# Minkowski and Galilei/Newton Fluid Dynamics: A Geometric 3 + 1 Spacetime Perspective
## 3 Fluid Dynamics
### Three-Momentum Tensor  
In contrast to the number vector of Equation (77) expressing the flux of a single quantity--particle number--the stress-inertia tensor of Equation (102) embodies the flux of four-momentum. Thus, while the divergence of the number vector in Equation (101) will ultimately reduce to a single partial differential equation, Equations (126) and (127) separate into four partial differential equations; and the divergence \(\mathbf{\
abla}\cdot\mathbf{T}\) being a coordinate-free geometric expression, there is freedom in deciding what information one would like to extract, i.e., which partial differential equations one wishes to generate. Begin the separation by defining a \((1,1)\) tensor \(\mathbf{M}\) expressing the flux of three-momentum. In particular, having chosen a fiducial frame \(\Theta\)--a global inertial frame, whose time coordinate \(t\) yields hyperplanes of simultaneity \(\mathcal{S}_{t}\), and whose associated observers have four-velocity \(\mathbf{w}\)--let it express the flux of the three momentum measured by \(\Theta\):  
\[\mathbf{M}=\int f\,\underline{\mathbf{q}}\otimes\mathbf{p}\;\mathrm{d}\mathrm{P}_{m} \tag{128}\]  
(note the first slot of this tensor is 'index down'). Because \(\underline{\mathbf{q}}\,=\,\mathbf{\gamma}\cdot\mathbf{p}\) (recall again Equations (34) and (50)), comparison with Equation (102) shows that  
\[\mathbf{M}=\mathbf{\gamma}\cdot\mathbf{T}. \tag{129}\]This applies to both \(\mathbb{M}\) and \(\mathbb{G}\); recall from the discussion in Section 2.4 that the underbar notation, applied here to \(q\), is allowed on \(\mathbb{G}\) because \(q\) is tangent to \(\mathcal{S}_{t}\). After having decomposed \(T\) in Equations (115) and (121) in terms of quantities defined with respect to the comoving frame \(\tilde{\mathcal{O}}\), the projection of momentum with respect to \(\Theta\) in Equation (129) entails some algebraic complications associated with the relationship between the two frames. (The more elegant geometry of \(\mathbb{M}\) relative to \(\mathbb{G}\) now begins to lead to messier practical consequences.) To help manage this, group the various terms in \(M\) as  
\[\mathbf{M}=\mathbf{M}_{\text{perfect}}+\mathbf{M}_{\text{dissipative}}, \tag{130}\]  
where \(\mathbf{M}_{\text{perfect}}\) includes terms involving \(n\), \(e\), and \(p\), properties intrinsic to a fluid element; and \(\mathbf{M}_{\text{dissipative}}\) involves the quantities \(\mathbf{J}\) and \(\mathbf{\Pi}\) associated with energy and momentum exchange between fluid elements. On \(\mathbb{M}\), inserting Equation (115) in Equation (129) yields  
\[M_{\text{perfect}}=\Lambda_{\mathbf{V}}\left(m\,n+\frac{e+p}{c^{2}}\right)\underline {\mathbf{V}}\otimes\mathbf{U}+p\overleftarrow{\mathbf{\gamma}}\,, \tag{131}\]  
in which Equations (82) and (110) have been used, and the left arrow in \(\overleftarrow{\mathbf{\gamma}}\) denotes that the _last_ index of a multilinear form has been raised; and  
\[\mathbf{M}_{\text{dissipative}}=\Lambda_{\mathbf{V}}\underline{\mathbf{V}}\otimes\frac{ \mathbf{J}}{c^{2}}+\frac{\mathbf{I}_{\mathbf{\gamma}}}{c^{2}}\otimes\mathbf{U}+\mathbf{\Pi}_\mathbf{{\gamma}}, \tag{132}\]  
in which the linear form \(\mathbf{J}_{\mathbf{\gamma}}\) and \((1,1)\) tensor \(\mathbf{\Pi}_{\mathbf{\gamma}}\)  
\[\mathbf{J}_{\mathbf{\gamma}} = \mathbf{\gamma}\cdot\mathbf{J}, \tag{133}\] \[\mathbf{\Pi}_{\mathbf{\gamma}} = \mathbf{\gamma}\cdot\mathbf{\Pi} \tag{134}\]  
have been defined. On \(\mathbb{G}\) things are simpler: inserting Equation (121) in Equation (129) yields  
\[\mathbf{M}_{\text{perfect}} = m\,n\,\underline{\mathbf{V}}\otimes\mathbf{U}+p\overleftarrow{\mathbf{ \gamma}}\,, \tag{135}\] \[\mathbf{M}_{\text{dissipative}} = \mathbf{\Pi}_{\mathbf{\gamma}}, \tag{136}\]  
reflecting the decompositions in Equations (83) and (118). (Again, the underbar notation is allowed because \(\mathbf{V}\) is tangent to \(\mathcal{S}_{t}\).)