Localizing Energy in Fierz-Lanczos theory
aa r X i v : . [ g r- q c ] A p r Localizing Energy in Fierz-Lanczos theory
Jacek Jezierski ∗ and Marian Wiatr † Department of Mathematical Methods in Physics,Faculty of Physics, University of Warsaw,ul. Pasteura 5, 02-093 Warsaw, PolandJerzy Kijowski ‡ Center for Theoretical Physics,Polish Academy of Sciences,Al. Lotnikw 32/46, 02-668 Warsaw, PolandApril 8, 2020
Abstract
We calculate energy carried by the massless spin-2 field using Fierz-Lanczos represen-tation of the theory. For this purpose Hamiltonian formulation of the field dynamic isthoroughly analyzed. Final expression for the energy is very much analogous to the Maxwellenergy in electrodynamics (spin-1 field) and displays the locality property. Known as a“super-energy” in gravity theory, this quantity differs considerably from the well understoodgravitational field energy (represented in linear gravity by the quadratic term in Taylorexpansion of the A.D.M. mass) which cannot be localized.
Linear gravity is a gauge-type field theory. The spacetime metric is split into a fixed “backgroundmetric” g µν and a “small perturbation” h µν playing a role of the configuration variable andadmitting gauge transformations: h µν −→ h µν + £ ξ g µν , (1)where the Lie derivative with respect to the vector field ξ describes an “infinitesimal coordinatetransformation” x µ → x µ + ξ µ ( x ). Linearized Einstein equations are second order differentialequations imposed on the metric variable h µν .A substantial, technical simplification of the theory is obtained if we formulate it in terms ofgauge-invariants. In case of the flat Minkowski background, an elegant gauge-invariant formula-tion is obtained in terms of components of the (linearized) Weyl tensor W λµνκ , i.e. the tracelesspart of the (linearized) curvature tensor R λκµν = ∇ µ Γ λκν − ∇ ν Γ λκµ , (2) ∗ E-mail: [email protected] † E-mail: [email protected] ‡ E-mail: [email protected] g + h :Γ λµν = 12 g λκ ( h κµ ; ν + h κν ; µ − h µν ; κ ) , (3)whereas both “ ∇ ” and “;” denote covariant derivative with respect to the background geometry g (see e.g. [1], [2] and [3]).Due to metricity condition (3), Riemann tensor satisfies the following identities : R λµνκ = − R µλνκ = − R λµκν = R νκλµ , (5) R [ λµνκ ] = 0 . (6)First identity leaves 21 independent components, so the Riemann tensor has 20 independentcomponents. Half of them is carried by the Ricci tensor R µν := R λµλν , (7)which is symmetric (again – due to metricity of the connection). Hence, the traceless part ofthe Riemann tensor: W λκµν = R λκµν −
12 ( g λµ R κν − g λν R κµ + g κν R λµ − g κµ R λν ) + 16 R ( g λµ g κν − g λν g κµ ) , (8)called Weyl tensor, has 10 independent components. The complete list of its identities is: W λµνκ = − W µλνκ = − W λµκν = W νκλµ , (9) W [ λµνκ ] = 0 , (10) W λµλκ = 0 . (11)It can be proved that the gauge-invariant content of linearized Einstein equations is equivalentto the “contracted 2-nd type Bianchi”: ∇ λ W λµνκ = 0 . (12)In particular, the existence of the metric field h µν , such that all the quantities arising here canbe obtained by its appropriate differentiation, is guaranteed by (12).Spin-two-particle quantum mechanics can also be formulated in a similar language (cf. [4]).Originally, the particle’s “wave function” is described by the totally symmetric, fourth order spin-tensor. However, there is a one-to-one correspondence between such spin-tensors and tensors W λµνκ satisfying identities (9–11) (the transformation between the two pictures can, e.g., befound in [5]). Moreover, evolution of a massless particle is governed by the same field equation(12). In this representation, the theory is often referred to as the Fierz-Lanczos theory. Here, Note that for tensors fulfilling (9), identity (6) is equivalent to first-type Bianchi identity: R λ [ µνκ ] = 0 . (4) More precisely, gauge-invariant part of vacuum Einstein metric h is equivalent to spin-2 field W , see [1] andTheorem 1 (formulae 2.15) in [2]. However, one has to remember that the operator h W [ h ] has non-trivialkernel which includes ‘cosmological solutions’. Typical example (in spherical coordinates) is h = r ( dt + dr )which corresponds to linearized de Sitter metric. It gives W [ h ] = 0 but its (linearized) Ricci is not vanishing. a priori no metrichere!) but are a straightforward consequence of the transformation from the spinorial to thetensorial language.Fierz-Lanczos theory can also be derived from a variational principle and the correspond-ing “potentials” are known as Lanczos potentials [6]–[17]. In the present paper we propose asubstantial simplification of this theory on both the Lagrangian and the Hamiltonian levels.Finally, we calculate the field energy equal to the value of the field Hamiltonian and prove itslocal character. This means that if the region V = V ∪ V is a union of two disjoint regions V and V then the corresponding field energies sum up: E V = E V + E V . (13)Our main result is: the energy of the Fierz-Lanczos field is entirely different from the wellunderstood (A.D.M.)-energy of the gravitational field. Linear expansion of the field dynamicsin a neighbourhood of the background metric g µν corresponds to the quadratic expansion of theA.D.M. energy (“mass”) which has been calculated by Brill and Deser (see [7]). Anticipatingresults which will be presented in the next paper, let us mention that gravitational energy cannotbe localized: identity (13) cannot be valid in gravity theory because the gravitational interactionenergy between the two energies (masses) has to be taken into account on the right-hand-side .We conclude that linear gravity and the Fierz-Lanczos theory differ considerably. They canbe described by the same field W and the same field equations (12), but the corresponding phasespaces carry entirely different canonical (symplectic) structures. Consequently, energy carriedby the field is entirely different in both theories. Graviton is not a simple “massless spin-twoparticle”. Quantum mechanics of a spin-two particle can be written either in the spinor or in the tensorlanguage. The relation between the two equivalent formalisms can be found e.g. in the Taubpaper [5]. Here, we shall use the tensor formalism. This means that the field configuration isdescribed by the “Weyl-like” tensor fulfilling identities (9–11) typical for the Weyl tensor of ametric connection.In what follows we describe properties of the theory on a flat four-dimensional Minkowskispace (signature ( − , + , + , +)) whose metric g µν is used to rise and lower tensor indices.Weyl-like tensor W can be nicely described in a (3+1)-decomposition. Denoting by t = x thetime variable and by ( x k ) , k = 1 , ,
3, the remaining space variables , 10 independent componentsof W are uniquely described by two three-dimensional symmetric, traceless tensors (cf. [16],[18]): D kl = W k l , B ji = 12 ε jkl W ikl . (14)Trace D ij g ij vanishes due to identity (11), whereas (4) implies vanishing of B ij g ij . Antisym-metric part of B is given by W kkl , so it vanishes because Weyl tensor is traceless. In Cartesian known in gravity theory as one of the so called “super-energies” This observation does not contradict the so called quasi-localization of gravitational energy. Here, we use Lorentzian linear coordinates. Similarly as in Maxwell electrodynamics, generalization to curvi-linear coordinates is obvious. ǫ jkl = √ det η mn ε jkl are equal to the correspondingcomponents of the Levi-Civita tensor ε jkl = ǫ jkl / √ det η kl because det η kl = 1.Field equations ∇ λ W λµνκ = 0 can be written in a way similar to Maxwell electrodynamics:div D = 0 , (15)div B = 0 , (16)˙ D = curl B , (17)˙ B = − curl D . (18)where “dot” denotes the time derivative ∂ . Moreover, the following differential operators ofrank 1, acting on symmetric, traceless tensor fields K ij have been introduced:(div K ) l = ∇ k K kl , (19)(curl K ) ij = 12 (cid:16) ε kli ∇ k K lj + ε klj ∇ k K li (cid:17) = ∇ k K l ( j ε kli ) . (20)It is obvious that curl K is also a symmetric, traceless tensor.For transverse-traceless tensors D i B (i.e. fulfilling constrains (15–16)), symmetrization informula (20) is not necessary because the antisymmetric part of ε kli ∇ k K lj vanishes: ε nij ε kli ∇ k K lj = ε ijn ε kli ∇ k K lj = (cid:16) g jk g nl − g jl g nk (cid:17) ∇ k K lj = ∇ k K nk − ∇ n K jj = 0 . (21) Similarly as in electrodynamics, field equations (15–18) can be derived from a variational princi-ple. For this purpose we use the following simple observation (see Appendix for an easy proof):
Lemma:
Given a symmetric, transverse-traceless field B on a 3D-Euclidean space (i.e. theCauchy surface { t = 0 } ), there is a symmetric, transverse-traceless field p such that B = curl p . (22)The field p is unique up to second derivatives ∂ i ∂ j ϕ of a harmonic function: ∆ ϕ = 0. Corollary:
Given field configuration (
D, B ) satisfying field equations (15–18) on Minkowskispacetime M , there is a symmetric, transverse-traceless field p on each Cauchy hypersurface { t = const . } which fulfills not only (22) but, moreover, D = − ˙ p . (23)The field p satisfies wave equation¨ p = ∆ p . (24) Proof.
