Logarithmic Correction to the Bekenstein-Hawking Entropy

Abstract

The exact formula derived by us earlier for the entropy of a four dimensional nonrotating black hole within the quantum geometry formulation of the event horizon in terms of boundary states of a three dimensional Chern-Simons theory is reexamined for large horizon areas. In addition to the semiclassical Bekenstein-Hawking contribution proportional to the area obtained earlier, we find a contribution proportional to the logarithm of the area together with subleading corrections that constitute a series in inverse powers of the area.

Testing Holographic Principle from Logarithmic and Higher Order Corrections to Black Hole Entropy

“…And, as was noted in Ref. [41] and explicitly shown in Ref. [60], the constant and the O(S −1 BH ) terms might be affected by taking the levelk away from the asymptotic value (∞), which we have assigned above in the integral (7.21), or by including the spin values higher than 1/2 such as we are away from the largest number of punctures (7.28), but the logarithmic term is not affected.…”

“…where S QG = lnN (E(p)), (7.26) which is the entropy defined in quantum geometry approach [10,57,41]. As we shall see, the additional term in (7.25) is the key ingredient which resolves the above mentioned confusions.…”

“…where A n is the area eigenvalue of the horizon with n punctures each of which carries a spin 1/2, and N (E(A n )) is the degeneracy of the energy level E(A n ) [10,41]. In the very large p limit, this may be approximated as an integral…”

“…b Quantum Geometry Approach of the Horizon For four-dimensional non-rotating black holes with or without a cosmological constant or electric, magnetic, and dilatonic charges, there is an alternative derivation of black hole entropy from the quantum geometry approach of Ashtekar et al that counts the boundary states of a three-dimensional Chern-Simons theory at the horizon [10,57,41]. There were some confusions about the logarithmic-correction term in this approach, by identifying the entropy in this approach with the usual (micro-canonical) entropy S = lnΩ(E) and generalizing this result in d = 4 to other dimensions [40]; as a result of this, a wrong behavior of central charge c (or P ) in the Cardy's formula (4.16) has been speculated in Ref.…”

“…by choosing γ = γ 0 ln2 [10,41] 16 . Note that this result applies to the four-dimensional SchwarzschildAdS as well as the Schwarzschild black holes since the formulation depends crucially on the local properties of the horizon [57].…”