Combining 3-momentum and kinetic energy on Galilei/Newton spacetime
aa r X i v : . [ phy s i c s . g e n - ph ] J a n ✐ ✐ “Continuum” — 2020/1/14 — 3:36 — page 1 — ✐✐✐ ✐ ✐✐ ADV. THEOR. MATH. PHYS.Volume 0, Number 0, 1–15, 20201
Combining 3-momentum and kineticenergy on Galilei/Newton spacetime
Christian Y. Cardall
Without the mass-energy equivalence available on Minkowskispacetime M , it is not possible on 4-dimensional non-relativisticGalilei/Newton spacetime G to combine 3-momentum and total mass-energy in a single tensor object. However, given a fiducialframe, it is possible to combine 3-momentum and kinetic energyinto a linear form (particle) or (1, 1) tensor (continuum) in a man-ner that exhibits increased unity of classical mechanics on flat rel-ativistic and non-relativistic spacetimes M and G . As on M , for amaterial continuum on G the First Law of Thermodynamics canbe considered a consequence of a unified dynamical law for energy-momentum rather than an independent postulate.
1. Introduction
Traditional points of departure for non-relativistic and relativistic classicalmechanics (e.g. [1–3]) feature distinct pictures of space and time. The tradi-tional non-relativistic picture is that tensor fields on 3-dimensional Euclideanposition space E evolve as functions of absolute time t . In contrast, under-lying the relativistic picture is a unified 4-dimensional spacetime; for presentpurposes let this be flat Minkowski spacetime M . Tensor fields on M embody This manuscript has been authored by UT-Battelle, LLC, under contract DE-AC05-00OR22725 with the US Department of Energy (DOE). The US governmentretains and the publisher, by accepting the article for publication, acknowledgesthat the US government retains a nonexclusive, paid-up, irrevocable, worldwidelicense to publish or reproduce the published form of this manuscript, or allowothers to do so, for US government purposes. DOE will provide public access tothese results of federally sponsored research in accordance with the DOE PublicAccess Plan (http://energy.gov/downloads/doe-public-access-plan). ✐ “Continuum” — 2020/1/14 — 3:36 — page 2 — ✐✐✐ ✐ ✐✐ M , the 3-momentum and energy that are separate in the non-relativistic case arecombined in a single geometric object, the 4-momentum.The attitude towards the traditional non-relativistic and relativistic for-mulations of classical mechanics is often limited to deriving the former asa limit of the latter. As 4-dimensional equations are split into 3 + 1 dimen-sions, unified balance of 4-momentum on M is decomposed into balanceof 3-momentum and balance of energy. Then the c → ∞ (infinite speed oflight) limit of the relativistic equations in 3 + 1 dimensions coincides withthe non-relativistic equations.It is intriguing to consider the extent to which the conceptual rela-tionship can be exploited in the reverse direction, by reformulating non-relativistic physics in light of the relativistic perspective: Can the non-relativistic evolution equations on position space also be understood as con-straint equations on spacetime?