At each hypersurface { t = const . } choose any e p satisfying (22). Due to field equationswe have:curl (cid:16) D + ˙ e p (cid:17) = curl D + ˙ B = 0 . D + ˙ e p ) differs from zero by ∂ i ∂ j ϕ , where ∆ ϕ = 0. Integratingwith respect to time, we can find α such that ˙ α = ϕ and ∆ α = 0. Whence: D + ˙ e p = ∂ i ∂ j ˙ α . We conclude that p := e p − ∂ i ∂ j α (25)fulfills (23). Taking into account that curl curl = − ∆ on symmetric, transverse-traceless fields,we obtain:¨ p = − ˙ D = − curl B = − curl curl p = ∆ p . Remark:
The object p is analogous to the vector potential A k in electrodynamics. Conditiondiv p = 0 plays a role of the Coulomb gauge. Condition (23) plays a role of the additional axialgauge A = 0, which can always be imposed on the Coulomb gauge.Similarly as in electrodynamics, we can assume that the first pair of “Maxwell equations”is satisfied a priori and derive the remaining equations from a variational principle. For thispurpose we treat p as a field potential, equations (22) and (23) as definition of D and B , andtake the following Lagrangian function : L ( p, ˙ p ) := α · D − B . (26)Indeed, we have: δ Z L = α Z ( ˙ pδ ˙ p − (curl p ) δ (curl p )) = α Z ( − Dδ ˙ p + ∆ pδp ) , (27)which implies (24) as the Euler-Lagrange equation for L . Moreover, quantity − αD = ∂ L ∂ ˙ p playsa role of the momentum canonically conjugate to p . To simplify notation, we shall skip theconstant α in what follows (e.g., using appropriate physical units in which α = 1).Formula (26) implies the following Hamiltonian density of the field: H := ( − D ) ˙ p − L = D − D − B D + B , (28)which generates the Hamiltonian field dynamics − ˙ p = δ H ∂ ( − D ) ; − ˙ D = δ H ∂p according to: δ H = DδD + Bδ (curl p ) = − ˙ pδ ( − D ) − (curl B ) δp + { boundary terms } (29) A constant α is necessary because, contrary to the case of electrodynamics, the quantity D − B does notcarry correct physical units. Actually, α must be calculated in ℓ -units – just an inverse to the cosmologicalconstant units. The physically correct value of α can be measured if we know how the field W interacts with anyrealistic field theory. Of course, the dimensional constant α could also be integrated a priori into definition of thefields D and B , but then W would not have the correct dimension of the curvature. emark: Quantity E V := Z V H (30)may be identified with amount of the field energy contained in V , provided the boundary termvanishes when integrating (29) over ∂V . For this purpose appropriate boundary conditions haveto be imposed (cf. [19]). Physically, control of boundary data ensures adiabatical insulationof the interior of V from its exterior. From the functional-analytic point of view boundaryconditions are necessary for the self-adjointness of the evolution operator (the Laplacian ∆ inour case) which guarantees the existence and uniqueness of the Cauchy problem within V .Hamiltonian description of the field evolution leads, therefore, to the phase space of initialdata parameterized by the configuration p and the canonical momentum − D . This means thatthe space carries the following symplectic structure:Ω = Z V δp ∧ δD , (31)and the Hamiltonian (30) generates field dynamics (23) – (24).Being correct from the Hamiltonian point of view, above Lagrangian version of the theoryis not satisfactory because it is not relativistic invariant. Indeed, field equations (12) are rela-tivistically invariant. Lorentz transformations of W λκµν uniquely imply transformation laws for D and B . But, like in electrodynamics, transformation law for the “Coulomb-gauged” potential p is not only non-relativistic but obviously non-local. In electrodynamics, Lorentz transforma-tions can be applied correctly to the four-potential A µ . They mix different gauges. Here, onecould relax the Coulomb gauge div p = 0 by adding a “symmetric-traceless part of a gradient”,namely: T S ( ∇ b ) ij := 12 ( ∂ i b j + ∂jb i ) − g ij ∂ k b k , (32)where b is a three-vector field. This would be an analog of the “gradient gauge” ∂ k ϕ in electro-dynamics which can be added to A k without changing the field B . If, moreover, we add ˙ ϕ to A , also the field D does not change. Unfortunately, here only divergence-free fields ∂ k b k = 0can be used in (32) if we want to keep equation curl p = B . Such a non-relativistic conditiondoes not allow us to organize both p and b into a single, local, fully relativistic object.The unique remedy for this disease which exists in the literature is the use of the so calledLanczos potentials, i.e. further relaxation of (22) and (23). The issue of energy localization will be thoroughly discussed in the next paper. Here, we limit ourselves todiscussion of the strongest possible boundary conditions: all the fields vanish in a neighbourhood of the boundary ∂V . This condition annihilates all the surface integrals arising during integration by parts. Consequently, theLaplacian operator ∆ arising here is a symmetric operator. In order to have field evolution correctly defined, itsappropriate self-adjoint extension has to be defined. For this purpose, correct boundary conditions are necessary.In case of the total field energy (i.e. when V = R ), boundary terms vanish due to the sufficiently fast fall-ofbehaviour of the field. Anticipating those results let us mention that, similarly to electrodynamics, the spin-two-particle theory admits the energy localization and the quantity (28) is a correct local energy density, whereaslinear gravity does not admit localization of energy. Lanczos potentials and the relativistic invariant variationalprinciple
Since Weyl tensor is obtained by differentiating connection coefficients Γ λµν , they are naturalcandidates for potentials describing Lanczos field. But – contrary to linear gravity – there is apriori no metric h here. Hence, what we obtain by this procedure from a generic connection: R λκµν = − Γ λκµ ; ν + Γ λκν ; µ (33)does not satisfy symmetry conditions (9) (to simplify further considerations we have lowered firstindex of the connection: Γ λµν = g λσ Γ σµν ). To produce Lanczos field we must use appropriatesymmetrization: r λκµν := R [ λκ ] µν + R [ µν ] λκ , (34)and finally eliminate traces: w αβµν := r αβµν −
