# Minkowski and Galilei/Newton Fluid Dynamics: A Geometric 3 + 1 Spacetime Perspective
## 2 Spacetime
### Galilei/Newton Spacetime
According to this convention, a speed measured by \(\Theta\) corresponds to the slope of a worldline relative to vertical, as was also the case in Figures 1 and 2. Galilei/Newton spacetime is invariant under inhomogeneous Galilei transformations. These leave the foliation of \(\mathbb{G}\) into hyperplanes of absolute simultaneity \(\mathcal{S}_{t}\) intact, and therefore preserve the causal structure. Inhomogeneous Galilei transformations also leave the position space line element within each \(\mathcal{S}_{t}\) unchanged, so that the components of \(\gamma\) are still given by Equation (12). Like the Poincare group, the inhomogeneous Galilei group includes 10 continuous symmetries: four spacetime translations; three position space rotations; and three Galilei boosts that leave time unchanged, but mix time into the position space coordinates. Again, the discrete symmetries are spatial inversion and time reversal. Galilei boosts are transformations between inertial frames in uniform relative motion, with different families of inertial observers. An inertial frame \(\Theta^{\prime}\) (blue) in relative motion to \(\Theta\) is also shown in Figure 3, in particular the worldlines of a few of its observers. The relativity of simultaneity, length contraction, and time dilation discernible in Figures 1 and 2 are absent from Figure 3.  
Figure 3: Galilei/Newton spacetime \(\mathbb{G}\) as charted in time coordinate \(t\) and position space coordinate \(x\) (other position space dimensions suppressed) by a global inertial frame \(\Theta\) (observer worldlines, red; surfaces of simultaneity, black). A second inertial frame \(\Theta^{\prime}\) (observer worldlines, blue; surfaces of simultaneity, black) with coordinates \(t^{\prime},x^{\prime}\) moving with positive velocity in the \(x\) direction relative to \(\Theta\) is also shown. In accordance with the absolute nature of time in Galilei/Newton spacetime, the surfaces of simultaneity of \(\Theta\) and \(\Theta^{\prime}\) coincide. As applied to \(\mathbb{G}\), the principle of special relativity means invariance under inhomogeneous Galilei transformations. For instance, the laws of physics as expressed in \(\mathcal{O}^{\prime}\), whose charting of \(\mathbb{G}\) is shown in Figure 4, must have the same form as in \(\mathcal{O}\). (Under a Galilei boost, the hyperplanes of simultaneity 'freely slide' past one another, like a beveling of a deck of cards, such that a 'tilted' family of worldlines can be brought to point'straight up'.) As with Minkowski spacetime, this requirement is satisfied mathematically in [PERSON]/Newton spacetime by expressing the laws of physics in terms of geometric objects, e.g., tensors on \(\mathbb{G}\). For instance, in describing the acceleration of a particle as  
\[m\,\mathbf{u}\cdot\mathbf{\
abla}\mathbf{u}=m\frac{\mathrm{d}\mathbf{u}}{\mathrm{d}\tau}= \frac{\mathrm{d}\mathbf{p}}{\mathrm{d}\tau}=f, \tag{43}\]  
the force \(\mathbf{f}\) must be a four-vector on \(\mathbb{G}\) in order that the the equation be covariant under inhomogeneous Galilei transformations. Because the derivative of the constant particle inertia \(m=p^{0}\) vanishes, \(\mathbf{t}\left(\mathbf{f}\right)=\mathbf{t}\cdot\mathbf{f}=t_{\mathrm{a}}f^{\mathrm{a}}=0\), that is, \(f\) must be tangent to \(\mathcal{S}_{\mathrm{I}}\)--it is an instantaneous force--and it can depend only on the _differences_ of particles' positions and velocities on \(\mathcal{S}_{\mathrm{I}}\). [PERSON]'s gravitational force law is the paradigmatic example satisfying this requirement. With use of the Galilei/Newton absolute time \(t\) as the affine parameter \(\tau\) of particle worldlines, as discussed above, the components of the \(\Theta\) observer four-velocity \(\mathbf{w}\) of Equation (9) are  
\[\left[w^{\mu}\right]_{G}= \begin{bmatrix}1\\ 0\\ 0\\ 0\end{bmatrix} _{G} \tag{44}\]  
as measured by \(\Theta\). Unlike Minkowski spacetime, because there is no spacetime metric on \(\mathbb{G}\), there is no one-form \(\underline{\mathbf{w}}\) associated with \(\mathbf{w}\) by metric duality.