Where Does Black Hole Entropy Lie? Some Remarks on Area-Entropy Law, Holographic Principle and Noncommutative Space-Time

In confrontation with serious and fundamental problems towards ultimate theory of quantum gravity and physics of Planck scale, 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 {\it 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 definite proportional relation, $ n^L_{\rm dof} (V_d^L)= {\cal A} (V_d^L) / G_d$, between the number of spatial degrees of freedom $n^L_{\rm dof} (V_d^L)$ inside of any $d-$dimensional spherical volume $V_d^L$ with radius $L $ and its boundary area ${\cal A} (V_d^L).$ 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: {\it Statistical} and {\it substantial} understanding of area-entropy law of black holes under a novel concept of noncommutative quantized space-time.