Hamiltonian Approach to GR – Part 1: Covariant Theory of Classical Gravity

Eur. Phys. J. C (2017) 77:329 DOI 10.1140/epjc/s10052-017-4854-1

Regular Article - Theoretical Physics

Hamiltonian approach to GR – Part 1: covariant theory of classical gravity

Claudio Cremaschini1,a, Massimo Tessarotto2,3 1 Faculty of Philosophy and Science, Institute of Physics and Research Center for Theoretical Physics and Astrophysics, Silesian University in Opava, Bezruˇcovo nám.13, 74601 Opava, Czech Republic 2 Department of Mathematics and Geosciences, University of Trieste, Via Valerio 12, 34127 Trieste, Italy 3 Faculty of Philosophy and Science, Institute of Physics, Silesian University in Opava, Bezruˇcovo nám.13, 74601 Opava, Czech Republic

Received: 6 January 2017 / Accepted: 21 April 2017 © The Author(s) 2017. This article is an open access publication

Abstract A challenging issue in General Relativity con- tonian (respectively classical and quantum) approaches are cerns the determination of the manifestly covariant con- developed. The two formulations will be referred to as theo- tinuum Hamiltonian structure underlying the Einstein field ries of Covariant Classical and, respectively, Quantum Grav- equations and the related formulation of the corresponding ity (CCG/CQG) or briefly CCG- and CQG-theory. covariant HamiltonÐJacobi theory. The task is achieved by As shown below CCG-theory is built upon the results adopting a synchronous variational principle requiring dis- presented in Refs. [5,6] about the variational formulation tinction between the prescribed deterministic metric tensor of GR achieved in the context of a DeDonder–Weyl-type g(r) ≡{gμν(r)} solution of the Einstein field equations approach and the corresponding possible realization of a which determines the geometry of the background space- super-dimensional and manifestly covariant Hamiltonian time and suitable variational fields x ≡{g,π} obeying an theory. In particular, based on a suitable identification of the appropriate set of continuum Hamilton equations, referred effective kinetic energy and the related Hamiltonian density to here as GR-Hamilton equations. It is shown that a pre- 4-scalars adopted in Ref. [6], the aim here is to prescribe requisite for reaching such a goal is that of casting the same a reduced-dimensional continuum Hamiltonian structure of equations in evolutionary form by means of a Lagrangian SF-GR, to be referred to here as Classical Hamiltonian Struc- parametrization for a suitably reduced canonical state. As a ture (CHS). The crucial goal of the paper is to show that CHS result, the corresponding HamiltonÐJacobi theory is estab- can be associated with arbitrary possible solutions of the Ein- lished in manifestly covariant form. Physical implications stein field equations corresponding either to vacuum or non- of the theory are discussed. These include the investigation vacuum conditions. In other words, this means that the same of the structural stability of the GR-Hamilton equations with Hamiltonian structure is coordinate-independent and occurs respect to vacuum solutions of the Einstein equations, assum- for arbitrary external source terms which may appear in the ing that wave-like perturbations are governed by the canon- variational potential density. ical evolution equations. Despite being intimately related to the one earlier considered in Ref. [6], the new Hamiltonian structure is achieved in fact by means of the parametrization of the correspond- 1 Introduction ing canonical state in terms of the proper time determined along arbitrary geodetics of the background metric field ten- This is the first paper of a two-part investigation dealing with sor. This feature turns out to be of paramount importance for the Hamiltonian theory of the gravitational field, and more the establishment of the corresponding canonical transforma- precisely, the one which is associated with the so-called Stan- tion and covariant HamiltonÐJacobi theories. The CHS deter- dard Formulation of General Relativity (SF-GR) [1Ð4], i.e., mined in this way is shown to be realized by the ensemble the Einstein field equations. In the second paper the corre- {xR, HR} represented by an appropriate variational canon- sponding quantum formulation will be presented. For this ical state xR ={g,π}, with both g and π being suitably purpose, in the two papers new manifestly covariant Hamil- identified tensor fields representing appropriate continuum Lagrangian coordinates and conjugate momenta and HR a corresponding variational Hamiltonian density. a e-mail: [email protected] 123 329 Page 2 of 16 Eur. Phys. J. C (2017) 77:329 √ Its basic feature is that of being based, in analogy with identifying in the action functional d ≡ d4r −|g|, with Ref. [6], on the adoption of the synchronous Hamiltonian d4r being the corresponding canonical measure expressed in variational principle for the variational formulation of GR. terms of the said parametrization and |g| denoting as usual However, in difference with the same reference, new fea- the determinant of the metric tensor g (r). In the context of tures are added which, as we intend to show, are mandatory the synchronous variational principle to GR a further require- for the construction of the corresponding canonical transfor- ment is actually included which demands that the prescribed mation and HamiltonÐJacobi theories. In particular for this field gμν(r) must determine, besides d,alsothegeomet- purpose, first, a reduced-dimensional representation is intro- ric properties of space-time. This means that gμν(r) should duced for the canonical momenta, which, however, leaves uniquely prescribe the tensor transformation laws of arbi- formally unchanged the corresponding variational Hamil- trary tensor fields, which may depend, in principle, besides tonian density. Second, an appropriate parametrization by gμν(r), both on the variational state xR and the 4-position μ μν means of a suitably defined proper time s is introduced so r ≡{r }. This requires in particular that gμν(r) and g (r), that the resulting EulerÐLagrange equations are now realized respectively, lower and raise tensor indices of the same ten- by means of Hamilton equations in evolution form. sor fields. In a similar way gμν(r) uniquely determines also Accordingly, variational and prescribed tensor fields are the standard Christoffel connections which enter both the introduced, with the prescribed ones, in contrast to the vari- Ricci tensor Rμν and the covariant derivatives of arbitrary ational fields, being left invariant by the synchronous varia- variational tensor fields. Therefore, in the context of syn- tions. In particular, the same Hamilton equations may reduce chronous variational principle to GR the approach known identically to the Einstein field equations, which are fulfilled in the literature as “background space-time picture” [8Ð10 ] by the prescribed fields, if suitable initial conditions are set. is adopted, whereby the background space-time Q4,g (r) is This occurs provided the Poisson bracket of the Hamiltonian considered defined “a priori” in terms of gμν(r),whileleav- density is a local function, i.e., it does not depend explicitly ing unconstrained all the variational fields xR ={g,π} and on proper time. In the realm of the classical theory the phys- in particular the Lagrangian coordinates g (r) ≡{gμν (r)}. ical behavior of variational fields provide the mathematical Indeed, consistent with Ref. [6], the physical interpreta- background for the establishment of a manifestly covariant tion which arises from CCG-theory exhibits a connection Hamiltonian theory of GR, and in particular the CCG-theory also with the so-called induced gravity (or emergent gravity) realized here. When passing to the corresponding covariant [11,12], namely the conjecture that the geometrical proper- quantum theory (i.e., in the present case the CQG-theory ties of space-time should reveal themselves as a mean field to be developed in the subsequent paper) variational fields description of microscopic stochastic or quantum degrees become quantum fields and inherit the corresponding tensor of freedom underlying the classical solution. In the present transformation laws of classical fields. Thanks to its intrinsic approach this is achieved by introducing the prescribed met- consistency with the principles of covariance and manifest ric tensor gμν (r) in the Lagrangian and Hamiltonian action covariance, the synchronous variational setting developed in functionals, which is held constant in the variational prin- Refs. [5,6] provides at the same time: ciples when performing synchronous variations and has to be distinguished from the variational field gμν (r).Inthis picture, gμν (r) should arise as a macroscopic prescribed Ð the natural framework for a Hamiltonian theory of clas- mean field emerging from a background of variational fields sical gravity which is consistent with SF-GR; gμν (r), all belonging to a suitable functional class. This per- Ð the prerequisite for the establishment of a covariant quan- mits one to introduce a new representation for the action func- tum theory of gravitational field which is in turn consis- tional in superabundant variables, depending both on gμν (r) tent with classical theory and SF-GR (see Ref. [7], herein andgμν (r). Such a feature, as explained above, is found to be Part 2). instrumental for the identification of the covariant Hamilto- nian structure associated with the classical gravitational field According to such an approach the 4-scalar, i.e., invariant, and provides a promising physical scenario where to develop 4-volume element of the space-time (d) entering the action a covariant quantum treatment of GR. functional is considered independent of the functional class In this reference, one has to acknowledge the fact that of variations, so that it must be defined in terms of a pre- the Hamiltonian description of classical systems is a manda- scribed metric tensor field g (r), represented equivalently tory conceptual prerequisite for achieving a corresponding either in terms of its covariant or counter variant compo- quantum description [14,15], i.e., in the case of continuum μν nent, i.e., either g (r) ≡{gμν (r)} or g (r) ≡{g (r)}.Here systems, the related relativistic quantum field theory. This r ≡{r μ} and g (r) denote, respectively, an arbitrary GR- task involves the identification of the appropriate Hamilto- frame parametrization and an arbitrary particular solution of nian representation of the continuum field, to be realized by the Einstein field equations. This is obtained therefore upon means of the following steps: 123 Eur. Phys. J. C (2017) 77:329 Page 3 of 16 329

