The Newtonian spacetime arena of the framework

The arena. The non-relativistic quantum framework of this category uses as its spacetime the affine time space of Affine time space with Euclidean simultaneous slices and adapted charts: a quadruple $(V,\tau ,g,A)$ consisting of a four-dimensional real vector space $V$, a surjective linear map $\tau :V\to \mathbb{R}$ (the absolute-time covector), the three-dimensional kernel $K=\ker \tau$ (the spatial displacement space), a symmetric positive definite bilinear form $g:K\times K\to \mathbb{R}$ (the spatial metric), and an affine space $A$ modelled on $V$ whose elements are called events. When the framework speaks of "a particle at position $x$ at time $t$", of "the time interval between two events" or of "the distance between two simultaneous events", these are read in this structure.

Spatial orientation. In addition to the quadruple, fix the orientation of $K$ required by Affine time space with Euclidean simultaneous slices and adapted charts: an equivalence class of ordered bases, where two bases are equivalent exactly when their change-of-basis map has positive determinant. Every adapted chart uses a $g$-orthonormal basis in this fixed class. This chooses an orientation, not a distinguished basis.

Absolute time and slices. The absolute time of a displacement $w\in V$ is $\tau (w)$; two events $p,q\in A$ are simultaneous when $\tau (q-p)=0$, and simultaneity is an equivalence relation whose classes are the simultaneous slices. For a fixed adapted chart and its coordinates $(t,x)$ of Existence and uniqueness of adapted coordinates, and the affine structure of simultaneous slices, the slices are the level sets of $t$ and the number $t(p)$ is the time coordinate of the event $p$. The pair of events of one slice carries the slice distance $d_g(p,q)=\sqrt{g(p-q,p-q)}$; no distance is defined between events of different slices by this structure.

Smooth structure. $A$ is read with the affine smooth structure determined by adapted coordinates: for a nonempty open interval $I\subseteq \mathbb{R}$, a curve $\gamma :I\to A$ is of class $C^k$ when its coordinate representation $s\mapsto (t(\gamma (s)),x(\gamma (s)))$ in one adapted chart is of class $C^k$; the transition law between adapted charts of Transformation between adapted charts and its kinematic consequences makes this reading independent of the chart. Regularity of curves, of potential functions and of apparatus descriptions of the framework is stated with respect to this smooth structure.

What the arena does not contain. The data contain no inner product on $V$, no Lorentzian metric, no light cone, no causal classification, no universal speed and no curvature; they contain no distinguished event, no distinguished time vector and no distinguished spatial basis. In particular the arena alone does not select a frame: the inertial frames of the model and the transformations between them are fixed by the separate definition of the frames on this page and the accompanying postulate.

Status. These clauses define the spacetime objects of the model and their elementary readings. They are not a physical postulate that nature is such an arena and not a mathematical theorem; the corresponding adoption is made by the kinematics postulate of this page. No dynamics, force law, potential or matter model is fixed here, and no empirical measurement is recorded.

• The coordinate space of the Galilean transformations of The Galilean transformations of the coordinate space is a model of the coordinate form of this arena: an adapted chart identifies $A$ bijectively with $\mathbb{R}\times {\mathbb{R}}^3$, and the transition law between two such charts is one of those transformations, as recorded in Transformation between adapted charts and its kinematic consequences.
• Nothing in this definition fixes units. The readings of the time coordinate in seconds, of spatial coordinates in metres and of slice distances in metres are fixed by the unit and quantity conventions of Dimensions, SI unit conventions and quantity types of the framework.