The answer has been yes, to a certain extent. At least by the 1920s, Weyl[4] and Cartan [5–7] considered the combination of Euclidean position space E and Euclidean absolute time E into a non-relativistic 4-dimensional space-time. Works by Toupin and Truesdell [8, 9], Trautmann [10, 11], and K¨untzle[12] bear mention as example entry points into what has been a rathermathematically-oriented literature across the intervening decades. A recentwork of mine [13] compares—with additional discussion and references—Minkowski spacetime M and what I call Galilei/Newton spacetime G , bothof which are flat 4-dimensional manifolds, and indeed 4-dimensional affinespaces. That work illustrates that kinetic theory on spacetime provides anintuitive understanding of fluid dynamics from a mostly 4-dimensional per-spective on both relativistic and non-relativistic spacetimes, expressed by thesame 4-dimensional equations for the fluxes of baryon number, 3-momentum,and kinetic+internal energy on both M and G .However, a perceived inability to unite 3-momentum and energy ina single tensor in the non-relativistic case has remained notable. In the1970s Duval and K¨unzle [14] used a variational principle to derive for anon-relativistic material continuum a tensor unifying stresses and internalenergy flux, but it excludes mass flow; its 4-divergence does not vanish in theabsence of an external 4-force, but is equated to a bulk acceleration term.More recently, de Saxc´e and Vall´ee constructed a tensor of vanishing diver-gence that includes the non-relativistic kinetic energy of bulk motion, andexhibits the transformation properties of the latter under Galilei boosts, byassembling an ‘energy-momentum-mass-tensor’ on a 5-dimensional extended ✐ “Continuum” — 2020/1/14 — 3:36 — page 3 — ✐✐✐ ✐ ✐✐ Combining 3-momentum and kinetic energy 3non-relativistic spacetime [15–17]. The extra dimension, and the associatedtransformation of this object in part under the Bargmann group (a centralextension of the Galilei group), are necessary to capture the transformationproperties of non-relativistic kinetic energy.Here I present a tensor equation on 4-dimensional spacetime that encom-passes both the 3-momentum and kinetic+internal energy of a material con-tinuum; consists of a vanishing divergence in the absence of external 4-forceper baryon; and does so in a conceptually unified way in both the relativisticand non-relativistic cases, that is, on both M and G . This is an equationfor the divergence of what I call the relative energy-momentum flux tensor S (the adjective ‘relative’ betrays the fact that it is defined in terms of afixed family of fiducial frames, in such a way that the transformation prop-erties of kinetic energy are not manifest). A key point is that S is a (1, 1)tensor, with components S µν , satisfying a linear form equation—in contrastto the (2, 0) total inertia-momentum flux tensor (a.k.a. energy-momentumor stress-energy tensor) T , with components T µν , satisfying a vector equa-tion. What happens is that Galilei invariance forbids energy contributionsto 4-momentum and a time component of 4-force when these are consideredas vectors. However, this restriction does not apply to 4-momentum and 4-force regarded as linear forms. In this work I motivate S and its governingequation on both M and G .
2. Relativistic classical mechanics on Minkowski spacetime
On Minkowski spacetime M the history of a particle of mass m is a worldline X ( τ ) parametrized by the proper time τ measured by a clock riding alongwith the particle. The tangent vector to the worldline, U = d X ( τ ) / d τ , isthe 4-velocity of such a comoving observer . It satisfies the normalization g ( U , U ) = − c , where g is the spacetime metric. The components of g andits inverse ←→ g are gathered by the 3 + 1 block matrices(2.1) g = (cid:18) − c
