=
281
INTERNATIONAL JOURNAL OF INNOVATION IN ENGINEERING RESEARCH & MANAGEMENT ISSN: 2348-4918 Peer Reviewed and Refereed Journal
VOLUME: 09, Issue 03, Paper id-IJIERM-IX-III, June 2022
ENTROPY OF A REGULAR BLACK HOLE BY ENTROPY FUNCTION FORMALISM
Dr. Birendra Kumar Yadav
Department of Physics, Shri Baldeo P.G. College Baragoan, Varanasi, India
Abstract - In this latter, we calculate the entropy of the regular black hole with respect to the corresponding parameters by using entropy function formalism. We show that the entropy function formalismagrees with the direct calculation of the entropy. We calculate the entropy of Bardeen and Heyward black holes and the entropy of these black hole resembles to the Bekenstein Hawking entropy.
Bardeen is first regular black hole model given by the Bardeen [1] based on the Sakharov [2] and Gliner proposal [3] and after 30 year Ayno, Beato and Gracia [4-6] give the exact black hole coupled to nonlinear electrodynamics (NLED). There are many exact regular black hole solutions are exists but all of these bases on Bardeen solution [7-22].The Einstein Hilbert action coupled to NLED is given by
√ , (1)
Where L(F) is the Lagrangian density of the NLED field. The black hole admits the following solution with magnetic monopole charge
, (2)
and d is the metric of the 2D sphere.The extremal Bardeen black hole solution corresponding to the and gives the near horizon metric of the form [23]
(3)
Where , and in the limit . Let us now apply the entropy function formalism [23] to obtain the entropy of the Bardeen
black hole. The near horizon solution is
(4)
with
The entropy function is written as , (5)
Where is the Lagrangian which is given by the
∫ (6) Substitute the value of L from Eq. (1) and substitute in Eq. (6) and then Eq.
(5) substitute in (7), we find the entropy of the black hole. We will consider the different black holes and calculate the entropy. These results are agreement with the Bekenstein Hawking area law. Using this formalism, we are able to calculate the entropy of the Bardeen black hole.
The source of Bardeen black hole is given by the following form [12, 14]
( √
√ )
(7) where g is the magnetic monopole charge, s is the parameter which is related to the mass and charge parameter by relation s = g/2M, F is the Faraday tensor ( ) and
is the electromagnetic field tensor.
=
282
INTERNATIONAL JOURNAL OF INNOVATION IN ENGINEERING RESEARCH & MANAGEMENT ISSN: 2348-4918 Peer Reviewed and Refereed Journal
VOLUME: 09, Issue 03, Paper id-IJIERM-IX-III, June 2022 The Bardeen black hole admits the
following black hole solution, the black hole metric is [12,14]
, (8)
Where M is the mass of the black hole and g is the magnetic monopole charge. This solution has a double horizon which is the contrast of Schwarzschild black hole. This is regular solution, it does not diverse including origin []. For the Bardeen black hole, we find
(9)
Where is the horizon radius which is the function of magnetic charge g.
The entropy of the black hole is calculated by the entropy formalism which is given by
(10)
Substitute the values of and in Eq. (10), the entropy of the Bardeen black hole is
(11)
Which is resemble to the Bekenstein Hawking entropy. Now we will calculate the entropy of the Heyward black hole which is my second example. The NLED source is given by [13]
⁄
⁄ (12) The Heyward black hole admits the following black hole solution, the black hole metric is [13]
, (13) For the Heyward black hole, we find
(14) The entropy of the black hole is calculated by the entropy formalism which is given by
(15) Substitute the values of and in Eq. (15), we calculate the entropy of the Heyward black hole which is resemble to the Bekenstein Hawking entropy.
The entropy function formalism is also good in regular black hole and agrees with the direct calculation of entropy.
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=
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INTERNATIONAL JOURNAL OF INNOVATION IN ENGINEERING RESEARCH & MANAGEMENT ISSN: 2348-4918 Peer Reviewed and Refereed Journal
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