# Minkowski and Galilei/Newton Fluid Dynamics: A Geometric 3 + 1 Spacetime Perspective
## 4 Conclusions
\begin{table}
\begin{tabular}{l l l} \hline \hline
**Symbol** & **Description** & **References** \\ \hline \(t\,\mathrm{d}V_{t}\) & Position space volume element associated with \(\mathcal{S}_{t}\) & Section 3.1 \\ \(\mathrm{d}P_{m}\) & Invariant three-momentum volume element & Equations (54) and (69) on \(\mathbb{M}\), \\  & & Equations (55) and (70) on \(\mathbb{G}\) \\ \(f\) & Particle distribution function & Equation (56) \\ \(\mathbf{N}\) & Particle number four-vector & Equation (77) \\  & (First momentum moment of \(f\)) & Equation (102) \\ \(\mathbf{T}\) & Stress–inertia tensor & Equation (128) \\  & ( \((2,0)\) Second momentum moment of \(f\)) & Equation (129) \\ \(\mathbf{M}\) & Three-momentum tensor & Equation (140) \\  & ( Related to \(\mathbf{T}\) and \(\mathbf{N}\) on \(\mathbb{M}\)) & Equation (141) on \(\mathbb{M}\) \\  & (Not simply related to \(\mathbf{T}\) on \(\mathbb{G}\)) & Equation (148) on \(\mathbb{G}\) \\ \(\mathbf{U}\) & Fluid four-velocity vector & Equations (82) and (83), \\  & (Obtained together with \(\mathbf{V}\) and \(N\)) & Equations (85) and (86) \\ \(\mathbf{V}\) & Fluid three-velocity vector measured by \(\Theta\) & Equations (82) and (83), \\  & (Obtained together with \(\mathbf{U}\) and \(N\)) & Equations (85) and (86) \\  & (Tangent to \(\mathcal{S}_{t}\)) & Equation (82) and (83), \\ \(N\) & Particle number density measured by \(\Theta\) & Equations (82) and (83), \\  & (Obtained together with \(\mathbf{U}\) and \(\mathbf{V}\)) & Equations (85) and (86) \\ \(n\) & Particle number density measured by \(\tilde{\Theta}\) & Equation (93) \\ \(e\) & Internal energy density measured by \(\tilde{\Theta}\) & Equations (105) \\ \(p\) & Pressure measured by \(\Theta\) & Equations (107) and (112) on \(\mathbb{M}\), \\  & & Equations (117) and (118) on \(\mathbb{G}\) \\ \(\mathbf{I}\) & Internal energy flux vector measured by \(\tilde{\Theta}\) & Equation (106) \\ \(\mathbf{\Pi}\) & Viscous stress tensor measured by \(\tilde{\Theta}\) & Equations (107) and (112) on \(\mathbb{M}\), \\  & (\((2,0)\)) & Equations (117) and (118) on \(\mathbb{G}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Geometric entities characterizing a simple fluid on Minkowski and Galilei/Newton spacetimes. A geometric spacetime perspective capable of embracing both Minkowski and Galilei/Newton fluid dynamics on a somewhat common footing must of necessity be at least partly a \(3+1\) formulation, as opposed to being fully four-dimensional. This is not because of the particle number four-vector \(\mathbf{N}\)--the first \(\mathbf{p}\) moment of the particle distribution function \(f\)--which is of essentially the same character on \(\mathbb{M}\) and \(\mathbb{G}\), as discussed in Section 3.2. The important difference appears in connection with the second \(\mathbf{p}\) moment of \(f\), the tensor \(\mathbf{T}\) addressed in Section 3.3. This is normally known as the stress-energy (or energy-momentum) tensor, but the present comparison with the Galilei/Newton case brings into sharp focus the realization that it is more properly understood as a stress-_inertia_ (or _inertia_-momentum) tensor. This has some impact on the nature of the \((1,1)\) three-momentum tensor \(\mathbf{M}\), obtained in Section 3.4 on both \(\mathbb{M}\) and \(\mathbb{G}\) as the projection of the first slot of \(\mathbf{T}\) onto the hyperplanes of simultaneity \(\mathcal{S}_{t}\).