Inertial frames of the framework and the class of frame changes

Inertial frames. Fix the arena $(V,\tau ,g,A)$ of The Newtonian spacetime arena of the framework. An inertial frame of the framework is an adapted chart

$F=(o,u,(e_1,e_2,e_3))$

of that arena in the sense of Affine time space with Euclidean simultaneous slices and adapted charts: an event $o\in A$ (the spatial and time origin of the frame), a vector $u\in V$ with $\tau (u)=1$ (the frame's time vector), and an ordered $g$-orthonormal positively oriented basis $(e_1,e_2,e_3)$ of $K$ (the frame's spatial triad). The associated frame coordinates are the functions $(t,x):A\to \mathbb{R}\times {\mathbb{R}}^3$ of Existence and uniqueness of adapted coordinates, and the affine structure of simultaneous slices, defined by $p=o+t(p)u+{\sum }_i{x}_i(p)e_i$.

What kind of object a frame is. A frame is a coordinate chart of $A$: a bijection of $A$ with $\mathbb{R}\times {\mathbb{R}}^3$ carrying the affine structure, and in particular a global chart, not a local one. It is deliberately not an observer congruence (no family of curves indexed by spatial position, no four-velocity field) and not a time-varying orthonormal basis field: the triad $(e_1,e_2,e_3)$ is a fixed basis of $K$ and the time vector $u$ is fixed once the frame is chosen. A time-dependent family of bases or a congruence of observers is a different geometric object, and statements about them are not statements about frames unless a separate construction relates them.

Allowed changes of frame. Let $F$ and $F^{\mathrm{\prime }}$ be two inertial frames with coordinates $(t,x)$ and $(t^{\mathrm{\prime }},x^{\mathrm{\prime }})$. The change of frame from $F$ to $F^{\mathrm{\prime }}$ is the bijection of $\mathbb{R}\times {\mathbb{R}}^3$ that sends the $F$-coordinates of an event to its $F^{\mathrm{\prime }}$-coordinates. By clause 1 of Transformation between adapted charts and its kinematic consequences there are $b\in \mathbb{R}$, $a,v\in {\mathbb{R}}^3$ and $R\in SO(3)$ (The orthogonal group $O(3)$ and the rotation group $SO(3)$) with

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

that is, the change of frame is the Galilean transformation $g_{(b,a,v,R)}$ of The Galilean transformations of the coordinate space applied to the coordinates. The class of transformations between inertial frames is therefore the set $G$ of the Galilean transformations, which by The Galilean transformations form a group of bijections of the coordinate space is a group of bijections of the coordinate space: the composite of two frame changes is again a frame change, and the inverse of a frame change is a frame change. The parameters are the time shift $b$, the spatial translation $a$, the boost velocity $v$ and the rotation $R$ of the two frames.

Conventions. The spatial triad of every frame is positively oriented for the fixed orientation of $K$ used in the definition of adapted charts, so all frame changes have $R\in SO(3)$ rather than $R\in O(3)$ with negative determinant. The reading of the coordinates in seconds and metres, and the physical meaning of the parameters as measured quantities, are fixed by the unit conventions recorded on this page; no measured value of any parameter is asserted here.

Status. This definition fixes the frames and the class of frame changes of the model. It asserts no property of forces or of the motion of any body, selects no distinguished frame, and makes no empirical claim. The adoption of these frames as the frames of the physical model, together with the reading of apparatus motion as classical, is part of the kinematics postulate of this page.

• The distinction between chart and congruence matters for later claims: a change of frame changes the coordinate description of a state or of a trajectory but does not move the apparatus, and an accelerated congruence of observers is not a frame of the model.
• Two inertial frames related by a pure time shift, a pure spatial translation, a pure boost or a pure rotation are the elementary cases listed in The Galilean transformations of the coordinate space, and a general frame change is the composite of these.
• Every Galilean map is realized by a pair of frames. Fix $F=(o,u,(e_i))$ and any $(b,a,v,R)\in \mathbb{R}\times {\mathbb{R}}^3\times {\mathbb{R}}^3\times SO(3)$. Define $e_j^{\mathrm{\prime }}:={\sum }_i{R}_{ji}{e}_i$, $u^{\mathrm{\prime }}:=u-{\sum }_i(R^{\mathsf{T}}v{)}_i{e}_i$, and $o^{\mathrm{\prime }}:=o-bu+{\sum }_i(R^{\mathsf{T}}(vb-a){)}_i{e}_i$. Then $\tau (u^{\mathrm{\prime }})=1$, and $(e_1^{\mathrm{\prime }},e_2^{\mathrm{\prime }},e_3^{\mathrm{\prime }})$ is a positively oriented $g$-orthonormal basis of $K$ because its change-of-basis matrix is orthogonal with determinant $\det R=1$. Thus $F^{\mathrm{\prime }}:=(o^{\mathrm{\prime }},u^{\mathrm{\prime }},(e_j^{\mathrm{\prime }}))$ is an inertial frame. Substitution into $p=o+tu+{\sum }_i{x}_i{e}_i=o^{\mathrm{\prime }}+t^{\mathrm{\prime }}{u}^{\mathrm{\prime }}+{\sum }_j{x}_j^{\mathrm{\prime }}{e}_j^{\mathrm{\prime }}$ gives $t^{\mathrm{\prime }}=t+b$ and $x^{\mathrm{\prime }}=Rx+vt+a$. Hence every parameter quadruple in the definition of $G$ occurs as the coordinate change of two inertial frames; combined with the adapted-chart lemma's forward direction, the class of changes is exactly $G$.