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The gravitational Hamiltonian, first order action, Poincaré charges and surface terms
Alejandro Corichi and Juan D Reyes 2015 Class. Quantum Grav. 32 195024
Ever since Einstein and Hilbert were racing to complete the general theory of relativity, almost 100 years ago, having a variational principle for it was at the forefront of the theoretical efforts. An action and the variational principle accompanying it are the preferred ways to describe a physical theory. At the classical level, all the information one can possibly ask about a physical system is conveniently codified into a single scalar function S. Additionally, in covariant approaches to quantum mechanics, the action S provides, through the path integral, a fundamental link between the classical and quantum descriptions. Ideally, the Hamiltonian structure of the theory itself -the starting point for canonical quantization- may too be extracted from the same action.
Needless to say, asymptotically flat solutions to Einstein’s equations, modelling isolated gravitating systems, play a prominent role in general relativity. Surely, variational principles for the theory must be formulated such that the aforementioned information and structures accommodating these solutions may be juiced out of the (properly defined) scalar functional of the gravitational degrees of freedom. But, are they?
In our paper we consider the issue of having well defined gravitational actions for which a 3+1 splitting yields a consistent Hamiltonian formulation for asymptotically flat configurations. We focus on a first order covariant action based on orthonormal tetrads and Lorentz connections, commonly referred to as the Holst action. The ‘Holst term’ added to the Einstein-Palatini action was originally introduced by Hojman et. al, but it was Sören Holst who first showed that its 3+1 decomposition plus partial gauge fixing gives, for compact spacetimes without boundaries, a Hamiltonian action for general relativity in terms of Ashtekar-Barbero variables. Our interest on this action does not merely rely on this last fact. Indeed, it is all but accidental. A variational principle based on tetrads instead of metric variables is necessary to incorporate Fermions, and the Holst action is the simplest one producing a canonical theory without the complications of second class constraints -as is the case of the simpler first order Einstein-Palatini action-. It is, furthermore, the classical starting point for Loop Quantum Gravity and some Spin-foam models.
On the technical side, the complications of considering asymptotically flat configurations arise from the requirement -at least on-shell- of a finite action integral over an infinite volume spacetime and its differentiability with respect to variations of non-compact support. Indeed, in our work we consider a generalized variational principle where variations at spatial infinity are not fixed but instead they are consistent with an interpretation as generalized tangent vectors on the phase space admitting such asymptotically flat solutions. While this generalization may not be strictly necessary from a purely classical point of view, with an eye towards quantization such condition is certainly required, particularly for the semiclassical approximation of the path integral. Using this variational principle, we extend Holst’s result to configuration spaces containing asymptotically flat solutions. We show that the 3+1 decomposition and so called time gauge fixing of the action, amended with a surface term, leads also to a well-defined Hamiltonian action in Ashtekar-Barbero variables. Now, the surface term recovers precisely the ADM energy and ADM momentum. This surface term had previously been shown by one of us and Wilson-Ewing to render a well-defined covariant variational principle for asymptotically flat spacetimes, leading also to a covariant phase space formulation with well-defined generators of asymptotic symmetries. Our results may sound at first as something to be expected, but they are non-trivial, especially when one takes into account all the technicalities involved in evaluating the asymptotic limit and the requirements for a well-defined variational principle.
Throughout our discussion, we draw a clear distinction between the different asymptotic or fall-off conditions of the fields necessary for a well-defined Hamiltonian action and those for a consistent Hamiltonian formulation admitting well-defined canonical generators of (asymptotic) Poincaré transformations in Ashtekar-Barbero variables. We furthermore contrast these conditions with the corresponding ones in covariant treatments. There is no known clean way to construct the phase space of general relativity accommodating the full class of asymptotically flat solutions while at the same time singling out a unique Poincaré group from the full and larger asymptotic symmetry group. Additional restrictions need to be imposed. For the covariant phase space formulation these restrictions eliminate the so called supertranslations and logarithmic translations. For the 3+1 Hamiltonian formulation, the Regge-Teitelboim parity conditions for the metric variables suffice to uniquely construct the canonical generators, leaving residual ‘odd’ supertranslations as gauge symmetries. By starting from the fall-off conditions that make the covariant variational principle and the covariant phase space well-defined, our strategy yields a derivation of the parity conditions for connection variables independent of the conditions given by Regge and Teitelboim for ADM variables.
For conguration spaces containing asymptotically flat solutions, the Holst action fulfils one’s original expectations. We can juice out (almost) all the relevant physical information from it, including the conserved energy-momentum charges, and we can extract the seed from which to grow the Hamiltonian structures to finally exhibit the complete set of Poincaré generators and conserved charges in Ashtekar-Barbero variables. Our treatment represents the first case in which one obtains a fully consistent Hamiltonian theory, starting from a well-defined action principle. The burden of switching to a description in terms of tetrads and connections pays off here. Unlike the case of the known second order formalisms based on more conventional metric variables, one can do all this without introducing in the action infinite surface counter terms which furthermore may only be defined implicitly.
In summary, it took almost 100 years to have a consistent formulation of 3+1 general relativity -for asymptotically flat configurations in first order variables-, after Hilbert wrote the first action principle for GR. It was about time.
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