# Minkowski and Galilei/Newton Fluid Dynamics: A Geometric 3 + 1 Spacetime Perspective
## 2 Spacetime
### Galilei/Newton Spacetime  
As described in Section 2.2, Galilei/Newton spacetime \(\mathbb{G}\) is a fiber bundle. This bundle has base space \(\mathbb{T}\), 'time', a one-dimensional flat manifold with Euclid distance expressing physical time intervals in terms of differences of a time coordinate \(t\). The bundle has typical fiber \(\mathbb{E}^{3}\), 'position space', a three-dimensional flat manifold with metric \(\mathbf{\gamma}\); in rectangular coordinates \((x,y,z)\) on \(\mathbb{E}^{3}\), \(\mathbf{\gamma}\) has the components displayed in Equation (12), giving Euclid geometry. Each fiber is a hyperplane of simultaneity \(\mathcal{S}_{t}\) corresponding to time \(t\). The position space coordinates can be set independently on each \(\mathcal{S}_{t}\), but, because of the affine structure of \(\mathbb{G}\), they can be chosen such that the coordinate curves of \(x,y,\) and \(z\)--considered as scalar functions of the events in \(\mathbb{G}\)--are infinitely extended and mutually orthogonal parallel lines, as discussed in Section 2.1. In this case, \((x^{\mu})=(t,x,y,z)\in\mathbb{R}^{4}\) are Galilei coordinates constituting a global inertial frame on \(\mathbb{G}\). There is no spacetime metric on \(\mathbb{G}\). There is no scalar product between four-vectors; physical time and distance are measured separately by the Euclid geometries of \(\mathbb{T}\) and \(\mathbb{E}^{3}\). There is no metric duality between vectors and linear forms, and more generally no raising and lowering of indices of tensors on \(\mathbb{G}\). However, raising and lowering of indices of tensors on \(\mathcal{S}_{t}\)_are_ allowed with respect to the three-metric \(\mathbf{\gamma}\). Considering also that on \(\mathbb{M}\) the actions of \(\mathbf{g}\) and \(\mathbf{\gamma}\) on tensors tangent to \(\mathcal{S}_{t}\) coincide, allow, for notational convenience--albeit with some trepidation--a limited use of the underbar and overarrow notation with tensors on \(\mathbb{G}\), but _strictly confined to tensors tangent to \(\mathcal{S}_{t}\)_, in which the associated raising and lowering is with respect to \(\mathbf{\gamma}\). As discussed further below, primary examples are what might loosely be regarded as variants of \(\mathbf{\gamma}\) itself: \(\mathbf{\gamma}\), \(\mathbf{\gamma}\), and \(\mathbf{\overline{\mathbf{\gamma}}}\). These are related by metric duality only on \(\mathcal{S}_{t}\); their four-dimensional extensions must be understood as independent tensors on \(\mathbb{G}\), as if they were represented by different symbols altogether. Let the universal time coordinate \(t\) serve as the affine parameter \(\tau\) of the particle four-velocity discussed in Section 2.1. That is,  
\[\mathrm{d}t=\mathrm{d}\tau, \tag{39}\]  
and the particle four-velocity of Equation (6) is simply  
\[\left[u^{\mu}\right]_{\Theta}= \begin{bmatrix}1\\ v^{i}\end{bmatrix} _{\Theta} \tag{40}\]  
as reckoned by \(\Theta\). The four-momentum \(\mathbf{p}=mu\) has components  
\[\left[p^{\mu}\right]_{\Theta}= \begin{bmatrix}m\\ q^{i}\end{bmatrix} _{\Theta}, \tag{41}\]with three-momentum \(q^{i}=mv^{i}\) measured by \(\Theta\). That is, once again \(\mathbf{q}\) is a three-vector on \(\mathcal{S}_{t}\), whose push-forward onto \(\mathbb{G}\) (denoted by the same symbol \(\mathbf{q}\)) has components  
\[\left[q^{\mu}\right]_{\Theta}= \begin{bmatrix}0\\ q^{i}\end{bmatrix} _{\Theta} \tag{42}\]  
according to \(\Theta\). In contrast to the relative time and relative particle energy--or rather, inertia \(\epsilon_{\mathbf{q}}/c^{2}\)--of Minkowski spacetime, the absolute time of Galilei/Newton spacetime induces an absolute particle inertia \(p^{0}=m\). A spacetime diagram of \(\mathbb{G}\) as charted by the inertial frame \(\Theta\) (red) is shown in Figure 3. As with Figures 1 and 2, by convention, the worldlines of the fiducial observers--relative to whose reference frame, in this case \(\Theta\), measurements are made--are vertical. The hyperplanes of simultaneity, universal for all observers, are horizontal.