[Paper Review] Counting the Apparent Horizon

Arundhati Dasgupta|ArXiv.org|Oct 7, 2003

Black Holes and Theoretical Physics1 references3 citations

TL;DR

This paper derives the Bekenstein-Hawking entropy of a Schwarzschild black hole as the logarithm of the degeneracy of the area operator in a semiclassical coherent state framework, showing that the entropy arises from discrete quantum numbers associated with radial edges at the apparent horizon. Using gauge coherent states in canonical quantum gravity, it identifies the horizon's area spectrum as equispaced and links the entropy to the counting of these quantum microstates, providing a holographic explanation for black hole entropy in terms of surface degrees of freedom.

ABSTRACT

Using a discrete spectrum proposed for expectation values of canonical variables in black hole coherent states, the semiclassical entropy associated with the Schwarzschild space-time is derived to be the area of the apparent horizon.

Motivation & Objective

• To explain the origin of black hole entropy in terms of quantum geometry and surface degrees of freedom.
• To resolve the holographic nature of black hole entropy by identifying where quantum degrees of freedom reside.
• To derive the semiclassical entropy of a Schwarzschild black hole using coherent states in canonical quantum gravity.
• To show that the entropy arises from degeneracy in the momentum operator at the apparent horizon, not from bulk or interior states.
• To provide a concrete realization of the black hole state via coherent states and link it to measurable geometric operators.

Proposed method

• Constructs gauge coherent states peaked on classical values of holonomy and momentum variables in canonical gravity.
• Uses the canonical pair $ h_e(A), P^I_e(A,E) $ defined by Thiemann, with $ P_e = \sqrt{P^I_e P^I_e} $ representing the gauge-invariant momentum for radial edges.
• Identifies the area operator with $ P_{e_r} $, showing its spectrum is equispaced: $ P_{e_r} = (j + \frac{1}{2})t $, where $ t \sim l_p^2 / a_N $.
• Splits the coherent state into outside, horizon, and inside components: $ \Psi = \Psi^{\text{out}} \Psi^{\text{hor}} \Psi^{\text{in}} $.
• Traces over the internal (inside) coherent state to obtain a reduced density matrix, leaving the horizon and outside as the observable system.
• Counts degeneracy of $ P_{e_r} $ at the horizon, leading to entropy $ S_{BH} = \frac{A_H}{l_p^2} \log 2 $, with Immirzi parameter $ \beta = 4\log 2 $.

Experimental results

Research questions

• RQ1 Why is black hole entropy associated with surface degrees of freedom, and how does this relate to holography?
• RQ2 Where are the quantum degrees of freedom that contribute to black hole entropy located—on the horizon, inside, or outside?
• RQ3 How can the Bekenstein-Hawking entropy formula emerge from a quantum gravity framework with a semiclassical limit?
• RQ4 What role do coherent states play in realizing classical spacetime from quantum geometry?
• RQ5 Why is the degeneracy counting only meaningful at the horizon and not elsewhere on a 2-sphere?

Key findings

• The entropy of the black hole is derived as $ S_{BH} = \frac{A_H}{l_p^2} \log 2 $, matching the Bekenstein-Hawking formula up to the Immirzi parameter.
• The area operator is identified with the gauge-invariant momentum $ P_{e_r} $, which has an equispaced spectrum $ (j + \frac{1}{2})t $ in the coherent state framework.
• Degeneracy arises from counting quantum numbers of $ P_{e_r} $ at the horizon, which are not visible in classical expectation values.
• The horizon acts as a boundary condition: fixing $ j_i $ at the horizon determines the state everywhere outside, confirming holography.
• Tracing over the internal coherent state removes access to bulk degrees of freedom, leaving only the horizon and outside as observable, thus defining the entropy.
• The Immirzi parameter is fixed to $ \beta = 4\log 2 $ to reproduce the correct entropy, though this value is not considered definitive due to the simplicity of the coherent state model.