12 ( r αµ η βν − r αν η βµ + η αµ r βν − η αν r βµ ) + 16 ( η αµ η βν − η αν η βµ ) r , (35)where we denoted: r αβ = r µαµβ , r = r µν η µν . (36)This object fulfills already identities (9–11) i.e. is a genuine Fierz-Lanczos field.Decomposing Γ λµν into irreducible parts, we see that only one of them enters into definition(35) of w . Taking into account its symmetry: Γ λµν = Γ λ ( µν ) , we first decompose it into thetotally symmetric part and the remaining part whose totally symmetric part vanishes:Γ λµν = Γ ( λµν ) + e Γ λµν , (37)with e Γ ( λµν ) = 0. This way 40 independent components of Γ split into 20 components of thetotally symmetric, rank 3 tensor and the remaining 20 components of e Γ. The first part dropsout from (33).Instead of e Γ, in most papers devoted to Lanczos potentials, the authors use its antisym-metrization in first indices: e A λµν := e Γ [ λµ ] ν . (38)Vanishing of the totally symmetric part of e Γ implies vanishing of the totally antisymmetricpart of the new object: e A [ λµν ] = 0. We stress, however, that both objects are equivalent: noinformation is lost during such an antisymmetrization, because there is a canonical isomorphismbetween both types of tensors. Indeed, it is easy to check that the inverse transformation (from e A to e Γ) is given by the symmetrization operator: e Γ λµν = 34 e A λ ( µν ) . (39)We see that (33) and (34) imply: r λκµν := − e A λκµ ; ν + e A λκν ; µ − e A µνλ ; κ + e A µνκ ; λ . (40)7inally, when passing to the Fierz-Lanczos field (35), the trace e A λ := e A λµν g µν drops out. Hence,we define the Lanczos potential as the traceless part of e A : A λµν := e A λµν − (cid:16) e A λ g µν − e A µ g λν (cid:17) . (41)This object fulfills the following algebraic identities: A λµν = − A µλν , (42) A [ λµν ] = 0 , (43) A λµµ = 0 (44)(see also [9] and [8]). It has 16 independent components, because 4 among the original 20 wascarried by the trace e A λ .The field w written explicitly in terms of A looks as follows (see [9]): w αβµν = 2 A αβ [ ν ; µ ] + 2 A νµ [ α ; β ] − ( A σ ( αµ ); σ η βν − A σ ( αν ); σ η βµ + A σ ( βν ); σ η αµ − A σ ( βµ ); σ η αν ) . (45)Let e Γ λ := e Γ λµµ . Observe that γ λµν defined as the traceless part of e Γ λµν : γ λµν = e Γ λµν − (cid:16)e Γ λ g µν − e Γ ( µ g ν ) λ (cid:17) , contains the same information as A λµν : A λµν = γ [ λµ ] ν ; γ λµν = 34 A λ ( µν ) . (46)This object fulfills the following algebraic identities: γ λµν = γ λνµ , (47) γ ( λµν ) = 0 , (48) γ λµµ = 0 , (49)and the corresponding expression for the Fierz-Lanczos field reads: w αβµν = 2 γ [ αβ ][ ν ; µ ] + 2 γ [ νµ ][ α ; β ] −
34 ( γ σαµ ; σ η βν − γ σαν ; σ η βµ + γ σβν ; σ η αµ − γ σβµ ; σ η αν ) . (50)Hence, there are two equivalent versions of potentials for the Fierz-Lanczos field. In what follows,we shall use A λµν – the version proposed by Lanczos, as being more popular in the literature. Take an invariant Lagrangian density L = L ( w ). It depends upon potentials and its firstderivatives via w , exclusively. Euler-Lagrange’a equations δLδA λµν = 0 (51)8an be written in a “symplectic” way δL ( A, ∂A ) = ∂ κ (cid:16) W λµνκ δA λµν (cid:17) = (cid:16) ∂ κ W λµνκ (cid:17) δA λµν + W λµνκ δA λµν,κ , (52)or, equivalently: ∂ κ W λµνκ = ∂L∂A λµν , (53) W λµνκ = ∂L∂A λµν,κ . (54)Canonical momentum W is a tensor density, because L was a scalar density and we can equiv-alently use tensor W , such that W = p | det g | W . These equations can be formulated in acovariant form. We observe for this purpose, that expression W λµνκ δA λµν is a vector density,so its (partial) divergence is equal to covariant divergence. Therefore, equation (52) can berewritten: δL ( A, ∂A ) = ∇ κ (cid:16) W λµνκ δA λµν (cid:17) = (cid:16) ∇ κ W λµνκ (cid:17) δA λµν + W λµνκ δA λµν ; κ . (55)But L does not contain components of A explicite but only covariant derivatives of A . Hence,we obtain field equations: ∇ κ W λµνκ = 0 , (56) W λµνκ = ∂L∂A λµν ; κ . (57)First equation is universal, but relation between w and its momentum W is implied by a specificform of the Lagrangian. Define derivative of L with respect to w by the following identity: δL = ∂L∂w λµνκ δw λµνκ . (58)The quantity ∂L∂w λµνκ belongs to the (vector) space of contravariant tensor densities. Due tothe spacetime metric g , it is equipped with the (pseudo-)Euclidean, non-degenerate structure.Splitting this vector space into a direct sum of tensors having the same symmetries as the Weyltensor and its orthogonal complement (we denote by P w and P ⊥ w , respectively, the correspondingprojections), we write ∂L∂w λµνκ = P w (cid:18) ∂L∂w λµνκ (cid:19) + P ⊥ w (cid:18) ∂L∂w λµνκ (cid:19) (59)and, consequently, δL = (cid:20) P w (cid:18) ∂L∂w λµνκ (cid:19) + P ⊥ w (cid:18) ∂L∂w λµνκ (cid:19)(cid:21) δw λµνκ = P w (cid:18) ∂L∂w λµνκ (cid:19) δw λµνκ . (60)We see that condition ∂L∂w λµνκ = P w (cid:16) ∂L∂w λµνκ (cid:17) is necessary to give an unambiguous meaning tothe definition (58): it must fulfil the same algebraic identities as w does. Whence: δL = ∂L∂w λµνκ δw λµνκ = ∂L∂w λµνκ δr λµνκ = 4 ∂L∂w λµνκ δA λµκ ; ν , W λµκν = 4 ∂L∂w λµνκ . (61)Taking (cf. [16]) L = 116 p | det g | w λµνκ w λµνκ (62)we obtain δL = 18 p | det g | w λµνκ δw λµνκ = 12 p | det g | w λµνκ δA λµκ ; ν , so finally: W λµνκ = −W λµκν = − p | det g | w λµνκ . (63) In (3+1)-decomposition the “velocity tensor” w can be represented by two 3 D symmetric, trace-less tensors , which we call E and B : E kl = w k l , B ji = 12 ǫ jkl w ikl . (64)In analogy with electrodynamics, the corresponding components of the “momentum tensor” W could be called D and H (cf. (14)), but the Lagrangian (62) implies the “constitutive equations”(63) equivalent to: D = E , H = B . It is easy to show (proof in the Appendix), that w λµνκ w λµνκ = 8 (cid:0) E − B (cid:1) = ⇒ L = 12 p | det g | (cid:0) E − B (cid:1) . (65)The Lanczos potential A , which has 16 independent components, splits into two symmetric,traceless, three-dimensional tensors P ij and S ij and two three-dimensional covectors a i and b i .The latter are defined via decomposition of the three-dimensional two-form A ij : a i = − A i , (66) b i = − ε ikl A kl ⇔ A ij = − b m ε mij , (67) For simplicity, we restrict ourselves to the flat case. This means that the Cauchy surface { t = const . } carriesthe flat Euclidean metric η kl and we use Cartesian coordinates. Consequently, components of the tensor density ǫ jkl are equal to the corresponding components of the Levi-Civita tensor ε jkl = ǫ jkl / √ det η kl since det η kl = 1.Generalization to the curved space is relatively straightforward. Introducing F λµνκ := − W λµνκ = p | det g | w λµνκ we can define D , H in a way analogous to (64): D kl := F k l and H kl := ǫ kij F lij . P and S are defined as a symmetric part of A kl and A ijk ε ij l , respectively. Antisym-metric parts of them are already given by a and b , due to identities fulfilled by A . More precisely,we have (proof in the Appendix): A kl = − P kl + 12 b j ε j kl , (68)12 A ijk ε ij l = − S kl + 12 a j ε j kl ⇔ A ijk = − S kl ε lij + 12 ( a i η jk − a j η ik ) . (69)Relation (35) between potentials A and the field w can be written in terms of these three-dimensional objects. We obtain (proof in the Appendix): E kl = w k l = − ∂ P kl + ∂ i S j ( k ε l ) ij + 34 ( ∂ l a k + ∂ k a l ) − η kl ∂ i a i , (70) B kl = 12 ε ij l w k ij = ∂ S kl + ∂ i P j ( k ε ijl ) −
34 ( ∂ l b k + ∂ k b l ) + 12 η kl ∂ i b i . (71)These relations can be written shortly as: E = − ˙ P + curl S + 32 T S ( ∇ a ) , B = ˙ S + curl P − T S ( ∇ b ) , (72)where by “ T S ( ∇ b )” we denote the traceless, symmetric part of ∇ b . Hence, in Lorentzian coor-dinates, Lagrangian density of the theory can be expressed in terms of potentials as: L = 116 p | det g | w λµνκ w λµνκ = 12 p | det g | (cid:0) E − B (cid:1) (73)= 12 ((cid:18) ˙ P − curl S − T S ( ∇ a ) (cid:19) − (cid:18) ˙ S + curl P − T S ( ∇ b ) (cid:19) ) . (74)We see, that constraints (15–16) are obtained from variation of L with respect to a and b ,whereas dynamical equations (17–18) from variation with respect to P and S . This equationsexpressed by potentials ( P, S, a, b ) have the following form:32
T S (cid:18) ∇ ( ˙ a + 12 curl b ) (cid:19) = ¨ P + curl curl P , (75)32
T S (cid:18) ∇ (˙ b −
12 curl a ) (cid:19) = ¨ S + curl curl S . (76)
In (3+1)-decomposition, Fierz-Lanczos theory shows a far reaching analogy with electrodynam-ics. The only difference is that in FL theory we have two “vector potentials” ( P and S ) insteadof one ( A k ) in electrodynamics, and two “scalar potentials” ( a and b ) instead of one ( A ) in elec-trodynamics. To clarify this structure, we show in this Section how to formulate here classicalelectrodynamics in a similar way, i.e. using two independent potentials.Conventionally, classical (linear or non-linear) electrodynamical field is described by twodifferential two-forms: f = f µν d x µ ∧ d x ν and F = F µν ǫ µναβ d x α ∧ d x β . First pair of Maxwellequations: d f = 0 and the second pair: d F = J are universal, whereas “constitutive equations”,i.e. relation between f and F depends upon a model. In particular, linear Maxwell theorycorresponds to the relation F = ∗ f , where by “ ∗ ” we denote the Hodge “star operator”.11sually, we derive the theory from the variational principle, where the first pair of Maxwellequations is assumed a priori . For this purpose we substitute: f = d A , or f µν = ∂ µ A ν − ∂ ν A µ = A ν,µ − A µ,ν in coordinate notation, where A = ( A µ ) is a four-potential one-form and A ν,µ := ∂ µ A ν . In(3 + 1)-decomposition, electric and magnetic fields are then defined by components of f :( f k ) = ~E = − ˙ ~A + ~ ∇ A , (cid:16) ǫ mkl f kl (cid:17) = ~B = curl ~A , (77)whereas inductions: ~D and ~H arise as corresponding canonical momenta. More precisely, vari-ational principle can be written as follows: δL ( A ν , A ν,µ ) = ∂ µ ( F νµ δA ν ) = ( ∂ µ F νµ ) δA ν + F νµ δA ν,µ , (78)equivalent to ∂ µ F νµ = ∂L∂A ν = J ν , F νµ = ∂L∂A ν,µ = 2 ∂L∂f µν , (79)where the components of the canonical momentum tensor F are: F k = −F k = D k = p det g mn D k , F kl = ǫ klm H m , H m = 12 ǫ mkl F kl . (80)For linear (Maxwell) theory the Lagrangian density of the theory equals: L = − p | det g | f µν f µν = 12 p | det g | (cid:0) E − B (cid:1) , (81)and, whence, F νµ = p | det g | f µν or, equivalently, F = ∗ f . Consequently, “momenta” are equalto “velocities”: D = E and H = B .In absence of currents (i.e. when J =0), both the electric and magnetic fields play a symmetricrole. This means that the Hodge-star operator “ ∗ ” is an additional symmetry of the theory and we could, as well, begin with a potential ( C µ ) = ( C , ~C ) for the dual form h = ∗ f :( h k ) = − ~B = − ˙ ~C + ~ ∇ C , (cid:16) ǫ mkl h kl (cid:17) = ~E = curl ~C . (82)Variational principle δL ( C ν , C ν,µ ) = ∂ µ ( H νµ δC ν ) = ( ∂ µ H νµ ) δC ν + H νµ δC ν,µ , (83)of the same Lagrangian density L = − p | det g | h µν h µν = 12 p | det g | (cid:0) E − B (cid:1) , (84)gives now the same field equations: ∂ µ H νµ = ∂L∂C ν = 0 , H νµ = ∂L∂C ν,µ = 2 ∂L∂ h µν , (85) In Lorentzian coordinates the Hodge operator “*” transforms: E → − B and B → E . Similarly, D → − H and H → D . D = E and H = B playing a role of the corresponding canonical momenta H = ∗ h = ∗ ∗ f = − f : H k = −H k = − p det g mn H k , H kl = ǫ klm D m , D m = 12 ǫ mkl H kl . (86)The sum of (81) and (84) would imply the theory of two independent copies of electromagneticfield, say f and e f , such that ∗ h = e f : δL = 12 [( ∂ µ F νµ ) δA ν + ( ∂ µ H νµ ) δC ν + F νµ δA ν,µ + H νµ δC ν,µ ] . (87)To have only one copy, we must impose constraint: H = ∗F . The constraint is equivalent tothe requirement that L depends only upon the sum “ f + ∗ h ” and not upon the two potentialsindependently. Indeed, due to constraint we have: F νµ δA ν,µ + H νµ δC ν,µ = F νµ δA ν,µ + ( ∗F ) νµ δC ν,µ = F νµ δ ( A ν,µ + ( ∗ C ) ν,µ )= 12 F νµ δ ( f + ∗ h ) µν . (88)Hence, for linear electrodynamics, we can take L ( A ν , C ν , A ν,µ , C ν,µ ) = 12 p | det g | (cid:0) E − B (cid:1) = − p | det g | ( f + ∗ h ) µν ( f + ∗ h ) µν = − p | det g | (d A + ∗ (d C )) µν (d A + ∗ (d C )) µν , (89)which leads to a single copy of Maxwell electrodynamics with the Faraday tensor ϕ := f + ∗ h defined in terms of the two