00 1 (cid:19) , ←→ g = g − = (cid:18) − /c
00 1 (cid:19) in any inertial frame.Select a fiducial frame, a global inertial frame on M . Associated withthis frame is a family of fiducial observers whose uniform 4-velocity field w = ∂/∂t (also normalized as g ( w , w ) = − c ) is the tangent vector fieldto the coordinate lines of the global time coordinate t . The level surfaces S t of the global time coordinate t foliate M into affine hyperplanes; these ✐ “Continuum” — 2020/1/14 — 3:36 — page 4 — ✐✐✐ ✐ ✐✐ γ , and are associated with a uniform linear form t = d t = ∇ t .With the fiducial frame components of w and t gathered by 4-column and4-row(2.2) w = (cid:18) (cid:19) , t = (cid:0) (cid:1) respectively (in 3 + 1 block form), it is clear from Eq. (2.1) that t is relatedto w by(2.3) t = − c g · w = − c w .In this work the dot operator ( · ) introduced in Eq. (2.3)—which reads t µ = − g µα w α /c in components—never denotes a scalar product of vectors, butonly contraction with the first available index. Instead, a scalar productof vectors will always be expressly given in terms of a metric tensor, forinstance g ( w , w ) or γ ( v , v ). The linear form w is the index-lowered metricdual of w , with components w µ = g µα w α .The 3-metric γ and its index-raised siblings ←− γ , −→ γ and ←→ γ on S t , withcomponents γ ij , γ j i = γ ij = δ ji , and γ ij respectively, can also be regardedas tensors on M (with components γ µν , γ ν µ , γ µν , and γ µν ) that behave asprojection tensors to S t . These can be expressed in terms of the metric tensor g , inverse ←→ g , and identity tensor δ = ←− g = −→ g on M : γ = g + 1 c w ⊗ w , ←→ γ = ←→ g + 1 c w ⊗ w ,(2.4) ←− γ = δ + 1 c w ⊗ w , −→ γ = δ + 1 c w ⊗ w .(2.5)The components of these tensors all transform differently under Lorentztransformations. However, in the fiducial inertial frame the single matrix(2.6) γ = ←− γ = −→ γ = ←→ γ = (cid:18)
00 1 (cid:19) gathers the components of all of them.Because the projection tensors to S t have vanishing contractions with w and/or w , they can be used to decompose tensors on M into spacelike and ✐ “Continuum” — 2020/1/14 — 3:36 — page 5 — ✐✐✐ ✐ ✐✐ Combining 3-momentum and kinetic energy 5timelike pieces. For instance, the 4-velocity U decomposes as(2.7) U = Λ v ( w + v ) ,in which Λ v = (cid:0) − γ ( v , v ) /c (cid:1) − / is the Lorentz factor following from thenormalization, and the 3-velocity v defined from the projection Λ v v = ←− γ · U = U · −→ γ is a vector on M that happens to be tangent to S t (and thereforeas needed could also be regarded as simply a vector on S t ).We will also need decompositions relative to U . Such decompositionsallow specification of material properties by defining quantities measured bya comoving observer. They are accomplished at any point X of M with theprojection tensors h = g + 1 c U ⊗ U , ←→ h = ←→ g + 1 c U ⊗ U ,(2.8) ←− h = δ + 1 c U ⊗ U , −→ h = δ + 1 c U ⊗ U (2.9)to a local hyperplane S U ( X ) that is the orthogonal complement of U ( X ).Consider now the dynamics of a relativistic material particle. For pur-poses of relating to the non-relativistic case, take care to distinguish betweenthe inertia-momentum vector I = m U of a particle and its total energy-momentum form , the metric dual I = m U , where U = g · U . The timelikecomponents of the 4-column and 4-row(2.10) I = (cid:18) m Λ v m Λ v v (cid:19) , I = (cid:0) − mc Λ v m Λ v v T (cid:1) gathering their components in the