Where does black-hole entropy Lie?

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.