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Brown-York charges with mixed boundary conditions

Gloria Odak, Simone Speziale

TL;DR

The paper investigates how gravitational energy, encoded in Hamiltonian surface charges, depends on conservative boundary conditions in general relativity. Using both the covariant phase space framework (FGP prescription) and canonical ADM methods, it derives explicit charges for Dirichlet, York mixed, and Neumann boundary conditions, including extensions to non-orthogonal corners. It demonstrates exact agreement between the two approaches, shows that the quasi-local and asymptotic energies differ with boundary choice (e.g., Kerr energies become $M$, $2M/3$, and $M/2$ after renormalization for Dirichlet, York, and Neumann, respectively), and clarifies how corner terms and boundary Lagrangians shape the energy definition. The results reinforce the FGP framework, highlight the physical relevance of boundary data in GR, and have implications for quasi-local energy concepts and potential quantum-gravity contexts.

Abstract

We compute the Hamiltonian surface charges of gravity for a family of conservative boundary conditions, that include Dirichlet, Neumann, and York's mixed boundary conditions defined by holding fixed the conformal induced metric and the trace of the extrinsic curvature. We show that for all boundary conditions considered, canonical methods give the same answer as covariant phase space methods improved by a boundary Lagrangian, a prescription recently developed in the literature and thus supported by our results. The procedure also suggests a new integrable charge for the Einstein-Hilbert Lagrangian, different from the Komar charge for non-Killing and non-tangential diffeomorphisms. We study how the energy depends on the choice of boundary conditions, showing that both the quasi-local and the asymptotic expressions are affected. Finally, we generalize the analysis to non-orthogonal corners, confirm the matching between the covariant and canonical results without any change in the prescription, and discuss the subtleties associated with this case.

Brown-York charges with mixed boundary conditions

TL;DR

The paper investigates how gravitational energy, encoded in Hamiltonian surface charges, depends on conservative boundary conditions in general relativity. Using both the covariant phase space framework (FGP prescription) and canonical ADM methods, it derives explicit charges for Dirichlet, York mixed, and Neumann boundary conditions, including extensions to non-orthogonal corners. It demonstrates exact agreement between the two approaches, shows that the quasi-local and asymptotic energies differ with boundary choice (e.g., Kerr energies become , , and after renormalization for Dirichlet, York, and Neumann, respectively), and clarifies how corner terms and boundary Lagrangians shape the energy definition. The results reinforce the FGP framework, highlight the physical relevance of boundary data in GR, and have implications for quasi-local energy concepts and potential quantum-gravity contexts.

Abstract

We compute the Hamiltonian surface charges of gravity for a family of conservative boundary conditions, that include Dirichlet, Neumann, and York's mixed boundary conditions defined by holding fixed the conformal induced metric and the trace of the extrinsic curvature. We show that for all boundary conditions considered, canonical methods give the same answer as covariant phase space methods improved by a boundary Lagrangian, a prescription recently developed in the literature and thus supported by our results. The procedure also suggests a new integrable charge for the Einstein-Hilbert Lagrangian, different from the Komar charge for non-Killing and non-tangential diffeomorphisms. We study how the energy depends on the choice of boundary conditions, showing that both the quasi-local and the asymptotic expressions are affected. Finally, we generalize the analysis to non-orthogonal corners, confirm the matching between the covariant and canonical results without any change in the prescription, and discuss the subtleties associated with this case.

Paper Structure

This paper contains 15 sections, 97 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Mixed boundary conditions
  3. Non-orthogonal corners
  4. Surface charges from covariant phase space
  5. Hamiltonians from conservative boundary conditions
  6. Corner symplectic potential
  7. Charges for Dirichlet, mixed and Neumann boundary conditions
  8. Subtraction terms and symplectic renormalization
  9. Residual diffeomorphisms
  10. Kerr example
  11. Covariant surface charges with non-orthogonal corners
  12. Surface charges from canonical methods
  13. Conclusions
  14. Notations and kinematics
  15. Volume forms and pull-backs

Figures (2)

  • Figure 1: Left panel: A finite region of spacetime and the notation used for the boundaries. Right panel: The four normals generically present at the corner, here the one in the future of $\cal T$. Unbarred quantities refer to the space-like hypersurface, and barred quantities to the time-like one.
  • Figure 2: If the time-like hypersurface at which the boundary conditions are imposed is tilted with respect to the time-like evolution of the corner of $\Sigma$, symplectic flux can a priori leak through it, as in the top (and darker) arrow. In that case, the procedure to get integrable charges applies only to the 'barred observers', namely those at rest with respect to a foliation of $\cal T$.