Entangled particles spinning on the black hole horizon

In this paper, we present a technique to unify the Reissner–Nordstr¨om metric and the Kerr–Newman metric. We construct a speciﬁc model and calculate the entanglement entropy of black horizon. We are interested in the entangled particle and antiparticle spinning on the black hole horizon. The two Reissner-Nordstr¨om horizons r ± , are the results of the rotation of several entangled particle-antiparticle on the real horizon. The energy absorbed by a black hole is transformed into a kinetic energy of the entangled particle-antiparticles. This study provides a new type of black hole metric. We show that the rotation of an entangled system of a particle and an antiparticle can create a extremal black hole. We also explore some of the implications of this point of view for the black hole entanglement.

this ER=EPR correspondence is only demonstrated in a very simplified universe model, where gravity is generated in the absence of mass [1]. The Hawking radiation of a black hole is a scrambled cloud of radiation entangled with the black hole [1]. In this paper, we are interested in studying a two entangled particles on the black hole horizon. It is well known that the "Kerr metric" [2] is a metric that describes a rotating black hole, which is static and axisymmetric. When the black hole has an electric charge, the Schwarzschild solution is no longer valid. A non-rotating black hole corresponds to an isotropic black hole of mass M and charge Q, which described by the "Reissner-Nordström metric" [3,4]. For a charged black holes with |Q| ≪ M , are similar to Schwarzschild black holes. The Reissner-Nordström black holes have two horizon, the innermost is a Cauchy horizon. It is believed that black holes with |Q| ≻ M don't exist in nature, since they would contain a naked singularity. Their existence would be in contradiction with the principle of cosmic censorship of the Roger Penrose [5].
The present paper is organized as follows: The second section introduces a concept Entanglement on the horizon. Section 3 involves the ntangled particle and antiparticles system on the horizon. In section 4 we introduce a unique description of the Reissner-Nordström metric and the Kerr-Newman metric. Section 5, is devoted to calculate the geodesic of a new metric which describes the entangled system on the horizon. We will conclude in the last section.

Entanglement on the horizon
We consider a C,auchy slice Σ of the black hole spacetime with Minkowski coordinates (t, ⃗ x) is divided into two parts Σ + and Σ − . We assume that the "horizon" is at x = −a, the thin region (of the order of the Planck length) near the horizon becomes is at The absolute value function χ −→ |χ| is continuous but is not differentiable in 0. We will use this property of the absolute function to describe the entangled states in the black hole horizon. We define a general field |χ| as a component of two fields: we consider that |χ| is free scalar field in a background spacetime. Next, we considere the entanglement entropy between the outside and the thin region of the inside the horizon, based on the new field |χ| this the derivative of |χ|, seems to describe a two entangled states. It is possible to find both cases −1 and 1 at the same time at point O, i.e. we can find two states at a single point on the surface Σ.
there is a different tangent to the left and a tangent to the right at the point O, when We can represent the creation of a right particle and a left particle in the horizon by the lines |χ|. In this case, Eq.(2. 3) also has a T -symmetry, For spin systems, the Eq.(2.3) obeys ∂ χ |χ| ⊂ exp (i4πs), where s is the spin of the state in the defect Hilbert space [7]. We suppose that a field χ describes a particle which creates outside the horizon, and that −χ describes the particle created inside. While the field |χ| can describe both particles at the same time.  Clearly, ρ θ satisfies the von Neumann-Landau equation in terms of the eigenvectors |Ψ θ ⟩ and the associated eigen values λ θ depend on θ, therefore, the quantum Fisher information is zero. We propose that the particles created with θ = 0, don't escape the horizon but are trapped. We propose that these particles which corresponds to θ = 0, spins on the black hole horizon [10].  The orbital angular momentum operator of a particle in the horizon can then be defined as the vector operator L = −ix µ × ∂ µ . Moreover, we assume that we can't distinguish between the two particles χ and −χ, during the rotation in the horizon.
We propose that that the orbital angular momentum of the particles which turn in the horizon, has a quantum and classical aspect at the same time we start with the classical orbital angular momentum in the quantum frame the eigenvalue of the orbital angular momentum operator L zQ in the state Ψ 0 is expressed as below we compare the two orbital angular momentums (2.9a) and (2.10), we obtain a new state Ψ 0 , is a conformal transformation of vaccum state |Ψ 0 ⟩ as to obtain a classical and quantum description of horizon particles (θ = 0), the state |Ψ 0 ⟩ changes to new vacuum state Ψ 0 (t) by the phase exp ix µ p µ (t). We propose a new notation of the conformal state of vacuum |φ + ⟩ := Ψ 0 (t) and |φ − ⟩ := Ψ * 0 (t) , we will use this notation later. To study the behavior of particles spinning on the horizon, we propose to start studying first the rotation of the Kerr Newman black hole.

