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Considering the possibility of ‘renormalization’ of the gravitational constant on the horizon, leading to a dependence on the level of the associated Chern-Simons theory, a rescaled area spectrum is proposed for the nonrotating black hole horizon in loop quantum gravity. The statistical mechanical calculation leading to the entropy provides a unique choice of the rescaling function for which the Bekenstein-Hawking area law is yielded without the need to choose the Barbero-Immirzi parameter

Loop quantum gravity (LQG) provides a platform for the calculation of entropy for nonrotating (assumed henceforth) black holes from the first principles, albeit in the kinematic framework [

The entire procedure of the entropy calculation for black holes in LQG consists of the following steps:

(1)

(2)

(3)

Now, let me focus on the second step. It implies, in principle, the quantum area of an arbitrary geometric 2-surface of topology

The field equations on a black hole horizon are that of a CS theory coupled to sources:

Now, in quantum field theory (QFT), the physical parameters like mass, charge, etc. associated with free particles get renormalized due to their coupling with fields, consequently affecting the mode spectrum. Analogously, in the present scenario, there is a possibility of

Although this heuristic ‘renormalization’ argument is only at the level of an analogy made from a QFT viewpoint, the possibility of the scenario cannot be completely ruled out unless one gets to know the full dynamics of the theory.

As I have just argued,

Based on these arguments I propose that the area contribution from a single puncture with quantum number

Since the ‘bare’ spectrum needs to be positive definite, one has

Hence, I consider a

I shall consider here black holes with classical area

The microstate count for a spin configuration

Now, a sum over

The blue curve shows the variation of

It is manifested from the Figure

From (

However,

All standard QFTs are some effective field theories valid until some energy scale. Only the renormalized quantities are calculable and measurable. The bare values of those physical quantities cannot be theoretically calculated. This is not unexpected because one does not have access to the most fundamental theory from which the corresponding QFT has come out to be an effective one. Taking quantum electrodynamics (QED) as an example, the bare electron charge is never measurable because one cannot decouple the electron from its field. However, if one

In the present scenario, one has the ‘bare’ area spectrum of an arbitrary 2-sphere and the rescaled (‘renormalized’) one on the black hole horizon. This is because one is now dealing with LQG which is one of the candidates of the fundamental theory of quantum gravity. Hence, the ‘bare’ quantities are expected to be known in this theory. Therefore, it seems quite logical to demand that

One should be aware of the fact that this is an asymptotic limit and

Now, using (

Taking into account (

While the above explanation can be a possibility of the real physical dynamics behind the ‘renormalization’, one can pose the question that why the correlation causes ‘screening’ and not ‘antiscreening’. The answer to this question can only come from the study of the true dynamics which is hitherto unknown. Hence, the above discussed possibility cannot be ruled out yet.

Whatever I have discussed here is purely based on heuristic arguments that rely on some observations of the field theoretic structure that effectively describes the black hole horizon degrees of freedom in LQG and some analogies. This by no means is anything mathematically rigorous. However, having the knowledge of the full dynamics of quantum black holes in LQG yet out of reach, such a possibility of a ‘renormalized’ gravitational constant governing the microscopic physics on the horizon and giving rise to the BHAL irrespective of the value of

I hope this work may give a possible hint towards a more fundamental calculation of black hole entropy from LQG involving the underlying dynamics of quantum black hole horizons leading to the BHAL irrespective of the choice of

The full article can be made available.

The author declares that they have no conflicts of interest.

I am grateful to Benito Juarez Aubry and Tim Koslowski for carefully going through the manuscript and offering critical comments that have helped me in improving the same to a large extent. Most of the work was supported by DGAPA postdoctoral fellowship of UNAM, Mexico.