# Minkowski and Galilei/Newton Fluid Dynamics: A Geometric 3 + 1 Spacetime Perspective
## 2 Spacetime
### Minkowski Spacetime
For instance, in describing the acceleration of a particle as  
\[m\,\mathbf{u}\cdot\mathbf{\
abla}\mathbf{u}=m\frac{\mathrm{d}\mathbf{u}}{\mathrm{d}\tau}=\frac {\mathrm{d}\mathbf{p}}{\mathrm{d}\tau}=\mathbf{f}, \tag{24}\]  
the force \(\mathbf{f}\) must be a four-vector on \(\mathbb{M}\) in order that the the equation be covariant under Poincare transformations. Moreover, because \(\mathbf{g}\left(\mathbf{p},\mathbf{p}\right)=-m^{2}c^{2}\) is a constant, Equation (24) requires \(\mathbf{g}\left(\mathbf{p},\mathbf{f}\right)=0\). The Lorentz force law on a charged particle due to the electromagnetic field is the paradigmatic example satisfying these requirements. Figure 1: Minkowski spacetime \(\mathbb{M}\) as charted in time coordinate \(t\) and position space coordinate \(x\) (other position space dimensions suppressed) by a global inertial frame \(\Theta\) (red), in units with \(c=1\). A second inertial frame \(\Theta^{\prime}\) (blue) with coordinates \(t^{\prime},x^{\prime}\) moving with positive velocity in the \(x\) direction relative to \(\Theta\) is also shown, as are the past and future null cones (black) emanating from the common origin of \(\Theta\) and \(\Theta^{\prime}\). Curves of constant \(x,y,z\) and \(x^{\prime},y^{\prime},z^{\prime}\) can be regarded as the worldlines of families of inertial observers (solid lines with arrows; those passing through the origin are labeled and distinguished with heavier line weight), whose four-velocities are given by timelike tangent vectors \(\mathbf{w}\) and \(\mathbf{w^{\prime}}\) to the \(t\) and \(t^{\prime}\) coordinate curves. Hyperplanes of simultaneity \(\mathcal{S}\) and \(\mathcal{S}^{\prime}\) of \(\Theta\) and \(\Theta^{\prime}\) (dashed lines; those passing through the origin are labeled and distinguished with heavier line weight) are defined by constant values of \(t\) and \(t^{\prime}\), respectively, and are submanifolds with Euclid geometry charted by the position space coordinates. According to the relation between time coordinate \(t\) and worldline affine parameter \(\tau\) discussed above, the time coordinate \(t\) of a global inertial frame \(\Theta\) is equal to \(\tau\) for the \(\Theta\) observers. Thus, the components of the \(\Theta\) observer four-velocity \(\mathbf{w}\) of Equation (9) and its associated linear form \(\underline{\mathbf{w}}\) are  
\[\left[w^{\mu}\right]_{\Theta}= \begin{bmatrix}1\\ 0\\ 0\\ 0\end{bmatrix} _{\Theta},\quad\left[w_{\mu}\right]_{\Theta}= \begin{bmatrix}-c^ {2}&0&0&0\\ 0&0\end{bmatrix} _{\Theta} \tag{25}\]  
as measured by \(\Theta\). Comparing with Equation (2) for the time one-form reveals that  
\[\underline{\mathbf{w}}=-c^{2}\,\mathbf{t}, \tag{26}\]  
an association made possible by the Minkowski metric. The \(\Theta\) observer four-velocity \(\mathbf{w}\) is normal to \(\mathcal{S}_{t}\), in the sense that it has a vanishing scalar product with the natural basis vectors \(\partial_{x}\), \(\partial_{y}\), and \(\partial_{z}\) tangent to the coordinate curves charting \(\mathcal{S}_{t}\). However, that visually \(\mathbf{w}\) makes a right angle with \(\mathcal{S}_{t}\) in Figure 1, looking like a normal in Euclid geometry, is an accident of convention. The \(\Theta^{\prime}\) observer four-velocity \(\mathbf{w}^{\prime}\) is normal to \(\mathcal{S}^{\prime}_{t^{\prime}}\), but visually they do not make a right angle in Figure 1. For both \(\Theta\) and \(\Theta^{\prime}\), the normality according to the Minkowski metric is manifest graphically in the positioning of the null cone at an angle midway between the observer four-velocity and the hyperplanes of simultaneity.