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The problem of the dynamical structure and definition of energy for the classical general theory of relativity is considered on a formal level~As in a previous paper, the technique used is the Schwinger action principle, Starting with the full Einstein Lagrangian in first order Palatini form, an action integral is derived in which the algebraic constraint variables have been eliminated. This action possesses a "Hamiltonian" density which, however, vanishes due to the differential constraints. If the differential constraints are then substituted into the action, the true, nonvanishing Hamiltonian of the theory emerges. From an analysis of the equations of motion and the constraint equations, the two pairs of dynamical variables which represent the two independent degrees of freedom of the gravitational field are explicitly exhibited. Four other variables remain in theory; these may be arbitrarily specified, any such specification representing a choice of coordinate frame. It is shown that it is possible to obtain truly canonical pairs of variables in terms of the dynamical and arbitrary variables. Thus a statement of the dynamics is meaningful only after a set of coordinate conditions have been chosen. In general, the true Hamiltonian will be time dependent even for an isolated gravitational field. There thus arises the notion of a preferred coordinate frame, i.e. , that frame in which the Hamiltonian is conserved. In this special frame, on physical grounds, the Hamiltonian may be taken to define the energy of the field. In these respects the situation in general relativity is analogous to the parametric form of Hamilton's principle in particle mechanics. ' R. Arnowitt and S. Deser, Phys. Rev. 113, 745 (1959). This paper will be referred to as I. We use, as in I, natural units: c= 1, It;=167ryc 4= 1 (y is the Newtonian gravitational constant). In general relativity the diTiculty in carrying out the above program resides in the invariance under the function group of coordinate transformations.In I, this difficulty was overcome for the linearlized theory.It was seen there that the process of obtaining the correct canonical variables involved making a "gauge" (i.e. , linearized coordinate) transformation from an arbitrary gauge to a "radiation" gauge. In this paper we shall extend the analysis to consider the full theory in this light. Of the two types of constraints mentioned in I, the algebraic constraint variables can be handled quite simply in this formalism. At the beginning of the next section their explicit elimination will be carried out. In the process certain combinations of the remaining variables appear in the equations of motion. These combinations remain redundant until the differential constraints are utilized. However, they are physically significant in that the specification of the fields on a given spacelike surface can be given in terms of these combinations. From these considerations it is suggestive to restate the theory in.terms of variables that possess the geometrical properties of decomposin...

A change of variables from k to E and two angle variables 0 gives (f, *, e) = exp ( where J is the Jacobian. By Schwarz's inequality, the product of two square-integrable functions f and C is absolutely integrable with the weight function J. The Riemann-Lebesgue lemma then asserts the asymptotic vanishing of the norm llaf&&ll.The "asymptotic vanishing of destruction operators" was used previously by Coester and Kummel. '8 'sF. Coester and H.The general theory of relativity is cast into normal Hamiltonian form in terms of two pairs of independent conjugate Geld variables. These variables are explicitly exhibited and obey ordinary Poisson bracket relations. This form is reached by imposing a simple set of coordinate conditions. It is shown that those functionals of the metric used as invariant coordinates do not appear explicitly in the Hamiltonian and momentum densities, so that the standard differential conservation laws hold. The bearing of these results on the quantization problem is discussed.

The invariance of various de6nitions proposed for the energy and momentum of the gravitational 6eld is examined. We use the boundary conditions that g""approaches the Lorentz metric as 1/r, but allow g", to vanish as slowly as 1/r. This permits "coordinate waves. " It is found that none of the expressions giving the energy as a two-dimensional surface integral are invariant within this class of frames. In a frame containing coordinate waves they are ambiguous, since their value depends on the location of the surface at infinity (unlike the case where g"", vanishes faster than 1/r). If one introduces the prescription of space-time averaging of the integrals, one finds that the de6nitions of Landau-Lifshitz and Papapetrou-Gupta yield (equal) coordinate-invariant results. However, the definitions of Einstein, Mliller, and Dirac become unambiguous, but not invariant. The averaged Landau-Lifshitz and Papapetrou-Gupta expressions are then shown to give the correct physical energy-momentum as determined by the canonical formulations Hamiltonian involving only the two degrees of freedom of the 6eld. It is shown that this latter de6nition yields that inertial energy for a PI" should transform as a four-vector. Our boundary conditions have not stated that g",, should behave as 1/rs at infinity; in fact, it is most natural simply to require that g"., also vanish as 1/r. For example, we thus allow the metric to decrease as (e"*)/r. The asymptotic domain, as employed here, is de6ned by letting r approach in6nity for 6xed time, and is a region in which there is a negligible Qux of gravitational radiation. By contrast, in a previous paper, IVb, a wave zone was defined in which asymptotic relations were also studied. There, however, since the Aux of radiation was the object of interest, one had to verify that asymptotic expressions were good approximations for values of r where radiation was significant. To achieve desired accuracy, it was often required to move out along the light cone, which meant increasing t as well as r. In this paper we are dealing with the region beyond the wave front, this boundary representing the largest distance at which the Aux is appreciable. Such a boundary must, of course, exist in order that the energy of the system be finite. Let us now examine the various proposed expressions' ' for PI" within the above framework. A common characteristic of all' these is that they can be cast into the involve Lorentz transformations at In6nity. Of course,~S ee reference (6) for the surface integral form of the Einstein under rigid Lorentz rotations of the asymptotic frame, expressipn