fiducial frame confirm the appropriatenessof this nomenclature (see Eqs. (2.7), (2.2), and (2.1); v is the 3-columngathering the components of v on S t ). Linear form and vector versions ofNewton’s Second Law for particles on M read(2.11) dd τ I = Υ , dd τ I = −→ Υ in terms of the 4-force linear form Υ and vector −→ Υ = Υ · ←→ g (that is, Υ µ = Υ α g αµ ). The 4-force vector can be decomposed relative to either U or w ,(2.12) −→ Υ = θc U + −→ f = Θ c w + −→ F ,with heating rates per baryon and 3-force vectors θ , −→ f or Θ, −→ F measuredby comoving or fiducial observers respectively, projected out by contraction ✐ “Continuum” — 2020/1/14 — 3:36 — page 6 — ✐✐✐ ✐ ✐✐ U , −→ h or w , −→ γ . Note that θ = − U · −→ Υ = 0 for an ‘elementary’ particleof constant rest mass m .Turning to a material continuum, its classical mechanics on M are gov-erned by the spacetime constraints ∇ · N = 0,(2.13) ∇ · T = n −→ Υ (2.14)on the baryon number flux vector N and total inertia-momentum flux tensor T , a (2, 0) tensor, where ∇ is the spacetime covariant derivative. The 4-velocity U of the material continuum is defined by its alignment with thebaryon number flux N :(2.15) N = n U ,where n is the baryon number density measured by a comoving observer.Referring to the definition above of I = m U as the inertia-momentum perparticle , T can be decomposed relative to U as(2.16) T = n U ⊗ I − −→ Σ ,where the index-raised 4-stress −→ Σ = Σ · ←→ g (that is, Σ µν = Σ µα g αν ) is(2.17) −→ Σ = − n U ⊗ ǫc n U − n U ⊗ q c n − q c n ⊗ n U + −→ ς .These last two equations suggest the flux of inertia per baryon; note inparticular the factors of c . The internal energy density ǫ and 3-flux q , and3-stress −→ ς , are measured by a comoving observer and projected out bycontractions with U , −→ h . Alternatively, −→ Σ can be decomposed as(2.18) −→ Σ = − w ⊗ Ec w − w ⊗ Q c − Q c ⊗ w + −→ σ in terms of internal energy density E and 3-flux Q , and 3-stress σ , measuredby a fiducial observer and projected out by contractions with w , −→ γ .Decomposed to 3 + 1 dimensions and in the limit c → ∞ , Eqs. (2.13) and(2.14) reduce to various non-relativistic formulations. These are obtained Here the baryon mass m may reflect a variable average, or may be set to a ref-erence constant, with energy associated with nuclear composition changes includedin ǫ [18]. ✐ “Continuum” — 2020/1/14 — 3:36 — page 7 — ✐✐✐ ✐ ✐✐ Combining 3-momentum and kinetic energy 7with the decompositions in Eqs. (2.7), (2.12) and (2.15)-(2.18), and pro-jections along w or U and perpendicular to them using Eqs. (2.4)-(2.5) or(2.8)-(2.9) respectively. Note also the decomposition(2.19) ∂ ( ) ∂t = w · ∇ , D ( ) = −→ γ · ∇ ( )relative to w of the spacetime covariant derivative on M , where D is thecovariant derivative on S t associated with its Euclidean 3-metric γ . Alter-natively(2.20) d( )d τ = U · ∇ ( ), D ( ) = −→ h · ∇ ( )is the decomposition relative to U . In the non-relativistic limit as c → ∞ ,(2.21) d( )d τ → d( )d t , D ( ) → D ( ) − t ( v · D )( ),where(2.22) d( )d t = ∂ ( ) ∂t + v · D ( ).is the non-relativistic material derivative.
3. Galilei/Newton spacetime, baryon conservation, and massconservation
