The Rest-Frame Instant Form of Metric Gravity May Be Defined
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The Rest-Frame Instant Form of Metric Gravity. Luca Lusanna Sezione INFN di Firenze L.go E.Fermi 2 (Arcetri) 50125 Firenze, Italy E-mail [email protected] Abstract In a special class of globally hyperbolic, topologically trivial, asymp- totically flat at spatial infinity spacetimes selected by the requirement of absence of supertranslations (compatible with Christodoulou-Klainermann spacetimes) it is possible to define the rest-frame instant form of ADM canon- ical gravity by using Dirac’s strategy of adding ten extra variables at spatial infinity and ten extra first class constraints implying the gauge nature of these variables. The final canonical Hamiltonian is the weak ADM energy and a discussion of the Hamiltonian gauge transformations generated by the eight first class ADM constraints is given. When there is matter and the Newton constant is switched off, one recovers the description of the matter on the Wigner hyperplanes of the rest-frame instant form of dynamics in Minkowski spacetime. May 1, 2019 arXiv:gr-qc/0101048v1 12 Jan 2001 Typeset using REVTEX 1 I. INTRODUCTION Our description of the four (gravitational, electromagnetic, weak, strong, with or without supersymmetry) interactions is based on action principles which, due to manifest Lorentz invariance, to local gauge invariance (minimal coupling) and/or diffeomorphism invariances make use of singular Lagrangians. This implies the use of Dirac-Bergmann theory of con- straints [1–4] for their Hamiltonian formulation. An open problem is the identification of the physical degrees of freedom hidden behind manifest gauge invariance and/or general covariance (Dirac observables). When this can be done in a global way, classical physics can be reformulated only in terms of a canonical basis of Dirac observables and then quantized. This alternative to the traditional sequence first quantize then reduce 1 is still unexplored. In particular with it one would avoid to quantize any timelike degree of freedom. See Refs. [5,6] for the status of these topics. As a consequence of these researches [5], in special relativity it is now possible to de- scribe isolated systems (particles, strings, field configurations) in a way which implements the separation of the relativistic canonical center of mass from the relative degrees of freedom and allows to make the canonical reduction to Dirac observables in a Wigner-covariant way. Since it is known that the reduction to a completely fixed gauge breaks manifest Lorentz covariance, it turns out that in this approach the breaking can be concetrated in the non- covariance of the canonical center of mass (viz. that of the Newton-Wigner 3-position operator) independently from the system under consideration2. A new form of dynamics [8], the one-time Wigner-covariant rest-frame instant form [9], emerges from these investigations and it is reviewed in Appendix A. In it each configuration of an isolated system with timelike conserved 4-momentum is described on the spacelike Wigner hyperplanes orthogonal to the 4-momentum (equal time Cauchy surfaces), leaves of the foliation associated with a 3+1 splitting of Minkowski spacetime identified by the configuration itself. While the decoupled center of mass can be identified with a point particle clock for the mathematical time de- scribing the evolution (the Hamiltonian is the invariant mass of the system configuration), all the dynamics is in the relative degrees of freedom (weak form of Mach principle). Then the principal tool for the canonical reduction of every theory to a canonical basis of Dirac observables is the Shanmugadhasan canonical transformation [10] together with the associated multitemporal equations (see the paper e) in Ref. [3]). These canonical transformations allow to find (in general only local) Darboux bases in which a subset of the new momenta carry the same information of the first class constraints, namely their vanishing identifies the same presymplectic submanifold [11,12] of phase space as the orig- 1Based on BRST observables invariant under infinitesimal gauge transformations modulo problems like the Gribov ambiguity. 2As shown in Refs. [5,6] the region of non-covariance of the canonical center of mass identifies an intrinsic classical unit of length, the Møller radius [7] (ratio of the spin to the mass of the isolated system), which is a natural candidate for a ultraviolet cutoff in quantization. 