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

Alejandro Corichi, Juan D. Reyes

TL;DR

The authors develop a consistent Hamiltonian formulation for general relativity in Holst form using Ashtekar-Barbero variables for asymptotically flat spacetimes, rooted in a well-posed boundary-augmented action. Through a $3+1$ decomposition and time gauge, they derive a finite, differentiable Hamiltonian action whose surface term reproduces the ADM energy and momentum, and they establish parity conditions for connection variables independent of ADM parity. They construct renormalized Poincaré generators on the extended phase space that reproduce ADM charges on-shell and satisfy the hypersurface deformation algebra, thereby integrating the covariant and canonical pictures in the real connection framework. The results clarify the role of boundary terms and cut-offs (cylindrical vs hyperbolic) in the Hamiltonian formulation and set the stage for further exploration of diffeomorphism generators and Dirac analysis without fixing the gauge prematurely.

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.

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

TL;DR

The authors develop a consistent Hamiltonian formulation for general relativity in Holst form using Ashtekar-Barbero variables for asymptotically flat spacetimes, rooted in a well-posed boundary-augmented action. Through a decomposition and time gauge, they derive a finite, differentiable Hamiltonian action whose surface term reproduces the ADM energy and momentum, and they establish parity conditions for connection variables independent of ADM parity. They construct renormalized Poincaré generators on the extended phase space that reproduce ADM charges on-shell and satisfy the hypersurface deformation algebra, thereby integrating the covariant and canonical pictures in the real connection framework. The results clarify the role of boundary terms and cut-offs (cylindrical vs hyperbolic) in the Hamiltonian formulation and set the stage for further exploration of diffeomorphism generators and Dirac analysis without fixing the gauge prematurely.

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.

Paper Structure

This paper contains 13 sections, 208 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries: asymptotically flat spacetimes and the variational principle
  3. First order action with boundary terms
  4. 3+1 splitting
  5. Hamiltonian formulation and Poincaré charges
  6. Phase space
  7. Gauge vs asymptotic symmetries: constraints and Poincaré generators
  8. Gauss constraint and $SU(2)$-gauge generator
  9. Proper diffeomorphism gauge generators and improper gauge Poincaré generators
  10. Summary and discussion
  11. Comparison of hyperbolic and cylindrical asymptotic expansions
  12. 3+1 decomposition with gauge fixing
  13. Generators in Ashtekar-Barbero variables

Figures (4)

  • Figure 1: Generic boundary $\partial M=\tau\cup\Sigma_1\cup\Sigma_2$ for spacetime region of integration. Variations may be 'fixed' only at the Cuachy surfaces $\Sigma_1$ and $\Sigma_2$ and must remain compatible with asymptotic conditions for the original fields.
  • Figure 2: Hyperbolic and cylindrical cut-offs
  • Figure 3: Hyperbolic and cylindrical temporal cut-offs together with hyperbolic cuts defining (asymptotic) spacetime volume integration region $M$ and boundary $\partial M$.
  • Figure 4: Orthogonal $3+1$ decomposition of tetrad $e^\mu_I$.