriented basis of }K,$

with adapted coordinates $(t,x)$ as in Affine time space with Euclidean simultaneous slices and adapted charts: the time coordinate is $t(p)=\tau (p-o)$ and the spatial coordinates are the coefficients of $p-o-tu\in K$ in the basis $(e_1,e_2,e_3)$. Existence and uniqueness of the coordinates, and the affine structure of the slices, are the mathematical content of Existence and uniqueness of adapted coordinates, and the affine structure of simultaneous slices and Affine time space with Euclidean simultaneous slices and adapted charts; no physical claim beyond the reading of clause (iii) is attached to them here.

An inertial frame is a coordinate chart: the triple $(o,u,(e_i))$. It is definitively not an observer congruence and not a time-varying basis field. In the model the basis $(e_i)$ is fixed once and for all; a family of bases depending on the time coordinate is a different structure, whose kinematics is defined separately where it is used, and is not an inertial frame. The physical reading of the frame is that $t$ is the model's time coordinate in seconds and $x$ the spatial position in metres, by the conventions of Dimensions and SI unit conventions of the non-relativistic classical mechanics framework.

(iii) Stationary observers. For $x_0\in {\mathbb{R}}^3$ the curve ${\alpha }_{x_0}:\mathbb{R}\to A$, ${\alpha }_{x_0}(t):=o+tu+{\sum }_{j=1}^3(x_0{)}_j{e}_j$, is the observer at rest in the frame at position $x_0$; the family $\{{\alpha }_{x_0}{\}}_{x_0\in {\mathbb{R}}^3}$ is the stationary observer congruence of the frame. Each curve has $t({\alpha }_{x_0}(s))=s$, is $C^{\mathrm{\infty }}$ in adapted coordinates, and in the frame's own spatial coordinates has constant coordinate triple $x_0$. Here an observer worldline means precisely a $C^2$ curve $\alpha :I\to A$ on a nondegenerate open real interval with $t(\alpha (s))=s$, used to describe an observer; the stationary curves just defined have $I=\mathbb{R}$. This kinematic definition assigns neither a particle label nor a mass or a dynamical law to an observer. A particle worldline will separately be assigned to a particle label in the particle postulate. The congruence is a derived object, determined by the frame and the spacetime data, not an additional primitive. Distinct frames can have the same family of parametrized curves: if $o^{\mathrm{\prime }}=o+{\sum }_j{c}_j{e}_j$ while $u^{\mathrm{\prime }}=u$ and $e_j^{\mathrm{\prime }}=e_j$, then ${\alpha }_{x_0}^{\mathrm{\prime }}(t)={\alpha }_{x_0+c}(t)$ for all $t$. Thus shifting only the spatial origin relabels this family. Changing $u$ changes the curve family, since every curve in it has the affine derivative $u$.

(iv) Allowed changes between frames. The model adopts the class of transformations between inertial frames

$t^{\mathrm{\prime }}=t+b,x^{\mathrm{\prime }}=Rx+vt+a,$

where $b\in \mathbb{R}$ is a time shift ($[b]=\tau$), $a\in {\mathbb{R}}^3$ a spatial translation ($[a_i]=\mathrm{\ell }$), $v\in {\mathbb{R}}^3$ a boost velocity ($[v_i]=\mathrm{\ell }{\tau }^{-1}$) and $R\in SO(3)$ a rotation ($[R_{ij}]=1$). Every pair of inertial frames is related by exactly one such transformation: that two adapted charts of the same data differ by a transformation of this form, with $R\in SO(3)$ because both bases are positively oriented, is the mathematical transformation law proved in Transformation between adapted charts and its kinematic consequences. Uniqueness follows by comparing the coordinate maps on all $(t,x)\in \mathbb{R}\times {\mathbb{R}}^3$: evaluating the time map fixes $b$; evaluating the spatial map at $(0,0)$ fixes $a$, at $(1,0)$ fixes $v$, and at $(0,e_j)$ for the three standard coordinate vectors fixes each column of $R$. The kinematic consequences of the law — the transformation of coordinate velocity and acceleration, and the frame-independence of zero coordinate acceleration — are proved as a statement about the model in Galilean transformation of worldlines: velocity and acceleration laws, frame-independence of zero coordinate acceleration, and the group of frame changes.

(v) Free motion is not asserted here. The framework's dynamics enters through the force and equation-of-motion postulates of the pages that follow; those pages adopt, as the content of Newton's first law, that there exist inertial frames in the sense of clause (ii) in which a particle with zero applied interaction has zero coordinate acceleration. This postulate introduces the frames and their changes but asserts no dynamics: it does not state that any particular motion has zero acceleration, and it does not privilege any one inertial frame — each of them is a legitimate arena for the formulation of the model's equations.

(vi) Kinematic symmetry is not conservation. The class (iv) is a class of coordinate changes. A quantity is conserved along a motion when its value as a function of the frame time is constant on the interval of that motion. No such constancy is asserted here, nor is it asserted that every mechanical system's description is invariant in form; a force law must also transform appropriately under (iv) to be a law of the model in every frame, and an external apparatus or background field can single out frames for the description of a concrete experiment. Conservation statements, when they hold, are theorems of later pages with their own hypotheses, not consequences of this postulate.

• Adopted, not deduced. Clauses (i)-(iv) adopt the frame notion and the transformation class; clauses (v)-(vi) delimit what is not asserted. The mathematical transformation law between adapted charts is cited, not reproved, and is used only through its exact statement and hypotheses.

• Domain of intended use. Inertial frames are the frames of the non-relativistic model of Adopted Newtonian spacetime structure: absolute time, Euclidean spatial slices and their smooth structure: spatial Euclidean geometry and absolute time. The transformations of clause (iv) are Galilean; no Lorentz transformation, no boost composition law of special relativity and no light-speed limit appears in this model.

• Empirical content. The classification of an apparatus frame as inertial, and the adequacy of the Galilean transformation class for a concrete experiment, are application-level judgements with stated approximations; this postulate asserts no measurement and no numerical value. Agreement of model predictions with observations is evidence about the model, not a proof of these assumptions.