2002/02/28 by Jacob D. Bekenstein, Gilad Gour
Mathematics · Physics and Astronomy · #Algebraic number #Black Holes and Theoretical Physics #Black hole (networking) #Black hole thermodynamics #Cosmology and Gravitation Theories #Degeneracy (biology) #Degenerate energy levels #Eigenvalues and eigenvectors #Entropy (arrow of time) #Extremal black hole #Logarithm #Mathematical analysis #Mathematics #Noncommutative and Quantum Gravity Theories #Observable #Physics #Pure mathematics #Quantum mechanics #astro-ph #gr-qc #hep-th #quant-ph
paper · pdf · doi:10.1103/physrevd.66.024005
published as Phys.Rev. D66 (2002) 024005 · PhysRevTeX, 14 pages
arxiv created 2002/05/02 · openalex publication_date 2002/06/28 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
What is the nature of the energy spectrum of a black hole? The algebraic approach to black hole quantization requires the horizon area eigenvalues to be equally spaced. As stressed long ago by Mukhanov, such eigenvalues must be exponentially degenerate with respect to the area quantum number if one is to understand black hole entropy as reflecting degeneracy of the observable states. Here we construct the black hole stationary states by means of a pair of ``creation operators'' subject to a particular simple algebra, a slight generalization of that for a pair of harmonic oscillators. This algebra reproduces the main features of the algebraic approach, in particular the equally spaced area spectrum. We then prove rigorously that the nth area eigenvalue is exactly 2n-fold degenerate. Thus black hole entropy qua logarithm of the number of states for a fixed horizon area is indeed proportional to that area.