Step #1 : Establishment of underlying Lagrangian and schemes. In fact, GR can be formulated in any GR-frame Hamiltonian variational action principles. (i.e., coordinate system) by introducing a suitable local point Step #2 : Construction of the corresponding EulerÐ transformation r μ → r μ = f μ(r) leading to a decom- Lagrange equations, realized, respectively, in terms position of this type. In particular, the 3 + 1 approach is of appropriate continuum Lagrangian and Hamiltonian convenient for purposes related, for example, to the defi- equations. nition of conventional energy-momentum tensors, thermo- Step #3 : Determination of the corresponding set of con- dynamic and kinetic values, and to provide corresponding tinuum canonical transformations and formulation of the methods of quantization [38Ð40]. The latter are exempli- related HamiltonÐJacobi theory. fied by the well-known approach developed by Arnowitt, Deser and Misner (1959Ð1962 [28]), usually referred to as The proper realization of these steps remains crucial. In ADM theory in the literature. The same theory is based on actual fact, the last target appears as a prerequisite of fore- the introduction of the so-called 3 + 1 decomposition of most importance for being able to reach a consistent for- space-time, which by construction is foliation dependent, mulation of relativistic quantum field theory for General in the sense that it relies on a peculiar choice of a family Relativity, i.e., the so-called Quantum Gravity. The con- of GR frames for which time and space transform sepa- clusion follows by analogy with Electrodynamics. In fact, rately so that space-time is effectively split into the direct as it emerges from the recent investigation concerning the product of a 1-dimensional time and a 3-dimensional space axiomatic foundations of relativistic quantum field theory subsets, respectively (ADM-foliation) [41]. Instead, differ- for the radiation-reaction problem associated with classical ent types of 2 + 2 splitting (or with double 3 + 1 and relativistic extended charged particles (see Refs. [16Ð21]), it 2 + 2 splitting) are considered, for instance, to find new is the HamiltonÐJacobi theory which naturally provides the classes of GR exact solutions [42Ð44], to develop the the- formal axiomatic connection between classical and quantum ory of geometric flows related to classical gravity, quantum theory, to be established by means of a suitable realization gravity and geometric thermodynamics [45,46], or to elab- of the quantum correspondence principle. orate some approaches based on deformation quantization Prerequisite for reaching such goals in the context of rela- of GR and modified gravity theories [47,48]. In comparison tivistic quantum field theory is the establishment of a theory with these approaches, the manifestly covariant Lagrangian fulfilling at all levels both the Einstein general covariance and Hamiltonian formulations of GR reported in Refs. [5,6] principle and the principle of manifest covariance. Such a and developed below mainly differ because, first, there is no viewpoint is mandatory in order that the axiomatic construc- introduction of foliation of space-time, so that the 4-tensor tion method of SF-GR makes any sense at all [22]. Indeed, formalism is preserved at all stages of investigation. Sec- in order that physical laws have an objective physical char- ond, in contrast to the Hamiltonian theory of GR obtained acter they cannot depend on the choice of the GR reference from the ADM decomposition [4], both Lagrangian and frame. This requisite can only be met provided all classi- Hamiltonian dynamical variables and the canonical state are cal physical observables and the corresponding mathemati- expressed in 4-tensor notation and satisfy as well the mani- cal relationships holding among them, i.e., the physical laws, fest covariance principle. Third, in the context of CCG-theory can actually be expressed in tensorial form with respect to the Hamiltonian flow associated with the Hamiltonian struc- the group of transformations indicated above. In the context ture {xR, HR} (see Eq. (4) below) is defined with respect of SF-GR the adoption of the same strategy requires there- to an invariant proper time s, and not a coordinate-time as fore the realization of Steps #1–#3 in manifest covariant in ADM theory. Finally, it must be stressed that, in such form. As far as the actual identification of Steps #1 and a context for the proper implementation of the DeDonderÐ #2 for SF-GR is concerned, the candidate is represented by Weyl formalism, besides the customary 4-scalar curvature the variational theory reported in Refs. [5,6]. The distinctive term of the EinsteinÐHilbert Lagrangian, 4-tensor (i.e., man- features of such a variational theory, which sets it apart from ifestly covariant) momenta must be adopted in the action previous Hamiltonian formulations in the literature [23Ð28], functional. This property which can be fulfilled only adopt- lie in its consistency with the criteria indicated above and the ing a synchronous variational principle is missing in the DeDonderÐWeyl classical field theory approach [29Ð37]. ADM Hamiltonian theory, where field variables and conju- Nevertheless, well-known alternative approaches exist in gate momenta are identified only after performing the 3 + 1 the literature which are based on non-manifestly covariant foliation on the EinsteinÐHilbert Lagrangian of the associ- approaches. For the purpose of formal comparison let us ated asynchronous variational principle. Despite this differ- briefly mention some of them, a detail analysis being left ence the two approaches are complementary in the sense that to future developments. For definiteness, approaches can they exhibit distinctive physical properties associated with be considered which are built upon space-time foliations, the two canonical Hamiltonian structures underlying SF-GR namely based on so-called 3 + 1 and/or 2 + 2 splitting (see again Ref. [6]). 123 329 Page 4 of 16 Eur. Phys. J. C (2017) 77:329

