_Galilean spacetime_ refers to a model for space that precedes the introduction of [spacetime as we would view it in relativistic systems](Spacetime.md). Informally this means being able to model everything with points in 3-dimensional space where time is considered as a separate one-dimensional _parameter_ (as opposed to a dimension as it is in relativistic spacetime). We describe exactly what that means to a mathematical physicist or a mathematician [below.](Galilean%20spacetime.md#Formal%20definition%20of%20Galilean%20spacetime) # Formal definition of [[Galilean spacetime]] Galilean spacetime has the following qualities: 1. It models the universe as a four-dimensional [[Affine space]] we will call $\mathcal{A}^4.$ Points in this space are called _world points_ or simply _events_. * Given this, _parallel displacements of the universe_ (i.e. the universe at any moment) refer to [[Vector spaces]]s in $\mathbb{R}^4.$ 2. Time is modeled as a [[Linear map]], $t:\mathbb{R}^4\rightarrow\mathbb{R},$ from parallel displacements of the universe to the _number line._ * With this we define the notion of a _time interval_ between events $a\in\mathcal{A}^4$ to $b\in\mathcal{A}^4$ as the real number $t(b-a)$ where if $t(b-a)=0,$ two events are _simultaneous._ * The set of simultaneous events in $\mathcal{A}^4$ form an [affine subspace](Affine%20space.md#Affine%20subspace), $\mathcal{A^3}.$ * Below we present the notion of a _time interval_ relative to $\mathcal{A}^4$ and $\mathcal{A}^3.$  (image adapted from Arnold, V.I, Weinstein A., Vogtmann K., _Mathematical Methods of Classical Mechanics_) 3. The _distance between simultaneous events_ is given by the [Euclidean norm](2-norm). This implies that every space of simultaneous is a 3D [Euclidean space](Euclidean%20space.md). --- # Recommended Reading For an introduction of Galilean spacetime as given [above](Galilean%20spacetime.md) see * [Arnold, V.I, Weinstein A., Vogtmann K., _Mathematical Methods of Classical Mechanics_, Springer, 2nd edition](%5BGraduate%20Texts%20in%20Mathematics%5D%20V.%20I.%20Arnold,%20A.%20Weinstein,%20K.%20Vogtmann%20-%20Arnold%20V%20I%20Mathematical%20Methods%20Of%20Classical%20Mechanics%20(1989,%20Springer)%20-%20libgen.lc.pdf), pgs. 5 and 6. #Mechanics #MathematicalFoundations/Geometry