# Minkowski and Galilei/Newton Fluid Dynamics: A Geometric 3 + 1 Spacetime Perspective
## 3 Fluid Dynamics
### 3+ 1 Fluid Dynamics  
To summarize the current section to this point: the previous subsections have motivated three principal geometric objects characterizing a simple fluid on both Minkowski spacetime \(\mathbb{M}\) and Galilei/Newton spacetime \(\mathbb{G}\)--the particle number vector \(\mathbf{N}\), the \((1,1)\) three-momentum tensor \(\mathbf{M}\) (first index down), and the energy vector \(\mathbf{E}\). These names derive from their defining expressions in Equations (77), (128) and (140), which exhibit the quantities whose spacetime flux is generated by the flow of particles with four-momenta \(\mathbf{p}\). Note that separate mention of three-momentum and energy already presupposes some chosen fiducial frame--in this paper, on \(\mathbb{M}\) and \(\mathbb{G}\), a global inertial frame \(\Theta\)--with which are associated a family of observers with constant four-velocity \(\mathbf{w}\), and a foliation of spacetime (whether \(\mathbb{M}\) or \(\mathbb{G}\)) into hyperplanes of simultaneity \(\mathcal{S}_{t}\). In addition to these spacetime fluxes of quantities defined with respect to fiducial observers, crucial for allowing the equations governing them to form a closed system are the definitions of a family of comoving observers and quantities measured by them. This begins with a specification of a 'fluid velocity'. Associated with this is a notion of macroscopically small but microscopically large 'fluid elements' of a fixed number of particles, which can be regarded as a system to which thermodynamics can be applied in order to provide the necessary closures. The average flow of particle number defines a fluid velocity vector \(\mathbf{U}\), taken to be the four-velocity of comoving observers, whose reference frame is denoted \(\hat{\mathcal{O}}\). The fluid four-velocity \(\mathbf{U}\), three-velocity \(\mathbf{V}\), and particle density \(N\) are defined by Equations (82), (83), (85) and (86). (The three-velocity \(\mathbf{V}\) can be regarded as a four-vector on \(\mathbb{M}\) or \(\mathbb{G}\) that happens to be tangent to \(\mathcal{S}_{t}\).) Even in coordinate-free notation, these definitions reference the fiducial frame \(\Theta\), in the following sense: they rely on a decomposition, in Equations (35), (51), (79) and (80), of particle four-momentum \(\mathbf{p}\) that separates the portion tangent to \(\mathcal{S}_{t}\) from \(\mathbf{w}\), i.e., the particle three-momentum \(\mathbf{q}\) and associated particle velocity \(\mathbf{v}_{\mathbf{q}}\). The definition of fluid velocity \(\mathbf{U}\) allows quantities measured by comoving observers to be defined. In particular, an alternative decomposition of \(\mathbf{p}\) can be given, see Equations (67), (68), (90) and (91), furnishing the particle three-momentum \(\mathbf{s}\) and velocity \(\mathbf{v}_{\mathbf{s}}\) according to \(\bar{\Theta}\). (Notice that reference to a different set of observers does not require use of components and a change of coordinates; use of coordinate-free notation is fully viable, and even conceptually simplifying, for this purpose.) Then, the particle density \(n\) according to \(\bar{\Theta}\) can be given (Equation (93)), and the vanishing average velocity in the comoving frame confirmed (Equation (94)). The internal energy density \(e\) (Equation (105)) and pressure \(p\) (Equations (107), (112), (117) and (118)) round out the quantities characterizing a fluid element. The quantities responsible for dissipation--which are non-zero only if the distribution function \(f\) is asymmetric as a function of the comoving frame rectangular components of \(\mathbf{v}_{\mathbf{s}}\)--are the internal energy flux \(\mathbf{J}\) (Equation (106)) and viscous stress \(\mathbf{\Pi}\) (Equations (107), (112), (117) and (118)), representing internal energy and momentum exchange _between_ fluid elements. (Ultimately, by definition--and with the caveat that it is not the only possible convention, see e.g., Ref. [5]--in the present treatment of a simple fluid there is no net particle number exchange between fluid elements.)