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Physics & Astronomy International Journal

Research Article Volume 4 Issue 1

Gibb’s free energy of the spinning black holes

Alok Ranjan,1 Dipo Mahto2

1Department of Physics, Research Scholar University, India
2Associate Professor, Department of Physics, Marwari College, India

Correspondence: Dipo Mahto, Associate Professor, Department of Physics, Marwari College, TMBU Bhagalpur, India , Tel 9006187234

Received: January 31, 2020 | Published: February 12, 2020

Citation: Ranjan A, Mahto D. Gibb’s free energy of the spinning black holes. Phys Astron Int J. 2020;4(1):23-28. DOI: 10.15406/paij.2020.04.00199

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Abstract

The present gives a model for the change in Gibb’s free energy of spinning black holes with corresponding change in the event horizon using the first law of black hole mechanics, mass-energy equivalence relation applied to the Gibb’s free energy of the Reissnner-Nordstrom black hole and concludes that the magnitude of change in free energy with corresponding change in the event horizon is approximately similar to the magnitude of change in temperature of spinning black holes with corresponding change in the mass in XRBs.

Keywords: free energy, surface gravity, XRBs

Introduction

David Hochberg1 computed the O(h) corrections to the mass, thermal energy, entropy and free energy of the black hole due to the presence of hot conformal scalars, massless spinors and U(1) gauge quantum fields in the vicinity of the black hole using the recent solutions of the semi-classical back-reaction proble.1 You Gen Shen and Chang-Jun Gao calculated free energy and entropy of diatomic black hole due to arbitrary spin fields using the membrane model based on the brick-wall model and showed that the energy of scalar field and the entropy of Fermionic field have similar formulas containing only a numerical coefficient between them.2 David Kastor et al.3 analysed the free energy and also specific heat in the small and large black hole limits and comment upon the Hawking-Page phase transition for generic Ads-Lovelock black holes.3 Hugues Beauchesne and Ariel Edery has shown that the negative of the total Lagrangian approaches the Helmholtz free energy of a Schwarzschild black hole at the time of collapse. He also computed the numerical value of the interior Lagrangian to the expected analytical value of the interior Gibb’s free energy for different initial states.4 In the present work, we have proposed a model for the change in free energy with corresponding change in the event horizon using the first law of black hole mechanics, mass-energy equivalence relation applied to the Gibb’s free energy of the Reissnner-Nordstrom black hole.

Theoretical discussion

Black holes are boost machines processing the high frequency input and deliver it as low frequency output, owing to the gravitational shift and also provide a glimpse of the world at very short distance scales. This world consists of nothing, but vacuum fluctuations.5 Black hole is one of new physical phenomena predicted by General relativity6 and defined as the solution of Einstein’s gravitational field equations in the absence of matter that describes the space-time around a gravitationally collapsed star.7 The gravity of a black hole is so abnormal that nothing can escape from it. The generalised form for entropy of Reissnner-Nordstrom black holes in commutative space is given by the following equation8

S= A s 4 π Q 2 ln A s 4 + n=1 c n ( 4 A s )+C MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGtb Gaeyypa0tcfa4aaSaaaOqaaKqzGeGaamyqaKqbaoaaBaaaleaajugW aiaadohaaSqabaaakeaajugibiaaisdaaaGaeyOeI0IaeqiWdaNaam yuaKqbaoaaCaaaleqabaqcLbmacaaIYaaaaKqzGeGaciiBaiaac6ga juaGdaWcaaGcbaqcLbsacaWGbbqcfa4aaSbaaSqaaKqzGeGaam4Caa WcbeaaaOqaaKqzGeGaaGinaaaacqGHRaWkjuaGdaGfWbGcbeWcbaqc LbmacaWGUbGaeyypa0JaaGymaaWcbaqcLbmacqGHEisPa0qaaKqzaj GaeyyeIuoaaKqzGeGaam4yaKqbaoaaBaaaleaajugWaiaad6gaaSqa baqcfa4aaeWaaOqaaKqbaoaalaaakeaajugibiaaisdaaOqaaKqzGe GaamyqaKqbaoaaBaaaleaajugWaiaadohaaSqabaaaaaGccaGLOaGa ayzkaaqcLbsacqGHRaWkcaWGdbaaaa@66DA@    (1)

Where C=constant

cn=quantum gravity model dependent coefficients, As=surface area of the black holes & Q=charge on the black holes.

