1. Introduction
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. Regarded as an application of mechanics and thermodynamics to an infinitesimal ‘fluid element’, the equations express evolution of the element’s particle density (due to its changing volume), its velocity (due to Newton’s Second Law), and its internal energy density (due to the First Law of Thermodynamics). Alternatively, the non-relativistic fluid equations can be expressed in conservative form as balance equations for particle number, momentum, and ‘total’ (internal plus kinetic) energy in a fixed infinitesimal ‘control volume’.
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 (inhomogeneous Lorentz) transformations in Minkowski (special relativistic) spacetime, or general coordinate transformations in Einstein (general relativistic) spacetime. The velocity three-vector of the non-relativistic case, denoting the rate of change of a fluid element’s position in space, is augmented and modified to a four-vector on spacetime, tangent to a fluid element’s worldline. Thermodynamic state variables are scalar fields defined in a ‘comoving frame’ (in which the First Law of Thermodynamics applies) riding along with a fluid element. The spatial particle flux of the non-relativistic formulation is augmented and modified to a particle number four-vector whose spacetime four-divergence expresses conservation of particles. The non-relativistic stress tensor expressing three-momentum flux is generalized to a stress-energy tensor on spacetime whose four-divergence expresses local conservation (or balance) of energy-momentum.
In the vast literature on fluid dynamics and continuum mechanics more generally (considering also elastic solids), it is virtually universal practice to adopt these separate viewpoints as distinct—even disjoint—conceptual points of departure. In a non-relativistic treatment, time-dependent fields on position space are employed, with momentum and energy treated separately: energy conservation is obtained by an ad hoc combination of the First Law of Thermodynamics (applied to a mass element) with the equations for mass and momentum conservation. In a relativistic formulation, a unified stress-energy tensor on spacetime is typically postulated from the outset, whose spacetime divergence gives the equations of motion (whose reduction to the non-relativistic equations is then argued post hoc in the limit of an infinite speed of light). These different perspectives are typically taken even when non-relativistic and relativistic approaches are discussed within the same work, for instance the classic text by Landau and Lifshitz [
1], a recent very thorough monograph by Ferrase and Bini [
2], and an even more recent paper by the present author [
3], to pluck to just a few examples from what is, again, a vast amount of literature. These disjoint conceptual points of departure are also taken by de Groot and collaborators in two separate works on kinetic theory, one non-relativistic [
4] and one relativistic [
5], in which the fluid dynamical limit is obtained in the course of the exposition. Two examples that include a geometric and less familiar aspect—the deployment of exterior calculus (different from the usual tensor calculus more familiar to physicists), to more easily prove certain theorems such as conservation of vorticity and helicity—but that are still standard in their separate viewpoints on non-relativistic space and time or relativistic spacetime respectively, are a non-relativistic variational approach of Prix [
6,
7]; (A series of papers by Carter and collaborators [
8,
9,
10,
11] addresses similar territory, but with the four-dimensional perspective on non-relativistic spacetime discussed below. However, an object they call a ‘stress momentum energy tensor’ seems in fact to be what is called below a ‘mass-momentum’ or ‘stress–inertia’ or ‘inertia–momentum’ tensor, as its divergence provides only three momentum equations, with the fourth independent (‘time’) component of its divergence providing no new information. In fact, unlike the non-covariant presentation of Prix [
6,
7], these papers do not present a conservative ‘total’ energy equation. Indeed, they apparently see no need to discuss an energy equation at all, presumably not regarding as necessary for their purposes anything beyond the laws of thermodynamics applied to a fluid element.) and an introduction to relativistic fluid dynamics by Gourgoulhon [
12] that shows a reduction to the non-relativistic limit only after the relativistic equations have been discussed.