2 inal constraints3. The variables conjugate to this subset of momenta (the Abelianized first class constraints) describe the gauge degrees of freedom of the system, namely they give a parametrization of the Hamiltonian gauge orbits. To find this set of Abelianized gauge variables one has to solve the multitemporal equations, namely the equations describing the Hamiltonian gauge transformations viewed as functional equations in the gauge parameters. Their solution shows how the original canonical variables describing the system depends on the Abelianized gauge variables. The remaining pairs of conjugate variables in these Darboux bases form a canonical basis of Dirac observables associated to the given Abelian- ization of the first class constraints: they are in strong involution (namely they have zero Poisson brackets) with both the Abelianized constraints and gauge variables and in weak involution with respect to the original constraints. In flat spacetime one has also to take into account the stratification of the constraint presymplectic manifold induced by the Hamilto- nian action of the Poincar´egroup: there are as many strata as allowed Poincar´eorbits for the conserved total 4-momentum of the isolated system. Each stratum will have different Darboux-Shanmugadhasan bases adapted to the little group of the associated Poincar´eorbit. In general this procedure works only locally, but when the configuration space is non-compact there can exist global Shanmugadhasan canonical transformations. For finite dimensional systems there are general theorems [13] connected with the Lie theory of functions groups [14] which ensure the existence of local Shanmugadhasan canonical transformations. How- ever till now there is no extension of these theorems to gauge field theories, mainly because now some of the first class constraints can be interpreted as elliptic equations, so that, for most (but not all) of the choices of the function space for the fields, they can have zero modes (the Gribov ambiguity of Yang-Mills theories). Notwithstanding that, the heuristic search of Shanmugadhasan canonical trnasformations for gauge theories is the only existing method for the individuation of possible canonical bases of Dirac observables. See Ref. [5] for a full account of what is known on these topics. After these developments the main question is whether the ADM Hamiltonian formu- lation [15] of gravity plus matter can be put in a form which reproduces this instant form of dynamics for the matter when the Newton constant is switched off4. To try to imple- ment this program, the allowed pseudo-Riemannian spacetimes must be restricted to be orientable, globally hyperbolic, topologically trivial (with the leaves of each 3+1 splitting diffeomorphic to R3, so that they admit global coordinate charts) and asymptotically flat at spatial infinity. This last requirement implies the existence of the asymptotic ADM Poincar´echarges [17,18], which should reduce to the ten Poincar´egenerators of the isolated system (whose existence is fundamental for particle physics) when the Newton constant is switched off. However, it is known that at spatial infinity the group of asymptotic symmetries is the infinite dimensional SPI group [19,20]. Besides an invariant 4-dimensional subgroup of translations it contains an infinite number of Abelian supertranslations. This forbids the 3If second class constraints are present, they are replaced by pairs of conjugate variables. 4This is the deparametrization problem of general relativity, only partially solved in Ref. [16] by using coordinate gauge conditions. 3 identification of a unique Lorentz subgroup5. The presence of supertranslations is an ob- struction to the definition of angular momentum in general relativity [21,22] and there is no idea how to measure this infinite number of constants of motion if they are allowed to exist. Therefore, suitable boundary conditions at spatial infinity have to be assumed to kill the supertranslations. In this way the SPI group is reduced to a well defined asymptotic Poincar´egroup. As it will be shown in this paper, a convenient set of boundary conditions is obtained by assuming that the coordinate atlas of spacetime is restricted in such a way that the 4-metric always tends to the Minkowski metric in Cartesian coordinates at spatial infinity with the 3-metric on each spacelike hypersurface associated with the allowed 3+1 splittings becoming Euclidean at spatial infinity in a direction independent way6. Then, this last property is assumed also for all the other Hamiltonian variables like the lapse and shift functions. These latter variables are assumed to be the sum of their asymptotic part (growing linearly in the 3-coordinates on the leave [17,18]) plus a bulk part with the quoted property. The final result of all these requirements is a set of boundary conditions compatible with Christodoulou-Klainermann spacetimes [23]. As a consequence, the allowed 3+1 splittings of spacetime have all the spacelike leaves approaching Minkowski spacelike hyperplanes at spatial infinity in a direction-independnt way. It will be shown that these asymptotic hyperplanes are orthogonal to the weak (viz. the volume form of the) ADM 4-momentum, for those spacetimes for which it is timelike. Therefore these hyperplanes reduce to the Wigner hyperplanes in Minkowski spacetime when the Newton constant is switched off. To arrive at these results Dirac’s strategy [24,1] of adding ten extra degrees of freedom at spatial infinity and then to add ten first class constraints so that the new degrees of freedom are gauge variables, will be followed. In this way the rest-frame instant form of metric gravity may be defined.