# Minkowski and Galilei/Newton Fluid Dynamics: A Geometric 3 + 1 Spacetime Perspective
## 1 Introduction
Fourth, especially when it comes to initial value problems, there are usually no practical reasons to consider a four-dimensional perspective on particular systems. Once physical laws are determined (based in part on the postulated nature and symmetries of spacetime), most often solutions in particular cases are found by splitting spacetime into'space' and 'time'--even in general relativity. Nevertheless, formulating non-relativistic fluid dynamics in terms of tensors on four-dimensional [PERSON]/Newton spacetime--so that covariance with respect to inhomogeneous Galilei transformations is manifest--seems like a worthwhile enterprise, for it is as true of non-relativistic physics as it is of relativistic physics that contemplation of spacetime structure and symmetries gives a deeper feel for some of the constraints on the structure of physical law. Such a program has been undertaken in more heavily mathematically-oriented literature beginning with [PERSON] and [PERSON] [25; 26]. One might wish for an analogue of the relativistic stress-energy tensor on non-relativistic four-dimensional spacetime, with vanishing divergence in the absence of external forces; but the non-equivalence of non-relativistic mass and energy implies that the most straightforward non-relativistic analogue--a'mass-momentum' tensor [26; 27] (herein called the'stress-inertia' or 'inertia-momentum tensor', see Section 3.3)--cannot accommodate internal energy, and so the timelike projection of its spacetime divergence is merely redundant with conservation of mass (or, herein, baryon number). An alternative rank-2 tensor on non-relativistic four-dimensional spacetime unifying stresses and energy flow (while excluding mass flow) is obtained by [PERSON] and [PERSON] [28] with a variational method, but it does not have vanishing divergence in the absence of external fields. An alternative view that somewhat unifies non-relativistic and relativistic spacetimes is opened by [PERSON] et al. [29], who show that both non-relativistic and Minkowski four-dimensional spacetimes \(\mathcal{M}\) can be regarded as embedded in a five-dimensional extended spacetime \(\hat{\mathcal{M}}\) with a Lorentz metric. On this extended spacetime, the Bargmann group--an extension of the [PERSON] group--encodes the non-relativistic physics. Recently, [PERSON] and [PERSON] adopted this five-dimensional perspective in developing their geometric vision of non-relativistic continuum physics [30; 31; 32], defining a rank-2 'energy-momentum-mass tensor' \(\hat{\mathbf{T}}\) that is a one-form with respect to five-dimensional \(\hat{\mathcal{M}}\) and a vector with respect to four-dimensional \(\mathcal{M}\), whose divergence on \(\mathcal{M}\) vanishes in the absence of external forces. Represented as a \(4\times 5\) matrix \(\hat{\mathcal{I}}^{\mu}_{\hat{\varrho}}\) in 'Bargmann coordinates' on \(\hat{\mathcal{M}}\), the columns represent flux vectors of energy, three-momentum, and mass on \(\mathcal{M}\). Despite the heavily geometric spacetime approach, they focus on the non-relativistic case and briefly mention only in passing the relationship of their formalism to relativistic physics. In pondering the difficulty of 'geometrizing' non-relativistic fluid dynamics due to the inequivalence of the mass and internal energy of fluid elements, it may be useful to step back and consider three 'levels' of geometrization, and how much is required for a given purpose. The lowest level of geometrization is essentially none: one simply works with partial differential equations in a particular coordinate system expressing the time evolution of physical fields in position space. This is concrete, and ultimately necessary for numerical simulations on computers; but it can obscure the underlying structure of the theory, and the coordinatetransformations required to compare Eulerian (static 'lab' frame), Lagrangian (comoving'material' frame), and arbitrary-Lagrangian-Eulerian (something in between) formulations are messy and confusing.