# Minkowski and Galilei/Newton Fluid Dynamics: A Geometric 3 + 1 Spacetime Perspective
## 2 Spacetime
### Minkowski Spacetime
Consider the projection tensor  
\[\gamma=g+c^{2}\,\mathbf{t}\otimes\mathbf{t}=g+\frac{1}{c^{2}}\,\underline{\mathbf{w}} \otimes\underline{\mathbf{w}}, \tag{27}\]  
Figure 2: Minkowski spacetime \(\mathbb{M}\) as charted in time coordinate \(t^{\prime}\) and position space coordinate \(x^{\prime}\) (other position space dimensions suppressed) by an inertial reference frame \(\Theta^{\prime}\) (blue), in units with \(c=1\), showing also \(\Theta\) (red) moving with negative velocity in the \(x^{\prime}\) direction relative to \(\Theta^{\prime}\). This figure is a Lorentz boost or pseudo-rotation of Figure 1, under which the null cones remain unchanged. with components  
\[\left[\gamma_{\mu\
u}\right]_{\Theta}= \begin{bmatrix}0&0_{j}\\ 0_{i}&\gamma_{ij}\end{bmatrix} _{\Theta}= \begin{bmatrix}0&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1\end{bmatrix} _{\Theta} \tag{28}\]  
in inertial frame \(\Theta\). Use of the same symbol \(\mathbf{\gamma}\) as for the three-metric is justified by the fact that for vectors \(\mathbf{a}\) and \(\mathbf{b}\) on \(\mathbb{M}\) but tangent to \(\mathcal{S}_{t}\), the result \(\mathbf{\gamma}\left(\mathbf{a},\mathbf{b}\right)\) is the same as if \(\mathbf{\gamma}\), \(\mathbf{a}\), and \(\mathbf{b}\) were all tensors on \(\mathcal{S}_{t}\). With first index raised, this projection tensor reads  
\[\overrightarrow{\mathbf{\gamma}}=\delta+c^{2}\overrightarrow{\mathbf{t}}\otimes\mathbf{ t}=\delta+\frac{1}{c^{2}}\,\mathbf{w}\otimes\underline{\mathbf{w}}=\delta-\mathbf{w} \otimes\mathbf{t}, \tag{29}\]  
also with components  
\[\left[\gamma^{\mu}{}_{\
u}\right]_{\Theta}= \begin{bmatrix}0&0_{j}\\ 0^{i}&\delta^{i}{}_{j}\end{bmatrix} _{\Theta}= \begin{bmatrix}0&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1\end{bmatrix} _{\Theta} \tag{30}\]  
with respect to \(\Theta\). This raised version \(\overrightarrow{\mathbf{\gamma}}\) projects a vector on \(\mathbb{M}\) into a vector tangent to \(\mathcal{S}_{t}\). The other mixed version,  
\[\overleftarrow{\mathbf{\gamma}}=\delta+c^{2}\,\mathbf{t}\otimes\overrightarrow{\mathbf{ t}}=\delta+\frac{1}{c^{2}}\,\underline{\mathbf{w}}\otimes\mathbf{w}=\delta-\mathbf{t} \otimes\mathbf{w} \tag{31}\]  
with its second index raised, i.e., \(\gamma_{\mu}{}^{\
u}\), also has the same components according to \(\Theta\). Finally, the projection tensor with both indices raised is  
\[\overrightarrow{\mathbf{\gamma}}=\overrightarrow{\mathbf{\delta}}+c^{2}\overrightarrow {\mathbf{t}}\otimes\overrightarrow{\mathbf{t}}=\overrightarrow{\mathbf{\delta}}+\frac{ 1}{c^{2}}\,\mathbf{w}\otimes\mathbf{w}, \tag{32}\]  
yet again with components  
\[\left[\gamma^{\mu\
u}\right]_{\Theta}= \begin{bmatrix}0&0^{i}\\ 0^{i}&\gamma^{ij}\end{bmatrix} _{\Theta}= \begin{bmatrix}0&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1\end{bmatrix} _{\Theta} \tag{33}\]  
with respect to \(\Theta\). The particle four-velocity \(\mathbf{u}\) and four-momentum \(\mathbf{p}\) can be decomposed in terms of the \(\Theta\) observer four-velocity \(\mathbf{w}\). Comparison of Equations (8), (20) and (25) reveals  
\[\mathbf{u}=\Lambda_{\mathbf{p}}\left(\mathbf{w}+\mathbf{v}\right). \tag{34}\]  
As for the four-momentum,  
\[\mathbf{p}=\frac{\epsilon_{\mathbf{q}}}{c^{2}}\,\mathbf{w}+\mathbf{q}, \tag{35}\]  
thanks to Equations (21), (22) and (25). In each case, contraction with \(\mathbf{t}\) or \(\underline{\mathbf{w}}\) projects out the timelike component proportional to \(\mathbf{w}\) (i.e., \(\Lambda_{\mathbf{p}}\) and \(\epsilon_{\mathbf{q}}\), up to a sign and factors of \(c^{2}\)), and contraction with \(\overrightarrow{\mathbf{\gamma}}\) projects out the portion tangent to \(\mathcal{S}_{t}\) (i.e., \(\Lambda_{\mathbf{p}}\mathbf{v}\) and \(\mathbf{q}\)). A couple of final comments on the particle energy are in order. First, define a kinetic energy  
\[e_{\mathbf{q}}=\epsilon_{\mathbf{q}}-mc^{2}=\left(\Lambda_{\mathbf{p}}-1\right)mc^{2} \tag{36}\]by subtracting from the total particle energy the portion associated with the rest mass. Second, determine the evolution of the kinetic energy from Equation (24).