# Minkowski and Galilei/Newton Fluid Dynamics: A Geometric 3 + 1 Spacetime Perspective
## Appendix B Reduction to Partial Differential Equations  
The \(3+1\) geometric formulation of the equations of fluid dynamics in Equations (190)-(192) is closely tied to partial differential equations in conservative form useful for numerical simulations. See, for instance, Ref. [33] for an accessible introduction to the necessary differential geometry--in this case, Lie derivatives, and covariant derivatives on the position space slices \(\mathcal{S}_{t}\), which are in themselves three-dimensional flat Riemann manifolds endowed with a metric \(\gamma\). As a simple example, use a reference frame \(\bar{\Theta}\) that differs from our fiducial frame \(\Theta\), a global inertial frame, only in the use of curvilinear spatial coordinates \((x^{r})\) on the position space slices \(\mathcal{S}_{t}\) instead of the rectilinear \((x^{i})\). Because \(\mathbf{w}\) is a constant vector (being the four-velocity of inertial observers), and, thanks to Equations (25) and (44) on \(\mathbb{M}\) and \(\mathbb{G}\), respectively, the Lie derivative in (for instance) Equation (190) becomes  
\[\mathcal{L}_{\mathbf{w}}N=w^{\mathbf{n}}\frac{\partial N}{\partial x^{\mathbf{n}}}=\frac{ \partial N}{\partial t},\] (A9)  
and similarly  
\[\mathcal{L}_{\mathbf{w}}S_{\mathbf{\gamma}}=\frac{\partial S_{\mathbf{\gamma}}}{\partial t}\] (A10)  
for the components of Equation (191). Using the natural basis associated with the coordinates \((x^{\mathbf{\gamma}})\), and for the moment writing \(\mathbf{F}_{N}=\mathbf{F}\), the position space divergence in (for instance) Equation (190) is  
\[\mathbf{D}\cdot\mathbf{F} = D_{\mathbf{\gamma}}F^{\mathbf{\gamma}}=\frac{\partial F^{\mathbf{\gamma}}}{ \partial x^{\mathbf{\gamma}}}+\Gamma^{\mathbf{\gamma}}_{\mathbf{\gamma}}F^{\mathbf{\gamma}}\] (A11) \[= \frac{\partial F^{\mathbf{\gamma}}}{\partial x^{\mathbf{\gamma}}}+\frac{ \partial\ln\sqrt{\gamma}}{\partial x^{\mathbf{\gamma}}}F^{\mathbf{\gamma}}=\frac{1}{ \sqrt{\gamma}}\frac{\partial}{\partial x^{\mathbf{\gamma}}}\left(\sqrt{\gamma}F^ {\mathbf{\gamma}}\right).\] (A12)  
Here, the Christoffel symbols  
\[\Gamma^{\mathbf{\gamma}}_{\mathbf{\gamma}\mathbf{\gamma}}=\frac{1}{2}\gamma^{\mathbf{\gamma} \mathbf{\gamma}}\left(\frac{\partial\gamma_{\mathbf{\gamma}\mathbf{\gamma}}}{\partial x^ {\mathbf{\gamma}}}+\frac{\partial\gamma_{\mathbf{\gamma}\mathbf{\gamma}}}{\partial x^\mathbf{{\gamma}}}-\frac{\partial\gamma_{\mathbf{\gamma}\mathbf{\gamma}}}{\partial x^{\mathbf{ \gamma}}}\right)\] (A13)  
account for the variation of the non-constant curvilinear basis vectors. From these results, the identity \(\Gamma^{\mathbf{\gamma}}_{\mathbf{\gamma}\mathbf{\gamma}}=\Gamma^{\mathbf{\gamma}}_{\mathbf{\gamma }\mathbf{\gamma}}=\partial\ln\sqrt{\gamma}/\partial x^{\mathbf{\gamma}}\) for a contracted Christoffel symbol (where \(\bar{\gamma}=\det\left[\gamma_{\mathbf{\gamma}\mathbf{\gamma}}\right]\) is the determinant of the matrix of metric components) found in divergences can be derived. However, each free index gets a Christoffel symbol correction term; for a \((1,1)\) tensor as in Equation (191), it turns out that  
\[D_{\mathbf{\gamma}}F^{\mathbf{\gamma}}_{\mathbf{\gamma}}=\frac{1}{\sqrt{\gamma}}\frac{ \partial}{\partial x^{\mathbf{\gamma}}}\left(\sqrt{\bar{\gamma}}F^{\mathbf{\gamma}}_{ \mathbf{\gamma}}\right)-\frac{1}{2}F^{\mathbf{\gamma}}\frac{\partial\gamma_{\mathbf{\gamma }\mathbf{\gamma}}}{\partial x^{\mathbf{\gamma}}}\] (A14)  
for the symmetric case \(F^{\mathbf{\gamma}}=F^{\mathbf{\gamma}}\). Now, a metric  
\[\left[\gamma_{\mathbf{\gamma}\mathbf{\gamma}}\right]_{\mathbf{\mathcal{O}}}= \begin{bmatrix} 1&0&0\\ 0&a(x^{\mathbf{\gamma}})&0\\ 0&0&b(x^{\mathbf{\gamma}})\,c(x^{\mathbf{\gamma}})\end{bmatrix}