1.1 Goals of the paper spaces. The global prescription should include also the validity of suitable regularity properties of the corre- Going beyond the considerations discussed above, the con- sponding Hamiltonian density HR. struction of CCG-theory involves a number of questions, • GOAL #5: Identification of the gauge properties of the closely related to the continuum Hamiltonian theory reported classical GR-Lagrangian and Hamiltonian densities. in Ref. [6], which remain to be addressed. This involves pos- The related issue concerns the identification of the pos- ing the following distinct goals: sible gauge indeterminacies, in terms of suitable gauge functions, characterizing the Lagrangian and Hamilto- • GOAL #1: Reduced continuum Hamiltonian theory for nian densities. SF-GR Ð The search for a reduced-dimensional realiza- • GOAL #6: Dimensionally-normalized form of CHS. In tion of the continuum Hamiltonian theory for the Ein- particular, the goal here is to show that a suitable stein field equations, which still satisfies the principle of dimensional normalization of the Hamiltonian structure manifest covariance. In fact, as a characteristic feature of {xR, HR} can be reached so that the canonical momenta the DeDonderÐWeyl approach, in the Hamiltonian the- acquire the physical dimensions of an action, a feature ory given in Ref. [6] the canonical variables defining the required for the establishment of a quantum theory of canonical state have different tensorial orders, with the GR in terms of HamiltonÐJacobi theory. More precisely, momenta being realized by third-order 4-tensors. In con- this involves the construction of a non-symplectic canon- trast, the new approach to be envisaged here should pro- ical transformation for the GR-Hamiltonian density HR. vide a realization of the canonical state xR ≡{gμν,πμν} The issue is to show that this can be taken to be of the in which both generalized coordinates and correspond- form ing momenta have the same tensorial dimension and are ⎧ μν → μν = μν, represented by second-order 4-tensor fields. ⎨ g g g π μν → π μν = αL π μν, , • GOAL #2: Evolution form of the reduced continuum ⎩ k (1) → ≡ + = αL , Hamilton equations Ð A further problem is whether HR H R T R V k HR the same reduced continuum Hamilton equations can κ κ = c3 be given a causal evolution form, namely they can be where is the dimensional constant 16πG , L is a cast as canonical evolution equations. Since originally 4-scalar scale length to be defined, α is a suitable dimen- the continuum Hamilton equations are realized by PDE, sional 4-scalar, while T R, and V denote the correspond- this means that some sort of Lagrangian representation ing transformed effective kinetic and potential densities should be determined. Hence, by introducing a suitable defining the transformed Hamiltonian density H R. Then Lagrangian Path (LP) parametrization of the canonical the question arises whether α can be prescribed in such a μν state xR ≡{gμν,πμν} in terms of the proper time asso- way that the transformed canonical momentum π has ciated with the prescribed tensor field gμν(r) indicated the dimensions of an action. above, the corresponding continuum canonical equations • GOAL #7: Structural stability of the GR-Hamilton equa- are found to be realized by means of evolution equations tions of CCG-theory Ð The final issue concerns the study advancing in proper time the canonical state. These will of the structural stability which in the framework of CCG- be referred to as GR-Hamilton equations of CCG-theory: theory the canonical equations exhibit with respect to they generate the evolution of the corresponding canon- their stationary solutions, i.e., the solutions of the Ein- ical fields by means of a suitable canonical flow. stein equations. In fact, depending on the specific realiza- • GOAL #3: Realization of manifestly covariant contin- tion of CHS considered here, infinitesimal perturbations uum Hamilton–Jacobi theory Ð A related question which whose dynamics is governed by the said canonical evolu- arises involves, in particular, the determination of the tion equations may exhibit different stability behaviors, canonical transformation, which generates the flow cor- i.e., be stable/unstable or marginally stable, with respect responding to the continuum canonical evolution equa- to arbitrary solutions of the Einstein field equations. For tions. This concerns, more precisely, the development of definiteness, the case of vacuum solutions with a non- a corresponding HamiltonÐJacobi theory applicable in vanishing cosmological constant is treated. It is shown the context of CCG-theory and the investigation of the that the stability analysis provides a prescriptions for the canonical transformation generated by the correspond- gauge functions indicated above which characterize the ing Hamilton principal function. GR-Hamiltonian density. • GOAL #4: Global prescription and regularity properties of the corresponding GR-Lagrangian and Hamiltonian In view of these considerations and of the results already densities. The Lagrangian and Hamiltonian formulations achieved in Refs. [5,6], in this paper the attention will be should be globally prescribed in the appropriate phase focused on the investigation of GOALS #1–#7. These topics, 123 Eur. Phys. J. C (2017) 77:329 Page 5 of 16 329 together with the continuum Lagrangian and Hamiltonian the variational Hamiltonian 4-scalar density to be suitably D theories proposed in Refs. [5,6], have potential impact in the determined. Finally, Ds is the covariant s-derivative context of both classical and quantum theories of General ∂ Relativity. D α = + t (s)∇α, (6) Ds ∂s

α 2 Evolution form of Hamilton equations for SF-GR while t (s) and ∇α are respectively the tangent 4-vector to the geodesics r(s) ≡ {r μ(s)} and the covariant derivative In this section the problem of the determination of a reduced evaluated at the same position in terms of the prescribed continuum Hamiltonian theory for GR is addressed for a metric tensor gμν(r). prescribed Hamiltonian system. This is represented by the The GR-Hamilton equations are covariant with respect CHS {xR, HR} which is formed by an appropriate 4-tensor to arbitrary canonical transformations. This property implies canonical state xR and a suitable 4-scalar Hamiltonian den- that the same equations are covariant with respect to an arbi- sity HR (xR,xR(r), r, s). In particular, the target requires one trary local point transformation (LPT) which leaves invariant to find a realization of the variational canonical momentum a given space-time represented by the differential manifold in such a way that, in the corresponding reduced canonical of the type Q4,g(r) , so that the general covariance princi- state, fields and reduced momenta form a couple of second- ple and the principle of manifest covariance are necessarily rank conjugate 4-tensors. The requisite is that such a Hamil- both fulfilled by construction. tonian theory should warrant the validity of the non-vacuum The realization of the evolution form of the GR-Hamilton Einstein field equations to be achieved as realizations of suit- equations represents a requirement for the construction of a able reduced continuum Hamilton equations set in evolution corresponding manifestly covariant HamiltonÐJacobi theory form and referred to as GR-Hamilton equations of CCG- of GR. The construction of these equations is based on the theory. More precisely, these are realized by the initial-value following steps. problem represented by the canonical equations: ⎧ 2AÐStep #1: prescription of the reduced-dimensional ( ) ∂ ( , ( ), , ) ⎨ Dgμν s = HR xR xR r r s , Hamiltonian density Ds ∂πμν (s) π ( ) (2) ⎩ D μν s ∂ HR (xR ,xR (r),r,s) =− μν , Ds ∂g (s) In the first step the Hamiltonian density HR(xR,xR(r), r, s) is identified, extending the treatment given in Ref. [6]. In and the initial conditions of the type terms of the reduced canonical variables this yields ( ) ≡ (o)( ), gμν so gμν so HR (x,x, r) ≡ TR (xR,xR) + V (g,x, r), (7) (o) (3) πμν(so) ≡ πμν (so). where the effective kinetic and potential densities TR (xR,xR) Then the solution of the initial-value problem (2) and (3) and V (g,x, r, s) can be taken, respectively, of the general generates the Hamiltonian flow form ( , ) ≡ 1 π π μν, ( ) → ( ), TR xR xR κ ( ) μν xR so xR s (4) 2 f h (8) V (g,x, r, s) ≡ σ Vo (g,x) + σ VF (g,x, r, s) . which is associated with the Hamiltonian structure {xR, HR} . Here the notation is as follows. First, s denotes the proper Here f (h) and σ denote suitable multiplicative gauge time prescribed along an arbitrary geodesic curve r(s) ≡ functions which remain in principle still arbitrary at this {r μ(s)}. This is associated with the prescribed metric tensor point. More precisely, f (h) identifies an “a priori” arbi- gμν(r) of the background space-time. Second, trary non-vanishing and smoothly-differentiable real gauge function depending on the variational weight-factor h = − 1 αβ xR(s) ≡{gμν(r(s)), πμν(r(s))} (5) 2 4 g gαβ introduced in Ref. [5] and prescribed in such a way that identifies the s-parametrized reduced-dimensional varia- μν tional canonical state, with gμν(r) and πμν(r) being the cor- f (g (r)) = 1. (9) responding continuum Lagrangian coordinates and the conjugate momenta,xR(s) ≡ gμν(r(s)), πμν(r(s)) ≡ 0 being We anticipate here that the function f (h) will be shown in the corresponding prescribed state and HR(xR,xR(r), r, s) Part 2 to be identically f (h) = 1, as required by quantum 123 329 Page 6 of 16 Eur. Phys. J. C (2017) 77:329 theory of GR. Furthermore, σ denotes the additional constant for greater generality, such a representation shall be taken of gauge function σ =±1. Finally, in Eq. (8) the two 4-scalars the form

