^{1}

^{3}.

We examine the logarithmic corrections to the black hole (BH) entropy product formula of outer horizon and inner horizon by taking into account the

There has been considerable ongoing excitement in physics of BH thermodynamic product formula [particularly area (or entropy) product formula] of inner horizon (

It has been suggested by Larsen [

It should be noted that this is the continuation of our earlier investigation [

The logarithmic correction of BH entropy of

In the present work, we shall compute the general logarithmic correction to BH entropy of

It has been known that BHs in Einstein’s gravity as well as other theories of gravity are much larger than the Planck scale length where the Bekenstein-Hawking entropy is precisely proportional to the horizon area [

It should be emphasized that logarithmic corrections to the Bekenstein-Hawking formula are very interesting and a great deal about such corrections is known in string theory and beyond. Logarithmic corrections arise from various sources, the simplest of which are the statistical fluctuations around thermal equilibrium. These are always present because they arise from saddle point Gaussian corrections to the integral that computes the density of states from the partition function. In some cases, such as the BTZ BH in pure 3D gravity, these are the only logarithmic corrections to the Bekenstein-Hawking entropy. However, more generally the logarithm of the partition function,

Moreover, it must be noted that a given theory of quantum gravity will assign a Hilbert space to

We have started with the partition function [

The density of states of the said thermodynamic system may be expressed by taking an inverse Laplace transformation (keeping

Now if we consider the system to be in equilibrium, then the inverse temperature is defined to be

Substituting (

Let us put

Thus, the logarithm of the density of states gives the corrected entropy of

Assume that the exact entropy function

Now the above function has an extremum value at

Therefore, the product becomes

For completeness, we further compute the logarithmic correction of

Now we apply this formula for specific BHs in order to calculate the logarithm correction to the Bekenstein-Hawking entropy of

The BH entropy and BH temperature [

Therefore, the entropy correction is given by

The second example we take is the Kehagias-Sfetsos (KS) BH [

The entropy for KS BH [

Therefore, the entropy correction for KS BH is given by

Now we take the AdS space. First, it should be Schwarzschild-AdS space-time (in the limit

The only physical horizon [

Now we take the RN-AdS case [

The quartic Killing horizon equation becomes

The entropy should read

Therefore, the logarithmic correction becomes

Now we take the spinning BH.

The BH entropy and BH temperature [

Now the logarithmic correction is computed to be

Next we take charged rotating BH.

The BH entropy and BH temperature [

The logarithmic correction is derived to be

The horizon equation [

The logarithmic correction for KN-AdS BH should read

The BH horizon is at

The BH horizons for rotating BTZ BH [

Therefore, the BH entropy correction is calculated to be

The horizons of charged dilation BH [

The BH temperature of

The entropy for both the horizons is

Thus, the BH entropy correction for

The horizon for Kerr-Sen [

The BH entropy and BH temperature for Sen BH are

Therefore, the logarithmic correction is calculated to be

This is an example of a dynamical cosmological BH [

The entropy (the surface area at

The logarithmic correction is found to be

The

The horizon radii for 5D Gauss-Bonnet BH [

The entropy of

The BH temperature of

The logarithmic correction of entropy

To sum up, we computed the general logarithmic corrections to the BH entropy product formula of inner horizon and outer horizon by taking into consideration the effects of statistical quantum fluctuations around the thermal equilibrium and also via CFT. We showed, followed by our earlier work [

The author declares that they have no conflicts of interest.