It seems widely underappreciated that the time and position space of non-relativistic physics also form a four-dimensional spacetime (e.g., [
13,
14,
15,
16,
17,
18,
19,
20,
21,
22,
23])—call it, say, Galilei/Newton spacetime. (Projection of a conception of spacetime as a four-dimensional manifold back onto the thinking of Galileo and Newton is ahistorical and anachronistic, but useful for purposes of comparison with Minkowski and Einstein spacetimes. The designation ‘Galilei/Newton’ serves two purposes. First, it is a reminder that Newton, not having the spacetime and fiber bundle concepts available, assumed absolute position space (e.g., [
18]), with Galilei invariance being an accidental consequence of the postulated Second Law rather than a symmetry of spacetime itself. Second, this moniker allows Galilei/Newton spacetime with its global inertial frames to be distinguished from the phenomenologically equivalent Newton/Cartan spacetime—a low-velocity/low-energy limit of general relativity that features absolute time and Euclid geometry on each surface of simultaneity, but with the gravitational force eliminated, via the equivalence principle, in favor of space time curvature (albeit via an affine connection not derivable from a spacetime metric; see e.g., Refs. [
14,
15,
18,
24]).) There are probably several reasons for this. First, the notion of spacetime as a four-dimensional manifold was conceived shortly after the publication of special relativity, and subsequently played a major role in the development of general relativity, so that the whole notion seems inextricably connected with these theories. Conception of a non-relativistic spacetime was only retrospective, coming a few years later. Second, when one approaches a problem in a non-relativistic context, the comfortable familiarity of separate time and position space is a mindset difficult to resist, as it accords more closely with both lived experience and early physics instruction. One feels as if one is avoiding various complications and counterintuitive notions. Third—and ironically, in light of the previous point—in geometric terms, Galilei/Newton spacetime is less elegant than Minkowski or Einstein spacetimes. This is because there is no spacetime metric: while it is a differentiable manifold, Galilei/Newton spacetime is not a (pseudo)Riemann manifold, and must be given different structure. Each (hyper)surface of simultaneity is a Riemann manifold, with a flat three-metric for position space; but without metric duality there is no raising or lowering of indices of spacetime tensors. 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 Galilei/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 Toupin and Truesdell [
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 Duval and Künzle [
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 Duval et al. [
29], who show that both non-relativistic and Minkowski four-dimensional spacetimes
can be regarded as embedded in a five-dimensional extended spacetime
with a Lorentz metric. On this extended spacetime, the Bargmann group—an extension of the Galilei group—encodes the non-relativistic physics. Recently, de Saxcé and Vallée 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’
that is a one-form with respect to five-dimensional
and a vector with respect to four-dimensional
, whose divergence on
vanishes in the absence of external forces. Represented as a
matrix
in ‘Bargmann coordinates’ on
, the columns represent flux vectors of energy, three-momentum, and mass on
. 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 coordinate transformations required to compare Eulerian (static ‘lab’ frame), Lagrangian (comoving ‘material’ frame), and arbitrary-Lagrangian-Eulerian (something in between) formulations are messy and confusing. 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 ‘’ 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 approach is not only natural, but essentially compulsory, given absolute time. However, for the computer simulation of relativistic systems, the perspective is ultimately also natural, since time evolution from initial conditions is basically the only game in town. In this sense, a geometric spacetime perspective provides for as much unification between the non-relativistic and relativistic cases as needed or desired for this particular purpose.
Modulo the ‘
’ qualifier, however, the approach presented in this paper is otherwise a
geometric spacetime perspective. (An excellent monograph by Gourgoulhon [
33], while focused on general relativity, provides the necessary background in differential geometry (including coordinate-free notation) and a conceptual understanding of the geometric
spacetime approach that are helpful in back-porting these ideas to Galilei/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 de Saxcé and Vallée [
31] are also helpful.) Particle momenta
, and the four-velocities
and
of ‘fiducial’ (lab frame) and ‘comoving’ (material) observers respectively, are four-vectors on spacetime. In a first sub-level of the
perspective, spacetime divergences of spacetime tensors
,
, and
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
because of the separation of three-momentum four-flux
from energy four-flux
. 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
,
, and the momentum-inertia tensor
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
perpsective, the spacetime divergences of
,
, and
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
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.) Also 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.
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
and Galilei/Newton spacetime
.
