Planck-Scale Corrections to the Cardy–Verlinde Formula in SAdS Black Hole

Abstract

Possible corrections to the thermodynamic quantities of a 4D Schwarzschild–anti–de Sitter (SAdS) black hole are investigated by considering the generalized uncertainty principle (GUP) and the modified dispersion relation (MDR) separately. The quantum gravitational corrections to the Hawking temperature, energy and entropy of the black hole are calculated based on both the GUP and the MDR analysis. The explicit form of the corrections is worked out up to the sixth power of the Planck length. The quantum-corrected thermodynamic quantities due to GUP and MDR are used separately to obtain the quantum-gravitational corrections to the Cardy–Verlinde (C–V) formula. It is found that the C–V formula receives some new corrections in either of approaches. Through comparison of the corrections obtained from GUP and MDR approaches, it is shown that the results of these two alternative approaches should be identical if one uses the suitable expansion coefficients. Finally, the renormalized form of the C–V formula is introduced by redefining the Virasoro operator and the central charge within both the GUP and the MDR.