View of ENTROPY OF A QUANTUM FIELDS IN CHARGED BTZ BLACK HOLE SPACE TIME

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ENTROPY OF A QUANTUM FIELDS IN CHARGED BTZ BLACK HOLE SPACE TIME

Dr. Birendra Kumar Yadav

Department of Physics, Shri Baldeo P.G. College Baragoan, Varanasi, India

Abstract - Black Holes are the classical solution of general theory of relativity. Classically the black holes are characterized by the No hair theorem. In the presence of quantum mechanics, the black holes radiate but this radiation does not carry any information this is known as information loss paradox. The quantum fields (scalar and fermion) in presence of charged BTZ black hole are studied in the work. By applying the entanglement entropy approach entropy is numerically computed. In addition, our work is extended to calculate the pre-factor of logarithmic corrections. The pre-factors of the entropy are -0.124 and - 0.320 for the scalar and fermion fields respectively which resemble to the Mann and Solodukhin results.

1 INTRODUCTION

The origin of black hole entropy is remained mysterious after its proposal.

The understanding of black hole entropy can be identified by using the microstates counting which is responsible for black hole thermodynamics. The first indication of the entropy was given by Bekenstein and Hawking.[1]They showed that entropy of black hole is related to area of event horizon, as area increases the entropy of black hole also increases and vice-versa and the surface gravity of the event horizon is responsible for the analysis of the black hole’s temperature. By considering the quantum mechanical effect it is shown that black hole radiates but the thermal radiation does not carry any information regarding the black hole.

This is known as the information loss paradox.

‘t Hooft proposed a model in which quantum fields are considered in the presence of black hole and in to control the divergences, he introduced a boundary, which is placed at a small distance from the horizon known as the Brick wall. Many attempts have been made to understand the microscopic origin of black hole entropy by using the Euclidian entropy method, brick wall model [2], D-brane statistical method [3], Holographic entanglement entropy method [4] and entanglement entropy method [5,6]. Von Neumann’s

entanglement entropy gives the inter- relationship between two subsystems which depends upon the curvature of the entangling surface. These correlations give rise to the black hole entropy [7-10].

In this paper we study the entropy of the quantum field in black hole numerically by using the entanglement entropy method, which is based on Bombelli and Srednicki model [5,6]. The entanglement entropy is given by the equation (S = Tr(rA ln rA), here A is any system whose reduced density matrix is defined by symbol rA. Reduced density matrix is shown to be influenced by short range correlations across the entangling surface []. This method is applied to the scalar and fermion field in the presence of charged BTZ black hole and calculate the entropy of the black hole. Further, we also analyse our results for the study of logarithmic correction to the entropy and calculate the pre-factor of the logarithmic correction numerically. The numerical value of pre-factor of the scalar field resembles to the Mann and Solodukhin results [7].

This remaining part of the work is arranged as follows. We have studied scalar field (scalar and fermions) in the presence of charged BTZ black hole in Section II. Fermion fields in the presence of charged BTZ black hole is presented in Section III. Entanglement entropy of

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massless quantum fields (scalar and

fermions) are studied numerically by taking into the consideration of logarithmic correction of entropy are given in Section IV. Finally in Section V we peruse our results with physical significance of entropy for quantum fields in BTZ black hole.

2 SCALAR FIELDS IN CHARGED BTZ BLACK HOLE

If a negative cosmological constant (1/l²) is taken into the consideration then it is found that the solution for the case of (2+1) dimensions Einstein’s gravity in the presence electromagnetic field is the charged BTZ black hole. The metric of charged BTZ black hole is given by [12];

2 2 2

2 2 2 2 2

2 2 2 2 2

ln

ln 1 dr rd

l r r Q l M r l dt r r Q l M r

ds 



 



 



 



 





 



 



 



 



 









In the presence of charged BTZ black hole the scalar field is given as [13-17]

)) (

2 (

1     



  ^{dt g g}

^{  }

S

where the modes are











^im

m

m

t e

t ^ 

 ( , , ) ( , )

Now, the action of the scalar field in the presence of charged BTZ black hole becomes





























 



 



 



 









 



 



 



 







 ² ₂² ₂^{2 2 2}

2 2 2 2

) 1 ( ln 2 ln

1

m m

m l r

r r Q l M r r l r r Q l M r dt r

S  _ 

and the corresponding the Hamiltonian is

. ln ln

2 ln 1 2

1 2

2 2 2 2 2 2 2

2 2 2 2

2 2 2 2 2

m

m r

l r r Q l M r m r

l r r Q l M r

l r r Q l M r r d d

H   _ ^

 



 



 



 

























 



 



 









 



 



 



 









 

where



_mis the conjugate momentum. By using the following relation we can discretize the Hamiltonian of the scalar field of the black hole.