Entangled particle and antiparticle on the horizon
In spherical coordinates (t, r, θ, φ), the Kerr-Newman metric is with r s = 2M is the Schwarzschild radius. We take the speed of light c = the gravitational constant G N = the vacuum permittivity 4πε 0 = 1.
The Kerr-Newman metric [8] describes a black hole if and only if where M is the mass of the black hole, Q is the electric charge and J is the angular momentum. The case J 2 ≼ M 4 − Q 2 M 2 describes an extremal black hole. In the case where (Q = 0, J ̸ = 0), we will have a Kerr black metric. For (Q ̸ = 0, J ̸ = 0), we get a Kerr-Newman black hole. For (Q ̸ = 0, J = 0), we get a Reissner-Nordström black hole. Finally for (Q = 0, J = 0), we obtain a Schwarzschild metric. We know that the Schwarzschild radius is written as r s = 2M . We can rewrite Eq.(3.4) by a more general form: the objective behind this general form, is to make appear the Kruskal-Szekeres coordinates to cover the entire spacetime manifold around the horizon. We take u = − r 2 s 4Q 2 , one cane obtain the term ue u is the lightlike Kruskal coordinate, where this last equation makes it possible to find the charge of the particles rotating in the horizon Q ± = ±M (3.8) every particle on the horizon, has a charge which depends directly on the black hole mass; if the black hole mass larger, the charge of a horizon particle will be more important.
This equation shows that there are two types of spinning particles in the horizon. And The black hole has a Bekenstein-Hawking entropy checked this relation presents a minimum value for the entropy of a black hole. We will use this entropy in the next section.

Mirage horizon of the entangled particle-antiparticle
In this section, we want to find a unique description for the Reissner-Nordström metric and the Kerr-Newman metric. The Reissner-Nordström metric [12] reads where dω 2 = dθ 2 + sin 2 θdφ 2 .
The event horizons for the spacetime are located where 1 g rr = 0, which gives two solutions, i.e. we will have two event horizons are located in these concentric event horizons become degenerate for (3.8), which corresponds to an extremal black hole, which explains the result found by ( Since r ± depends on J ± , then, r ± describe the two particles. Therefore, we propose to replace r s in Eq.
we start with a region near to singularity 0 r ≼ r s , in this case we propose to use the Taylor series for this metric: this last expression explains the presence of the entanglement in the horizon. When n −→ ∞, we can't separate precisely between two cases; when n even or odd. Clearer, Substituting Eqs. (3.8,4.5) into Eq.(5.1) one can obtain where dω 2 = dθ 2 + sin 2 θdφ 2 .
we first provided the equations of motion of the particle-antiparticle system. Now, we consider the spherical coordinates one can compute the geodesics of a two-entangled particles system |J ± |. The Euler-Lagrange equations of motion for |J ± | are then given in local coordinates by we find differential equations, (see appendix), then we calculate their solution where ω is the angular frequency for both particles, the rotation of the two particles begins at time t 0 . For t = t 0 , we obtain θ (t 0 ) = ±1; this result is equivalent to the rest of particle-antiparticle.  The local velocity is therefore for t = t 0 , the velocity of the entangled particles will be zero. If |J ± | = 0, v(r, t) will be zero, which implies that the entanglement between the particle and the antiparticle is essentially to generate by the angular momentum.

Conclusion
In the present work, we used an absolute field to describe two entangled states in the horizon. Then we studied the entangled particle-antiparticle that revolve around on the horizon. We have studied these particles for Kerr Newman black hole. We have shown that the rotation of an entangled system of a particle and an antiparticle, can create a extremal black hole. The present study was designed to determine the black hole effect on entangled particles on the horizon. This study has shown that the system of two entangled particles has a single metric, which derives the motion of the electrically charged particle and antiparticle on the horizon. This study has raised important questions about the nature of the entanglement, since we have shown that the two angular momentums of the two entangled particles are perfectly connected with a Taylor by an operator (−1) ∞ at infinite. This connection leaves the two entangled particles in rotation between them.
The analysis of the entangled particles geodesic, has extended our knowledge of more about the rotations of the entangled particle-antiparticle.