Effective loop quantum gravity framework for vacuum spherically symmetric space-times

TL;DR

This paper develops an effective loop quantum gravity framework for vacuum spherically symmetric space-times, deriving exact solutions that incorporate quantum corrections, bounded curvature, and horizon structures depending on black hole mass.

Contribution

It introduces a new effective framework for loop quantum gravity in spherical symmetry, providing explicit solutions with quantum corrections and analyzing horizon properties.

Findings

Curvature scalars are bounded by the Planck scale.

Solutions have the correct classical limit and decay at large distances.

For large mass, the space-time has both outer and inner horizons; for small mass, no horizons exist.

Abstract

We develop an effective framework for the $\bar\mu$ scheme of holonomy corrections motivated by loop quantum gravity for vacuum spherically symmetric space-times. This is done by imposing the areal gauge in the classical theory, and then expressing the remaining components of the Ashtekar-Barbero connection in the Hamiltonian constraint in terms of holonomies of physical length $\ell_{\rm Pl}$. The stationary solutions to the effective Hamiltonian constraint can be found exactly, and we give the explicit form of the effective metric in Painlev\'e-Gullstrand coordinates. This solution has the correct classical limit, the quantum gravity corrections decay rapidly at large distances, and curvature scalars are bounded by the Planck scale, independently of the black hole mass $M$. In addition, the solution is valid for radii $x \ge x_{\rm min} \sim (\ell_{\rm Pl}^2 M)^{1/3}$ indicating the need for a matter field, with an energy density bounded by the Planck scale, to provide a source for the curvature in the space-time. Finally, for $M \gg m_{\rm Pl}$, the space-time has an outer and also an inner horizon, within which the expansion for outgoing radial null geodesics becomes positive again. On the other hand, for sufficiently small $M \sim m_{\rm Pl}$, there are no horizons at all in the effective metric.