Deformed General Relativity and Quantum Black Holes Interior

Abstract

Effective models of black holes interior have led to several proposals for regular black holes. In the so-called polymer models, based on effective deformations of the phase space of spherically symmetric general relativity in vacuum, one considers a deformed Hamiltonian constraint while keeping a non-deformed vectorial constraint, leading under some conditions to a notion of deformed covariance. In this article, we revisit and study further the question of covariance in these deformed gravity models. In particular, we propose a Lagrangian formulation for these deformed gravity models where polymer-like deformations are introduced at the level of the full theory prior to the symmetry reduction and prior to the Legendre transformation. This enables us to test whether the concept of deformed covariance found in spherically symmetric vacuum gravity can be extended to the full theory, and we show that, in the large class of models we are considering, the deformed covariance can not be realized beyond spherical symmetry in the sense that the only deformed theory which leads to a closed constraints algebra is general relativity. Hence, we focus on the spherically symmetric sector, where there exist non-trivial deformed but closed constraints algebras. We investigate the possibility to deform the vectorial constraint as well and we prove that non-trivial deformations of the vectorial constraint with the condition that the constraints algebra remains closed do not exist. Then, we compute the most general deformed Hamiltonian constraint which admits a closed constraints algebra and thus leads to a well-defined effective theory associated with a notion of deformed covariance. Finally, we study static solutions of these effective theories and, remarkably, we solve explicitly and in full generality the corresponding modified Einstein equations, even for the effective theories which do not satisfy the closeness condition. In particular, we give the expressions of the components of the effective metric (for static and spherically symmetric black holes interior) in terms of the functions that govern the deformations of the theory.

Critical Evaluation of Common Claims in Loop Quantum Cosmology

“…A possible solution to this problem is to implement holonomy modifications in an anomaly-free way which does not break any gauge transformations but may deform the classical structure of hypersurface deformations given in [63,64]. Consistent deformations are possible in spherically symmetric models with holonomy modifications [71,72,73,74,75,76,77], but they imply a non-classical space-time structure which is related to slicing independence only in some cases, and after field redefinitions [78,79]. The latter feature not only resolves the contradiction between holonomy modifications and covariance pointed out in [11], it also shows why singularities can be resolved in loop quantum cosmology even for matter obeying the usual energy conditions: Not only the dynamics but also space-time structure become non-classical as a consequence of holonomy modifications, unhinging the mathematical foundation of singularity theorems.…”

Bouncing compact objects. Part I. Quantum extension of the Oppenheimer-Snyder collapse

“…See also Refs. [61][62][63] and Refs. [64][65][66] for extended discussions regarding the problem of covariance in such polymer constructions.…”

Space–Time Physics in Background-Independent Theories of Quantum Gravity

“…Time-like homogeneity with modified dynamics leads to a formal line element: ds 2 =ñ(t) 2 dt 2 −Ã(t) 2 dx 2 +b(t) 2 dϑ 2 + sin 2 ϑdϕ 2 (42) if solutionsñ,Ã andb are simply inserted in the classical line element. Since properties of space-time transformations have not been checked at this point, there is no guarantee that (42) presents a proper effective line element.…”