# Minkowski and Galilei/Newton Fluid Dynamics: A Geometric 3 + 1 Spacetime Perspective
## 4 Conclusions
However, the much larger difference is in connection with an energy four-vector \(\mathbf{E}\) introduced in Section 3.5, due to the conceptual difference in particle kinetic energy \(e_{\mathbf{q}}\) discussed in the paragraph before last: on \(\mathbb{M}\), because kinetic energy contributes to inertia, the vector \(\mathbf{E}\) can readily be obtained in terms of the projection of \(\mathbf{T}\) along the inertial observer four-velocity \(\mathbf{w}\); but on \(\mathbb{G}\), the absolute nature of inertia implies that a comparable projection of \(\mathbf{T}\) (via the time one-form \(\mathbf{t}\)) would be effectively redundant with the particle number vector \(\mathbf{N}\). Thus there is no simple relation between \(\mathbf{E}\) and \(\mathbf{T}\) on \(\mathbb{G}\), despite the fact that \(\mathbf{E}\) on \(\mathbb{G}\) turns out to be the \(c\to\infty\) limit of \(\mathbf{E}\) on \(\mathbb{M}\). While this ab initio distinction between three-momentum and energy is necessary to accommodate the [PERSON]/Newton case, the geometric objects \(\mathbf{N}\), \(\mathbf{M}\), and \(\mathbf{E}\)--the spacetime fluxes of particle number, three-momentum, and energy--nevertheless obey balance equations involving spacetime divergences on both \(\mathbb{M}\) and \(\mathbb{G}\):  
\[\mathbf{\
abla}\cdot\mathbf{N} = 0, \tag{187}\] \[\mathbf{\
abla}\cdot\mathbf{M} = \mathbf{Q}_{S},\] (188) \[\mathbf{\
abla}\cdot\mathbf{E} = \mathbf{Q}_{E}, \tag{189}\]  
where the sources \(\mathbf{Q}_{S}\) and \(\mathbf{Q}_{E}\) are for instance those given in Sections 3.4 and 3.5 for electromagnetism on \(\mathbb{M}\) and gravity on \(\mathbb{G}\). However, a fully \(3+1\) formulation is desirable for practical solution of initial value problems in any case, and this also can be expressed in a coordinate-free geometric way, as given in Section 3.6. See also Table 4. On the affine spacetimes \(\mathbb{M}\) and \(\mathbb{G}\),  
\[\mathcal{L}_{\mathbf{w}}N+\mathbf{D}\cdot\mathbf{F}_{N} = 0, \tag{190}\] \[\mathcal{L}_{\mathbf{w}}\mathbf{S}+\mathbf{D}\cdot\mathbf{F}_{\mathbf{S}} = \mathbf{Q}_{S},\] (191) \[\mathcal{L}_{\mathbf{w}}E+\mathbf{D}\cdot\mathbf{F}_{E} = \mathbf{Q}_{E}, \tag{192}\]  
where \(N\), \(\mathbf{S}\), and \(E\) are are volume densities of particle number, three-momentum, and internal-plus-kinetic energy measured by the fiducial observers of \(\mathbb{G}\), with \(\mathbf{F}_{N}\), \(\mathbf{F}_{S}\), \(\mathbf{F}_{E}\) being the corresponding spatial fluxes; and \(\mathcal{L}_{\mathbf{w}}\) and \(\mathbf{D}\) are, respectively, the Lie derivative along the fiducial observer four-velocity \(\mathbf{w}\) and the three-covariant derivative on the hyperplanes of simultaneity \(\mathcal{S}_{t}\). In coordinates associated with the fiducial (and, on \(\mathbb{M}\) and \(\mathbb{G}\), global inertial) observers, these equations reduce immediately to the familiar conservative formulations of special relativistic and non-relativistic fluid dynamics (see Appendix B). In addition, in this geometric form, they will provide a useful conceptual bridge to arbitrary-Lagrange-Euler and general relativistic formulations. **Funding:** This work was supported by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics under contract number DE-AC05-00 OR22725. **Acknowledgments:** This manuscript has been authored by UT-Battelle, LLC, under contract DE-AC05-00 OR22725 with the US Department of Energy (DOE). The US government retains and the publisher, by accepting the article for publication, acknowledges that the US government retains a nonexclusive, paid-up, irrevocable, worldwide license to publish or reproduce the published form of this manuscript, or allow others to do so, for US government purposes.