μν V (g,x) ≡ κh[g Rμν − 2], ( ) ≡ ( ( ), ( ), ( ), ) , o (10) HR s HR xR s xR r r s s (15) VF (g,x, r) ≡ hLF (g,x, r) , i.e., including also a possible explicit dependence in terms identify, respectively, the gravitational and external-field of the proper time s. Specific examples in which explicit s- source contributions defined in Ref. [5], with L being asso- F dependences may occur in the theory include: ciated with a non-vanishing stress-energy tensor. (1) Continuum canonical transformations and in particular 2BÐStep #2: Lagrangian path parametrization canonical transformations generating local or nor local point transformations (see Ref. [13]). In this case explicit In the second step we introduce the notion of Lagrangian s-dependences may arise in the transformed Hamiltonian path (LP) [14,15]. For this purpose, preliminarily the real γ density due to explicit s-dependent generating functions. 4-tensor t (g(r), r) is introduced such that identically (2) HamiltonÐJacobi theory (see Sect. 3), where in a similar α γ way the explicit s-dependence in the Hamiltonian density t (g(r), r)∇αt (g(r), r) = 0, (11) may be generated by the canonical flow. ( ) γ (( ), ) δ(( ), ) = , gγδ r t g r r t g r r 1 (3) Stability theory for wave-like perturbations where explicit = γ s-dependences may appear in the variational fields xR so that by construction t (g(r), r) is tangent to an arbitrary ( ) μ xR s (see Sect. 5). geodetics belonging to an arbitrary 4-position r ≡ {r } of the space-time Q4,g(r) [2]. Then the LP is identified with 2CÐStep #3: the reduced Hamiltonian variational principle the geodetic curve

μ μ μ μ Given these premises, in the context of CCG-theory the {r (s)}≡ r (s) ∀s ∈ R, r (s ) = r , (12) o o explicit construction of the GR-Hamilton equations (2)follows in analogy with the extended Hamiltonian theory which is solution of the initial-value problem achieved in Ref. [6]. The goal also in the present case is μ( ) in fact the development of a manifestly covariant variational dr s = tμ(s), ds (13) approach, i.e., in which at all levels all variational fields, μ( ) = μ. r so ro including the canonical variables, the Hamiltonian density, as well as their synchronous variations and the related EulerÐ Here the 4-scalar proper time s is defined along the same Lagrange equations, are expressed in 4-tensor form. To this { μ( )} 2 = ( ) μ( ) ν( ) curve r s so that ds gμν r dr s dr s . Further- end in the framework of the synchronous variational principle μ( ) μ( ) ≡ more, t s identifies the LP-parametrized 4-vector t s developed there Ð and in agreement with the DeDonderÐWeyl μ(( ( )), ( )) d ≡ ∂ t g r s r s .InEq.(13) ds ∂s identifies the ordi- approach Ð the variational functional is identified with a real nary derivative with respect to s. In the following we shall 4-scalar call implicit s-dependences the dependences on the proper time s appearing in the variational fields through the LP S (x,x) ≡ L (x,x, r, s) , parametrization of the fields. In contrast, we shall denote as R d R (16) explicit s-dependences the proper-time dependences which enter either explicitly on s itself or through the dependence with L R (x,x, r, s) being the variational Lagrangian density on r(s) ≡ {r μ(s)}. Let us now introduce the parametrization obtained replac- D μν μ L (x,x, r, s) ≡ πμν g − H (x,x, r, s) . (17) ing everywhere, in all the relevant tensor fields, r ≡ {r } with R Ds R r(s) ≡ {r μ(s)}, namely obtained identifying ( ,, , ) ⎧ Thus, L R x x r s is identified with the Legendre trans- ⎨ gμν(s) ≡ gμν(r(s)), form of the corresponding variational Hamiltonian density D μν π μν( ) ≡ π μν( ( )), H (x,x, r, s) defined above, with πμν g denoting the ⎩ s r s (14) R Ds x(r) ≡ x(r(s)). so-called exchange term. Then the variational principle associated with the functional SR (x,x) is prescribed in terms of This yields for the Hamiltonian density HR the so-called the synchronous-variation operator δ (i.e., identified with the LP-parametrization, in terms of which the reduced Hamilton Frechet derivative according to Ref. [5]), i.e., by means of equations (2) can in turn be represented. In the remainder, the synchronous variational principle 123 Eur. Phys. J. C (2017) 77:329 Page 7 of 16 329

δSR (x,x) = 0 (18) In this case, one furthermore notices that the identities ( )μν( ) = δμ D ( ) ≡ gμν s g s μ and Ds gμν s 0 hold, so that by obtained keeping constant both the prescribed state x and construction πμν(s) ≡ 0 and hence the canonical equation the 4-scalar volume element d. This yields the 4-tensor for πμν(s) (or equivalently Eq. (23)) yields for the prescribed EulerÐLagrange equations cast in symbolic form fields ⎧ δ ( ,) π ⎨ SR x x = , 1 αβ 8 G δ μν 0 Rμν − gμν(s) [g (s)Rαβ − ]= Tμν, g 2 2 (25) δ ( ,) (19) 2 c ⎩ SR x x = , δπμν 0 which coincides with the Einstein field equations. Therefore, which are manifestly equivalent to the Hamilton equations in this framework the latter are obtained by looking for a (2). These equations can be written in the equivalent Poisson- stationary solution of the GR-Hamilton equation (2), i.e., bracket representation requiring the initial conditions D ( ) = , ( , ( ), , ) , gμν(so) ≡ gμν(so), xR s [xR HR xR xR r r s ](xR ) (20) (26) Ds πμν(so) ≡ πμν(so) = 0, with [, ]( ) denoting the Poisson bracket evaluated with xR while requiring furthermore for all s ∈ I respect to the canonical variables xR, namely , ( , ( ), , ) π ( ) = . [xR HR xR xR r r s ](xR ) μν s 0 (27) ∂ ∂ ( ) ∂ ∂ ( ) = xR HR s − xR HR s . μν μν (21) ∂g ∂πμν ∂πμν ∂g Notice that, in principle, additional extrema may exist for the ∂V (g,xR (r),r,s) effective potential, i.e. such that μν = 0. One can Then, after elementary algebra, the PDE’s (20) yield the GR- ∂g (s) show that this indeed happens, for example, in the case of vac- Hamilton equations in evolution form given above by Eq. (2). uum, namely letting Tμν ≡ 0. Thus, besides gμν(s) ≡ gμν In particular, invoking Eqs. (8)Ð(10) it follows that 2 additional extrema include gμν(s) ≡−gμν and the case ∂ ( , ( ), , ) 3 V g xR r r s μν( ) = σκh(s)Rμν in which g s satisfies identically the constraint equations μν 1 μν ∂g (s) h(s) = 0 and 1 − gμν(s)g = 0. However, once the ini- π 2 1 αβ 8 G tial conditions (26) are set the stationary solution is unique. − σκgμν(s) (g (s)Rαβ − 2) − σκ Tμν, (22) 2 c2 The prerequisite for the existence of such a particular solu- where Rμν ≡ Rμν(s) and Tμν ≡ Tμν(s) denote the tion is, however, the validity of the constraint condition (24), LP-parametrizations of the Ricci and stress-energy tensors. i.e., the requirement that the GR-Hamilton equations (2)are Hence, in the case the gauge function f (h) is prescribed as autonomous. Such a property is non-trivial. In fact, it might f (h) = 1, the canonical equations (2) reduce to the single be in principle violated if non-local effects are taken into equivalent Lagrangian evolution equation for the variational account (see for example Refs. [17,21]). Analogous circum- field gμν(s) in the LP-parametrization: stance might arise due to possible quantum effects. The issue will be further discussed in Part 2. D D gμν(s) + σ h(s)Rμν Finally, for completeness, we mention also the connection Ds Ds between the reduced Hamiltonian system {xR, HR} defined π 1 αβ 8 G according to Eqs. (5) and (7) and the representation given − σgμν(s) [g (s)Rαβ − 2]−σ Tμν = 0. (23) 2 c2 in Ref. [6] in terms of the “extended” Hamiltonian system This concludes the proof that the GR-Hamilton equations {x, H} and based on the adoption of the “extended” canonical μν α (2), as well as the equivalent Lagrangian equation (23)areÐ state x ≡ g , μν . More precisely, the connection is as expected Ð both variational. obtained, first, by the prescription H = HR, and, second, α α upon identifying μν = t πμν. In fact it then follows that α 2DÐStep #4: connection with Einstein field equations πμν = tα μν, so that πμν (r) represents the projection of α μν (r) along the tangent vector tα(s) to the background The connection of the canonical equations (2) with the Ein- geodesic curve. stein theory of GR can be obtained under the assumption that the Hamiltonian density does not depend explicitly on proper 2EÐStep #5: alternative Hamiltonian structures time s, i.e., it is actually of the form As indicated above, Eq. (22) together with the GR-Hamilton HR = HR (xR,xR(r), r) . (24) equations (2) provides the required connection with the Ein- 123 329 Page 8 of 16 Eur. Phys. J. C (2017) 77:329 ⎧ μν stein field equations. In practice this means that any suitably- ⎨ V1 (g,xR(r), r, s) ≡ κhg Rμν + VF (g,x, r) , (,, ) =− κ, smooth 4-scalar function such that ⎩ Uo g x s 2 Pμν(xR) =−σκgμν(s), ∂V (g,x (r), r, s) R (32) ∂ μν( ) g s gμν (s)=gμν (s) 1 αβ with V (g,x, r) being given by Eq. (10). The present = σκRμν − σκgμν(s) (g (s)Rαβ − 2) F 2 example means that the contribution of the cosmologi- π 8 G cal constant in the Einstein field equations can also be −σκ Tμν = 0 (28) c2 interpreted as arising due to a non-vanishing canonical realizes an admissible Hamiltonian structure of GR. The momentum of the form given by Eq. (30). Alternatively, choice corresponding to Eq. (8) with the functions V (g,x) a realization of the Einstein field equations with vanish- o ≡ and V (g,x, r) prescribed according to Eq. (10) corre- ing cosmological constant ( 0) can be achieved in F ( , ( ), , ) sponds to the lowest-order polynomial representation (but terms of the potential density V g xR r r s of the it is still non-linear, and thus non-trivial) in terms of the vari- form given above by Eq. (10), while setting at the same time ational field gμν(s) for the variational Hamiltonian. However, alternative possible realizations of the Hamil- Pμν(xR) = σκgμν(s), (33) tonian structure {xR, HR} can be readily identified. In fact, once the initial conditions (26) are set, alternative possible realizations of the GR-Hamilton equations (2), leading to where now can be interpreted as an arbitrary real 4- the correct realization of the Einstein field equations, can be scalar. achieved. These are obtained introducing a transformation of the type From the previous considerations it follows, however, that if a solution of the type (30) is permitted the actual identifi- ⎧ cation of the variational potential density remains essentially ⎨⎪ gμν(s) → gμν(s), μν μν μν undetermined. It is, however, obvious that once the require- π (s) → π (s) − (s − so)P (xR), (29) ( ) ≡ ⎩⎪ ment Pμν xR 0, or equivalently the constraint condition V (g,x, r, s) → V1 (g,x, r, s) + Uo (g,x, s) . introduced above (27) are set, the transformations (29) reduce necessarily to the trivial one, namely Notice that here the function Uo (g,x, s) remains in prin- ⎧ ciple arbitrary, so that it can always be determined so that ⎨ gμν(s) → gμν(s), π μν( ) → π μν( ), the extremal value of the potential density is preserved, ⎩ s s (34) ( ,, , ) → ( ,, , ) + (,, ) , namely V (g,x, r, s) = V1 (g,x, r, s) + Uo (g,x, s).How- V g x r s V1 g x r s Uo g x s μν ever, π (s), Pμν(xR) and V1 (g,x, r, s) can always be determined so that: which leaves unaffected the CHS. As a consequence, in validity of the constraint (27), the same Hamiltonian structure remains uniquely determined. (1) the extremal momentum π μν(s) is prescribed so that

