Abstract
A kinetic theory of classical particles serves as a unified basis for developing a geometric 3 + 1 spacetime perspective on fluid dynamics capable of embracing both Minkowski and Galilei/Newton spacetimes. Parallel treatment of these cases on as common a footing as possible reveals that the particle four-momentum is better regarded as comprising momentum and inertia rather than momentum and energy; and, consequently, that the object now known as the stress-energy or energy-momentum tensor is more properly understood as a stress-inertia or inertia-momentum tensor. In dealing with both fiducial and comoving frames as fluid dynamics requires, tensor decompositions in terms of the four-velocities of observers associated with these frames render use of coordinate-free geometric notation not only fully viable, but conceptually simplifying. A particle number four-vector, three-momentum (1, 1) tensor, and kinetic energy four-vector characterize a simple fluid and satisfy balance equations involving spacetime divergences on both Minkowski and Galilei/Newton spacetimes. Reduced to a fully 3 + 1 form, these equations yield the familiar conservative formulations of special relativistic and non-relativistic fluid dynamics as partial differential equations in inertial coordinates, and in geometric form will provide a useful conceptual bridge to arbitrary-Lagrange–Euler and general relativistic formulations.
Highlights
In contemplating fluid dynamics—whether purely for deeper theoretical understanding and appreciation, or for the practical purpose of developing an extensible simulation code—it may be useful to abstract certain notions that are common to both the relativistic and non-relativistic cases.Non-relativistic fluid dynamics is normally formulated as a set of evolution equations for time-dependent fields on position space
In the case of relativistic fluid dynamics, spacetime allows a different perspective, as reflected in a formulation that is manifestly covariant with respect to Poincaré transformations in Minkowski spacetime, or general coordinate transformations in Einstein spacetime
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.) by way of maintaining focus on the basic geometric objects characterizing fluid dynamics: while external gravitational and electromagnetic forces are considered as the paradigmatic examples consistent with the flat spacetimes of Galilei/Newton and Minkowski, respectively, the present work includes no further discussion of the Poisson and Maxwell equations determining these fields, nor possible reformulations that would move the gravitational potential and electric and magnetic fields from source terms into the definitions of fluid densities and fluxes
Summary
In contemplating fluid dynamics—whether purely for deeper theoretical understanding and appreciation, or for the practical purpose of developing an extensible simulation code—it may be useful to abstract certain notions that are common to both the relativistic and non-relativistic cases. The simple relationship between E, N, and the momentum-inertia tensor 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.) In the second and final sub-level of the 3 + 1 perpsective, the spacetime divergences of N, M, and E are decomposed into a Lie derivative of a density from one spacelike slice to the (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. 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.) by way of maintaining focus on the basic geometric objects characterizing fluid dynamics: while external gravitational and electromagnetic forces are considered as the paradigmatic examples consistent with the flat spacetimes of Galilei/Newton and Minkowski, respectively, the present work includes no further discussion of the Poisson and Maxwell equations determining these fields, nor possible reformulations that would move the gravitational potential and electric and magnetic fields from source terms into the definitions of fluid densities and fluxes
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