# Minkowski and Galilei/Newton Fluid Dynamics: A Geometric 3 + 1 Spacetime Perspective
## 3 Fluid Dynamics
### Kinetic Theory  
Kinetic theory is a link between microscopic particle dynamics and a macroscopic fluid description of a physical system. It considers volume elements that are macroscopically small--in practical terms, infinitesimal--yet contain a very large number of particles, enabling a statistical description. Consider a position space volume element \(t\,\mathrm{dV}_{t}\) in spacetime, where \(\mathbf{t}\) is the one-form defined in Section 2.1 and \(\mathrm{dV}_{t}\) is a three-volume in the hypersurface \(\mathcal{S}_{t}\). For a single particle type of mass \(m\), consider also a thin tube of particle worldlines, with four-momenta such that their three-momenta fall within an invariant three-momentum range \(\mathrm{dP}_{m}\) of the three-momentum determining the four-momentum \(\mathbf{p}\). The Lorentz-invariant three-momentum element associated with \(\mathbb{M}\), i.e., restricted to the mass shell, can be represented schematically for example as  
\[\mathrm{dP}_{m}=\frac{c^{2}}{(2\pi\hbar)^{3}}\frac{\mathrm{d}\mathbf{q}}{\mathbf{c}_{ \mathbf{q}}} \tag{54}\]  
with reference to the decomposition of \(\mathbf{p}\) in Equation (35). The corresponding Galilei-invariant three-momentum element associated with \(\mathbb{G}\) is  
\[\mathrm{dP}_{m}=\frac{1}{(2\pi\hbar)^{3}}\frac{\mathrm{d}\mathbf{q}}{m} \tag{55}\]  
with reference to Equation (51). The number of such worldlines piercing \(\mathbf{t}\,\mathrm{dV}_{t}\) is  
\[f\,\left(\mathbf{t}\cdot\mathbf{p}\right)\,\mathrm{dP}_{m}\,\mathrm{dV}_{t}=f\,t_{ \alpha}\,p^{\alpha}\,\mathrm{dP}_{m}\,\mathrm{dV}_{t}, \tag{56}\]  
which defines the single-particle distribution function \(f\) as a scalar field on phase space (i.e., spacetime plus momentum space restricted to the mass shell). Slice off a short section of the tube of wordlines whose timelike extent is \(\mathbf{p}\,\mathrm{d}\lambda\), where \(\lambda\) is the affine parameter of the worldlines. The Liouville theorem and an assumption of uncorrelated particle collisions leads to the Boltzmann equation  
\[\frac{\mathrm{d}f}{\mathrm{d}\lambda}=C(f), \tag{57}\]  
expressing the change--due to microscopic interactions as given by the phase space density of collisions \(C(f)\)--in worldline occupancy along the length of the tube segment. The left-hand side is a directional derivative along the phase flow:  
\[\mathbf{L}_{m}(f)=C(f), \tag{58}\]  
where \(\mathbf{L}_{m}\) is the Liouville vector, restricted to the mass shell, arising from the equations determining the worldlines \(\mathcal{X}(\lambda)\):  
\[\frac{\mathrm{d}\mathcal{X}(\lambda)}{\mathrm{d}\lambda} = \mathbf{p}, \tag{59}\] \[\frac{\mathrm{d}\mathbf{p}}{\mathrm{d}\lambda} = m\mathbf{f}. \tag{60}\]  
(The force vector \(\mathbf{f}\) and its components \(f^{\mu}\) are not to be confused with the distribution function \(f\), a scalar. The factor of \(m\) appears in Equation (60) so that \(\mathbf{f}\) has the same definition as in Equations (24) and (43), recalling that \(\mathrm{d}\tau=m\,\mathrm{d}\lambda\).) Clearly a plausible account in terms of kinetic theory requires a distinction between long-range forces \(\mathbf{f}\), responsible for the deviation of particle worldlines from geodesics; and short-range forces responsible for particle collisions, over whose microscopic effective range particle worldlines can be regarded as rectilinear. An explicit expression for the Liouville vector \(\mathbf{L}_{m}\) based on Equations (59) and (60), and the concomitant rendering of Equation (58) into a partial differential equation, depends on a choice of phase space coordinates--four spacetime coordinates, and three momentum space coordinates on the mass shell. Approach this task in a geometric spirit by choosing momentum space coordinates in accordance with a decomposition of \(\mathbf{p}\) in terms of the four-velocity of observers associated with a reference frame in which the momentum components are to be reckoned.