Comparison and contrast begin with the spacetimes themselves, discussed in
Section 2. See
Table 1. Both
and
are four-dimensional differentiable manifolds; more particularly, they are both affine spaces admitting global inertial reference frames. Selection of a particular inertial frame
as ‘fiducial’ provides an associated family of observers with constant four-velocity
and hyperplanes of simultaneity
. Minkowski spacetime
is a pseudo-Riemann manifold with spacetime metric
embodying an invariant null cone structure, with the pull-back
of
onto each ‘position space’ hyperplane
constituting the flat three-metric thereon. In contrast,
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
, upon which
is the flat three-metric. While a pseudo-Riemann manifold and a fiber bundle are qualitatively different, the correspondence between the invariant structures of
and
can be pictured in a spacetime diagram as the limit
pressing a null cone of
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
into an absolute hyperplane of simultaneity of
. Moreover, the (push-forward of the) inverse three-metric
on
is the
limit of the inverse spacetime metric
on
, with the Galilei invariance of the components of
recognized as a geometric remnant of the Lorentz invariance of the components of
.
A classical particle worldline is a curve on spacetime, whose suitably parametrized tangent vector is the particle four-momentum
. See
Table 2. The zeroth component
is the particle inertia; on
, the inertia
depends on the particle three-momentum
, while on
the inertia is simply the particle rest mass
m—an absolute inertia resulting from absolute simultaneity. Because kinetic energy
contributes to inertia on
, it can readily be extracted from
. In contrast, the absolute nature of inertia on
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
in such a way that it is the
limit of
on
, the kinetic energy turns out to obey the same geometric equation on both
and
(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. See also
Table 3. Two types of reference frames must be invoked. First, a fiducial frame
is necessary to cover an extended region of spacetime with a single coordinate system. (On
and
, as addressed in this paper, the fiducial frame is a global inertial one in which the paradigmatic forces of electromagnetism and gravity can be defined respectively.) Second, a comoving frame
is also necessary, in order to define quantities related by thermodynamics, so that a closed system of equations can be obtained. Defining and working with quantities related to these two frames does not require use of components and changes of coordinates; use of coordinate-free geometric notation is fully viable, and even conceptually simplifying, for this purpose.
A geometric spacetime perspective capable of embracing both Minkowski and Galilei/Newton fluid dynamics on a somewhat common footing must of necessity be at least partly a
formulation, as opposed to being fully four-dimensional. This is not because of the particle number four-vector
—the first
moment of the particle distribution function
f—which is of essentially the same character on
and
, as discussed in
Section 3.2. The important difference appears in connection with the second
moment of
f, the tensor
addressed in
Section 3.3. This is normally known as the stress-energy (or energy-momentum) tensor, but the present comparison with the Galilei/Newton case brings into sharp focus the realization that it is more properly understood as a stress–
inertia (or
inertia–momentum) tensor. This has some impact on the nature of the
three-momentum tensor
, obtained in
Section 3.4 on both
and
as the projection of the first slot of
onto the hyperplanes of simultaneity
. However, the much larger difference is in connection with an energy four-vector
introduced in
Section 3.5, due to the conceptual difference in particle kinetic energy
discussed in the paragraph before last: on
, because kinetic energy contributes to inertia, the vector
can readily be obtained in terms of the projection of
along the inertial observer four-velocity
; but on
, the absolute nature of inertia implies that a comparable projection of
(via the time one-form
) would be effectively redundant with the particle number vector
. Thus there is no simple relation between
and
on
, despite the fact that
on
turns out to be the
limit of
on
.
While this ab initio distinction between three-momentum and energy is necessary to accommodate the Galilei/Newton case, the geometric objects
,
, and
—the spacetime fluxes of particle number, three-momentum, and energy—nevertheless obey balance equations involving spacetime divergences on both
and
:
where the sources
and
are for instance those given in
Section 3.4 and
Section 3.5 for electromagnetism on
and gravity on
.
However, a fully
formulation is desirable for practical solution of initial value problems in any case, and this also can be expressed in a coordinate-free geometric way, as given in
Section 3.6. See also
Table 4. On the affine spacetimes
and
,
where
N,
, and
E are are volume densities of particle number, three-momentum, and internal-plus-kinetic energy measured by the fiducial observers of
, with
,
,
being the corresponding spatial fluxes; and
and
are, respectively, the Lie derivative along the fiducial observer four-velocity
and the three-covariant derivative on the hyperplanes of simultaneity
. In coordinates associated with the fiducial (and, on
and
, global inertial) observers, these equations reduce immediately to the familiar conservative formulations of special relativistic and non-relativistic fluid dynamics (see
Appendix B). In addition, in this geometric form, they will provide a useful conceptual bridge to arbitrary-Lagrange–Euler and general relativistic formulations.