μν μν π (s) = (s − so)P (xR); (30) 3 Manifestly covariant Hamilton–Jacobi theory

μν (2) P (xR) and V1 (g,x, r, s) are such that From the results established in the previous section, it follows that, thanks to the realization introduced here for the GR-

∂ ( , ( ), , ) Hamilton equations of CCG-theory (i.e. Eq. (2)), the same V1 g xR r r s − ( ) μν Pμν xR take the form of dynamical evolution equations. This fol- ∂g (s) μν μν g (s)=g (s) lows as a consequence of the parametrization in terms of the 1 8πG = σκRμν − σκgμν(s) (R − 2) − σκ Tμν. proper time s adopted for all geodetics belonging to the back- 2 c2 ground space-time. This feature permits one to develop in a (31) standard way, in close analogy with classical Hamiltonian mechanics, the theory of canonical transformations. Given Hence, a particular possible realization which leads to a these premises, in this section the problem of constructing a functionally-different prescription of the potential den- HamiltonÐJacobi theory of GR is addressed. Such a theory sity, and hence of the same Hamiltonian structure, is should describe a dynamical flow connecting a generic phase- provided, for example, by the setting space state with a suitable initial phase-space state charac- 123 Eur. Phys. J. C (2017) 77:329 Page 9 of 16 329 ∂2 βγ , , , , terized by identically-vanishing (i.e., stationary with respect S g Pμν xR r s = determinant det ∂ ρσ ∂ = 0ismet. to s) coordinate fields and momenta. In view of the similar- g Pιξ X R xR Under such a condition the direct canonical equation (41) ity of the LP formalism for GR with classical mechanics, it βγ βγ determines g as an implicit function of the form g = is expected that also in the present context the HamiltonÐ βγ βγ g (G , Pμν,xR, r, s). Jacobi theory follows from constructing a symplectic canon- The following statement holds on the relationship between ical transformation associated with a mixed-variable gener- βγ the GR-HamiltonÐJacobi and the GR-Hamilton equations. ating function of type S(g , Pμν,xR, r, s). Accordingly, the transformed canonical state X R ≡ THM.1 – equivalence of GR-Hamilton and {Gμν, Pμν} must satisfy the constraint equations GR-Hamilton–Jacobi equations D The GR-Hamilton–Jacobi equation (42) subject to the con- Pμν(s ) = 0, (35) Ds o straint (40) is equivalent to the set of GR-Hamilton equations D μν expressed in terms of the initial canonical variables, as given G (s ) = 0, (36) Ds o by Eq. (2). which imply the Hamilton equations Proof Without loss of generality and avoiding possible mis- =[ , ( , , , )] , understandings, the compact notation S (g, P,x , r, s) will 0 Pμν K R X R X R r s (X R ) (37) R μν be used in the following proof to denote the GR-Hamilton 0 =[G , K R(X R, X R, r, s)](X ), (38) R principal function. To start with, we evaluate first the par- where K R(X R, X R, r, s) is the transformed Hamiltonian tial derivative of Eq. (42) with respect to gik, keeping both ∂ ( , , , , ) given by S g P xR r s r μ ∂gιξ and constant. This gives ( , , ) = ( , , , ) K R X R X R r HR xR xR r s ∂ ∂ S (g, P,x , s) ∂ H gβγ , R ,x , r, s + ( βγ , , , , ). ∂gik R ∂gιξ R S g Pμν xR r s (39) ∂s ∂ ∂ + S (g, P,x , r, s) = 0. (43) Thanks to Eqs. (37) and (38), the transformed Hamiltonian ∂ ∂ ik R s g (g,P) is necessarily independent of X R. As a consequence, K R identifies an arbitrary gauge function, i.e., in actual fact Then let us evaluate in a similar manner the partial derivative ∂ ( , , , ) = ( , ) S g P xR s gμν r μ K R K R xR r , which can always be set equal to zero with respect to ∂gik , keeping and constant. (K R = 0). On the other hand, canonical transformation the- This gives ory requires that it must be ∂ ∂ ( , , , ) βγ , S g P xR s , , , βγ ∂ S(g,P,x ,r,s) HR g ιξ xR r s ∂ S(g , Pμν,xR, r, s) ∂ R ∂g πιξ = , (40) ∂gik ∂gιξ ⎡ ⎤ βγ ∂ ∂ ιξ ∂ S(g , Pμν,xR, r, s) + ⎣ ( , , , , )⎦ = . = . ∂ ( , , , , ) S g P xR r s 0 (44) G (41) ∂ S g P xR r s ∂s ∂ Pιξ ∂ ik g (g,P) s Then, introducing the -parametrization, it follows that Eq. ∂ S(g,P,xR ,s) With the identification πιξ = ιξ provided by Eq. (39) yields ∂g (40) it follows that Eq. (43) becomes βγ βγ ∂ S(g , Pμν,xR, s) HR g , ,xR, r, s ∂gιξ ∂ βγ D H (g ,πιξ ,x , r) + π = 0, (45) ∂ ik R R ik ∂ βγ g Ds + S(g , Pμν,x , r, s) = 0, (42) ∂s R which coincides with the second Hamilton equation in (2). which realizes the GR-HamiltonÐJacobi equation for the To prove also the validity of the Hamilton equation for gβγ ( βγ , , , , ) mixed-variable generating function S g Pμν xR r s . we first invoke the following identity: Due to its similarity with the customary HamiltonÐJacobi ⎡ ⎤ equation, well known in Hamiltonian classical dynamics, in ∂ ∂ ⎣ ( , , , )⎦ S classical GR-Hamilton ∂ ( , , , ) S g P xR s the following will refer to the ( ) ∂ S g P xR s ∂s ∂ ik principal function. The canonical transformations generated g (g,P) βγ by S(g , Pμν,x , s) are then obtained by the set of Eqs. ∂ ∂ R = ( , , , ) ∂ ( , , , ) S g P xR s (40)Ð(42). ∂ S g P xR s ∂s ∂ ik Now we notice, in view of the discussions given above, g D βγ ∂ ∂ S (g, P,xR, s) that the inverse canonical transformation X R → xR locally − , g ∂ ( , , , ) βγ (46) Ds ∂ S g P xR s ∂g exists provided the invertibility condition on the Hessian ∂gik 123 329 Page 10 of 16 Eur. Phys. J. C (2017) 77:329 where refers to a formal difference arising between the HamiltonÐ Jacobi theory for continuum fields built on the DeDonderÐ ∂ ∂ ( , , , , ) S g P xR r s = δi δk . Weyl covariant approach and the HamiltonÐJacobi theory ∂ ( , , , , ) βγ β γ (47) ∂ S g P xR r s ∂g holding in classical mechanics for particle dynamics. This ∂gik concerns, more precisely, the dimensional units to be adopted for the Hamilton principal function S and hence also the The first term on the r.h.s. of Eq. (46) vanishes identically μν ∂ canonical momentum π . Indeed, as is well known, in parti- because ∂ S (g, P,xR, r, s) must be considered as indepen- s cle dynamics S retains the dimension of an action (and there- dent of πik. Therefore, Eq. (44)gives fore of the action functional), so that [S] = [h¯ ]. In the present −3 ∂ case instead (see Eq. (7)) one has [S] = hL¯ , namely the βγ D ik HR(g ,πιξ ,xR, r, s) − g = 0, (48) dimension of S differs from that of an action by the cubic ∂π Ds ik length L−3. This arises because for continuum fields the action functional is an integral over the 4-volume element which coincides with the Hamilton equation for gik and of the Hamiltonian density, while for particle mechanics it gives also the relationship of the generalized velocity D gik Ds is expressed as a line integral over the proper-time length. with the canonical momentum, since here no explicit s- One has to notice, however, that, first, the dimensions of dependence appears. This proves the equivalence between S may be changed by the introduction of a non-symplectic the GR-HamiltonÐJacobi and GR-Hamilton