Surface terms, Asymptotics and Thermodynamics of the Holst Action

Abstract

We consider a first order formalism for general relativity derived from the Holst action. This action is obtained from the standard Palatini-Hilbert form by adding a topological-like term and can be taken as the starting point for loop quantum gravity and spin foam models. The equations of motion derived from the Holst action are, nevertheless, the same as in the Palatini formulation. Here we study the form of the surface terms of the action for general boundaries as well as the symplectic current in the covariant formulation of the theory. Furthermore, we analyze the behavior of the surface terms in asymptotically flat space-times. We show that the contribution to the symplectic structure from the Holst term vanishes and one obtains the same asymptotic expressions as in the Palatini action. It then follows that the asymptotic Poincare symmetries and conserved quantities such as energy, linear momentum and relativistic angular momentum found here are equivalent to those obtained from the standard Arnowitt, Deser and Misner formalism. Finally, we consider the Euclidean approach to black hole thermodynamics and show that the on-shell Holst action, when evaluated on some static solutions containing horizons, yields the standard thermodynamical relations.

16 pages

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• Palatini Approach to Modified Gravity: f(R) Theories and Beyond
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• Black hole entropy and Lorentz-diffeomorphism Noether charge
• The First Law of Black Hole Mechanics for Fields with Internal Gauge Freedom
• The twistorial structure of loop-gravity transition amplitudes
• Imaginary action, spinfoam asymptotics and the 'transplanckian' regime of loop quantum gravity
• Isolated Horizons and Black Hole Entropy in Loop Quantum Gravity
• Actions, topological terms and boundaries in first order gravity: A review
• Complex Ashtekar variables and reality conditions for Holst's action
• Static isolated horizons: SU(2) invariant phase space, quantization, and black hole entropy
• Hamiltonian and Noether charges in first order gravity
• A note on dual gravitational charges
• Hamiltonian spinfoam gravity
• Gravitational $SL(2,\mathbb{R})$ Algebra on the Light Cone
• First law of black hole mechanics with fermions
• The gravitational Hamiltonian, first order action, Poincaré charges and surface terms
• Barbero-Immirzi parameter, manifold invariants and Euclidean path integrals
• Gravity Asymptotics with Topological Parameters
• Energy in first order 2+1 gravity
• Weakly Isolated Horizons: First order actions and gauge symmetries
• On the on-shell equivalence of general relativity and Holst theories with nonmetricity, torsion, and boundaries
• Note on the phase space of asymptotically flat gravity in Ashtekar-Barbero variables
• Modifications in the Spectrum of Primordial Gravitational Waves Induced by Instantonic Fluctuations
• Note on Kerr/CFT correspondence in a first order formalism
• Boundary terms, branes and AdS/BCFT in first-order gravity
• Supertranslations at Spatial and Timelike Infinities in the First-Order Formalism
• Mechanics of Isolated Horizons in Scalar-Tensor Theories