While Galilei/Newton spacetime G —the non-relativistic analogue ofMinkowski spacetime M —has a qualitatively distinct geometric character,in many ways it can be understood as the c → ∞ limit of the latter [13]. Theabsolute object on M governing causality is the metric g (with inverse ←→ g ),which embodies the lightcones. As c → ∞ these lightcones are ‘pressed down’into fixed spatial hyperplanes S t with a unique linear form field t embodyingabsolute time. A spacetime metric no longer makes sense (see Eq. (2.1))— G is not a pseudo-Riemann manifold—but the inverse metric ←→ g limits sen-sibly to the degenerate inverse ‘metric’ ←→ γ (compare Eqs. (2.1), (2.6), and(2.4)), whose Galilei invariance is the remnant of Lorentz invariance thatsurvives the limit. The projection tensors γ , ←− γ , and −→ γ also exist (nowregarded as separate tensors on G unrelated by spacetime metric duality),and the 4-vector field w has the same role associated with a fiducial inertialframe. While the contraction t · w = 1 still holds, the metric relationship ←→ g · t = − c − w on M degenerates to ←→ γ · t = on G . ✐ “Continuum” — 2020/1/14 — 3:36 — page 8 — ✐✐✐ ✐ ✐✐ G , withtangent vector field U . This 4-velocity is still related to baryon number fluxby Eq. (2.15), but is now related to the 3-velocity by(3.1) U = w + v .Baryon number conservation is still expressed by Eq. (2.13).Even without the notions of a (pseudo-)norm or orthogonality affordedby a spacetime metric, spacelike projections and in particular decomposi-tions with respect to timelike vector fields are still available on G , but caremust be taken to understand their geometric implications.On M the tensors ←− γ and ←− h and their siblings project to spacelike hyper-planes S t and S U ( X ) that are orthogonal to w and U respectively. The cor-responding availability of orthogonal decompositions allows some flexibilityin how these are expressed—various combinations of up and down indices,without information loss.On G the situation is different. Absolute time means that S t are the onlyspacelike hypersurfaces, and the degeneracy of ←→ γ means that projectionsto S t are not unique; ←→ h and its siblings also project to S t . While on G the tensors ←− γ , −→ γ and ←− h , −→ h with vanishing contractions with w and U respectively can be expressed ←− γ = δ − w ⊗ t , −→ γ = δ − t ⊗ w ,(3.2) ←− h = δ − U ⊗ t , −→ h = δ − t ⊗ U ,(3.3)their siblings(3.4) ←→ h = ←→ γ , h = γ − t ⊗ v − v ⊗ t exist but cannot be expressed in terms of a spacetime metric or identitytensor. (Note Eq. (2.3); and also, that while linear forms on M such as U dual to vectors with time components do not exist on G , the particularcombination(3.5) − c U = Λ v (cid:18) t − c v (cid:19) → t does limit sensibly as c → ∞ . The index-lowered v is v = γ · v .) Due to theidentity tensor in Eqs. (3.2) and (3.3), information-preserving decomposi-tions of (1, 1) tensors are possible on G , while projections involving (2, 0) or(0, 2) tensors entail information loss. ✐ “Continuum” — 2020/1/14 — 3:36 — page 9 — ✐✐✐ ✐ ✐✐ Combining 3-momentum and kinetic energy 9Decomposition of Eq. (2.14) on G for inertia-momentum balance pro-vides an instructive example. In the c → ∞ limits of Eqs. (2.12) and (2.17)-(2.18), the index-raised 4-force −→ Υ and 4-stress −→ Σ lose their timelike compo-nents. The index-raising −→ Σ = Σ · ←→ g on M that limits to −→ Σ = Σ · ←→ γ on G (and similarly for Υ ) has a projective character that nullifies information oninternal energy and heating. Spacelike projections of Eqs. (2.16) and (2.14)give the usual non-relativistic momentum balance. But the only timelikeprojection available on G —contraction with t —produces(3.6) ∇ · ( m N ) = 0,which together with Eq. (2.13) implies the conservation of mass that heldsway until Einstein.Thus inertia of a continuum, represented in the (2, 0) tensor T , has beendecoupled from its energy in the passage from M to G . The apparent conse-quence, long assumed, has been that a complete picture of the energy of acontinuum on G requires the First Law of Thermodynamics as an indepen-dent postulate (e.g. [2]).
