Abstract.
In confrontation with serious and fundamental problems towards ultimate theory of quantum gravity and Planck scale physics, we emphasize the importance of underlying noncommutative space-time such as Snyder’s or Yang’s Lorentz-covariant quantized space-time. The background of Bekenstein-Hawking’s area-entropy law and holographic principle is now substantially understood in terms of Kinematical Holographic Relation (KHR), which holds in Yang’s quantized space-time as the result of the kinematical reduction of spatial degrees of freedom caused by its own nature of noncommutative geometry. KHR implies a proportional relation, \( n_{dof}(V_{d}^{L})=A (V_{d}^{L})/G_{d}\), between the number of spatial degrees of freedom \( n_{dof} (V_{d}^{L})\) inside any d -dimensional spherical volume \( V_{d}^{L}\) with radius L and its boundary area \( A(V_{d}^{L})\) . It yields a substantial basis for our new area-entropy law of black holes and further enables us to connect “the first law of black hole mechanics” with “the thermodynamics of black holes” towards our final goal: A statistical and substantial understanding of the area-entropy law of black holes under a novel concept of noncommutative quantized space-time.
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Tanaka, S. Where does black-hole entropy Lie?. Eur. Phys. J. Plus 129, 11 (2014). https://doi.org/10.1140/epjp/i2014-14011-9
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DOI: https://doi.org/10.1140/epjp/i2014-14011-9