Abstract
Dynamical black-hole scenarios have been developed in loop quantum gravity in various ways, combining results from mini and midisuperspace models. In the past, the underlying geometry of space-time has often been expressed in terms of line elements with metric components that differ from the classical solutions of general relativity, motivated by modified equations of motion and constraints. However, recent results have shown by explicit calculations that most of these constructions violate general covariance and slicing independence. The proposed line elements and black-hole models are therefore ruled out. The only known possibility to escape this sentence is to derive not only modified metric components but also a new space-time structure which is covariant in a generalized sense. Formally, such a derivation is made available by an analysis of the constraints of canonical gravity, which generate deformations of hypersurfaces in space-time, or generalized versions if the constraints are consistently modified. A generic consequence of consistent modifications in effective theories suggested by loop quantum gravity is signature change at high density. Signature change is an important ingredient in long-term models of black holes that aim to determine what might happen after a black hole has evaporated. Because this effect changes the causal structure of space-time, it has crucial implications for black-hole models that have been missed in several older constructions, for instance in models based on bouncing black-hole interiors. Such models are ruled out by signature change even if their underlying space-times are made consistent using generalized covariance. The causal nature of signature change brings in a new internal consistency condition, given by the requirement of deterministic behavior at low curvature. Even a causally disconnected interior transition, opening back up into the former exterior as some kind of astrophysical white hole, is then ruled out. New versions consistent with both generalized covariance and low-curvature determinism are introduced here, showing a remarkable similarity with models developed in other approaches, such as the final-state proposal or the no-transition principle obtained from the gauge-gravity correspondence.
Highlights
Black-hole models have recently regained considerable attention in loop quantum gravity.A certain consensus seems to have formed according to which the singularity in a classical black hole is replaced by a non-singular phase in which the density and curvature are not infinite but large, such that infalling matter might bounce back and re-emerge after the initial horizon has evaporated.Such models are used in conceptual discussions about a possible non-singular fate of black holes, but even in phenomenological descriptions that aim to derive potentially observable effects from the re-emergence of matter.It is important to note that none of these models are based on consistent embeddings of possible effects from loop quantum gravity in a covariant space-time theory
Well before [7] was published, modified brackets of the form (27) had already been found by analyzing space-time structure directly in terms of generators of hypersurface deformations [10,11,12,13]. These studies followed the usual reasoning of effective field theories, in which potential quantum effects are explored in a theory of classical type by including quantum modifications as well as other terms of the same order that are consistent with all required symmetries, here given by general covariance as represented by hypersurface deformations
The final-state scenario already shows that models that take signature change seriously lead to unexpected similarities between what has been suggested in loop quantum gravity and other approaches
Summary
Black-hole models have recently regained considerable attention in loop quantum gravity. They are independent of any difference between approaches that have been put forward in order to formulate inhomogeneous models of loop quantum gravity, such as hybrid models [5], the dressed-metric approach [6], partial Abelianization in spherically symmetric models [7], or using timelike homogeneous slices in static spherically symmetric space-times [8], just to name those to which the recent no-go results about covariance directly apply Even before these results became available, it had been found that covariance can be preserved by some loop effects at least in a deformed way, which respects the number of classical symmetries underlying general covariance but may change their algebraic relationships [9,10,11,12,13,14,15,16,17]. The condition that physics at low curvature be deterministic rules out certain black-hole models, including bouncing ones, and suggests new ones that are compatible with determinism as well as generalized covariance These scenarios show interesting relationships with other proposals unrelated to loop quantum gravity. There is a refreshing contrast with bounce-based black holes in loop quantum gravity, which are often put in opposition to other approaches
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