, )

(

^A

m

  q

 

_m

(  )  p

_A

a ,

and

)

2

' ,

( V a

V   

_AB

The system can be discretized as

a ) 2 / 1 1 ( 

 

and

 (    ' )  

_AB

/ A

, here A,B=1,2…N and a is the cut-off length.

The reduced density matrix of the system is



_V

 Ke

^H, the Hermitian matrix H is analysed along the Hamiltonian of the system. The two point correlators of the system are,

We can write the density matrix as [18,19],

where,

where

C  XP

and

XP  1 / 4 .

The entropy of the system is given by [],

We have two cases for the value of C, one is ≥ 1/2 and other is ≤ 1/2. The total entropy of the scalar field is calculated which is found to be the of all the modes m in the presence of charged BTZ black hole.

3 FERMION FIELDS IN BTZ BLACK HOLE SPACETIME

The fermion field in the presence of charged BTZ black hole is given as [20, 21],

where

r

l r r Q l M r u



 



 







2 ln

2 2 2

Pauli matrices are given by



₁

, 

₂ ^and



3. The wave function



of the system

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can be expressed in the form of two

component spinors.

where ^{1 ( 1/2)}

2

1 ^

 ^{i m}

m e^

  ^and

) 2 / 1 ( 2

2

1

^



^{i m}

m

e

^

 

which is the

function of the azimuthal quantum number m and the corresponding Hamiltonian is written in term of two component spinor.

The explicit form of Hm can be evaluated by using the (8) and is given by,

where

 ~

₀

 

₃

,  ~

₁

 

₁

,  ~

₂

 

₂ and to change the signature from Lorentzian to Euclidean



₃ is to be multiplied by imaginary unit. The discretized Hamiltonian can be expressed for the case of general quadratic as discretized Hamiltonian can written as

For N lattice the discrete Hamiltonian for these variables becomes

Here the indices

i and j

represent the discrete variables for the radial distance



and

M

_ijrepresents the (2 X 2) matrix for constant values of the indices

j and

i

.

where

i , j  1 , 2 .... N .

For a quadratic case, the correlator is directly related to the

M

ijby the relation

C   (  M )

^{, where}

)

 (x

is unit step function. We can explain (NxN) matrix

~

²k^2,2l²

M

ij , for

. ....

2 , 1

, l N

k 

and

 ,   1 , 2

. The matrix of

M ~

is given as;

The matrix element for BTZ black hole are clearly written as

As

M

_ijis related to the fermion correlator of the entanglement entropy. Therefore, for the case of general quadratic the matrix emerge in the Hamiltonian. Hence its entropy is written as

here c can take two one is 1 and other is 0. Total entropy of the system has been calculated by considering the sum of all the modes m.

4 NUMERICAL COMPUTATION FOR ENTANGLEMENT ENTROPY

Numerical estimation of the entanglement entropy of the scalar fields and fermion field in the presence of charged BTZ black hole space time has been done. The result of the entanglement entropy of the scalar and fermion field are;

here the Cs is proportionality constant which is calculated numerically and the results are shown in Fig. 1. The proportionality constant Cs are 0:297 and 0:30 for scalar and fermion field

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respectively which we calculated by the

fitting method.

We can also be able to calculate the logarithmic contribution of the entropy [9, 16] of the fermion field by using our developed model.

By adopting the fitting method the numerical value of the co-efficient is determined and is given as a =0:303; b = - 0.124; c = -0.186 for scalar and a=0.320, b= -0.320, c= -0.327 for fermion fields.

5 RESULTS AND CONCLUSION

The work entanglement entropy approach has been used to examine the entanglement entropy of quantum fields (scalar and fermion) in charged BTZ black hole. The entropy of both scalar and fermion fields are numerically estimated by using the entanglement entropy and we continue our findings to evaluate the sub-leading corrections to the black hole entropy and they are of the form:

3 2

ln( r / a ) c

c

_



, which is logarithmic in nature. The pre-factors of the entropy are -0.124 and -0.320 for the scalar and fermion fields respectively, which is same as the Mann and Solodukhin results for scalar fields in charged BTZ black hole space time [22].

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