# Minkowski and Galilei/Newton Fluid Dynamics: A Geometric 3 + 1 Spacetime Perspective
## 3 Fluid Dynamics
### 3+ 1 Fluid Dynamics
Again, the assumption of fluid dynamics is that, given some microscopic model, all of these quantities defined in the comoving frame are related by (non-equilibrium, if necessary) thermodynamics. In the course of defining both the primary geometric objects associated with fiducial observers--the spacetime fluxes \(\mathbf{N}\), \(\mathbf{M}\), and \(\mathbf{E}\)--and the fluid velocity characterizing comoving observers and the physical quantities measured by them, relationships between them have also been given in the preceding subsections. For \(\mathbf{N}\), these include Equations (87)-(89). For \(\mathbf{M}\), the relevant equations are Equation (130), followed by Equations (131) and (132) on \(\mathbb{M}\), and by Equations (135) and (136) on \(\mathbb{G}\). For \(\mathbf{E}\), the relationships are given in Equation (145), followed by Equations (146) and (147) on \(\mathbb{M}\), and by Equations (153) and (154) on \(\mathbb{G}\). For both \(\mathbf{M}\) and \(\mathbf{E}\), it has seemed worthwhile to separate contributions labeled 'perfect' (pertaining to quantities characterizing individual fluid elements) and 'dissipative' (involving quantities expressing 'leakage' between fluid elements)--both to help manage the algebraic complications associated with relating two sets of observers, and because the limit of a perfect fluid is commonly used in practice. Having completed this summary view to this point, the next step is to convert the balance equations obeyed by \(\mathbf{N}\), \(\mathbf{M}\), and \(\mathbf{E}\) in Equations (101), (137), (138), (155) and (156), involving spacetime divergences, into \(3+1\) equations suitable for initial value problems. As alluded to above, an important step towards a \(3+1\) perspective has already been taken in the separation of momentum and energy embodied in \(\mathbf{M}\) and \(\mathbf{E}\), defined with reference to the fiducial observer four-velocity \(\mathbf{w}\) and hyperplanes of simultaneity \(\mathcal{S}_{t}\); what remains is to similarly decompose the spacetime divergences. On the affine spacetimes \(\mathbb{M}\) and \(\mathbb{G}\), and for the inertial fiducial observers \(\Theta\), rectangular coordinates give an instant \(3+1\) separation \[\left[\
abla_{\mu}\right]_{\Theta}=\left[\frac{\partial}{\partial t},\frac{ \partial}{\partial x^{i}}\right]_{\Theta}. \tag{157}\]  
However, in more geometric terms, the time derivative can be expressed (thanks to the constancy of \(\mathbf{w}\)) as the Lie derivative \(\mathcal{L}_{\mathbf{w}}=\mathbf{w}\cdot\mathbf{\
abla}\), and the spatial derivative in terms of a covariant derivative \(\mathbf{D}\) on \(\mathcal{S}_{t}\). These operate on the corresponding elements of a decomposition of the slots of \(\mathbf{N}\), \(\mathbf{M}\), and \(\mathbf{E}\) which are contracted with the spacetime operator \(\mathbf{\
abla}\). Such a decomposition of \(\mathbf{N}\) is given already in Equation (89), such that Equation (101) takes the \(3+1\) form  
\[\mathcal{L}_{\mathbf{w}}N+\mathbf{D}\cdot\mathbf{F}_{N}=0 \tag{158}\]  
on both \(\mathbb{M}\) and \(\mathbb{G}\), where  
\[\mathbf{F}_{N}=N\mathbf{V} \tag{159}\]  
is the'spatial' flux, i.e., the portion of the spacetime particle flux \(\mathbf{N}\) tangent to \(\mathcal{S}_{t}\), and the fiducial particle density \(N\) in \(\Theta\) is given in terms of the comoving density \(n\) in \(\bar{\Theta}\) by Equations (87) and (88). Similar decompositions of \(\mathbf{M}\) and \(\mathbf{E}\) remain to be given. For this purpose, the decompositions of the fluid four-velocity \(\mathbf{U}\) in Equations (82) and (83) are needed.