equations, both canonical transformation. This means that, by a suitable expressed in manifestly covariant form. choice of the same transformation, S can actually recover the dimension of an action. Second, the relationship between This conclusion recovers the relationship between Hamil- the Hamilton principal function S and the Hamiltonian func- ton and HamiltonÐJacobi equations holding in Hamiltonian tion itself remains in all cases the same, with the two func- Classical Mechanics for discrete dynamical systems. The tions differing by the dimension of a length (see also Sect. 4 connection is established also in the present case for the con- below). tinuum gravitational field thanks to the manifestly covari- Before concluding, the following additional remarks are ant LP-parametrization of the theory and the representa- in order: tion of the Hamiltonian and HamiltonÐJacobi equations as dynamical evolution equations with respect to the proper time s characterizing background geodetics. The physical (1) The GR-HamiltonÐJacobi description permits one to interpretation which follows from the validity of THM.1 is construct explicitly canonical transformations mapping remarkable. This concerns the meaning of the HamiltonÐ in each other the physical and virtual domains. The Jacobi theory in providing a wave mechanics description of generating function determined by the GR-HamiltonÐ the continuum Hamiltonian dynamics. This follows also in Jacobi equation is a real 4-scalar field. the present context by comparing the mathematical struc- (2) The generating function obtained in this way realizes ture of the HamiltonÐJacobi equation (42) with the well- the particular subset of canonical transformations which known eikonal equation of geometrical optics. In fact, Eq. map the physically-observable state xR into a neighbor- βγ ∂ S(g ,Pμν ,xR ,r,s) ing admissible virtual canonical state xR. (42) contains the squared of the derivative ιξ , ∂g (3) The virtue of the approach is that it preserves the validity ( βγ , , , , ) so that the Hamilton principal function S g Pμν xR r s of the Einstein equation in the physical domain. In other is associated with the eikonal (i.e., the phase of the wave), words, the canonical transformations do not affect the while the remaining contributions due to the geometrical and physical behavior. physical properties of the curved space-time formally play (4) A further issue concerns the connection between the the role of a non-uniform index of refraction in geometrical same prescribed metric tensor g(r) ≡{gμν(r)} and the optics [49]. The outcome pointed out here proves that the variational/extremal state x (s) = {g(s), π(s)}.This ( ) R dynamics of the field gμν s in the virtual domain of varia- can be obtained by establishing a proper statistical the- tional fields where the Hamiltonian structure is defined and ory achieved by considering the initial state the HamiltonÐJacobi theory (42) applies must be characterized by a wave-like behavior and can therefore be given a x (s ) = x(s ) + δx (s ) (49) geometrical optics interpretation. This feature is expected R o o R o to be crucial for the establishment of the corresponding manifestly covariant quantum theory of the gravitational as a stochastic tensor and thus endowed with a suit- field. able phase-space probability density. The topic can be An important qualitative feature must be pointed out developed in the framework of a statistical description regarding the HamiltonÐJacobi theory developed here. This of classical gravity to be discussed elsewhere in detail. 123 Eur. Phys. J. C (2017) 77:329 Page 11 of 16 329

4 Properties of CHS equation in (50)) remains in principle essentially indeter- minate. In fact the regularity condition (51) requires only In this section some properties are discussed which charac- that terize the manifestly covariant Hamiltonian theory of GR. ∂2 TR = 1 = , αβ 0 (52) 4A Ð global prescription and regularity ∂πμν∂π κ f (h) implying that the function f (h) can be realized by an This refers, first of all, to the global prescription and reg- arbitrary non-vanishing (for example, strictly positive) ularity of the GR-Lagrangian and GR-Hamiltonian densi- and suitably smooth dimensionless real function. In addi- ties L R ≡ L R(y,g, r, s) and HR ≡ HR(x,g, r, s) defined tion, in order that both T (y,g) and V (y,g) (see again according to Eqs. (17) and (7) which are associated with R Eq. (50)) are realized by means of integrable functions the corresponding Hamiltonian structure {xR, HR} indicated in the configurations space U , accordingly the functions above. For this purpose we notice that in Eq. (17)the g f (h) and 1/f (h) should be summable too. As a conse- effective kinetic and potential densities expressed in terms D μν quence the prescription of f (h) remains in principle still of the Lagrangian state y = gμν, g and the LP- Ds free within the classical theory of SF-GR developed in parametrization (20) can be taken to be of the general type Sect. 2. This means that f (h) should be intended in such f (h) D D μν a context as a gauge indeterminacy affecting the Hamil- TR (y,g) = gμν g , 2κ Ds Ds (50) tonian density H , i.e., with respect to the multiplicative ( ,, , ) ≡ σ ( ,) + σ ( ,, , ) . R V g g r s Vo g g VF g g r s gauge transformation Here, f (h) and σ identify the two distinct multiplicative ⎧ ( ,) ≡ 1 π π μν → ( ,) gauge functions introduced above (see Sect. 2), κ is the ⎨⎪ TR xR g 2κ μν TR xR g c3 1 μν 1 κ = ≡ πμνπ = TR (xR,g) , (53) dimensional constant 16πG , while the rest of the ⎪ 2κ f (h) f (h) ⎩ notations is expressed in standard form according to Refs. πμν → πμν = f (h)πμν. [5,6]. More precisely, in the second equation Vo (g,g) and VF (g,g, r, s) ≡ hLF (g,g, r, s) are defined as in Eq. (10) • (B) The second indeterminacy is related to the so- and must be expressed here in Lagrangian variables, with called multiplicative gauge transformation of the effec- the field Lagrangian L F (g,g, r, s) being prescribed accord- tive potential density V (g,g, r, s) (see again Sect. 2). ingtoRef.[5]. Furthermore,TR (y,g) identifies the generic More precisely the indeterminacy is related to the con- form of the effective kinetic density. It follows that a suffi- stant gauge factor σ =±1. cient condition for the global prescription of the canonical • (C) The third indeterminacy is related to the so-called state, i.e., the existence of a smooth bijection connecting the additive gauge transformation. Indeed, one can readily Lagrangian and Hamiltonian states is the so-called regu- show (see Ref. [6]) that L R(y,g, s) is prescribed up to larity condition of the GR-Hamiltonian (and corresponding an arbitrary additive gauge transformation of the type GR-Lagrangian) density. This requires more precisely that in the whole Hamiltonian phase space L R(y,g, r, s) → L R(y,g, r, s) + V (54)