In the case of black holes having charge (Q=0), then the equation (1) takes its form

S= A s 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGtb Gaeyypa0tcfa4aaSaaaOqaaKqzGeGaamyqaKqbaoaaBaaaleaajugW aiaadohaaSqabaaakeaajugibiaaisdaaaaaaa@3EA2@    (2)

The equation (2) is known as standard Bekenstein entropy of black hole. For the spherically symmetric and stationary or Schwarzschild black hole, its surface area is naturally given by the following equation9,10

A=4π R bh 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGbb Gaeyypa0JaaGinaiabec8aWjaadkfajuaGdaqhaaWcbaqcLbmacaWG IbGaamiAaaWcbaqcLbmacaaIYaaaaaaa@4155@    (3)

Where the radius of event horizon for non-spinning and spinning black holes are given by equations (4) and (5) respectively.

R bh = 2GM c 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGsb qcfa4aaSbaaSqaaKqzadGaamOyaiaadIgaaSqabaqcLbsacqGH9aqp juaGdaWcaaGcbaqcLbsacaaIYaGaam4raiaad2eaaOqaaKqzGeGaam 4yaKqbaoaaCaaaleqabaqcLbmacaaIYaaaaaaaaaa@446F@    (4)

And  

R bh = GM c 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGsb qcfa4aaSbaaSqaaKqzadGaamOyaiaadIgaaSqabaqcLbsacqGH9aqp juaGdaWcaaGcbaqcLbsacaWGhbGaamytaaGcbaqcLbsacaWGJbqcfa 4aaWbaaSqabeaajugWaiaaikdaaaaaaaaa@43B3@    (5)

The entropy of black holes (S) can be obtained by putting (3) into eqn (2) as:

S=π R bh 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGtb Gaeyypa0JaeqiWdaNaamOuaKqbaoaaDaaaleaajugWaiaadkgacaWG ObaaleaajugWaiaaikdaaaaaaa@40A9@    (6)

The above equation is differentiated, we have

dS=2π R bh d R bh MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGKb Gaam4uaiabg2da9iaaikdacqaHapaCcaWGsbqcfa4aaSbaaSqaaKqz adGaamOyaiaadIgaaSqabaqcLbsacaWGKbGaamOuaKqbaoaaBaaale aajugWaiaadkgacaWGObaaleqaaaaa@4679@    (7)

The Gibb’s free energy of the Reissnner-Nordstrom black hole is given by the following equation3

G=ETS Φ H .Q MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGhb Gaeyypa0JaamyraiabgkHiTiaadsfacaWGtbGaeyOeI0IaeuOPdyuc fa4aaSbaaSqaaKqzadGaamisaaWcbeaajugibiaac6cacaWGrbaaaa@42FD@    (8)

Where E=ADM mass of Reissner–Nordström black hole which gives the total energy of a space-time as defined by an observer at spatial infinity, using the Hamiltonian formalism, for an asymptotically flat space-time and The ADM mass consists of two contributions: black hole horizon and solitonic residue. It is always greater than the Schwarzschild black holes.

 T=Temperature of black hole.

 S=Entropy of black hole.

 Q=Charge of the black hole.

  Φ H MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqqHMo GrjuaGdaWgaaWcbaqcLbmacaWGibaaleqaaaaa@3ABF@ =Electrostatic potential at the outer horizon of black hole.

Actually the astronomical black hole is not likely to have any significant charge, because it will usually neutralised by surrounding plasma.11 Hence the charge Q=0, the equation (8) becomes

G=ETS MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGhb Gaeyypa0JaamyraiabgkHiTiaadsfacaWGtbaaaa@3BBF@    (9)

The product of temperature (T) and entropy (S) for the Reissnner-Nordstrom black hole is given by.3

TS= 1 2 M 2 Q 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGub Gaam4uaiabg2da9KqbaoaalaaakeaajugibiaaigdaaOqaaKqzGeGa aGOmaaaajuaGdaGcaaGcbaqcLbsacaWGnbqcfa4aaWbaaSqabeaaju gWaiaaikdaaaqcLbsacqGHsislcaWGrbqcfa4aaWbaaSqabeaajugW aiaaikdaaaaaleqaaaaa@4633@    (10)