independent four-potentials A and C : ϕ = d A + ∗ d C . (90)Moreover, F νµ = − p | det g | ϕ µν (91)and L = − p | det g | ϕ µν ϕ µν . Equation (90) in (3 + 1)-decomposition, reads: ~E = − ˙ ~A + curl ~C + ~ ∇ A , ~B = ˙ ~C + curl ~A − ~ ∇ C . (92)Unlike in the standard variational formulation of electrodynamics: 1) the variation is performedwith respect to two independent potentials: A µ and C µ , and 2) the first pair of Maxwell equationsis not imposed a priori but obtained from the variational principle. So, the complete set ofMaxwell equationsdiv D = 0 (93)div B = 0 (94)˙ D = curl B (95)˙ B = − curl D , (96)13s derived , not imposed a priori. Expressed in terms of potentials ( ~A, ~C, A , C ), these equationsread: ∇ ˙ A = ¨ ~A + curl curl ~A , (97) ∇ ˙ C = ¨ ~C + curl curl ~C . (98)The gauge group of such a theory is much bigger than the usual “gradient gauge”: it iscomposed of all the transformations of the four-potentials which do not change the value of thefield ϕ . Hence, not only “ A → A + d φ ” and “ C → C + d ψ ”, with two arbitrary functions φ and ψ but, more generally, any transformation of the type A → A + ξ ; C → C + η , (99)where the four-covector fields ξ = ( ξ µ ) and η = ( η µ ) satisfy equation:d ξ + ∗ d η = 0 . (100)It is obvious that both such d ξ and d η fulfill free Maxwell equations. In particular, the cased ξ = d η = 0 corresponds to the standard “gradient gauge”.We show in the sequel that, from the Hamiltonian point of view, such an exotic formulation ofelectrodynamics is perfectly equivalent to the standard formulation, using a single four-potential( A µ ). Field energy is defined as the Hamiltonian function generating time evolution of the field. Tocalculate its value, a (3 + 1)-decomposition has to be chosen and the Legendre transformationbetween “velocities” and “momenta” must be performed in the Lagrangian generating formula.In conventional formulation of electrodynamics we begin, therefore, with formula (78): δL = ∂ µ ( F νµ δA ν ) = ∂ ( F ν δA ν ) + ∂ k ( F νk δA ν )= ∂ ( F k δA k ) + ∂ k ( F k δA + F lk δA l )= − ∂ ( D k δA k ) + ∂ k ( D k δA + F lk δA l )= − ˙ D k δA k − D k δ ˙ A k + ∂ k ( D k δA + F lk δA l )= ˙ A k δ D k − ˙ D k δA k − δ (cid:16) D k ˙ A k (cid:17) + ∂ k ( D k δA + F lk δA l ) . (101)Putting the complete derivative δ (cid:16) D k ˙ A k (cid:17) on the left hand side, we obtain − δ (cid:16) −D k ˙ A k − L (cid:17) = ˙ A k δ D k − ˙ D k δA k + ∂ k ( D k δA + F lk δA l ) , (102)which is analogous to the Hamiltonian formula − δ ( p ˙ q − L ) = ˙ pδq − ˙ qδp in mechanics, where − ~ D is the momentum canonically conjugate to ~A and H = −D k ˙ A k − L is the Hamiltoniandensity. The boundary term ∂ k ( D k δA + F lk δA l ) is usually neglected by sufficiently strongfall-off conditions at infinity. We stress, however, that the above symplectic approach enables14ne to localize energy within a (not necessary infinite) 3D volume V with boundary ∂V . Forthis purpose we integrate (102) over V and obtain − δ H V = Z V (cid:16) ˙ A k δ D k − ˙ D k δA k (cid:17) + Z ∂V (cid:16) D ⊥ δA − F ⊥ l δA l (cid:17) , (103)where by “ ⊥ ” we denote the component perpendicular to the boundary and H V = R V H . Im-posing boundary conditions for A and for A k (components of ~A tangent to ∂V ), we obtainan infinitely dimensional Hamiltonian system generated by the Hamiltonian functional equal tothe “Noether energy” H V . Whereas controlling A k at the boundary means to control B ⊥ , thecontrol of the scalar potential A means “electric grounding” of the boundary. This is not anadiabatic insulation of the field from the external World but rather a “thermal bath”, with theEarth and its fixed scalar potential playing a role of the “thermostat”. Hence, H V is not theinternal energy of the physical system: “electro-magnetic field contained in V ”, but rather itsfree energy: the uncontrolled flow of electric charges between ∂V and the Earth plays the samerole as the uncontrolled heat flow between the body and the thermostat during the isothermalprocesses. To avoid exchange of energy between the thermostat and the system, we must in-sulate it adiabatically. For this purpose we perform an extra Legendre transformation between D ⊥ and A at the boundary (cf. [19]): D ⊥ δA = δ (cid:16) D ⊥ A (cid:17) − A δ D ⊥ and we obtain: − δ e H V = Z V (cid:16) ˙ A k δ D k − ˙ D k δA k (cid:17) + Z ∂V (cid:16) − A δ D ⊥ − F ⊥ k δA k (cid:17) , (104)where e H V = H V + Z ∂V D ⊥ A = Z V − L − D k ˙ A k + ∂ k (cid:16) D k A (cid:17) (105)= Z V − L + D k (cid:16) − ˙ A k + ∂ k A (cid:17) = Z V D k E k − L . (106)In linear Maxwell electrodynamics we obtain the standard, local, Maxwell energy density : − L + D k E k = − p | det g | (cid:0) E − B (cid:1) + p | det g | E = 12 p | det g | (cid:0) D + B (cid:1) . (107)The boundary term in (104) vanishes if we control D ⊥ and B ⊥ on ∂V . Cauchy data are,therefore, described by : 1) electric induction ~D satisfying constraints (93), and: 2) equivalenceclass of ~A modulo the gradient gauge ~ ∇ A (each class uniquely represented by the magnetic field ~B satisfying constraints (94)). These Cauchy data form the phase space of the system equippedwith the symplectic formΩ = Z V δA k ∧ δD k , (108) The time-time component of the so called “canonical” energy-momentum tensor. The time-time component of the symmetric or Maxwell energy-momentum tensor. δD ⊥ | ∂V = 0. Due to this gauge-invariance, each class of equivalent field configurations can be uniquely represented by, i.e.,the Coulomb-gauged potential e A k fulfilling the Coulomb gauge condition: div e A = 0. Such arepresentant is unique if we impose the boundary condition δ e A ⊥ | ∂V = 0. It can be proved thatboundary conditions transform the Hamiltonian (107) into a genuine self-adjoint operator e H V ,governing the field evolution on an appropriately chosen Hilbert-K¨ahler space of Cauchy datain V , and the symplectic form becomes: Ω = R V δ e A k ∧ δD k . The same conclusion may be obtained if we work directly with the field Cauchy data. To simplifynotation, we use Lorentzian coordinates ( p | det g | = 1). According to (77), we have: L = 12 (cid:16) ~E − ~B (cid:17) = 12 (cid:26)(cid:16) ~ ∇ A − ˙ ~A (cid:17) − (cid:16) curl ~A (cid:17) (cid:27) . (109)We see that A is a gauge variable because its momentum vanishes identically. Moreover,momentum canonically conjugate to 3D vector potential ~A equals: − ~D := ∂L∂ ˙ ~A = − ~E . (110)Consequently, variation of L with respect to A implies constraints: − δLδA = ∂ k D k = 0 . (111)Hence, we have: δL = D k δ (cid:16) − ˙ A k + ∂ k A (cid:17) − B k δ (cid:16) ǫ ijk ∂ i A j (cid:17) = (112) δ n D k (cid:16) − ˙ A k + ∂ k A (cid:17)o + (cid:16) ˙ A k − ∂ k A (cid:17) δD k + ∂ i (cid:16) ǫ ikj B k δA j (cid:17) − (cid:16) ǫ jik ∂ i B k (cid:17) δA j . Putting the complete divergence δ (cid:0) D k E k (cid:1) on the left-hand side, we obtain: − δ (cid:16) D k E k − L (cid:17) = ˙ A k δD k − ˙ D k δA k + ∂ i (cid:16) − A δD i + ǫ ikj B k δA j (cid:17) , (113)which finally implies (107) and (104). The boundary term vanishes if we control D ⊥ and B ⊥ = curl A k on ∂V . In Fierz-Lanczos formulation we have more potentials, but also the gauge group (99–100) ismuch bigger. In this Section we prove that – when reduced with respect to constraints – bothformulations are perfectly equivalent. Hence, the Hamiltonian formulation and the notion offield energy does not depend upon a choice of a particular variational principle. Indeed, consider16agrangian density (84) and the corresponding Euler-Lagrange equations (85):div ~D = 0 (114)div ~B = 0 (115)˙ ~D = curl ~H (116)˙ ~B = − curl ~E (117) ~D = ~E (118) ~H = ~B . (119)For fields satisfying these equations (i.e. on shell ), integration by parts implies: δ Z V L = Z V n ~Dδ (cid:16) − ˙ ~A + curl ~C + ~ ∇ A (cid:17) − ~Hδ (cid:16) ˙ ~C + curl ~A − ~ ∇ C (cid:17)o (120)= Z V n − ~Dδ ˙ ~A + curl ~Dδ ~C − ~Hδ ˙ ~C − curl ~Hδ ~A o (121)= − Z V (cid:16) ~Dδ ˙ ~A + ˙ ~Dδ ~A + ~Hδ ˙ ~C + ˙ ~Hδ ~C (cid:17) = − Z V ∂ (cid:16) ~Dδ ~A + ~Hδ ~C (cid:17) . (122)Here, we have neglected the boundary integrals. They vanish because of appropriate boundaryconditions which assure the adiabatic insulation of V . Hence, fields ~D and ~H play a roleof (minus) momenta canonically conjugate to ~A and ~C , respectively. To perform correctlyLegendre transformation and obtain the value of the Hamiltonian function, we must reduce thissymplectic structure to independent, physical degrees of freedom. For this purpose we use theHodge decomposition of the space of three-dimensional vector fields ~X into two subspaces: ~X = ~X v + ~X s , (123)where ~X v is sourceless (i.e. div ~X v = 0) and curl ~X s = 0. In particular, assuming trivialtopology of the region V , we obtain that there exist a vector field ~W and a function f such that ~X v = curl ~W and ~X s = ~ ∇ f .Putting aside all the functional-analytic issues, consider field configuration having compactboundary in V . Integrating by parts, we see that ~X v and ~X s are mutually orthogonal in theHilbert space L :( ~X v | ~Y s ) = Z V ~X v · ~Y s = 0 . From (114–115) and (118–119) we have ~D v = ~D = ~E = ~E v , (124) ~H v = ~H = ~B = ~B v . (125) The boundary conditions are necessary for the complete functional-analytic formulation of the Hamiltonianevolution. These issues (the appropriate definition of the Hilbert space of Cauchy data and the correct self-adjointextension of the Hamiltonian) will be discussed in another paper. From the functional-analytic point of view the subspace of sourceless fields is defined as the L -closure ofsmooth, sourceless fields, having compact support in V and the remaining subspace as its orthogonal complementin the Hilbert space L . ~A v and ~C v . Definea sourceless vector potential W for C v , i.e. curl W = C v . Applying again the curl to thisequation, we conclude that ¨ W = − curl curl W , i.e. (cid:3) W = 0.Now, integrating by parts and using orthogonality relations, we reduce (122) as follows: δ Z V L = − Z V ∂ (cid:16) ~Dδ ~A + ~Hδ ~C (cid:17) = − Z V (cid:16) ~Dδ ˙ ~A + ˙ ~Hδ ~C + ~Hδ ˙ ~C + ˙ ~Dδ ~A (cid:17) = − Z V n ~Dδ ˙ ~A − curl ~Dδ ~C + ~Hδ ˙ ~C + curl ~Hδ ~A o = − Z V n ~Dδ ˙ ~A v − ~Dδ curl ~C v + ~Hδ ˙ ~C v + ˙ ~Dδ ~A v o = − Z V n ~Dδ ˙ ~A v − ~Dδ curl curl ~W + ~Hδ curl ˙ ~W + ˙ ~Dδ ~A v o = − Z V n ~Dδ ˙ ~A v + ~Dδ ¨ ~W + curl ~Hδ ˙ ~W + ˙ ~Dδ ~A v o = − Z V n ~Dδ ˙ ~A v + ~Dδ ¨ ~W + ˙ ~Dδ ˙ ~W + ˙ ~Dδ ~A v o = − Z V n ~Dδ ( ˙ ~A v + ¨ ~W ) + ˙ ~Dδ ( ~A v + ˙ ~W ) o = − Z V (cid:18) ~Dδ ˙ ~ e A + ˙ ~Dδ~ e A (cid:19) = − Z V ∂ (cid:18) ~Dδ~ e A (cid:19) , where we have defined the following, source-free, field: ~ e A := ~A v + ˙ ~W .Hence, our original phase space ( ~A, ~C, ~D, ~H ) of Cauchy data, equipped with a symplecticform ω = δ ~A ∧ δ ~D + δ ~C ∧ δ ~H , reduces on shell to ( ~ e A, ~D ) with a symplectic form e ω = δ~ e A ∧ δ ~D , identical with the structure (108) derived in Section 8.1 from the conventional variationalprinciple. We see that the reduced (with respect to constraints) phase space in Fierz-Lanczos formulationcan be described by pair ( ~ e A, ~D ), where ~ e A = ~A V + ˙ ~W plays a role of the field configuration,whereas − ~D plays a role of its canonically conjugate momentum. It is, therefore equivalent tothe corresponding phase space in the conventional formulation. Hence, Legendre transformationto the Hamiltonian picture goes exactly as in Section 8.1: H = − L − ~D · ˙ ~ e A = −