4. A more unified view of classical mechanics on Minkowskiand Galilei/Newton spacetimes
While the inertia-momentum vector I = m U of a particle exists on G , thetotal energy-momentum form I = m U does not because of the absence of aspacetime metric (note that the first and second equations of Eq. (2.10) doand do not make sense respectively as c → ∞ ). Thus at first glance it looksas though the vector version of Newton’s Second Law in Eq. (2.11) can existon G , but the linear form version cannot.However, information on internal energy and external heating need notbe regarded as completely lost in the passage from M to G . To motivate thisI introduce the concepts of relative 4-velocity V and relative 4-momentum P of a particle as a vector and linear form respectively on both M and G .Give the relative 4-velocity the unified definition(4.1) V = U − w ,which from Eqs. (2.7) and (3.1) results in the more specific expressions V = (Λ v − w + Λ v v (on M ),(4.2) V = v (on G ).(4.3) ✐ “Continuum” — 2020/1/14 — 3:36 — page 10 — ✐✐✐ ✐ ✐✐
10 Christian Y. CardallDefine the relative 4-momentum as P = − mc (Λ v − t + m Λ v v (on M ),(4.4) P = − m γ ( v , v ) t + m v (on G ),(4.5)where γ · U = Λ v v on M and v on G . While the relation P = m V = m g · V on M (see Eqs. (4.2) and (2.3)) does not exist on G , Eq. (4.5)is the perfectly sensible c → ∞ limit of Eq. (4.4). Thanks to the constancyof t on M , the dynamical law(4.6) d P d τ = Υ for an ‘elementary’ particle of constant mass m is equivalent to the linearform version of Eq. (2.11). This equation also applies on G , where thanksto Eq. (3.5) and (2.3) the 4-force linear form limits to(4.7) Υ = − θ t + f = − Θ t + F .Thus on G , contraction of Eq. (4.6) with ←− γ gives Newton’s Second Law(4.8) d p d t = F in terms of the non-relativistic 3-momentum p = m v . Also on G , contractionof Eq. (4.6) with U vanishes, and contraction with V or w gives the Work-Energy Theorem(4.9) d e v d t = F · v in terms of the particle kinetic energy e v = m γ ( v , v ) /
2. (Beware that unliketheir vector counterparts −→ F = −→ f since ←→ γ = ←→ h on G , the linear forms F = Υ · ←− γ and f = Υ · ←− h are not equal!)The case of a material continuum is a straightforward generalization. Inits natural (1, 1) incarnation, the 4-stress of Eqs. (2.17) and (2.18) limits tothe alternative decompositions Σ = n U ⊗ ǫn t + q ⊗ t + ς (4.10) = w ⊗ E t + Q ⊗ t + σ (4.11)on G . As a (1, 1) tensor, information on internal energy density and fluxsurvive the c → ∞ limit. A unified 4-dimensional version of Newton’s Second ✐ “Continuum” — 2020/1/14 — 3:36 — page 11 — ✐✐✐ ✐ ✐✐ Combining 3-momentum and kinetic energy 11Law for an infinitesimal continuum element on M and G , including both anexternal force and internal stresses, reads(4.12) n dd τ (cid:18) Π n (cid:19) = n Υ + ∇ · Σ with Π = n P . Thanks to Eqs. (2.13), (2.15), and (2.20) this is equivalentto(4.13) ∇ · S = n Υ ,where(4.14) S = n U ⊗ P − Σ is the (1, 1) relative energy-momentum flux tensor ; compare Eq. (2.16). Onboth M and G , contraction of Eq. (4.12) or (4.13) with ←− γ , U , V , and w respectively yield balance of 3-momentum, internal energy (First Lawof Thermodynamics), kinetic energy (Work-Energy Therorem), and inter-nal+kinetic energy (in conservative form). On G , the first three respectivelyturn out to be the familiar non-relativistic relations n dd t (cid:16) π n (cid:17) = n F + D · σ ,(4.15) n dd t (cid:16) ǫn (cid:17) = n θ − D · q + σ : Dv ,(4.16) n dd t (cid:16) ǫ v n (cid:17) = n F · v + ( D · σ ) · v ,(4.17)where π = n p is the 3-momentum density and ǫ v = n e v is the bulk kineticenergy density, while σ is the Cauchy 3-stress, defined here as a (1, 1) ten-sor field with components σ ij , so that σ : Dv = σ ab D a v b ; and finally thecontraction with w yields(4.18) ∂ǫ kin ∂t + D · ( ǫ kin v + q − σ · v ) = n ( θ + F · v ) ,where ǫ kin = ǫ + ǫ v . This last equation, for bulk kinetic plus internalenergy—that is, macroscopic and microscopic kinetic energy—also followsfrom the sum of Eqs. (4.16) and (4.17), as obtained in the traditionalapproach when one regards Eqs. (4.15) and (4.16) as independent postu-lates on E . ✐ “Continuum” — 2020/1/14 — 3:36 — page 12 — ✐✐✐ ✐ ✐✐
