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We study a spherically symmetric setup consisting of a Schwarzschild metric as the background geometry in the framework of classical polymerization. This process is an extension of the polymeric representation of quantum mechanics in such a way that a transformation maps classical variables to their polymeric counterpart. We show that the usual Schwarzschild metric can be extracted from a Hamiltonian function which in turn gets modifications due to the classical polymerization. Then, the polymer corrected Schwarzschild metric may be obtained by solving the polymer-Hamiltonian equations of motion. It is shown that while the conventional Schwarzschild space-time is a vacuum solution of the Einstein equations, its polymer-corrected version corresponds to an energy-momentum tensor that exhibits the features of dark energy. We also use the resulting metric to investigate some thermodynamical quantities associated with the Schwarzschild black hole, and in comparison with the standard Schwarzschild metric the similarities and differences are discussed.

One of the most important arenas that show the power of general relativity in describing the gravitational phenomena is the classical theory of black hole physics. However, when we introduce the quantum considerations to study of a gravitational systems, general relativity does not provide a satisfactory description of the physics of the system. The phenomena such as black hole radiation and all kinds of cosmological singularities are among the phenomena in which the use of quantum mechanics in their description is inevitable. This means that although general relativity is a classical theory, in its most important applications, the system under consideration originally obeys the rules of quantum mechanics. Therefore, any hope in the accurate description of gravitational systems in high energies depends on the development of a complete theory of quantum gravity. That is why the quantum gravity is one of the most important challenges in theoretical physics which from its DeWitt’s traditional canonical formulation [

Among the effective theories that also use a minimal length scale in their formalism, we can mention the polymer quantization [

To polymerize a dynamical system one usually begins with a classical system described by Hamiltonian

In this paper, we are going to study how the metric of the Schwarzschild black hole gets modifications due to the classical polymerization. Since the thermodynamical properties of the black hole come from its geometrical structure, the corrections to the black hole’s geometry yield naturally modifications to its thermodynamics. To do this, we begin with a general form of a spherically symmetric space-time and then construct a Hamiltonian in such a way that the Schwarzschild metric is resulted from the corresponding Hamiltonian equations of motion. We then follow the procedure described above and by applying it to the mentioned Hamiltonian we get the classical polymerized Hamiltonian, by means of which we expect to obtain the polymer-corrected Schwarzschild metric. The paper is organized as follows. In Section

As is well known, in Schrödinger picture of quantum mechanics, the coordinates and momentum representations are equivalent and may be easily converted to each other by a Fourier transformation. However, in the presence of the quantum gravitational effects the space-time may take a discrete structure so that such well-defined representations are no longer applicable. As an alternative, polymer quantization provides a suitable framework for studying these situations [

According to the mentioned above form of the Hilbert space of the polymer representation of quantum mechanics, the position space (with coordinate

We start with the general spherically symmetric line element as (it can be shown that, by introducing of new radial and time coordinates as

From now on we focus on Schwarzschild black hole metric and to justify the meaning of the constant

As explained in the second section the method of polymerization is based on the modification of the Hamiltonian to get a deformed Hamiltonian

At first, let us take a look at the horizon(s) radius of the metric (

Now, let us investigate the properties of the matter corresponding to the metric (

The energy-momentum tensor (

Finally, let us take a quick look at thermodynamics of the metric (

Left: Temperature versus mass. The solid line shows the qualitative behavior of the relation (

The entropy versus mass. The figure is plotted for

In this section we are going to study how light and particles will move in the geometrical background given by metric (

Left: the outgoing (dashed line) and incoming (solid line) geodesics for

To complete our geodesics analysis, let us now consider the radial trajectory of a falling free particle. It moves along the time-like geodesics which results the following equations of motion [

The trajectory of an infalling particle in terms of the proper time

In this paper we have studied the classical polymerization procedure applied on the Schwarzschild metric. This procedure is based on a classical transformation under which the momenta are transformed like their polymer quantum mechanical counterpart. After a brief review of the polymer representation of quantum mechanics, we have introduced the classical polymerization by means of which the Hamiltonian of the theory under consideration gets modification in such a way that a parameter

No data were used to support of the author’s study in the manuscript: Classical Polymerization of the Schwarzschild Metric.

The author declares that they have no conflicts of interest.

This work has been supported financially by Research Institute for Astronomy and Astrophysics of Maragha (RIAAM) under research project no. 1/5237-107.