∂2 ∂2 = HR ≡ TR = 1 = . being V a gauge scalar field of the form V αβ αβ 0 (51) DF(g,g,r,s) ( ,, ) ∂πμν∂π ∂πμν∂π κ f (h) Ds , with F g g s being an arbitrary, suitably- smooth real gauge function of class C(2) with respect to 4B Ð gauge indeterminacies of CHS the variables (g, s) (see also related discussion in Ref. [5]). As shown in Ref. [6] at the classical level the Hamilto- nian structure {xR, HR} of SF-GR remains intrinsically non- 4C Ð dimensional normalization of CHS unique, with the Hamiltonian density HR being characterized by suitable gauge indeterminacies. Leaving aside the treat- In this section it is shown that the Hamiltonian structure ment of additive gauge functions earlier discussed in Refs. {xR, HR} can be equivalently realized in such a way that xR, [5,6], these refer more precisely to the following properties: and consequently also HR, can be suitably normalized, i.e., so that to achieve prescribed physical dimensions. Granted • (A) The first one is the so-called multiplicative gauge the non-symplectic canonical nature of the transformation transformation of the effective kinetic density. To iden- indicated above ( i.e., Eq. (1)) one can always identify the 4- tify it we notice that the scalar factor f (h) appearing in scalar α with a classical invariant parameter, i.e., both frame- the prescriptions of the effective kinetic density (see first independent and space-time independent. In particular, in the 123 329 Page 12 of 16 Eur. Phys. J. C (2017) 77:329 framework of a classical treatment it should be identified with so that T R(x R,g, r, s) identifies the normalized effective the classical parameter α ≡ αClassical, being kinetic density and V by analogy is the corresponding normalized effective potential density, with V o (g,g) and ( ,, , ) αClassical = mocL > 0, (55) V F g g r s now being prescribed, respectively, in terms of Vo (g,g) and VF (g,g, r, s) as with c being the speed of light in vacuum, m a suitable rest- o μν mass (to be later identified with the non-vanishing graviton V o (g,g) ≡ hαL g Rμν − 2 , (61) mass in the framework of quantum theory of GR) and L a ( ,, , ) ≡ hαL ( ,, , ) . VF g g r s 2k L F g g r s characteristic scale length to be considered as an invariant non-null 4-scalar. Without loss of generality this can always From the canonical equations (58) it is obvious that by con- be assumed of the form struction the transformed canonical momentum π μν takes the μν dimensions of an action, i.e., [π ]=[].Theset{x R, H R} L = L(mo), (56) thus provides an admissible representation of CHS. In particular it follows that the GR-Hamilton equations in normalized with mo itself being regarded as an invariant 4-scalar. Here form become, respectively, L is regarded as a classical invariant parameter, so that it ⎧ ⎨ Dgμν πμν should remain independent of all quantum parameters, i.e., = α , Ds L (62) in particular . In addition, in view of the covariance property ⎩ Dπμν ∂V (g,g,r,s) =− ∂ μν , of the theory, whereby the choice of the background space- Ds g time Q4,g(r) is in principle arbitrary, the 4-scalars m and o with the operator D/Ds being prescribed again in terms of L should be universal constants, namely also invariant with the corresponding prescribed metric tensors gμν(r). respect to the action of local and non-local point transfor- For completeness, we mention here also the normalized mations [13]. As an example, a possible consistent choice HamiltonÐJacobi equation corresponding to the canonical for the invariant function L(m ) is realized by means of the o equations (62). This is reached introducing the correspond- so-called Schwarzschild radius, i.e., ing normalized Hamilton principal function S(g, P,g, r, s), 2m G i.e., the mixed-variable generating function for the canonical L(m ) = o . (57) o c2 transformation ( ) ≡ ( , μν) ⇔ ( ) ≡ ( ( ), π μν( )), The invariant rest-mass mo remains, however, still arbitrary xR so Gμν P x s gμν s s (63) at this level, its prescription being left to quantum theory. It μν follows that the transformed GR-Hamilton equations (2) can with xR(so) ≡ (Gμν, P ) denoting the initial canonical GR- always be cast in the dimensional normalized form state. Then S(g, P,g, r, s) is prescribed in such a way that the π μν( ) π = ⎧ normalized canonical momentum s is given by μν ∂ S(g,P,g,r,s) ⎨ Dgμν ∂ ∂ = H R ≡ HR , ∂gμν , while the initial canonical coordinate Gμν is Ds ∂πμν ∂πμν (58) determined by the inverse canonical transformation Gμν = ⎩ Dπμν ∂ α ∂ =− H R ≡− L HR , ∂ S(g,P,g,r,s) Ds ∂gμν k ∂gμν ∂Pμν . It follows that the corresponding dimensionally normalized HamiltonÐJacobi equation which is equivalent to where the transformed Hamiltonian H R identifies the nor- Eqs. (62) is provided by malized GR-Hamiltonian density ∂ S(g, P,g, r, s) ∂ S(g, P,g, r, s) + H g, π ≡ ,g, r, s ∂s R ∂g 1 H (x ,g, r, s) = T (x ,g, r, s) + V (g,g, r, s) . = 0, (64) R R f (h) R R (59) with H R being prescribed by Eq. (59).

