# Minkowski and Galilei/Newton Fluid Dynamics: A Geometric 3 + 1 Spacetime Perspective
## 3 Fluid Dynamics
### Particle Number and Fluid Velocity Vectors  
The first momentum moment of the particle distribution function provides a macroscopic particle number flux, the four-vector  
\[\mathbf{N}=\int f\,\mathbf{p}\ \mathrm{dP}_{m}. \tag{77}\]  
This can be used to define a fluid four-velocity \(\mathbf{U}\), which can be regarded as the four-velocity of 'comoving observers' that ride along with the fluid, associated with a reference frame \(\tilde{\mathcal{O}}\), by writing also  
\[\mathbf{N}=n\,\mathbf{U} \tag{78}\]  
and analyzing Equation (77) to define the particle number density \(n\) as well as \(\mathbf{U}\). For this purpose begin with the decomposition of \(\mathbf{p}\) as given in Equations (35) and (51) on \(\mathbb{M}\) and \(\mathbb{G}\) respectively, regarding \(\mathbf{w}\) as the four-velocity of observers--call them 'fiducial observers'--associated with a global inertial frame \(\Theta\). Define the three-velocity \(\mathbf{v}_{\mathbf{q}}\) associated with the three-momentum \(\mathbf{q}\) measured by \(\Theta\) as  
\[\mathbf{v}_{\mathbf{q}} = \frac{c^{2}\,\mathbf{q}}{\epsilon_{\mathbf{q}}}\ \ \ (\mathrm{on}\ \mathbb{M}), \tag{79}\] \[\mathbf{v}_{\mathbf{q}} = \frac{\mathbf{q}}{m}\ \ \ (\mathrm{on}\ \mathbb{G}). \tag{80}\]  
Then, referring to Equations (54) and (55), the number flux vector reads as  
\[\mathbf{N}=\int f\,\left(\mathbf{w}+\mathbf{v}_{\mathbf{q}}\right)\ \frac{\mathrm{d}\mathbf{q}}{ \left(2\pi\hbar\right)^{3}} \tag{81}\]  
on both \(\mathbb{M}\) and \(\mathbb{G}\). Next, recognize that just as the four-velocity \(\mathbf{u}\) of an individual particle can be decomposed in terms of a three-velocity \(\mathbf{v}\) measured by the fiducial observers of \(\Theta\) as in Equations (34) and (50) on \(\mathbb{M}\) and \(\mathbb{G}\), respectively, so also the fluid four-velocity \(\mathbf{U}\) can be decomposed as  
\[\mathbf{U} = \Lambda_{\mathbf{V}}\left(\mathbf{w}+\mathbf{V}\right)\ \ \ (\mathrm{on}\ \mathbb{M}), \tag{82}\] \[\mathbf{U} = \mathbf{w}+\mathbf{V}\ \ \ (\mathrm{on}\ \mathbb{G}), \tag{83}\]  
in which \(\mathbf{V}\) is the fluid three-velocity, tangent to \(\mathcal{S}_{t}\), measured in \(\Theta\). Using this to substitute for \(\mathbf{w}\) in Equation (81) gives  
\[\mathbf{N}=\frac{\mathbf{U}}{\Lambda_{\mathbf{V}}}\int\frac{f\,\mathrm{d}\mathbf{q}}{\left(2 \pi\hbar\right)^{3}}-\mathbf{V}\int\frac{f\,\mathrm{d}\mathbf{q}}{\left(2\pi\hbar \right)^{3}}+\int\frac{f\,\mathbf{v}_{\mathbf{q}}\,\mathrm{d}\mathbf{q}}{\left(2\pi\hbar \right)^{3}} \tag{84}\]on \(\mathbb{M}\), and similarly with \(\Lambda_{\mathbf{V}}\to 1\) on \(\mathbb{G}\). Define, therefore, the particle number density \(N\) and average three-velocity \(\mathbf{V}\) measured by the fiducial observers of \(\Theta\) as  
\[N = \int\frac{f\,\mathrm{d}\mathbf{q}}{\left(2\pi\hbar\right)^{3}}, \tag{85}\] \[\mathbf{V} = \frac{1}{N}\int\frac{f\,\mathbf{v}_{q}\,\mathrm{d}\mathbf{q}}{\left(2\pi \hbar\right)^{3}} \tag{86}\]  
on both \(\mathbb{M}\) and \(\mathbb{G}\). Then, the last two terms of Equation (84) cancel, and comparison of the first term with Equation (78) yields  
\[\mathbf{N}=n\,\mathbf{U} \tag{78}\]0 \[\mathbf{N}=n\,\mathbf{U} \tag{78}\]1  
Note that the particle number vector decomposes as  
\[\mathbf{N}=n\,\mathbf{U} \tag{78}\]2  
on both \(\mathbb{M}\) and \(\mathbb{G}\). While \(N\) has been recognized in Equation (85) as the particle number density measured by the fiducial observers associated with \(\Theta\), the meaning of \(n\) as the number density measured by the comoving observers associated with \(\tilde{\Theta}\) should be more directly shown. Now that the fluid four-velocity \(\mathbf{U}\) has been defined in Equations (82), (83) and (86), consider again Equation (77) defining \(\mathbf{N}\) in light of the alternative decomposition of \(\mathbf{p}\) given by Equations (67) and (68). In this decomposition, \(\mathbf{s}\) is recognized as the particle three-momentum measured by the comoving observers, associated with \(\tilde{\Theta}\), having four-velocity \(\mathbf{U}\).