[Paper Review] Gravitational Energy-Momentum in MAG

James M. Nester, Chiang-Mei Chen+1 more|ArXiv.org|Nov 28, 2000

Black Holes and Theoretical Physics2 references3 citations

TL;DR

This paper develops a Hamiltonian framework for defining quasilocal energy-momentum in Metric-Affine Gravity (MAG), emphasizing the critical role of the boundary term in determining conserved quantities. By applying covariant boundary term choices and Hamiltonian variational principles, it derives consistent quasilocal expressions that recover ADM, Bondi, and special limits in GR and Poincaré Gauge Theory, and provides a generalized first law of black hole thermodynamics with a covariant entropy expression.

ABSTRACT

Energy-momentum (and angular momentum) for the Metric-Affine Gravity theory is considered from a Hamiltonian perspective (linked with the Noether approach). The important roles of the Hamiltonian boundary term and the many choices involved in its selection-which give rise to many different definitions-are emphasized. For each choice one obtains specific boundary conditions along with a value for the quasilocal, and (with suitable asymptotic behavior) total (Bondi and ADM) energy-momentum and angular momentum. Applications include the first law of black hole thermodynamics-which identifies a general expression for the entropy. Prospects for a positive energy proof are considered and quasilocal values for some solutions are presented.

Motivation & Objective

• To establish a consistent, covariant definition of quasilocal energy-momentum in Metric-Affine Gravity (MAG) that respects boundary conditions and conservation laws.
• To resolve the ambiguity in gravitational energy-momentum localization by identifying the Hamiltonian boundary term as the key determinant of quasilocal values.
• To ensure differentiability of the Hamiltonian under field variations by selecting boundary terms that vanish under controlled boundary conditions.
• To recover known limits such as ADM and Bondi energy-momentum in asymptotically flat spacetimes.
• To derive a generalized first law of black hole thermodynamics with a covariant entropy expression in MAG.

Proposed method

• Formulates the gravitational Hamiltonian in first-order formalism using differential forms, with a spatial hypersurface integral and a boundary 2-form term.
• Applies the Noether procedure via Lie derivatives to derive conserved currents, identifying the Hamiltonian 3-form as a conserved current on-shell.
• Introduces two covariant boundary term choices—based on field and momentum deviations from reference values—that ensure Hamiltonian differentiability and covariance.
• Imposes boundary conditions by requiring the boundary term in the Hamiltonian variation to vanish when fields are held fixed on the boundary surface.
• Uses asymptotic flatness conditions to derive finite, well-defined total energy-momentum at spatial and null infinity.
• Applies the formalism to exact MAG solutions, computing quasilocal energy under Dirichlet and Neumann boundary conditions with Minkowski reference geometry.

Experimental results

Research questions

• RQ1 How can quasilocal energy-momentum be consistently defined in Metric-Affine Gravity using a Hamiltonian approach?
• RQ2 What role does the Hamiltonian boundary term play in determining both boundary conditions and conserved quantities?
• RQ3 Can a covariant, differentiable Hamiltonian formulation be constructed for MAG that yields physically meaningful energy-momentum and angular momentum?
• RQ4 How do the quasilocal expressions in MAG reduce to known results in General Relativity and Poincaré Gauge Theory?
• RQ5 What is the form of the first law of black hole thermodynamics in MAG, and how is entropy expressed?

Key findings

• The quasilocal energy-momentum in MAG is fully determined by the choice of Hamiltonian boundary term, which also fixes the required boundary conditions.
• Two covariant boundary term choices—based on field and momentum deviations from reference values—ensure Hamiltonian differentiability and covariance.
• The formalism yields finite, well-defined ADM and Bondi energy-momentum in asymptotically flat spacetimes with standard fall-off conditions.
• For the first exact MAG solution with nonvanishing torsion and nonmetricity, the quasilocal energy is computed as E = a₀r(1−f⁻¹) for Dirichlet conditions and E = a₀r(f−1) for Neumann conditions in the f=1−m/r limit.
• The generalized first law of black hole thermodynamics is derived as TδS = ∫ₕ κε^{αβ}δρ_{αβ}, with surface gravity and binormal to the horizon.
• The formalism shows promising prospects for a positive energy proof in restricted MAG models with R² terms such as R^{α}_{α} or scalar curvature squared.