For Q=0,

TS= M 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGub Gaam4uaiabg2da9Kqbaoaalaaakeaajugibiaad2eaaOqaaKqzGeGa aGOmaaaaaaa@3C9A@    (11)

According to Einstein well-known mass-energy equivalence relation, we know that

E=M c 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGfb Gaeyypa0JaamytaiaadogajuaGdaahaaWcbeqaaKqzadGaaGOmaaaa aaa@3CB4@    (12)

Putting (11) and (12) into equation (9), we have

G=M c 2 M 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGhb Gaeyypa0JaamytaiaadogajuaGdaahaaWcbeqaaKqzadGaaGOmaaaa jugibiabgkHiTKqbaoaalaaakeaajugibiaad2eaaOqaaKqzGeGaaG Omaaaaaaa@4190@    (13)

Putting c=1 throughout our research work, the equation becomes

G=M M 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGhb Gaeyypa0JaamytaiabgkHiTKqbaoaalaaakeaajugibiaad2eaaOqa aKqzGeGaaGOmaaaaaaa@3D74@

G= M 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGhb Gaeyypa0tcfa4aaSaaaOqaaKqzGeGaamytaaGcbaqcLbsacaaIYaaa aaaa@3BB5@    (14)

The change in free energy of the Reissnner-Nordstrom black hole due to change in the mass of black hole can be obtained by differentiating the above equation

dG= 1 2 dM MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGKb Gaam4raiabg2da9KqbaoaalaaakeaajugibiaaigdaaOqaaKqzGeGa aGOmaaaacaWGKbGaamytaaaa@3E42@    (15)

The first law of black hole mechanics is simply an identity relating the change in mass M, angular momentum J, horizon area A and charge Q, of a black hole. The first order variations of these quantities in the vacuum satisfy.12,13

δM= k 8π δA+ΩδJυδQ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH0o azcaWGnbGaeyypa0tcfa4aaSaaaOqaaKqzGeGaam4AaaGcbaqcLbsa caaI4aGaeqiWdahaaiabes7aKjaadgeacqGHRaWkcqqHPoWvcqaH0o azcaWGkbGaeyOeI0IaeqyXduNaeqiTdqMaamyuaaaa@4BBF@    (16)

Where Ω MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGHPo Wvaaa@3814@ =Angular velocity of the horizon. υ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHfp qDaaa@384C@ =difference in the electrostatic potential between infinity and horizon.

For Q=0, dQ=0 and using the relation (2), the equation (16) becomes

δM= κ 2π δS+ΩδJ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH0o azcaWGnbGaeyypa0tcfa4aaSaaaOqaaKqzGeGaeqOUdSgakeaajugi biaaikdacqaHapaCaaGaeqiTdqMaam4uaiabgUcaRiabfM6axjabes 7aKjaadQeaaaa@475E@    (17)

Putting (7) in the above equation, we have

δM=κ R bh δ R bh +ΩδJ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH0o azcaWGnbGaeyypa0JaeqOUdSMaamOuaKqbaoaaBaaaleaajugWaiaa dkgacaWGObaaleqaaKqzGeGaeqiTdqMaamOuaKqbaoaaBaaaleaaju gWaiaadkgacaWGObaaleqaaKqzGeGaey4kaSIaeuyQdCLaeqiTdqMa amOsaaaa@4C97@    (18)

For maximum spin of black hole (a*=1), the angular momentum of the black hole is given by.12

J= M 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGkb Gaeyypa0JaamytaKqbaoaaCaaaleqabaqcLbmacaaIYaaaaaaa@3BD1@    (19)

This condition corresponds to spinning black holes

Or  

δJ=2MδM MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH0o azcaWGkbGaeyypa0JaaGOmaiaad2eacqaH0oazcaWGnbaaaa@3E04@    (20)

Putting the above value in eqn (18), we have

δM=κ R bh δ R bh +2ΩMδM MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH0o azcaWGnbGaeyypa0JaeqOUdSMaamOuaKqbaoaaBaaaleaajugWaiaa dkgacaWGObaaleqaaKqzGeGaeqiTdqMaamOuaKqbaoaaBaaaleaaju gWaiaadkgacaWGObaaleqaaKqzGeGaey4kaSIaaGOmaiabfM6axjaa d2eacqaH0oazcaWGnbaaaa@4E28@    (21)

