From Planck area to graph theory: Topologically distinct black hole microstates

Abstract

We postulate a Planck scale horizon unit area, with no bits of information locally attached to it, connected but otherwise of free form, and let $n$ such geometric units compactly tile the black hole horizon. Associated with each topologically distinct tiling configuration is then a simple, connected, undirected, unlabeled, planar, chordal graph. The asymptotic enumeration of the corresponding integer sequence gives rise to the Bekenstein-Hawking area entropy formula, automatically accompanied by a proper logarithmic term, and fixes the size of the horizon unit area, thereby constituting a global realization of Wheeler's "it from bit" phrase. Invoking Polya's theorem, an exact number theoretical entropy spectrum is offered for the 2+1 dimensional quantum black hole.

arXiv comment6 pages, 5 figures; v2: minor changes, ref added; v3: version published in PRD (RC)

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• Friedmann equations and cosmic bounce in a modified cosmological scenario
• Thermodynamics as a tool for (quantum) gravitational dynamics
• D-bound and Bekenstein bound for McVittie solution surrounded by dark energy cosmological fields
• Quantized Area of the Schwarzschild Black Hole: A non-Hermitian Perspective
• Black Holes, Complex Curves, and Graph Theory: Revising a Conjecture by Kasner
• Gravity from thermodynamics in vacuum: Lorentz invariance and the role of Bel-Robinson tensor