12 ( E − B ) − ~D · ( ˙ ~A v + ¨ ~W ) = 12 ( B − D ) − ~D · ( ˙ ~A v − curl curl ~W )= 12 ( B − D ) + ~D ( − ˙ ~A v + curl C v ) = 12 ( B − D ) + ~D · ~E v = 12 ( D + B ) , where we used the sourceless part of the first equation in (92): ~E v = − ˙ ~A v + curl C v .Reduction of the Fierz-Lanczos Lagrangian proposed in [16] (see our formula (74)) can beobtained in a way entirely analogous to what was done above.18 .5 Symplectic reduction of the spin-2 Fierz-Lanczos theory Take L = 116 p | det g | w λµνκ w λµνκ = 12 p | det g | (cid:0) D − B (cid:1) = 12 ((cid:18) ˙ P − curl S − T S ( ∇ a ) (cid:19) − (cid:18) ˙ S + curl P − T S ( ∇ b ) (cid:19) ) . Euler-Lagrange equations (cf. (15) and (63)) implied by L read:div D = 0 (126)div B = 0 (127)˙ D = curl H (128)˙ B = − curl E (129) D = E (130) H = B . (131)For fields contained in a region V , satisfying proper boundary conditions, we can integrate δL by parts and obtain on shell : δ Z V L = Z V n Dδ (cid:16) − ˙ P + curl S + T S ( ∇ a ) (cid:17) − Hδ (cid:16) ˙ S + curl P − T S ( ∇ b ) (cid:17)o (132)= Z V n − Dδ ˙ P + curl DδS − Hδ ˙ S − curl HδP o (133)= − Z V (cid:16) Dδ ˙ P + ˙ DδP + ~Hδ ˙ S + ˙ ~Hδ ~C (cid:17) = − Z V ∂ (cid:16) DδP + ~HδS (cid:17) . (134)Hence, fields D and H play a role of (minus) momenta canonically conjugate to P and S ,respectively. However, to perform correctly Legendre transformation and obtain Hamiltonian,we must reduce this symplectic structure to independent, physical degrees of freedom. For thispurpose, we use decomposition of three-dimensional tensors of rank 2. Following Straumann(see [20]), an arbitrary 3D symmetric, traceless tensor t kl can be decomposed into three parts(called: tensor , vector and scalar parts, respectively): t kl = t tkl + t vkl + t skl , wherediv t t = 0 , tr( t t ) = 0 ; t vkl = T S ( ∇ ξ ) kl , div ξ = 0 ; t skl = f ,kl −
13 ∆ f (135)for some function f and a covector ξ . For field configuration having compact boundary in V (more generally: for fields fulfilling appropriate boundary conditions on ∂V ), the decompositionis unique and the three components: t t , t v and t s are mutually orthogonal with respect to the L -scalar product: ( t | s ) = R V t · s .From (126)-(127) and (130)-(131) we have D t = D = E = E t , H t = H = B = B t . (136)19y taking transverse-traceless part of equations (75) and (76), we have that P t and S t fulfillwave equations. So, if we define h as a tensor, such thatcurl h = S (137)than h fulfills (cid:3) h = 0, too. (Existence and uniqueness of such h is proved in Appendix A.) Thisequation is obviously equivalent to ¨ h = − curl curl h .Now, we reduce expression (134), integrating by parts and using orthogonality relations: δL = − Z V ∂ ( DδP + HδS ) = − Z V ( Dδ ˙ P + ˙ DδP + Hδ ˙ S + ˙ HδS )= − Z V ( Dδ ˙ P t + ˙ DδP t + Hδ ˙ S t + ˙ HδS t )= − Z V ( Dδ ˙ P t + ˙ DδP t + curl Hδ ˙ h − Dδ curl S t )= − Z V ( Dδ ˙ P t + ˙ DδP t + ˙ Dδ ˙ h − Dδ curl curl h )= − Z V (cid:16) Dδ ( ˙ P t + ¨ h ) + ˙ Dδ ( P t + ˙ h ) (cid:17) = − Z V (cid:16) Dδ ˙ p + ˙ Dδp (cid:17) = − Z V ∂ ( Dδp ) , where we denoted p := P t + ˙ h . Hence, our symplectic structure ( P, S, D, H ) with a symplecticform ω = δP ∧ δD + δS ∧ δH , became reduced to ( p, D ) with a symplectic form e ω = δp ∧ δD ,derived in Section 3 from our naive variational principle (cf. (31)). In this formulation the transition to the Hamiltonian picture is straightforward and gives resultsidentical with the ones obtained in Section 3. If p = P t + ˙ h is the configuration field, and − D its canonical momentum then the Legendre transformation reads: H = − L − D · ˙ p = −
12 ( E − B ) − D · ( ˙ P t + ¨ h )= 12 ( B − D ) − D · ( ˙ P t − curl curl h ) = 12 ( B − D ) + D ( − ˙ P t + curl S t )= 12 ( B − D ) + D · E t = 12 ( D + B ) , where we have used the tensor part of the first equation in (72): curl S t − ˙ P t = E t . Similarly as in electrodynamics, the energy flux can also be localized. For this purpose we definethe Poynting vector: S k = ( E × B ) k := ǫ klm E li B mi , (138)fulfilling the following identity:div S = ∂ k (cid:16) ǫ klm E li B mi (cid:17) = (cid:16) ǫ klm ∂ k E li (cid:17) B i m + E li (cid:16) ǫ klm ∂ k B i m (cid:17) = (curl E | B ) − ( E | curl B ) = − (cid:16) ˙ B (cid:12)(cid:12)(cid:12) B (cid:17) − (cid:16) E (cid:12)(cid:12)(cid:12) ˙ E (cid:17) = − ∂ (cid:18) E + B (cid:19) = − ˙ H , S + ˙ H = 0 . (139)Integrating over any volume V , we obtain˙ H V = dd t Z V H = − Z ∂V S ⊥ . (140)Hence, we are able to control the energy transfer through each portion of the boundary ∂V . In this paper we were able to calculate the amount of energy E V carried by the massless spin-twofield and contained within a space region V ⊂ R . For this purpose we have used consequentlydefinition of energy as the Hamiltonian function generating field evolution within V . A priori ,evolution within V is not unique because can be arbitrarily influenced by exterior of V . Tomake the system autonomous, we must insulate it adiabatically from this influence: appropriateconditions have to be imposed on the behaviour of the field at the boundary ∂V . Mathematically,control of boundary conditions select among possible self-adjoint extensions of the evolutionoperator (typically: the Laplace operator) a single one which is positive. Moreover, it enablesus to organize the phase space of the field Cauchy data into a strong Hilbert-K¨ahler structure,where the “well-posedness” of the initial value problem is equivalent to the self-adjointness of theevolution operator. The use of specific representations of the theory (tensorial Fierz-Lanczos versus spinorial one, symplectic reduction by means of the Straumann decomposition versus imposing “Coulomb gauge” etc.) is irrelevant in this context: two such representations areisomorphic in a strong, functional-analytic sense. This way we have shown that the theoryadmits the “local energy density” H = D + B such that E V = Z V H .
Moreover, the flux of energy through boundary can also be localized by means of the Poyntingvector (138). We stress that – contrary to the common belief – such a local character of thefield energy is rather exceptional. In particular, theories of gravitation (both the completeEinstein theory and its linearized version) do not exhibit any such “energy density”(or localflux represented by Poynting vector). Nevertheless, in both versions of the theory, energy E V and its flux can be uniquely defined by our procedure, even if the locality property (13) is notvalid. The complete functional-analytic framework of our approach will be presented in the nextpaper. Acknowledgements
This research was supported in part by Narodowe Centrum Nauki (Poland) under Grant No.2016/21/B/ST1/00940 and by the Swedish Research Council under grant no. 2016-06596 whileJJ was in residence at Institut Mittag-Leffler in Djursholm, Sweden during the Research Pro-gram: General Relativity, Geometry and Analysis: beyond the first 100 years after Einstein, 02September - 13 December 2019. 21
Existence of tensor potential for transverse-traceless tensors
Lemma 1.
