# Minkowski and Galilei/Newton Fluid Dynamics: A Geometric 3 + 1 Spacetime Perspective
## 4 Conclusions  
Taking fluid dynamics on Minkowski spacetime as a reference with which comparisons might be made and against which contrasts can be drawn, the purpose of this paper is to consider Galilei/Newton fluid dynamics in geometric terms from a spacetime perspective. In a different spirit than the remarks in the concluding section of Ref. [3], which emphasize distinct principles that might be taken to underlie non-relativistic and relativistic fluid dynamics, the goal here has been to start with a common framework--a generic kinetic theory of classical particles--in order to explore the extent to which concepts and geometric objects pertaining to fluid dynamics can be developed in parallel on both Minkowski spacetime \(\mathbb{M}\) and Galilei/Newton spacetime \(\mathbb{G}\). Comparison and contrast begin with the spacetimes themselves, discussed in Section 2. See Table 1. Both \(\mathbb{M}\) and \(\mathbb{G}\) are four-dimensional differentiable manifolds; more particularly, they are both affine spaces admitting global inertial reference frames. Selection of a particular inertial frame \(\Theta\) as 'fiducial' provides an associated family of observers with constant four-velocity \(\mathbf{w}\) and hyperplanes of simultaneity \(\mathcal{S}_{t}\). Minkowski spacetime \(\mathbb{M}\) is a pseudo-Riemann manifold with spacetime metric \(g\) embodying an invariant null cone structure, with the pull-back \(\mathbf{\gamma}\) of \(g\) onto each 'position space' hyperplane \(\mathcal{S}_{t}\) constituting the flat three-metric thereon. In contrast, \(\mathbb{G}\) has no spacetime metric; it is a fiber bundle with a one-dimensional base manifold, 'time,' and 'position space' fibers constituting invariant hyperplanes of simultaneity \(\mathcal{S}_{t}\), upon which \(\mathbf{\gamma}\) is the flat three-metric. While a pseudo-Riemann manifold and a fiber bundle are qualitatively different, the correspondence between the invariant structures of \(\mathbb{M}\) and \(\mathbb{G}\) can be pictured in a spacetime diagram as the limit \(c\to\infty\) pressing a null cone of \(\mathbb{M}\) down onto a fiducial observer position space hyperplane (visualized as the horizontal axis), squeezing the distinct hyperplanes of simultaneity of inertial observers in relative motion in \(\mathbb{M}\) into an absolute hyperplane of simultaneity of \(\mathbb{G}\). Moreover, the (push-forward of the) inverse three-metric \(\overrightarrow{\overrightarrow{\mathbf{\gamma}}}\) on \(\mathbb{G}\) is the \(c\to\infty\) limit of the inverse spacetime metric \(\overrightarrow{\overrightarrow{\mathbf{\delta}}}\) on \(\mathbb{M}\), with the Galilei invariance of the components of \(\overrightarrow{\overrightarrow{\mathbf{\gamma}}}\) recognized as a geometric remnant of the Lorentz invariance of the components of \(\overrightarrow{\overrightarrow{\mathbf{\delta}}}\). A classical particle worldline is a curve on spacetime, whose suitably parametrized tangent vector is the particle four-momentum \(\mathbf{p}\). See Table 2. The zeroth component \(p^{0}\) is the particle inertia; on \(\mathbb{M}\), the inertia \(e_{q}/c^{2}\) depends on the particle three-momentum \(\mathbf{q}\), while on \(\mathbb{G}\) the inertia is simply the particle rest mass \(m\)--an absolute inertia resulting from absolute simultaneity. Because kinetic energy \(e_{q}\) contributes to inertia on \(\mathbb{M}\), it can readily be extracted from \(\mathbf{p}\). In contrast, the absolute nature of inertia on \(\mathbb{G}\) means that inertia and energy are not linked, such that the notion of kinetic energy must be introduced by hand as a separate concept. However, once defined on \(\mathbb{G}\) in such a way that it is the \(c\to\infty\) limit of \(e_{q}\) on \(\mathbb{M}\), the kinetic energy turns out to obey the same geometric equation on both \(\mathbb{M}\) and \(\mathbb{G}\) (see Equations (38) and (52)). A kinetic theory of classical particles motivates certain geometric objects pertaining to fluid dynamics, as summarized more fully in Section 3.6.