Loop quantum gravity without the Hamiltonian constraint

Abstract

We show that under certain technical assumptions, including the existence of a constant mean curvature (CMC) slice and strict positivity of the scalar field, general relativity conformally coupled to a scalar field can be quantised on a partially reduced phase space, meaning reduced only with respect to the Hamiltonian constraint and a proper gauge fixing. More precisely, we introduce, in close analogy to shape dynamics, the generator of a local conformal transformation acting on both, the metric and the scalar field, which coincides with the CMC gauge condition. A new metric, which is invariant under this transformation, is constructed and used to define connection variables which can be quantised by standard loop quantum gravity methods. While it is hard to address dynamical problems in this framework (due to the complicated 'time' function), it seems, due to good accessibility properties of the CMC gauge, to be well suited for problems such as the computation of black hole entropy, where actual physical states can be counted and the dynamics is only of indirect importance. The corresponding calculation yields the surprising result that the usual prescription of fixing the Barbero-Immirzi parameter beta to a constant value in order to obtain the well-known formula S = a(Phi) A/(4G) does not work for the black holes under consideration, while a recently proposed prescription involving an analytic continuation of beta to the case of a self-dual space-time connection yields the correct result. Also, the interpretation of the geometric operators gets an interesting twist, which exemplifies the deep relationship between observables and the choice of a time function and has consequences for loop quantum cosmology.

8 pages. v2: Journal version. Black hole state counting based on physical states added. Applications to loop quantum cosmology discussed. Gauge condition used shown to coincide with CMC gauge. Minor clarifications. v3: Erroneous topology dependence of the entropy in journal version corrected, conclusions fixed accordingly. Main results unaffected

References in corpus (23)

• Loop Quantum Cosmology: A Status Report
• Quantum Nature of the Big Bang: Improved dynamics
• Real Ashtekar Variables for Lorentzian Signature Space-times
• Isolated Horizons: Hamiltonian Evolution and the First Law
• Uniqueness of diffeomorphism invariant states on holonomy-flux algebras
• Mechanics of Isolated Horizons
• Time and a physical Hamiltonian for quantum gravity
• Black hole entropy and SU(2) Chern-Simons theory
• Einstein gravity as a 3D conformally invariant theory
• Black hole entropy from an SU(2)-invariant formulation of Type I isolated horizons
• Gravity quantized
• Topological black holes dressed with a conformally coupled scalar field and electric charge
• Are the spectra of geometrical operators in Loop Quantum Gravity really discrete?
• Non-minimally coupled scalar fields and isolated horizons
• Classical Scalar Fields and the Generalized Second Law
• Thermodynamical properties of hairy black holes in n spacetimes dimensions
• Non-minimal couplings, quantum geometry and black hole entropy
• Phase-space and Black Hole Entropy of Higher Genus Horizons in Loop Quantum Gravity
• Conformal geometrodynamics: True degrees of freedom in a truly canonical structure
• Coupling Shape Dynamics to Matter Gives Spacetime
• Non-Minimally Coupled Scalar Field and Ashtekar Variables
• A Note on the Symmetry Reduction of SU(2) on Horizons of Various Topologies
• Towards conformal loop quantum gravity

Cited by in corpus (11)

• Scalar Material Reference Systems and Loop Quantum Gravity
• Geometric temperature and entropy of quantum isolated horizons
• Black hole entropy from loop quantum gravity in higher dimensions
• Testing the role of the Barbero-Immirzi parameter and the choice of connection in Loop Quantum Gravity
• Loop quantum cosmology with self-dual variables
• Spectra of geometric operators in three-dimensional LQG: From discrete to continuous
• A quantum reduction to spherical symmetry in loop quantum gravity
• The Wald entropy formula and loop quantum gravity
• Anisotropic loop quantum cosmology with self-dual variables
• The algebra of observables in Gaußian normal spacetime coordinates
• Linking shape dynamics and loop quantum gravity