( 12ΩM )δM=κ R bh δ R bh MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqzGeGaaGymaiabgkHiTiaaikdacqqHPoWvcaWGnbaakiaawIca caGLPaaajugibiabes7aKjaad2eacqGH9aqpcqaH6oWAcaWGsbqcfa 4aaSbaaSqaaKqzadGaamOyaiaadIgaaSqabaqcLbsacqaH0oazcaWG sbqcfa4aaSbaaSqaaKqzadGaamOyaiaadIgaaSqabaaaaa@4EA2@

δM= κ ( 12ΩM ) R bh δ R bh MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH0o azcaWGnbGaeyypa0tcfa4aaSaaaOqaaKqzGeGaeqOUdSgakeaajuaG daqadaGcbaqcLbsacaaIXaGaeyOeI0IaaGOmaiabfM6axjaad2eaaO GaayjkaiaawMcaaaaajugibiaadkfajuaGdaWgaaWcbaqcLbmacaWG IbGaamiAaaWcbeaajugibiabes7aKjaadkfajuaGdaWgaaWcbaqcLb macaWGIbGaamiAaaWcbeaaaaa@5072@    (22)

Putting the above value in equation (15), we have

δG= κ 2( 12ΩM ) R bh δ R bh MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH0o azcaWGhbGaeyypa0tcfa4aaSaaaOqaaKqzGeGaeqOUdSgakeaajugi biaaikdajuaGdaqadaGcbaqcLbsacaaIXaGaeyOeI0IaaGOmaiabfM 6axjaad2eaaOGaayjkaiaawMcaaaaajugibiaadkfajuaGdaWgaaWc baqcLbmacaWGIbGaamiAaaWcbeaajugibiabes7aKjaadkfajuaGda WgaaWcbaqcLbmacaWGIbGaamiAaaWcbeaaaaa@51B7@    (23)

The above equation gives the change in Gibb’s free energy with corresponding change in the event horizon in terms of surface gravity, mass, angular velocity and event horizon of spinning black holes. In the case of spinning black holes, the surface gravity of a black hole is given by the Kerr solution.12,13

κ= ( M 4 J H 2 ) 1/2 2M{ M 2 + ( M 4 J H 2 ) 1/2 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH6o WAcqGH9aqpjuaGdaWcaaGcbaqcLbsacaGGOaGaamytaKqbaoaaCaaa leqabaqcLbmacaaI0aaaaKqzGeGaeyOeI0IaamOsaKqbaoaaDaaale aajugWaiaadIeaaSqaaKqzadGaaGOmaaaajugibiaacMcajuaGdaah aaWcbeqaaKqzadGaaGymaiaac+cacaaIYaaaaaGcbaqcLbsacaaIYa GaamytaiaacUhacaWGnbqcfa4aaWbaaSqabeaajugWaiaaikdaaaqc LbsacqGHRaWkcaGGOaGaamytaKqbaoaaCaaaleqabaqcLbmacaaI0a aaaKqzGeGaeyOeI0IaamOsaKqbaoaaDaaaleaajugWaiaadIeaaSqa aKqzadGaaGOmaaaajugibiaacMcajuaGdaahaaWcbeqaaKqzadGaaG ymaiaac+cacaaIYaaaaKqzGeGaaiyFaaaaaaa@64E6@     (24)

Each black hole is characterised by just three numbers: mass M, spin parameter a* defined such that the angular momentum of the black hole is a*GM2/c.11,12

Hence we have

J H =a*G M 2 /c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGkb qcfa4aaSbaaSqaaKqzadGaamisaaWcbeaajugibiabg2da9iaadgga caGGQaGaam4raiaad2eajuaGdaahaaWcbeqaaKqzadGaaGOmaaaaju gibiaac+cacaWGJbaaaa@43AA@    (25)

Using G=c=h=1, we have

J H =a* M 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGkb qcfa4aaSbaaSqaaKqzadGaamisaaWcbeaajugibiabg2da9iaadgga caGGQaGaamytaKqbaoaaCaaaleqabaqcLbmacaaIYaaaaaaa@40B4@    (26)

