# Minkowski and Galilei/Newton Fluid Dynamics: A Geometric 3 + 1 Spacetime Perspective
## 2 Spacetime
### Causal Structure  
While Minkowski and Galilei/Newton spacetimes are both affine spaces allowing global inertial observers, they are distinguished by additional structure--in particular, structure related to causality--that constrains the relationships between global inertial reference frames. In particular, the symmetry of an affine space under general linear transformations is restricted to symmetry under Lorentz transformations (rotations and Lorentz boosts) in Minkowski spacetime and Galilei transformations (rotations and Galilei boosts) in Galilei/Newton spacetime. (As previously noted, the symmetry of affine spaces under translations is preserved in both cases.) However, there must be enough remaining symmetry to respect the principle of special relativity, anticipated already by [PERSON] in his famous description of the goings-on below decks in a uniformly moving ship (e.g., [22]): physical law takes the same mathematical form in all global inertial reference frames. At the very least, this requires that position space be relative: one cannot say whether two events at different times occur in the same place. That is, two families of global inertial observers in relative uniform motion, obeying the same laws of physics expressed in the same mathematical form, assign different position space coordinates to the two events. Upon reflection with [PERSON], this seems persuasive enough, at least to us, from common experience, but it is also subtle enough to apparently have eluded [PERSON], and to have made controversial (even setting aside religious controversy) the revolutionary perspective of [PERSON]. As discussed further in Section 2.4, the causal structure of [PERSON]/Newton spacetime \(\mathbb{G}\) accepts only this much relativity, combining relative position space with an absolute time on which all observers agree. This accords with observed mechanical phenomena at low speed (much smaller than the speed of light \(c\)) and low energy per particle (much smaller than \(mc^{2}\)). Mathematically, absolute time is implemented by a projection that maps each point of \(\mathbb{G}\) onto a one-dimensional manifold \(\mathbb{T}\)--'time'. The time manifold \(\mathbb{T}\) is homogeneous, so that physical law is invariant under time translations. Physical time intervals are simply differences of a time coordinate \(t\), that is, a one-dimensional Euclid 'distance' on \(\mathbb{T}\). The set of events in \(\mathbb{G}\) mapped to a particular value of \(t\) is a hyperplane of simultaneity \(\mathcal{S}_{t}\), a three-dimensional Riemann manifold \(\mathbb{E}^{3}\) governed by Euclid geometry, constituting 'position space' at time \(t\). Importantly, spacetime \(\mathbb{G}\) is _not_ simply the product space \(\mathbb{T}\times\mathbb{E}^{3}\). Instead, Galilei relativity is implemented by allowing position space at different times to be completely independent: \(\mathcal{S}_{t_{1}}\) at time \(t_{1}\) is _not the same space_ as \(\mathcal{S}_{t_{2}}\) at time \(t_{2}\). While they are both identical instantiations of the three-dimensional flat Euclid manifold \(\mathbb{E}^{3}\), there is no a priori relationship between them prior to selection of a particular global inertial reference frame as 'fiducial.' The mathematical structure that combines absolute time with relative space is comparatively exotic: \(\mathbb{G}\) is a fiber bundle (e.g., [16; 17; 21; 22]), with one-dimensional base space \(\mathbb{T}\) and three-dimensional typical fiber \(\mathbb{E}^{3}\). As for causality: the time ordering of events is absolute; but there is no upper speed limit, and indeed forces act instantaneously at any distance within a single time slice \(\mathcal{S}_{t}\). However, absolute time--or more specifically, absolute simultaneity--is given up in Minkowski spacetime \(\mathbb{M}\), as discussed in Section 2.3: one cannot say whether two events in different places occur at the same time.