12 Christian Y. Cardall
5. Conclusion
Greater conceptual unity of the relativistic and non-relativistic classicalmechanics of material particles and continua is achieved by combiningkinetic energy and 3-momentum in a linear form P (particles) or (1, 1)tensor S (continua) on Minkowski and Galilei-Newton spacetimes M and G . Defining P as a linear form instead of as a vector geometrizes the deepprinciple that momentum is conjugate to displacement (a vector). Also, asnoted by Weyl [4] it is natural that force be regarded as a linear form, sothat direct contraction—without a scalar product—with displacement (orvelocity) yields work (or power).As on M , this perspective allows the First Law of Thermodynamics tobe regarded on G as a consequence of a unified dynamical law, Eq. (4.12) or(4.13), rather than an independent postulate. Perhaps long familiarity withthe luxury of a spacetime metric and insufficient attention to a thoroughlygeometric perspective on the non-relativistic case have led to this possibilitybeing long overlooked.Nevertheless, this viewpoint retains some limitations inherent to the non-relativistic case. While the total (internal + bulk) kinetic energy is governedby Eq. (4.13), inertia remains separate and is governed by the more familiarEq. (2.14) with different implications on M and G . Moreover the definitionsof the relative energy-momentum form and tensor P and S , as indicatedby the adjective ‘relative,’ depend on the selection of a (family of) fiducialframes associated with w (cf. Eq. (4.1)). Thus, while P and S are tensorson M and G , as tensors defined in terms of a family of fiducial observers w their timelike components with respect to other frames do not manifest thenon-relativistic transformation rule for kinetic energy.In connection with this dependence of P (and therefore also S ) on theselection of w , it is worth mentioning again two works cited in Sec. 1 regard-ing the non-relativistic case. Exploring non-relativistic covariance in fourdimensions, Duval and K¨unzle found the internal 4-stress tensor Σ , the partof S that does not depend on a choice of reference observer w (see Eqs. (4.10)and (4.14)). Exhibiting the transformation of the non-relativistic bulk kineticenergy requires an additional dimension, as emphasized by de Saxc´e andVall´ee [15–17]; from the perspective of the present work, the extra dimensionin effect allows for variation of the reference observer w . In fact, the 4 × S appears in Chapter 12 ofRef. [16], and as a submatrix of the 4 × ✐ “Continuum” — 2020/1/14 — 3:36 — page 13 — ✐✐✐ ✐ ✐✐ Combining 3-momentum and kinetic energy 13 S in geometric terms and motivates its existence on G as an instantly rec-ognizable c → ∞ limit of an easily understood tensor on M (see Eqs. (4.4),(4.5) and (2.17), (4.10) for expressions on M and G , which enter a unifiedEq. (4.14)).Curved spacetime generalizations can be examined by allowing for non-constant fiducial fields associated with the 3 + 1 foliation ( t and w in thepresent work).A final remark is that, as on M [2], a free particle on G can be given aHamiltonian but not Lagrangian formulation. The free particle energy on G can be expressed e v = P · U = P · I /m , corresponding to the Hamiltonian(5.1) H = P · w + 1 m P · ←→ γ · P yielding the expected canonical relation(5.2) d X d τ = ∂ H ∂ P = w + 1 m P · ←→ γ = U .The absence of a corresponding Lagrangian formulation is signaled bydet ( ∂ H /∂P µ ∂P ν ) = det ( γ µν ) = 0. This is a reminder of the range of possi-bilities allowed by a symplectic view of physics on spacetime [19, 20]: thereis more to life, and perhaps to nature, than Lagrangians on pseudo-Riemannmanifolds. Acknowledgments
This work was supported by the U.S. Department of Energy, Officeof Science, Office of Nuclear Physics under contract number DE-AC05-00OR22725.
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Physics Division, Oak Ridge National Laboratory,Oak Ridge, TN 37831-6354, USA
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