The radius is smaller in the case of spinning black holes, tending to GM/c2 as a* tends to 111 and in the case of spinning black holes having spin parameter (a*=1), then we have,

where   

J H = M 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGkb qcfa4aaSbaaSqaaKqzadGaamisaaWcbeaajugibiabg2da9iaad2ea juaGdaahaaWcbeqaaKqzadGaaGOmaaaaaaa@3F20@    (27)

using the relation (27) into (24), we have

κ=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH6o WAcqGH9aqpcaaIWaaaaa@39F7@    (28)

The equation (23) becomes

dG=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGKb Gaam4raiabg2da9iaaicdaaaa@39FA@     (29)

This equation shows that the change in free energy is zero for the spinning black holes spinning at max. Spin. The above equation can be written as:

G= constant (30)

The above equation shows that the total Gibb’s free energy of spinning black holes spinning at max. Rate has constant free energy.

Wang, D has shown that the angular velocity () evolves in a non-monotonous way in the case of thin disk-pure-accretion attaining a maximum at a*=0.994 and turns out to depend on the radial gradient of  near the BH horizon.14 One black hole at the heart of galaxy NGC1365 is turning at 84% the speed light. It has reached the cosmic speed limit and cannot spin any faster without revealing its singularity.15

For convenience, let us assume a*=0.9

J H = a M 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGkb qcfa4aaSbaaSqaaKqzadGaamisaaWcbeaajugibiabg2da9iaadgga juaGdaWgaaWcbaqcLbsacqGHxiIkaSqabaqcLbsacaWGnbqcfa4aaW baaSqabeaajugWaiaaikdaaaaaaa@42D8@  or  J H = 9 10 M 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGkb qcfa4aaSbaaSqaaKqzadGaamisaaWcbeaajugibiabg2da9Kqbaoaa laaakeaajugibiaaiMdaaOqaaKqzGeGaaGymaiaaicdaaaGaamytaK qbaoaaCaaaleqabaqcLbmacaaIYaaaaaaa@4328@ or   J H 2 = 81 100 M 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGkb qcfa4aa0baaSqaaKqzadGaamisaaWcbaqcLbmacaaIYaaaaKqzGeGa eyypa0tcfa4aaSaaaOqaaKqzGeGaaGioaiaaigdaaOqaaKqzGeGaaG ymaiaaicdacaaIWaaaaiaad2eajuaGdaahaaWcbeqaaKqzadGaaGin aaaaaaa@4689@ and hence, we have

M 4 J H 2 = M 4 81 M 4 100 = 19 100 M 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGnb qcfa4aaWbaaSqabeaajugWaiaaisdaaaqcLbsacqGHsislcaWGkbqc fa4aa0baaSqaaKqzadGaamisaaWcbaqcLbmacaaIYaaaaKqzGeGaey ypa0JaamytaKqbaoaaCaaaleqabaqcLbmacaaI0aaaaKqzGeGaeyOe I0scfa4aaSaaaOqaaKqzGeGaaGioaiaaigdacaWGnbqcfa4aaWbaaS qabeaajugWaiaaisdaaaaakeaajugibiaaigdacaaIWaGaaGimaaaa cqGH9aqpjuaGdaWcaaGcbaqcLbsacaaIXaGaaGyoaaGcbaqcLbsaca aIXaGaaGimaiaaicdaaaGaamytaKqbaoaaCaaaleqabaqcLbmacaaI 0aaaaaaa@5A6F@

or   ( M 4 J H 2 ) 1/2 = 19 10 M 2 =0.4358 M 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaGGOa GaamytaKqbaoaaCaaaleqabaqcLbmacaaI0aaaaKqzGeGaeyOeI0Ia amOsaKqbaoaaDaaaleaajugWaiaadIeaaSqaaKqzadGaaGOmaaaaju gibiaacMcajuaGdaahaaWcbeqaaKqzadGaaGymaiaac+cacaaIYaaa aKqzGeGaeyypa0tcfa4aaSaaaOqaaKqbaoaakaaakeaajugibiaaig dacaaI5aaaleqaaaGcbaqcLbsacaaIXaGaaGimaaaacaWGnbqcfa4a aWbaaSqabeaajugWaiaaikdaaaqcLbsacqGH9aqpcaaIWaGaaiOlai aaisdacaaIZaGaaGynaiaaiIdacaWGnbqcfa4aaWbaaSqabeaajugW aiaaikdaaaaaaa@5AE5@    (31)