Here the notation is as follows. First, for an arbitrary curved 4 5 Structural stability of the GR-Hamilton equations space-time (Q ,g(r)) the functions T R and V now are identified with In this section we present an application of the GR- πμν π Hamiltonian theory for the Einstein field equations developed T (x ,g, r, s) = μν , R R 2αL in this paper, which is represented by Eq. (2) or equivalently V (g,g, r, s) ≡ σ V o (g,g, r, s) + σ V F (g,g, r, s) , by the HamiltonÐJacobi equation (42). This refers to the sta- (60) bility of the GR-Hamilton equations with respect to their 123 Eur. Phys. J. C (2017) 77:329 Page 13 of 16 329 stationary solution. As shown above the latter realizes by δgμν(s) and δπμν(s) and accurate through O (ε) is obtained construction a solution of the Einstein field equations (25)in thanks also to the requirement (9): its most general form, i.e., in the presence of arbitrary exter- D μν 1 μν nal sources. Therefore the task to be addressed concerns the δg (s) = δπ (s), (70) Ds κ so-called structural stability of the GR-Hamilton equations D σ (with respect to the Einstein field equations), namely the sta- δπμν (s) = κ(R + 2)δgμν (s) Ds 2 bility of stationary solutions of the GR-Hamilton equations σ αβ assuming that the perturbed fields realize particular solutions + κ(gμν(s)Rαβ + Rμνgαβ (s))δg . (71) of the same GR-Hamilton equations. 2 As a first illustration of the problem, here we consider the In particular, introducing the representation (69) and recall- D case of arbitrary vacuum solutions realized by setting a van- ing the definition of the differential operator Ds , the first δπμν( ) ishing stress-energy tensor (Tμν = 0) and possibly retaining equation yields a unique relationship between s and δ μν( ), δπμν( ) = κ[ ω − ]δ μν( ). also a non-vanishing cosmological constant as corresponds g s namely s i c K g s Then Eq. to de Sitter (>0 ) or anti-de Sitter (<0) space-times. (71) determines the algebraic linear equation for δgμν (s): Let us address the problem in the context of the reduced ω 2 σ continuum manifestly covariant Hamiltonian theory. The −κ − K − κ[R + 2] δgμν(s) c 2 study is supported by the conclusions concerning the wave σκ αβ mechanics interpretation of the reduced continuum Hamil- = (gμν(s)Rαβ + Rμνgαβ (s))δg (s) . (72) 2 tonian dynamics of the gravitational field determined by the HamiltonÐJacobi theory. For this purpose we shall consider To solve it explicitly one needs to determine the corre- ( ) sponding algebraic equations holding for the independent perturbations of the reduced canonical state xR s which are αβ αβ tensor products gμν(s)δg Rαβ and gαβ (s)δg Rμν appear- suitably close to xR(s), namely of the form ing on the r.h.s. of the same equation. Thus, by first multi- μν μν μν g (s) = g (s) + εδg (s), (65) plying tensorially term by term Eq. (72)byRμν it follows πμν(s) = εδπμν(s), (66) ω 2 μν − − K − σ [R + ] R δgμν (s) where ε 1 is an infinitesimal dimensionless parameter c σ identifying the perturbations of the canonical fields. In par- μν αβ = Rμν R gαβ (s)δg (s) . (73) ticular, consistent with the existence of a HamiltonÐJacobi 2 theory and its physical interpretation pointed out above, we Next, multiplying tensorially term by term Eq. (72)by are authorized to consider a wave-like form of the pertur- gμν (s) one obtains instead bations. These are assumed to propagate along field geode- ω 2 μν tics, namely the same perturbations of the canonical fields − − K − σ [R + ] g (s) δgμν (s) μν (δg (s), δπμν(s)) are taken of the form c αβ μν μν = 2σ Rαβ δg (s) . (74) δg (s) = δg (g(s)) exp {G(s)} , (67) δπμν(s) = δπμν(g(s)) exp {G(s)} . (68) Combining together Eqs. (73) and (74) one is finally left with the equation Here G(s) denotes the eikonal ω 2 2 ω μ − − − σ [+ ] − μν = , G(s) = i s − iKr (s)tμ(s), (69) K R Rμν R 0 (75) c c with ω and K being 4-scalar parameters which, by con- which identifies the dispersion relation between ω and K , struction, have, respectively, the dimensions of a frequency i.e., the condition under which in the context of the reduced and that of a wave number ( i.e., the inverse of a length). continuum Hamiltonian theory the infinitesimal perturba- Therefore, denoting K ≡ 1/λ, according to the represen- tions (67) and (68) can occur. tation (69), ω and λ identify the invariant frequency and To analyze in terms of Eq. (75) the conditions for the wave-length of the wave-like perturbations of the canon- existence of stable, marginally stable or unstable oscillatory μν ical fields. The invariant character of ω and λ is a char- solutions for the canonical fields δg (s), δπμν(s) ,itis acteristic feature of the manifestly covariant Hamiltonian convenient to classify the possible complex roots for ω which theory. can locally occur. Such a classification has therefore neces- It is then immediate to show that, in terms of the canonical sarily a local character. More precisely, single roots of this evolution equations (20), the following set of linear differen- equation such that locally (a) Im(ω) < 0, (b) Im(ω) = 0 tial equations advancing in proper time the perturbed fields or (c) Im(ω) > 0 will be referred to as (locally) stable, 123 329 Page 14 of 16 Eur. Phys. J. C (2017) 77:329 marginally stable and unstable, respectively. Correspond- ultimately emerge from quantum theory. Nevertheless, the μν ingly, the perturbation δg (s), δπμν(s) will be classified stability property pointed out here can be viewed as a pre- to be locally (a) decaying, (b) oscillatory, (c) growing. There- requisite for the consistent development of a covariant quan- fore, the equilibrium solution g(r) will be denoted as (a) tum gravity theory. For this reason the issue will be further locally stable, (b) locally marginally stable and (c) locally discussed in Part 2. unstable, respectively, if:

(a) all roots of ω are locally stable: namely for them 6 Conclusions Im(ω) < 0; (b) there is at least one root of ω which is locally marginally A common fundamental theoretical aspect laying at the foun- stable, namely such that Im(ω) = 0; dation of both General Relativity (GR) and classical field the- (c) there is at least one locally unstable root of ω, namely ory is the variational character of the fundamental dynamical such that Im(ω) > 0. laws which identify these disciplines. This concerns both the representation of the Einstein field equations and the covari- To investigate the stability problem we consider the vac- ant dynamics of classical fields as well as of discrete (e.g., test uum configuration in which Tμν ≡ 0, but still = 0, so that particles) or continuum systems in curved space-time. Issues Rμν = gμν and R = 4 is the constant Ricci scalar. Then related to the variational formulation of the Einstein equa- Eq. (75) yields the dispersion relation in the explicit form tions have been treated in Refs. [5,6], where the existence of a new type of Lagrangian and Hamiltonian variational ω 2 2 approaches has been identified in terms of synchronous varia- − − K − 5σ − 4 (−σ)2 = 0, (76) c tional principles realized in the framework of the DeDonderÐ Weyl formalism. As shown in Ref. [6], this leads to the real- which yields the roots, ization of a manifestly covariant Hamiltonian theory for the √ Einstein equations. ω ± −7σ In this paper new aspects of the Hamiltonian structure of − K = √ . (77) c ± −3σ GR have been displayed which is referred to here as CCG- theory. Therefore, two possible alternatives can be distinguished in In particular, we have shown that a reduced-dimensional which respectively: realization of the continuum Hamiltonian theory for the Ein- stein field equations, denoted here as Classical Hamiltonian (A) First case: −σ ≥ 0. Then the equilibrium solution Structure of GR (CHS), actually exists in which both gen- g(r) is marginally stable since all roots of the dispersion eralized coordinates and corresponding conjugate momenta relation (76) have vanishing imaginary part. are realized by means of second-order 4-tensors. The virtue (B) Second case: −σ < 0. Theng(r) is necessarily unsta- of such an approach lies precisely in its general validity. This ble (there exists always an unstable root of the same Eq. means in fact that the same Hamiltonian structure holds for (76)). arbitrary particular solutions of the Einstein field equations and arbitrary realizations of the external source terms appear- Hence, both for the case of de Sitter and anti-de Sitter ing in the variational potential density. As a result, a causal space-times (, respectively, > 0or< 0) the possible occur- form has been obtained for the corresponding continuum rence of stability or instability depends on the multiplicative Hamilton equations by introducing a suitable Lagrangian gauge parameter σ appearing in the definition of the effective parametrization prescribed in terms of the proper time s potential density V (see the second equation given above in defined along field geodetics of the curved space-time. As (60)). However, a physically admissible Hamiltonian theory a result, the same equations are cast in the equivalent form of GR should predict stable solutions, i.e., which are struc- of an initial-value problem for suitable canonical evolution turally stable in the sense indicated above. This should occur equations, referred to here as GR-Hamilton equations. This in principle not just for special realizations of the background provides a physical interpretation for the reduced Hamilto- space-time but Ð at least in the case of vacuum Ð for arbi- nian theory, according to which an arbitrary initial canonical trary vacuum background space-times Q4,g(r) . In partic- state is dynamically advanced by means of the canonical flow ular, if >0 Ð as most frequently invoked in the literature generated by the same Hamilton equations. (see for example [50,51]) Ð this happens provided the gauge Given validity to such Hamiltonian theory, the case of factor σ is uniquely identified with σ =−1. Although the canonical transformations which generate the flow corre- rigorous proof of the validity of such a choice still remains sponding to the continuum canonical equations has been con- a mere conjecture at this point, its full justification should sidered. This has been obtained by introducing the appro- 123 Eur. Phys. J. 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