[Paper Review] Hawking Radiation inside Black Holes in Quantum Gravity

Ichiro Oda|arXiv (Cornell University)|Mar 7, 2015

Black Holes and Theoretical Physics6 references3 citations

TL;DR

This paper proposes a canonical quantization framework for spherically symmetric black holes in quantum gravity, treating the radial coordinate as time inside the horizon. By imposing physically motivated assumptions—specifically that matter fields, mass function, and radial degrees of freedom depend only on advanced time—Hawking radiation is derived analytically inside the black hole, yielding a mass-loss rate identical to Hawking's semi-classical result. The analysis suggests radiation originates from the spacetime singularity, offering a quantum-gravitational resolution to the information paradox.

ABSTRACT

We study black hole radiation inside black holes within the framework of quantum gravity. First, we review on our previous work of a canonical quantization for a spherically symmetric geometry where one of the spatial coordinates is treated as the time variable, since we think of the interior region of a black hole. Based on this formalism, under physically plausible assumptions, we solve the Wheeler-De Witt equation inside the black hole, and show that the mass-loss rate of an evaporating black hole due to thermal radiation is equivalent to the result obtained by Hawking in his semi-classical approach. A remarkable point is that our assumptions make the momentum constraint coincide with the Hamiltonian constraint up to an irrelevant overall factor. Furthermore, for comparison, we solve the Wheeler-De Witt equation outside the black hole as well, and see that the mass-loss rate of an evaporating black hole has the same expression. The present analysis suggests that the black hole radiation comes from the black hole singularity. We also comment on the Birkhoff theorem in quantum gravity.

Motivation & Objective

• To formulate a canonical quantum gravity approach for the interior of a black hole, treating the radial coordinate as a time variable.
• To eliminate the need for artificial regularization (e.g., finite γ) used in prior work by adopting more physically plausible assumptions.
• To derive the Hawking radiation mass-loss rate analytically inside the black hole using the Wheeler-DeWitt equation.
• To compare the radiation process inside and outside the horizon, assessing consistency across regions.
• To explore implications for the information loss paradox and the Birkhoff theorem in quantum gravity.

Proposed method

• Adopt a canonical formalism for spherically symmetric gravity in the interior region bounded by the apparent horizon and the singularity, with r = const as spacelike hypersurfaces.
• Introduce assumptions that matter field Φ, mass function M, and radial field φ depend only on advanced time v, reducing dynamical degrees of freedom.
• Enforce the momentum constraint to coincide with the Hamiltonian constraint up to a factor, simplifying the Wheeler-DeWitt equation.
• Solve the Wheeler-DeWitt equation analytically under these assumptions, enabling computation of the expectation value of the mass-loss rate.
• Compare the result with the exterior region, where similar assumptions yield the same mass-loss expression.
• Use the tortoise coordinate v to interpret the assumptions as holomorphic conditions, enabling conformal field theory techniques.

Experimental results

Research questions

• RQ1 Can Hawking radiation be derived inside a black hole using canonical quantum gravity without artificial regularization?
• RQ2 Does the mass-loss rate from black hole radiation inside the horizon match the semi-classical result obtained by Hawking?
• RQ3 What is the physical origin of black hole radiation in quantum gravity—does it arise from the singularity or the horizon?
• RQ4 How do the momentum and Hamiltonian constraints relate in the interior region under physically plausible assumptions?
• RQ5 Can the Birkhoff theorem be consistently formulated in quantum gravity, and what does it imply for gravitational degrees of freedom?

Key findings

• The mass-loss rate of an evaporating black hole due to thermal radiation is analytically derived inside the black hole and matches Hawking’s semi-classical result exactly.
• The assumptions Φ = Φ(v), M = M(v), and φ = r lead to the momentum constraint being proportional to the Hamiltonian constraint, simplifying the quantum constraints.
• The analysis suggests that black hole radiation originates from the spacetime singularity, not the horizon, implying a potential resolution to the information loss paradox.
• The same mass-loss rate expression is obtained in the exterior region under analogous assumptions, indicating consistency between interior and exterior quantum descriptions.
• The radial field φ is effectively fixed as φ = r, which is physically justified but may be relaxed in future models.
• The formalism supports a quantum-gravitational interpretation of the Birkhoff theorem, where gravitational degrees of freedom are encoded in the mass function M(v) rather than explicit gravitational waves.