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

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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 / Published online: 19 May 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 Hamilton–Jacobi 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 Hamilton–Jacobi 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 consid- ered 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 Hamilton–Jacobi theories. The CHS deter- dard Formulation of General Relativity (SF-GR) [1–4], 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 Hamilton–Jacobi 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 Euler–Lagrange 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” [8–10 ] 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 – 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- – 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.
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