A Short Essay on Quantum Black Holes and Underlying Noncommutative Quantized Space-Time

TL;DR

This paper explores the role of noncommutative geometry in quantum gravity, focusing on a unified holographic relation applicable across scales and its implications for black hole entropy near the Planck scale.

Contribution

It introduces a unified Kinematical Holographic Relation applicable from macroscopic to microscopic scales in noncommutative space-time, offering new insights into black hole entropy.

Findings

Unified KHR valid across all scales in d=3.

Potential modification of black hole entropy law at Planck scale.

Highlights the significance of noncommutative geometry in quantum gravity.

Abstract

We emphasized the importance of underlying noncommutative geometry or Lorenz-covariant quantized space-time towards ultimate theory of quantum gravity and Planck scale physics. We focused there our attention on the statistical and substantial understanding of Bekenstein-Hawking's Area-Entropy Law of black holes in terms of Kinematical Holographic Relation (KHR). KHR manifestly holds in Yang's quantized space-time as the result of kinematical reduction of spatial degrees of freedom caused by its own nature of noncommutative geometry and plays an important role in our approach without any recourse to the familiar hypothesis, so-called Holographic Principle. In the present paper, we find out a {\it unified} form of KHR applicable to the whole region ranging from macroscopic to microscopic scales in spatial dimension $ d=3.$ We notice a possibility of nontrivial modification of Area-Entropy Law of Black Holes which becomes most remarkable in the extremely microscopic system close to Planck scale. We finally refer to the historical background of noncommutative geometry or quantized space-time.