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Recently, one- and two-parameter deformed Einstein equations have been studied for extremal quantum black holes which have been proposed to obey deformed statistics by Strominger. In this study, we give a deeper insight into the deformed Einstein equations and consider the solutions of these equations for the extremal quantum black holes. We then represent the implications of the solutions, such that the deformation parameters lead the charged black holes to have a smaller mass than the usual Reissner-Nordström black holes. This reduction in mass of a usual black hole can be considered as a transition from classical to quantum black hole regime.

Recently, one- and two-parameter deformed Einstein equations, which are thought to describe the gravitational fields of extremal quantum black holes, have been studied in the framework of entropic gravity proposal [

On the other hand, quantum mechanically, mass can only be localized into a region, reduced Compton wavelength

These extremal black holes with a possible minimum mass are quantum mechanically stable objects and are useful for studying the quantum mechanics of black holes [

Extremal black holes are used to study the quantum mechanics of black holes. For large

One of the ways of studying quantum mechanics of black holes is the scattering of black holes to investigate whether they are bosons, fermions, or something else [

Therefore, the extremal quantum black holes can be considered as deformed bosons or fermions and the statistics obeyed by the extremal quantum black holes is deformed statistics. Moreover, the statistical mechanics of the deformed bosons and fermions have been studied in the literature through recent years [

Here, we firstly give a brief summary of one- and two-parameter deformed Einstein equations and then the solutions of the deformed Einstein equations for charged black holes. Since the solutions of standard Einstein equations for charged black holes are the Reissner-Nordström solutions in classical gravity, the solutions of the deformed Einstein equations for charged black holes can be considered in quantum gravity. Lastly, the implications of the solutions are represented. These are that the deformation parameters lead the charged black holes to have a smaller mass than the usual Reissner-Nordström black holes. This reduction in mass of a usual black hole can be considered as a transition from classical to quantum black hole regime.

By using the entropy of the deformed gas models in Verlinde’s entropic gravity approach, the deformed Einstein equations are obtained to describe the gravitational fields of these deformed objects. For a

On the other hand, to obtain the two-parameter deformed Einstein equations, it is suitable to introduce the (

In order to construct the deformed Einstein equations from the entropies in (

The source of gravity is energy or matter and it is distributed evenly over the degrees of freedom in space-time. The existence of energy or matter in space-time causes a temperature in the space-time. The product of the change of entropy during the displacement of source and the temperature is in fact the work and this work is originally led by the force which is known to be gravity [

By using Verlinde’s idea, one- and two-parameter deformed Einstein equations are recently derived from the deformed entropies (

In the next section, we solve one- and two-parameter deformed Einstein equations for a charged extremal black hole and investigate the implications of the solutions.

Since the underlying statistics of the extremal quantum black holes is known to be the deformed statistics, we admit the particles forming deformed gas models to be the quantum black holes and the corresponding deformed Einstein equations for these deformed particles are assumed to describe the gravitational fields of the quantum black holes of these deformed particles.

We know that the extremal quantum black holes should be charged, because the mass of them should decrease to the minimum value proportional to the charge. The classical charged black holes are treated by the standard Einstein equations and the classical solutions of the standard Einstein equations for the charged black holes are known as the Reissner-Nordström solutions. Here, we obtain the quantum analogs of the solutions of the Einstein equations for these classical charged black holes.

Deformed version of the Einstein field equations is assumed to describe the geometry of the space-time surrounding a charged spherical quantum black hole. Therefore, we need to solve the deformed Einstein-Maxwell equations for the charged quantum black holes. Because of the spherical symmetry, the generic form for the metric in 4 dimensions is [

The first case

The second case

Finally, the third case

In our deformed case, this decrease in mass of black hole which is controlled by the term

We investigate the reduction of the mass with respect to the classical Reissner-Nordström case, for the

The

The

The

The

Recently,

We represent the true and coordinate singularities from the solutions for quantum black holes. Also, the event horizons for these singularities are mentioned briefly. We also investigate the possible decrease in mass via Hawking radiation to a minimum value which is determined by the charge of quantum black hole. The difference in the decrease in classical black holes and quantum black holes is obvious from (

We illustrate the decreases in quantum masses

Since the theoretical possibility of concentrating a mass into its reduced Planck mass gives a radius containing the Schwarzschild radius and the obtained quantum masses of the extremal black holes in (

The authors declare that there are no competing interests regarding the publication of this paper.