or    2M[ M 2 + ( M 4 J H 2 ) 1/2 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaaIYa GaamytaiaacUfacaWGnbqcfa4aaWbaaSqabeaajugWaiaaikdaaaqc LbsacqGHRaWkcaGGOaGaamytaKqbaoaaCaaaleqabaqcLbmacaaI0a aaaKqzGeGaeyOeI0IaamOsaKqbaoaaDaaaleaajugWaiaadIeaaSqa aKqzadGaaGOmaaaajugibiaacMcajuaGdaahaaWcbeqaaKqzadGaaG ymaiaac+cacaaIYaaaaKqzGeGaaiyxaaaa@4FB4@

=2M[ M 2 +0.4358 M 2 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGH9a qpcaaIYaGaamytaiaacUfacaWGnbqcfa4aaWbaaSqabeaajugWaiaa ikdaaaqcLbsacqGHRaWkcaaIWaGaaiOlaiaaisdacaaIZaGaaGynai aaiIdacaWGnbqcfa4aaWbaaSqabeaajugWaiaaikdaaaqcLbsacaGG Dbaaaa@482F@

=2M[ 1.4358 M 2 ] = =2.8716 M 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiabg2da9iaaikdacaWGnbqcfa4damaadmaakeaajugib8qa caaIXaGaaiOlaiaaisdacaaIZaGaaGynaiaaiIdacaWGnbqcfa4dam aaCaaaleqabaqcLbmapeGaaGOmaaaaaOWdaiaawUfacaGLDbaajugi b8qacaqGGaGaeyypa0Jaaeiiaiabg2da9iaaikdacaGGUaGaaGioai aaiEdacaaIXaGaaGOnaiaad2eajuaGpaWaaWbaaSqabeaajugWa8qa caaIZaaaaaaa@507D@    (32)

Putting (31) and (32) in equation (24), we have

κ= .4358 M 2 2.8716 M 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH6o WAcqGH9aqpjuaGdaWcaaGcbaqcLbsacaGGUaGaaGinaiaaiodacaaI 1aGaaGioaiaad2eajuaGdaahaaWcbeqaaKqzadGaaGOmaaaaaOqaaK qzGeGaaGOmaiaac6cacaaI4aGaaG4naiaaigdacaaI2aGaamytaKqb aoaaCaaaleqabaqcLbmacaaIZaaaaaaaaaa@4A16@

    κ= .1517 M MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH6o WAcqGH9aqpjuaGdaWcaaGcbaqcLbsacaGGUaGaaGymaiaaiwdacaaI XaGaaG4naaGcbaqcLbsacaWGnbaaaaaa@3F87@ (33)

Using above equation, eqn (23) becomes

dG d R bh = .1517 R bh 2M(12ΩM) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaaO qaaKqzGeGaamizaiaadEeaaOqaaKqzGeGaamizaiaadkfajuaGdaWg aaWcbaqcLbmacaWGIbGaamiAaaWcbeaaaaqcLbsacqGH9aqpjuaGda WcaaGcbaqcLbsacaGGUaGaaGymaiaaiwdacaaIXaGaaG4naiaadkfa juaGdaWgaaWcbaqcLbmacaWGIbGaamiAaaWcbeaaaOqaaKqzGeGaaG Omaiaad2eacaGGOaGaaGymaiabgkHiTiaaikdacqqHPoWvcaWGnbGa aiykaaaaaaa@5258@

dG d R bh = .07588 R bh M(12ΩM) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaaO qaaKqzGeGaamizaiaadEeaaOqaaKqzGeGaamizaiaadkfajuaGdaWg aaWcbaqcLbmacaWGIbGaamiAaaWcbeaaaaqcLbsacqGH9aqpjuaGda WcaaGcbaqcLbsacaGGUaGaaGimaiaaiEdacaaI1aGaaGioaiaaiIda caWGsbqcfa4aaSbaaSqaaKqzadGaamOyaiaadIgaaSqabaaakeaaju gibiaad2eacaGGOaGaaGymaiabgkHiTiaaikdacqqHPoWvcaWGnbGa aiykaaaaaaa@5264@    (34)

The equation (34) gives the change in Gibb’s free energy with corresponding change in the event horizon in terms of mass, angular velocity and event horizon of spinning black holes.

