Black holes and active galactic nuclei as phenomenological test objects for a theory of quantum gravity
Final Report Abstract
The research projects addressed several aspects and open problems in loop quantum gravity, a non-perturbative, background independent approach to quantum gravity, which is founded on and, therefore, preserves the fundamental principles and notions of both general relativity and quantum field theory. The main focus was on the mathematical and physical aspects of classical and quantum black holes, especially in form of type II isolated horizons, on quantum cosmology, and on the covariant spinfoam approach to loop quantum gravity. A heuristic algebraic quantization of Kerr-Newman black holes, providing an approximate description of quantum black holes, was conducted. An operator algebra of the fundamental black hole observables horizon area AH , charge Q, and angular momentum J was derived, and the spectral behavior analyzed. The eigenvalue spectra of these black hole observables indicate that a strict cosmic censorship bound on the extensive parameters, different from the relation arising in general relativity, holds, implying that the extremal black hole state can neither be measured nor can its existence be proven within the given setting. This is, as turned out, a result of the specific form of the chosen angular momentum operator and the corresponding eigenvalue spectrum, or rather the quantum measurement process of angular momentum. The eigenvalue spectra of the black hole observables also allowed the derivation of a discrete black hole mass spectrum. Together with quantum mechanical considerations, which lead to upper and lower bounds on the lowest, physical black hole mass eigenvalue, one could, on the one hand, find an ultraviolet (Planck scale) cutoff of the Hawking temperature law and, on the other hand, regulate the fundamental parameter of loop quantum gravity, the Immirzi parameter γ, to have a numerical value approximately between 0.013 and 1.154. This interval contains all the values for γ obtained in more sophisticated approaches. The second objective was a technical analysis of quasi-local classical and quantum (type II) Kerr isolated horizon geometries that define rotating black holes in equilibrium states, specifically, the calculations of the SL(2, C) and SU(2) horizon boundary connections and their corresponding curvatures, for the description of the horizon degrees of freedom, as well as the construction of a quantum theory of Kerr isolated horizons. Using the SU(2) connection variable formulation, a conserved presymplectic structure, based on a SU(2) Chern-Simons theory in addition to certain auxiliary fields, that restore the required diffeomorphism invariance of general relativity and loop quantum gravity, was set up for the study of dynamical aspects in the phase space. The SU(2) connection variable formulation of classical rotating Kerr isolated horizons, thus, allowed for a canonical quantization in the context of loop quantum gravity. A counting of the microscopic quantum-gravitational horizon degrees of freedom leads to a thermodynamical formula for black hole entropy. The third aim was the investigation of first-order loop quantum-gravitational corrections to a general relativistic Friedmannian cosmological model in an effective setting, emerging from the Euclidean BRV spinfoam model of quantum cosmology and its Lorentzian generalization, and its physical implications. The Euclidean and Lorentzian EPRL transition amplitudes of the cosmological spinfoam models between coherent boundary states, that are peaked on homogeneous, isotropic geometries, satisfy constraint equations that each include a contribution of the order of the Planck constant. These terms also appear in the corresponding semiclassical, symplectic theories, giving small quantum-gravitational corrections to the classical Friedmann Hamiltonian constraint and, thus, to the dynamics of the scale factor, yielding insignificantly small deceleration contributions in the expansion of the universe. The corrections can either be seen as being of quantum origin, or to be due to the extra degrees of freedom introduced in the cosmological spinfoam models, or both. The robustness of the physical interpretation was established for arbitrary refinements of the quantum cosmological boundary states. Mathematical equivalences between the semiclassical cosmological models and, on the one hand, a classical, flat Friedmann cosmology without cosmological constant, coupled to an ultralight, irrotational, stiff, perfect fluid matter distribution, or, on the other hand, a massless, real scalar field theory were explored, demonstrating that different matter couplings and quantum geometry can have the same physical effects. Last, the two terms of opposite orientation in the asymptotic expansion of the vertex amplitude and the large-spin divergences in the covariant loop approach to quantum gravity were studied. Some indirect evidence was presented, indicating that these issues might be strictly related, i.e., that the spinfoam divergences may be generated by the presence of the two terms of opposite orientation. In a suitably simplified context, the characteristic spike divergences of the Ponzano-Regge model disappear when the theory is restricted to just one of the two orientations appearing in the asymptotic limit of the vertex amplitude. Thus, the divergences appear to be generated by antispacetime fluctuations of the geometry. This result came a bit as a surprise, given the intuition that quantum Regge calculus diverges because of the spikes in general and not that the crucial spikes at the root of the divergences are those just formed by anti-spacetimes which do not exist in conventional Regge calculus. This observation is crucial since several spinfoam theories, and in particular Lorentzian loop quantum gravity in four dimensions, contain the same two terms in the asymptotic limit. Summarizing, the research projects provided models, methods, and solution strategies for the quantumgravitational phenomenological sector, especially for quantum black holes and quantum cosmological models, and successfully advanced the understanding of certain aspects in covariant loop quantum gravity.
Publications
- ”Divergences and orientation in spinfoams”, Classical and Quantum Gravity 30, 055009 (2013)
M. Christodoulou, M. Langvik, A. Riello, C. Röken, C. Rovelli
- ”First-order quantum-gravitational correction to Friedmannian cosmology from covariant, holomorphic spinfoam cosmology”, International Journal of Modern Physics D 22, 1350005 (2013)
C. Röken
- ”Modelling black holes with angular momentum in loop quantum gravity”, eprint (2013)
E. Frodden, A. Perez, D. Pranzetti, C. Röken
(See online at https://doi.org/10.1007/s10714-014-1828-6) - ”On the nature of black holes in loop quantum gravity”, Classical and Quantum Gravity 30, 015005 (2013)
C. Röken
(See online at https://doi.org/10.1088/0264-9381/30/1/015005) - ”SL(2, C) and SU(2) Connection Variable Formulations of Kerr Isolated Horizon Geometries for Loop Quantum Gravity”, Physical Review D (2013)
C. Röken
