A glimpse into quantum gravity: A noncommutative spacetime results in quantization of area, mass and entropy

It is shown how a noncommutative spacetime leads to an area, mass and entropy quantization condition which allows to derive the Schwarzschild black hole entropy [Formula: see text], the logarithmic corrections, and further corrections, from the discrete mass transitions taking place among different mass states in [Formula: see text]. The higher-dimensional generalization of the results in [Formula: see text] follows. The discretization of the entropy-mass relation [Formula: see text] leads to an entropy quantization of the form [Formula: see text], such that one may always assign [Formula: see text] “bits” to the discrete entropy, and in doing so, make contact with quantum information. The physical applications of mass quantization, like the counting of states contributing to the black hole entropy, black hole evaporation, and the direct connection to the black holes-string correspondence [G. Horowitz and J. Polchinski, A correspondence principle for black holes and strings, Phys. Rev. D 55 (1997) 6189.] via the asymptotic behavior of the number of partitions of integers, follows. To conclude, it is shown how the recent large [Formula: see text] Matrix model (fuzzy sphere) of C.-S. Chu [A matrix model proposal for QG and the QM of black holes, preprint, arXiv:2406.01466] leads to very similar results for the black hole entropy as the physical model described in this work which is based on the discrete mass transitions originating from the noncommutativity of the spacetime coordinates.