Using G=h=c=1 with equation (5), we have

R bh =M MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGsb qcfa4aaSbaaSqaaKqzadGaamOyaiaadIgaaSqabaqcLbsacqGH9aqp caWGnbaaaa@3D8A@    (35)

Using the above relation with equation (34) and solving, we have

dG d R bh = .07588 (12ΩM) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaaO qaaKqzGeGaamizaiaadEeaaOqaaKqzGeGaamizaiaadkfajuaGdaWg aaWcbaqcLbmacaWGIbGaamiAaaWcbeaaaaqcLbsacqGH9aqpjuaGda WcaaGcbaqcLbsacaGGUaGaaGimaiaaiEdacaaI1aGaaGioaiaaiIda aOqaaKqzGeGaaiikaiaaigdacqGHsislcaaIYaGaeuyQdCLaamytai aacMcaaaaaaa@4CF4@     (36)

Some parameters of computed 2D models for progenitor stars with different masses and angular velocity of the Fe-core prior to collapse is given by16 From the data in the Table 1, it is clear that the masses of the collapsed stars are within the limit of 20 of the solar mass. We also know that the mass of the astrophysical objects like the black holes existing in XRBs ranging from 5 Mʘ -20Mʘ.11

S. No

Model

Mass (Mʘ)

Angular velocity (in sec-1)

Polar angles

Reference

1

S11.2

11.2

0

[46.8, 133.2]

16

2

S11.2

11.2

0

[46.8, 133.2]

16

3

S15

S15

0

[46.8, 133.2]

16

4

S15p

15

0

[46.8, 133.2]

16

5

S15r

15

0.5

[0, 90]

16

6

S20

20

0

[46.8, 133.2]

16

Table 1 Some parameters of computed 2D models for progenitor stars with different masses and angular velocity of the Fe-core prior to collapse

S. No

Mass of (M)

Radius of event horizon

dG/dRbh   Joule

1

5Mʘ

7375

7.626x10-33

2

6 Mʘ

8850

6.355x10-33

3

7 Mʘ

10325

5.447x10-33

4

8 Mʘ

11800

4.766x10-33

5

9 Mʘ

13275

4.236x10-33

6

10 Mʘ

14750

3.813x10-33

7

11 Mʘ

16225

3.466x10-33

8

12 Mʘ

17700

3.177x10-33

9

13 Mʘ

19175

2.933x10-33

10

14 Mʘ

20650

2.723x10-33

11

15 Mʘ

22125

2.542x10-33

12

16 Mʘ

23600

2.383x10-33

13

17 Mʘ

25075

2.242x10-33

14

18 Mʘ

26550

2.118x10-33

15

19 Mʘ

28025

2.006x10-33

16

20 Mʘ

29500

1.906x10-33

Table 2 Change in free energy of spinning black holes corresponding change with event horizon in XRBs

From the data in the Table 1, it is also clear that the maximum of the collapsed stars have zero angular velocity except one having polar angle of lateral grid boundaries. Hence we can use zero angular velocity in equation (36) and solving, we have

dG d R bh =0.7588 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaaO qaaKqzGeGaamizaiaadEeaaOqaaKqzGeGaamizaiaadkfajuaGdaWg aaWcbaqcLbmacaWGIbGaamiAaaWcbeaaaaqcLbsacqGH9aqpcaaIWa GaaiOlaiaaiEdacaaI1aGaaGioaiaaiIdaaaa@4507@    (37)

dGαd R bh MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGKb Gaam4raiabeg7aHjaadsgacaWGsbqcfa4aaSbaaSqaaKqzadGaamOy aiaadIgaaSqabaaaaa@3F60@    (38)

Eqn (38) shows that the change in free energy of spinning black holes is directly proportional to the corresponding change in the radius of event horizon. This shows that the change in the free energy of spinning black holes w.r.t. the event horizon existing in XRBs remains the same except for those spinning black holes having polar angle [0, 90] degree of lateral grid boundaries. This change in the free energy of spinning black holes w.r.t. the event horizon existing in XRBs can be shown by the following graph as:

