Total Quantum Statistical Entropy of Reissner-Nordstrom BlackHoles: Scalar Field Case

The total quantum statistical entropy of Reissner-Nordstrom (RN) black holes is evaluated. The spacetime of the black holes is divided into three regions-region 1,( r > r_o), region 2, ( r_o > r > r_i), and region 3, (r_i > r > 0 ), where r_o is the radius of the outer event horizon, and r_i is the radius of the inner event horizon. The total quantum statistical entropy of RN black holes is S = S₁ + S₂ + S₃, where S_i, ( i = 1,2,3) is the entropy, contributed by region S_i (i = 1,2,3). The detailed calculation shows that S₂ ≈ 0. S₁ = (π²/45) k_bA_o/ε²β³, S₃ = -(π²/45) k_bA_i/ε²β³, where A_o and A_i are, respectively, the area of the outer and inner event horizons. Thus, as r_i approaches r_o, in the extreme case the total quantum statistical entropy of RN black holes approaches zero.