Asymptotics and Hamiltonians in a first-order formalism

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 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.