The gravitational Hamiltonian, first order action, Poincaré charges and surface terms

Abstract

We consider the issue of attaining a consistent Hamiltonian formulation, after a 3+1 splitting, of a well defined action principle for asymptotically flat gravity. More precisely, our starting point is the gravitational first order Holst action with surface terms and fall-off conditions that make the variational principle and the covariant phase space formulation well defined for asymptotically flat spacetimes. Keeping all surface terms and paying due attention to subtleties that arise from the different cut-offs at infinity, we give a derivation of the gravitational Hamiltonian starting from this action. The 3+1 decomposition and time gauge fixing results in a well defined Hamiltonian action and a well defined Hamiltonian formulation for the standard -and more general- asymptotic ADM conditions. Unlike the case of the Einstein-Hilbert action with Gibbons-Hawking-York or Hawking-Horowitz terms, here we {\it {do}} recover the ADM energy-momentum from the covariant surface term also when more general variations respecting asymptotic flatness are allowed. Additionally, our strategy yields a derivation of the parity conditions for connection variables independent of the conditions given by Regge and Teitelboim for ADM variables. Finally, we exhibit the other Poincaré generators in terms of real Ashtekar-Barbero variables. We complement previous constructions in self-dual variables by pointing out several subtleties and refining the argument showing that -on shell- they coincide with the ADM charges. Our results represent the first consistent treatment of the Hamiltonian formulation for the connection-tetrad gravitational degrees of freedom, starting from a well posed action, in the case of asymptotically flat boundary conditions.

56 pages, 4 figures. Minor typos corrected and references added. File matches published version

References in corpus (5)

• Covariant Counterterms and Conserved Charges in Asymptotically Flat Spacetimes
• Hamiltonian and Noether charges in first order gravity
• Surface terms, Asymptotics and Thermodynamics of the Holst Action
• Energy in first order 2+1 gravity
• Note on the phase space of asymptotically flat gravity in Ashtekar-Barbero variables

Cited by in corpus (12)

• New boundary variables for classical and quantum gravity on a null surface
• Conservation of asymptotic charges from past to future null infinity: Supermomentum in general relativity
• Spacetime-noncommutativity regime of Loop Quantum Gravity
• Actions, topological terms and boundaries in first order gravity: A review
• Conservation of asymptotic charges from past to future null infinity: Maxwell fields
• Covariant phase space for gravity with boundaries: metric vs tetrad formulations
• Rainbow-like Black Hole metric from Loop Quantum Gravity
• Weakly Isolated Horizons: First order actions and gauge symmetries
• Quasi-local gravitational angular momentum and centre of mass from generalised Witten equations
• Weakly Isolated Horizons: $3+1$ decomposition and canonical formulations in self-dual variables
• Supertranslations at Spatial and Timelike Infinities in the First-Order Formalism
• On covariant and canonical Hamiltonian formalisms: Weakly Isolated Horizons