Given a symmetric, transverse-traceless field B on a 3D-Euclidean space (i.e. theCauchy surface { t = 0 } ), there is a symmetric, transverse-traceless field p such that B = curl p . (141) The field p is implied by B up to second derivatives ∂ i ∂ j ϕ of a harmonic function: ∆ ϕ = 0 .Proof. Since for every k = 1 , , B • k is divergence-free, we can solve equationcurl a • k = B • k . This means that there is a matrix a ij satisfying equation: ǫ lij ∂ i a kj = B lk . (142)Each solution is given uniquely up to a gradient. This means that for any triple φ k of functions,the matrix e a kj := a kj + ∂ j φ k , is also a solution of (142). To make the matrix e a symmetric, we must fulfill three equations:0 = ǫ njk e a jk = ǫ njk ( a jk + ∂ j φ k ) , (143)or, equivalentlycurl ~φ = ~ψ , (144)where we have defined vector fields ~φ = (cid:0) φ k (cid:1) and ~ψ = (cid:0) ψ k (cid:1) , where ψ n := − ǫ njk a jk . A sufficientcondition for the solvability is: div ψ = 0. But, due to (142), we have: − div ~ψ = ∂ n ǫ njk a jk = ǫ knj ∂ n a jk = B kk = 0 , (145)and, whence, the condition is fulfilled and the solution of (144) is given uniquely, up to a gradientof a function, say ϕ . This means that φ k is given uniquely up to ∂ k ϕ . We conclude that thereis a solution of (142) which is symmetric. It is given up to ∂ j ∂ k ϕ . This non-uniqueness can beused to make the solution traceless. For this purpose we put p ij = e a ij + ∂ i ∂ j ϕ , (146)and impose condition0 = p ii = e a ii + ∆ ϕ , (147)which we solve for ϕ . This way we have p which is another solution of (142) and is: 1) symmetricand 2) traceless. But, it is also divergence-free because of the following identity:0 = ǫ nlk B lk = ǫ nlk ǫ lij ∂ i a kj = (cid:16) δ ik δ jn − δ in δ jk (cid:17) ∂ i a kj = ∂ k a kn − ∂ n a kk = ∂ k a kn . The Lemma is, therefore, proved and the solution p ij is given up to ∂ i ∂ j ϕ , where ∆ ϕ = 0.22 Square of the Weyl tensor in (3+1)-decomposition
Equalities (64): E kl = w k l , B ji = 12 ε klj w ikl imply also w k l = E kl , w kij = − B kl ε lij , w kij = B kl ε lij . (148)Weyl property: − ε γδαβ w αβµν ε µνπρ = w γδπρ implies w ijmn = − ε ijk E kl ε mnl . (149)Finally, we obtain w αβµν w αβµν = 4 w k l w k l + 2 w kij w kij + 2 w ij k w ij k + w ijkl w ijkl == 4 E kl E kl − ε lij B kl ε mij B km + ε ijm E mn ε kln ε ija E ab ε klb == 4 E kl E kl − B kl B kl + 4 E mn E mn = 8( E − B ) . C (3+1)-decomposition of the Lanczos potential
If we define P kl = − A kl ) (150) S kl = − A ij ( k ε ij l ) (151) a i = − A i (152) b i = − ε ikl A kl ⇔ A ij = − b m ε mij , (153)then we obtain A kl = A kl ) + A kl ] = − P kl + 12 ( A kl − A lk ) = (154)= − P kl + 12 ( A kl + A lk + A k l ) = − kl + 12 A lk = − P kl + 12 b m ε mkl . (155)Tensor A ij [ k ε l ] ij is antisymmetric, so there exists a vector c m such that A ij [ k ε l ] ij = c m ε mkl . Multiplying this equation by ε klm , we have A ijk ( η im η jk − η ik η jm ) = 2 c m , so c m = 12 ( A mj j − A jmj ) = − A jmj = A m = − A m = a m . A ijk ε ij l onto symmetric and antisymmetric part: A ijk ε ij l = A ij ( k ε ij l ) + A ij [ k ε ij l ] = − S kl + a j ε jkl . (156)Multiplying this equality by ε lmn leads to following result:2 A mnk = − S kl ε lmn + a [ m η n ] k . (157)Now, using (45), we can express E and B in terms of P , S , a and b : E kl = w k l = A kl ;0 − A k l + A l k − A l k ;0 − (cid:0) A i i η kl + A kl );0 η + A i ( kl ); i η (cid:1) = − P kl + 2 a ( k ; l ) − a ii η kl + ˙ P kl − ε ji ( k S l ) j ; i + 12 (cid:0) a i ; i η kl − a ( k ; l ) (cid:1) = − ˙ P kl + (curl S ) kl + 32 a ( k ; l ) − a i ; i η kl ,B kl = 12 ε ij l w k ij = 12 ε ij l (cid:0) A k j ; i − A k i ; j + A ij k − A ijk ;0 − A j );0 η ki − A m (0 j ); m η ki + A i );0 η kj + A m (0 i ); m η kj (cid:1) = − ε ij l A kj ; i − b l ; k + ˙ S kl − ε mkl ˙ a m + 12 ε ikl ˙ a i − ε ikl A m i ; m − ε ikl A mi m = ε ij l P kj ; i − ε ij l ε kjm b m ; i − b l ; k + ˙ S kl + ε ikl P mi ; m + 14 ε ikl ε nmi b n ; m − ε ikl ε nmi b n ; m = (curl P ) kl − ε ikl P mi ; m − b k ; l + 12 b i ; i η kl − b l ; k + ˙ S kl + ε ikl P mi ; m − b k ; l + 14 b l ; k = ˙ S kl + (curl P ) kl − b ( k ; l ) + 12 b i ; i η kl . References [1] J. Jezierski, General Relativity and Gravitation (1995) 821–843[2] J. Jezierski, Classical and Quantum Gravity (2002) 2463–2490[3] J. Jezierski, Classical and Quantum Gravity (2002) 4405–4429[4] Markus Fierz, ¨Uber die relativistische Theorie kr¨aftefreier Teilchen mit beliebigem Spin,Helvetica Physica Acta 12, I, 3-37 (1939); arXiv:1704.00662 [physics.hist-ph][5] A. H. Taub. Lanczos’ splitting of the Riemann tensor. Computers, Mathematics with Ap-plications, 1:377, 1975.[6] C. Lanczos. A remarkable property of the Riemann-Christoffel tensor in four dimensions.Annals of Math, 39:842, 1938;C. Lanczos. Lagrangian Multiplier and Riemannian Spaces. Reviews of Modern Physics,21:497-502, 1949;C. Lanczos. The splitting of the Riemann tensor. Rev.Mod. Phys., 34:379-389, 1962;C. Lanczos. The variation principles of mechanics. Dover Publications Inc, New York, 4thedition, 1970. 247] Dieter R Brill and Stanley Deser, Variational methods and positive energy in general rela-tivity , Annals of Physics (1968) 548-570.[8] F. Bampi and G. Caviglia. Third-order tensor potentials for the Riemann and Weyl tensors.General Relativity and Gravitation (1983) 375-386.[9] S. B. Edgar. Nonexistence of the Lanczos Potential for the Riemann Tensor in Higher Di-mensions. General Relativity and Gravitation (3) (1994) 329;S. B. Edgar. The wave equations for the lanczos tensor/spinor, and a new tensor identity.Modern Physics Letters A (1994) 479-482;S. B. Edgar and A. H¨oglund. The Lanczos Potential for the Weyl Curvature Tensor: Exis-tence, Wave Equation and Algorithms. Proceedings of the Royal Society of London. SeriesA: Mathematical, Physical and Engineering Sciences 453.1959 (1997): 835-851.F. Andersson and S. B. Edgar, Local Existence of Spinor Potentials, arXiv:gr-qc/9902080;S.B. Edgar, A. H¨oglund, Gen. Relativ. Gravit. (2000) 2307;S.B. Edgar, J.M.M. Senovilla, Class. Quantum Grav. (2004) L133;S.B. Edgar, J.M.M. Senovilla, J. Geom. Phys. (2006) 2135-2162.[10] K.S. Hammon and L.K. Norris. The affine geometry of the lanczos H-tensor formalism.Gen. Rel. Grav. (1993) 55.[11] M. Novello and A. L. Velloso. The connection between the general observers and Lanczospotential. General Relativity and Gravitation (1987) 1251;M. Novello and N. P. Neto, Einstein’s Theory of Gravity in Fierz Variables, Centro Brasileirode Pesquisas Fisicas preprint CBPF-NF-012/88 (1988);M. Novello and N. P. Neto, Theory of Gravity in Fierz Variables (The Linear Case),Fortschr. Phys. (1992) 173-194;M. Novello and R. P. Neves, arXiv:gr-qc/0204058[12] H. Takeno. On the spintensor of Lanczos. Tensor, N.S., 14:103-119, 1964.[13] R. Illge, On Potentials for Several Classes of Spinor and Tensor Fields in Curved Spacetimes,Gen. Rel. Grav. (1988) 551-564[14] P. O’Donnell, H. Pye, Electron. J. Theor. Phys. (2010) 327;[15] P. Dolan, C.W. Kim, Proc. R. Soc. Lond. A (1994) 557;P. Dolan, A. Gerber, J. Math. Phys. (2003) 3013.[16] Daniel Cartin, Linearized general relativity and the Lanczos potential, arXiv:gr-qc/9910082;Daniel Cartin, The Lanczos potential as a spin-2 field, arXiv:hep-th/0311185.[17] Ahmet Baykal, Burak ¨Unal, A derivation of Weyl-Lanczos equations,arXiv:1801.03296 [gr-qc][18] Jacek Jezierski and Szymon Migacz, The 3+1 decomposition of Conformal Yano-Killingtensors and “momentary” charges for spin-2 field , Class. Quantum Grav. (2015) 035016[19] J. Kijowski, Gen. Relativ. Gravit. (1997) 307.[20] N. Straumann, Annalen Phys.17