Adopted Newtonian spacetime structure: absolute time, Euclidean spatial slices and their smooth structure

Adopt as physical assumptions of the non-relativistic classical mechanics framework the following statements about spacetime.

(i) Spacetime data. Spacetime is the mathematical structure $(V,\tau ,g,A,\text{fixed orientation of }K)$ of Affine time space with Euclidean simultaneous slices and adapted charts: a four-dimensional real vector space $V$; a surjective linear map $\tau :V\to \mathbb{R}$, the absolute-time covector; the kernel $K=\ker \tau$, a three-dimensional real vector space, the spatial displacement space; a spatial metric $g$, a symmetric positive-definite bilinear form $g:K\times K\to \mathbb{R}$; a fixed orientation of $K$, that is, a chosen one of the two classes of ordered bases of $K$ under the equivalence generated by positive-determinant changes of basis; and an affine space $A$ modelled on $V$ (a nonempty set with a free and transitive $V$-action, with displacements as in Affine time space with Euclidean simultaneous slices and adapted charts). The elements of $A$ are the events of the model. The structure is the arena of the model; no equation of motion, force law or matter content is asserted by this clause.

(ii) Interpretation of the data. For a displacement $w\in V$, the number $\tau (w)$ is the elapsed time carried by $w$, read in seconds; for a displacement $k\in K$ inside a simultaneous slice, the number $g(k,k)$ is the squared spatial length of $k$, read in square metres; the square root $\sqrt{g(k,k)}$ is the length of $k$. The dimensions and units are those fixed by Dimensions and SI unit conventions of the non-relativistic classical mechanics framework, so that $[\tau (w)]=\tau$ and $[g(k,k)]={\mathrm{\ell }}^2$. The orientation of $K$ is the convention relative to which spatial bases are called positively oriented and against which orientation-preserving rotations are defined.

(iii) Absolute time, simultaneity and distance. Two events $p,q\in A$ are simultaneous when their displacement lies in $K$, that is, when $\tau (q-p)=0$. Simultaneity is an equivalence relation on $A$: it is reflexive because $\tau (0)=0$, symmetric because $\tau (p-q)=-\tau (q-p)$ and transitive because $\tau$ is additive. Its classes are the simultaneous slices $A_t=\{p\in A:\tau (p-o)=t\}$ of any adapted chart; the class does not depend on the choice of adapted chart, and the elapsed time between two events is the chart-independent number $\tau (q-p)$. Two events of a common slice carry the slice distance $d_g(p,q):=\sqrt{g(p-q,p-q)}$, which is defined because $p-q\in K$; distances are not defined across distinct slices by this structure.

(iv) Smooth structure. $A$ is read with the affine smooth structure determined by its affine coordinates: an adapted chart of Affine time space with Euclidean simultaneous slices and adapted charts is a bijection $A\to \mathbb{R}\times {\mathbb{R}}^3$ whose inverse is the map $(t,x)\mapsto o+tu+{\sum }_i{x}_i{e}_i$, and curves are declared $C^k$ when their coordinate functions in one (equivalently, every) adapted chart are $C^k$. The equivalence of the two readings is supplied by the affine transition law between adapted charts proved in Transformation between adapted charts and its kinematic consequences; this postulate adopts the resulting smooth structure as the one in which regularity of worldlines and of force maps is stated.

(v) What the structure excludes. The data contain no inner product on $V$, no Lorentzian metric, no light cone, no universal speed bound, no null cones and no causal classification of displacements other than the split into $\tau$-values; they contain no distinguished origin, no distinguished vector $u$ with $\tau (u)=1$ and no distinguished basis of $K$. Consequently the structure by itself distinguishes no inertial frame: the class of frames and the transformations between them are fixed separately, by the frame postulate Inertial frames, stationary observers and the adopted Galilean transformations between frames. No statement about the speed of any signal, about electromagnetic propagation or about gravitation is made here.

(vi) Status. These clauses are adopted assumptions of the model, not theorems and not empirical findings. They are not derived from the uniqueness theory of differential equations, from any symmetry principle or from experiment; the Newtonian tradition and Galilean relativity motivate them. The model makes no claim that the physical world is exactly a four-dimensional affine space with these data, and no numerical measurement is recorded.

• Adopted, not deduced. Clauses (i)-(iv) fix the arena and the reading of its data; clause (v) records what is deliberately absent. Nothing here selects a force law, an equation of motion, a matter model or a solution of any equation; those enter through the postulates and definitions of the pages that follow.

• Domain of intended use. The model is intended for the non-relativistic regime, in which speeds are small compared with the speed of light and a classical particle or continuum description, as appropriate to the application, and a classical description of the apparatus are adequate. The framework sets no numerical cutoff for that regime; an application must state the regime in which the model, its force law and its apparatus description are adequate. Relativistic, quantum and quantum-gravitational regimes are outside the scope of this model.

• Empirical content is downstream. Empirical significance lies in the comparison of predictions computed from a fully specified model with reported observations; those comparisons carry apparatus, calibration, sampling and uncertainty qualifications and are recorded where the framework treats them. This postulate itself makes no quantitative empirical claim, and agreement of a model with data would be evidence about the model, not a proof of this postulate.

• Absolute time, not absolute space. Absolute simultaneity and Euclidean slice geometry are part of the adopted model. The model does not contain a distinguished rest frame: two inertial frames are related by the transformations of Inertial frames, stationary observers and the adopted Galilean transformations between frames, and the structure alone cannot select one of them. This is a property of the adopted data, not a conclusion of relativistic kinematics, and no Lorentzian or light-cone structure is imported.