Black holes in effective loop quantum gravity: Hawking radiation

Emergent modified gravity provides a covariant framework for holonomy effects in models of loop quantum gravity with consistent black hole solutions coupled to a scalar field. Several independent studies of the Hawking thermal distribution are shown here to lead to the same final result. This internal consistency is a direct consequence of general covariance, which is analogous to the situation in classical general relativity but highly nontrivial in the context of modified canonical gravity. Holonomy corrections to the evaporation rate enter through the greybody factor, slowing down the evaporation process when the holonomy modification function decreases monotonically. Accounting for backreaction, corrected covariant semi-classical stress-energy tensors are computed in various vacuum states. Thanks to these results, the new concept of a net stress-energy tensor makes it possible to compute evaporation rates directly from energy conservation laws.

💡 Research Summary

The paper presents a comprehensive study of Hawking radiation from black holes that arise in the “Emergent Modified Gravity” (EMG) framework, a covariant approach to incorporating holonomy corrections inspired by loop quantum gravity (LQG). The authors first review the canonical structure of general relativity and explain how a modified Hamiltonian constraint (\tilde H) can be introduced while keeping the diffeomorphism constraint unchanged. By demanding that the Poisson brackets of the constraints close (Eqs. 1a‑1c) and that the resulting gauge transformations reproduce spacetime Lie derivatives (Eq. 3), they derive a new structure function (\tilde q_{ab}) and an emergent line element (\tilde g_{\mu\nu}). This guarantees full diffeomorphism covariance even after holonomy modifications, a feature lacking in earlier LQG-inspired black‑hole models where gauge fixing preceded polymerisation.

Two matter couplings are examined: a minimally coupled scalar field and a non‑minimally coupled one. The latter introduces direct interaction terms between the scalar and the gravitational degrees of freedom, which affect the effective potential felt by perturbations. Both couplings satisfy the covariance condition (Eq. 5), ensuring that the scalar field transforms as a scalar under the emergent spacetime diffeomorphisms.

The Hawking spectrum is derived using three independent methods:

1. Geometric‑optics approximation – The scalar field is expanded in mode functions on the static exterior region, and the Bogoliubov transformation yields a thermal occupation number with temperature (T_H = \kappa/2\pi), where (\kappa) is the surface gravity of the EMG black hole.

2. Hamilton–Jacobi tunnelling – By solving the on‑shell Hamilton–Jacobi equation for a massless (\phi) field in both minimally and non‑minimally coupled cases, the authors compute the imaginary part of the action across the horizon. The resulting tunnelling probability reproduces the same Boltzmann factor (e^{-\omega/T_H}), confirming gauge‑independence (the calculation is performed in both diagonal and Painlevé‑Gullstrand gauges).

3. ADM mass conservation – Pair creation is interpreted as a transfer of ADM mass from the black hole to the emitted quanta, leading again to the same thermal distribution.

All three approaches converge to the same Hawking temperature, demonstrating that the EMG framework respects the same covariance‑driven consistency that holds in classical GR.

The holonomy corrections enter the evaporation process solely through the greybody factor (\Gamma(\omega)). The authors consider a general holonomy function (\lambda(x)) (with (x) the areal radius) and analyse two representative behaviours:

• Constant (\lambda) – The greybody factor differs from the classical one only by an overall constant, reproducing earlier results in the literature (e.g., Refs.