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 cannot 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 spherically symmetric black holes interior) in terms of the functions that govern the deformations of the theory.
Highlights
Black holes are iconic predictions of general relativity
We show in the case of spherical symmetry that it is impossible to construct a non-trivial deformation of the vectorial constraint, which leaves the deformed diff-algebra closed
S does not commute with π N, and their Poisson bracket is non-local in the sense that it involves derivatives of delta distributions. Such a situation is pathological and makes the theory ill-defined with an undefined number of degrees of freedom. The conclusion of this analysis is that one cannot extend the condition of having a closed algebra of constraints associated to an effective loop quantum gravity theory for an arbitrary background which satisfies the following properties: first, it is invariant under spatial diffeomorphisms; second, the deformation does not involve the three-dimensional Ricci tensor but only the extrinsic curvature tensor; and, the effective theory is a local theory of the metric
Summary
Black holes are iconic predictions of general relativity. Their classical description is very well established, and strong evidence for their existence is regularly reported through gravitational wave astronomy. It was realized that, if one starts from the spherically symmetric vacuum gravity phase space in terms of the self-dual variables, for which the Barbero–Immirzi parameter γ = ±i, one obtains polymer-like modifications while keeping the algebra of constraints closed and undeformed [55] This construction was generalized to the Gowdy model and. There exist modifications of spherically symmetric general relativity, inspired from loop quantum gravity, which lead to a deformed but closed constraints algebra This property makes the effective theories interesting because they are invariant under a deformed covariance, there is no effective quantum anomalies, and the classical solutions have a clear geometrical meaning.
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