Asymptotics and Hamiltonians in a First order formalism

TL;DR

The paper tackles the problem of defining conserved quantities in asymptotically flat 4D spacetimes without infinite counterterms by formulating GR in a first-order tetrad-connection framework. It develops a covariant phase space with carefully chosen boundary conditions that remove logarithmic translations and super-translations, yielding a finite action and well-defined Hamiltonians for asymptotic Poincaré symmetries. The resulting expressions for energy-momentum and angular momentum are given as clean surface integrals at spatial infinity and are shown to agree with Spi-based charges derived from asymptotic field equations, linking the Lagrangian, Hamiltonian, and Weyl tensor perspectives. The work also situates these results in the broader context of higher dimensions and non-commutative spectral-action approaches, suggesting both conceptual and practical avenues for generalized gravity theories.

Abstract

We consider 4-dimensional space-times which are asymptotically flat at spatial infinity and show that, in the first order framework, action principle for general relativity is well-defined \emph{without the need of infinite counter terms.} It naturally leads to a covariant phase space in which the Hamiltonians generating asymptotic symmetries provide the total energy-momentum and angular momentum of the space-time. We address the subtle but important problems that arise because of logarithmic translations and super-translations both in the Langrangian and Hamiltonian frameworks. As a forthcoming paper will show, the treatment of higher dimensions is considerably simpler. Our first order framework also suggests a new direction for generalizing the spectral action of non-commutative geometry.