# Minkowski and Galilei/Newton Fluid Dynamics: A Geometric 3 + 1 Spacetime Perspective
## 1 Introduction
In the general relativistic case, a blizzard of indices and connection coefficients contributes to the obfuscation of the central physical ideas. At the opposite end, the highest level of geometrization involves the deployment of as few geometric objects (typically tensors) as possible on spacetime, fully embodying the underlying symmetries of the theory, and the use of coordinate-free expressions whenever possible. In the case of relativity this is quite appealing: for a simple fluid, a unified stress-energy tensor and number flux vector, each with vanishing divergence, express the conservation laws in a manifestly covariant way, making no commitments as to a particular time coordinate or slicing of spacetime. In the non-relativistic case, however, absolute time and the associated absence of a full metric structure for four-dimensional spacetime mean that one is driven to the unfamiliar (to many day-to-day practitioners of fluid dynamics) and ugly (by comparison with the relativistic case) Bargmann group on a five-dimensional extended spacetime in order to obtain, as described above, an energy-momentum-mass tensor with vanishing divergence. This paper contributes a happy medium, a'mid-level' geometrization of non-relativistic fluid dynamics, which might be called _a geometric 3 + 1 spacetime perspective_. The label '\(3+1\)' signals a primary concession made from the outset: a foliation of spacetime--into three-dimensional spacelike slices to be labeled by a single parameter (the time coordinate)--is to be chosen. In non-relativistic spacetime the \(3+1\) approach is not only natural, but essentially compulsory, given absolute time. However, for the computer simulation of relativistic systems, the \(3+1\) perspective is ultimately also natural, since time evolution from initial conditions is basically the only game in town. In this sense, a geometric \(3+1\) spacetime perspective provides for as much unification between the non-relativistic and relativistic cases as needed or desired for this particular purpose. Modulo the '\(3+1\)' qualifier, however, the approach presented in this paper is otherwise a _geometric spacetime perspective_. (An excellent monograph by [PERSON] [33], while focused on general relativity, provides the necessary background in differential geometry (including coordinate-free notation) and a conceptual understanding of the geometric \(3+1\) spacetime approach that are helpful in back-porting these ideas to [PERSON]/Newton and Minkowski spacetimes. A helpful treatment of Minkowski spacetime by the same author, also with a geometric approach, is Ref. [23]. The'mathematical tools' chapters of the book by [PERSON] and [PERSON] [31] are also helpful.) Particle momenta \(\mathbf{p}\), and the four-velocities \(\mathbf{w}\) and \(\mathbf{U}\) of 'fiducial' (lab frame) and 'comoving' (material) observers respectively, are four-vectors on spacetime. In a first sub-level of the \(3+1\) perspective, spacetime divergences of spacetime tensors \(\mathbf{N}\), \(\mathbf{M}\), and \(\mathbf{E}\) representing four-fluxes of particle number, three-momentum, and kinetic/internal energy are contemplated; see Equations (187)-(189). (While spacetime divergences of spacetime tensors are dealt with here, this is partly \(3+1\) because of the separation of three-momentum four-flux \(\mathbf{M}\) from energy four-flux \(\mathbf{E}\). This is the second primary accommodation to the non-relativistic case, related to the first: the absolute nature of inertia (mass) in non-relativistic physics is in a sense 'conjugate' to absolute time, and requires separate treatment of energy and momentum. The simple relationship between \(\mathbf{E}\), \(\mathbf{N}\), and the momentum-inertia tensor \(\mathbf{T}\) in the relativistic case--thanks to the unification of mass and energy--is noted in this paper after the fact rather than taken as primary.)