# Minkowski and Galilei/Newton Fluid Dynamics: A Geometric 3 + 1 Spacetime Perspective
## 1 Introduction
In the second and final sub-level of the \(3+1\) perspective, the spacetime divergences of \(\mathbf{N}\), \(\mathbf{M}\), and \(\mathbf{E}\) are decomposed into a Lie derivative of a density from one spacelike slice to the next (a geometric expression of a time derivative) and a covariant three-divergence of a flux within the spacelike slice (a geometric expression of position space derivatives); see Equations (190)-(192). All of this is done to a large extent with a coordinate-free mode of expression. Two payoffs of this \(3+1\) geometric perspective are that it (1) treats the non-relativistic and relativistic cases on a similar conceptual footing, illuminating the relationship between them, and (2) will provide, in future work, a conceptual bridge to arbitrary-Lagrange-Euler and general relativistic formulations useful to numerical practitioners interested in going beyond static Eulerian coordinates and/or flat spacetime. Two other somewhat unique aspects of this paper enhance one of its main contributions, the conceptually unified treatment of the Galilei/Newton (non-relativistic) and Minkowski (special relativistic) cases. First, the presentation proceeds in an interleaved parallel manner, with sections or subsections generally treating both the Minkowski and Galilei/Newtonian cases (typically in that order, since Minkowski spacetime is the more familiar arena for four-dimensional thinking) so as to enhance comparison and contrast. Second, kinetic theory on both Galilei/Newton and Minkowski spacetimes, based on the four-momenta of classical particles, is used as a unified basis for motivating the structure and interpretation of the geometric objects that characterize a fluid-dynamical approach common to both spacetimes. Ultimately, this must be regarded as something of a pedagogical device aimed at motivating intuition and interpretation: the kinetic theory of classical particles is not the fundamental theory of the world, and more generally fluid dynamics may be regarded as a generic long-wavelength, low-frequency continuum approximation to any physical system (e.g., [34]), regardless of the nature of its microscopic dynamics. However, a resort to kinetic theory serves its present purpose as a means of avoiding the usual disjoint conceptual perspectives on non-relativistic and relativistic fluid dynamics discussed early in this section. This paper is organized as follows: descriptions of Minkowski and Galilei/Newton spacetimes and the trajectories of classical particles thereon serve as a starting point in Section 2. Fluid dynamics, obtained from kinetic theory on spacetime, are then discussed in Section 3. Concluding remarks are given in Section 4, along with four tables summarizing the many entities appearing in the formalism, including brief descriptions and references to equations in the text. For simplicity in demonstrating the Galilei/Newton spacetime perspective, Minkowski spacetime is the only relativistic case treated; only a single-component fluid is contemplated; and no discussions are included of specific microscopic models, or the closures via specific constitutive relations that might be obtained from them. (A primary example of a constitutive relation--the only one required of a simple perfect fluid--is an equation of state giving the pressure in terms of the particle number density and entropy, temperature, or internal energy. Another prominent example of a constitutive relation yields the celebrated Navier-Stokes equation in the non-relativistic case: dissipative stresses are assumed to be expressed as a viscosity parameter multiplying the gradient of velocity. The relativistic analogue remains a subject of active research; further discussion is beyond the scope of this paper, but Refs. [1; 5; 34] provide some classic and recent points of entry to the literature.)