Effective loop quantum geometry of Schwarzschild interior

Jerónimo Cortez, William Cuervo, Hugo A. Morales-Técotl, and Juan C. Ruelas
Phys. Rev. D 95, 064041 – Published 24 March 2017

Abstract

The success of loop quantum cosmology to resolve classical singularities of homogeneous models has led to its application to the classical Schwarszchild black hole interior, which takes the form of a homogeneous Kantowski-Sachs model. The first steps of this were done in pure quantum mechanical terms, hinting at the traversable character of the would-be classical singularity, and then others were performed using effective heuristic models capturing quantum effects that allowed a geometrical description closer to the classical one but avoided its singularity. However, the problem of establishing the link between the quantum and effective descriptions was left open. In this work, we propose to fill in this gap by considering the path-integral approach to the loop quantization of the Kantowski-Sachs model corresponding to the Schwarzschild black hole interior. We show that the transition amplitude can be expressed as a path integration over the imaginary exponential of an effective action which just coincides, under some simplifying assumptions, with the heuristic one. Additionally, we further explore the consequences of the effective dynamics. We prove first that such dynamics imply some rather simple bounds for phase-space variables, and in turn—remarkably, in an analytical way—they imply that various phase-space functions that were singular in the classical model are now well behaved. In particular, the expansion rate, its time derivative, and the shear become bounded, and hence the Raychaudhuri equation is finite term by term, thus resolving the singularities of classical geodesic congruences. Moreover, all effective scalar polynomial invariants turn out to be bounded.

  • Received 2 February 2017

DOI:https://doi.org/10.1103/PhysRevD.95.064041

© 2017 American Physical Society

Physics Subject Headings (PhySH)

Gravitation, Cosmology & AstrophysicsParticles & Fields

Authors & Affiliations

Jerónimo Cortez1,*, William Cuervo1,2,†, Hugo A. Morales-Técotl2,‡, and Juan C. Ruelas2,§

  • 1Departamento de Física, Facultad de Ciencias, Universidad Nacional Autónoma de México, Ciudad de México 04510, México
  • 2Departamento de Física, Universidad Autónoma Metropolitana Iztapalapa, San Rafael Atlixco 186, CP 09340 Ciudad de México, México

  • *jacq@ciencias.unam.mx
  • wfcuervo@gmail.com
  • hugo@xanum.uam.mx
  • §j.carlos.ruelas.v@gmail.com

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Issue

Vol. 95, Iss. 6 — 15 March 2017

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