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

Understand

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 $M$.