[A.sub.h] and [A.sub.c] are area of outer horizon (OH) or event horizon (EH) and inner horizon (IH) or Cauchy horizon (CH).

Hennig, "Inner Cauchy Horizon of Axisymmetric and Stationary Black Holes with Surrounding Matter in Einstein-Maxwell Theory," Physical Review Letters, vol.

Since the function g(r) has at most two zero points, axially symmetric space-time can have at most two horizons, the event horizon [r.sub.+] and the internal

Cauchy horizon [r.sub.-] < [r.sub.+] [27, 28].

In the previous work, we derived the general logarithmic correction to the entropy product formula of event horizon and

Cauchy horizon for various spherically symmetric and axisymmetric BHs by taking into account the effects of quantum fluctuations around the thermal equilibrium.

Here [M.sup.*] is the patch between event horizon [H.sub.+] located at r = [r.sub.+] and inner

Cauchy horizon [H.sub.-] located at r = [r.sub.-] for the spherically symmetric black hole with two separate horizons.

The roots [r.sub.h] = M + [square root of ([M.sup.2] - [a.sup.2])] and [r.sub.h] = [square root of ([M.sup.2] - [a.sup.2])] represent external event horizon and internal

Cauchy horizon, respectively.

Here [H.sup.+] is called event horizon and H- is called the

Cauchy horizon. In four-dimensional space-time, such fixed point sets are of two types, isolated points or zero-dimensional which we call NUTs and two surfaces or two-dimensional which we call Bolts.

There is a

Cauchy horizon at T = [T.sub.0] = 0 for any such initial hypersurface T = [T.sub.0] < 0.