But for  =0.5 s-1, the equation (36) becomes

dG d R bh = 0.07588 M1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaaO qaaKqzGeGaamizaiaadEeaaOqaaKqzGeGaamizaiaadkfajuaGdaWg aaWcbaqcLbmacaWGIbGaamiAaaWcbeaaaaqcLbsacqGH9aqpcqGHsi sljuaGdaWcaaGcbaqcLbsacaaIWaGaaiOlaiaaicdacaaI3aGaaGyn aiaaiIdacaaI4aaakeaajugibiaad2eacqGHsislcaaIXaaaaaaa@4AF8@     (39)

In compared to the mass of the spinning black holes, the term 1 is negligible and hence can be neglected.

dG d R bh = 0.07588 M MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaaO qaaKqzGeGaamizaiaadEeaaOqaaKqzGeGaamizaiaadkfajuaGdaWg aaWcbaqcLbmacaWGIbGaamiAaaWcbeaaaaqcLbsacqGH9aqpcqGHsi sljuaGdaWcaaGcbaqcLbsacaaIWaGaaiOlaiaaicdacaaI3aGaaGyn aiaaiIdacaaI4aaakeaajugibiaad2eaaaaaaa@4950@    (40)

| dG d R bh |= 0.07588 M MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaqWaaO qaaKqbaoaalaaakeaajugibiaadsgacaWGhbaakeaajugibiaadsga caWGsbqcfa4aaSbaaSqaaKqzadGaamOyaiaadIgaaSqabaaaaaGcca GLhWUaayjcSdqcLbsacqGH9aqpjuaGdaWcaaGcbaqcLbsacaaIWaGa aiOlaiaaicdacaaI3aGaaGynaiaaiIdacaaI4aaakeaajugibiaad2 eaaaaaaa@4C27@     (41)

This shows the magnitude of change in free energy of spinning black hole of angular velocity 0.5 per second with about maximum spin is inversely proportional to the mass.

Result and discussion

In the present work, we have proposed a model for the change in free energy with corresponding change in the event horizon in terms of surface gravity, angular velocity, mass and radius of the event horizon of the spinning black holes given by the equation δG= κ 2( 12ΩM ) R bh δ R bh MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH0o azcaWGhbGaeyypa0tcfa4aaSaaaOqaaKqzGeGaeqOUdSgakeaajugi biaaikdajuaGdaqadaGcbaqcLbsacaaIXaGaeyOeI0IaaGOmaiabfM 6axjaad2eaaOGaayjkaiaawMcaaaaajugibiaadkfajuaGdaWgaaWc baqcLbmacaWGIbGaamiAaaWcbeaajugibiabes7aKjaadkfajuaGda WgaaWcbaqcLbmacaWGIbGaamiAaaWcbeaaaaa@51B7@  using the first law of black hole mechanics, mass-energy equivalence relation applied to the Gibb’s free energy of the Reissnner-Nordstrom black hole. From the equation, it is clear that the surface gravity has vital role for change in free energy because its values vary for different types of black holes existing either in XRBs. In the present work, we have applied the above formula for zero surface gravity to get the change in free energy of all black holes are zero. We also have applied this work to the spinning black holes of maximum spin and obtained the change in free energy with corresponding change in the event horizon is given by equation dG d R bh = .07588 R bh M(12ΩM) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaaO qaaKqzGeGaamizaiaadEeaaOqaaKqzGeGaamizaiaadkfajuaGdaWg aaWcbaqcLbmacaWGIbGaamiAaaWcbeaaaaqcLbsacqGH9aqpjuaGda WcaaGcbaqcLbsacaGGUaGaaGimaiaaiEdacaaI1aGaaGioaiaaiIda caWGsbqcfa4aaSbaaSqaaKqzadGaamOyaiaadIgaaSqabaaakeaaju gibiaad2eacaGGOaGaaGymaiabgkHiTiaaikdacqqHPoWvcaWGnbGa aiykaaaaaaa@5264@  . After this, the work is specialised for zero angular velocity and angular velocity (0.5 per second) and also obtained that the change in free energy of spinning black holes is directly proportional to the corresponding change in the radius of event horizon. This shows that the change in the free energy of spinning black holes w.r.t. the event horizon existing in XRBs remains the same except for those spinning black holes having polar angle [0,90] degree of lateral grid boundaries.

For the angular velocity (=0.5 sec-1), we have finally obtained the magnitude of change in free energy with corresponding change in the event horizon as: | dG d R bh |= 0.07588 M MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaqWaaO qaaKqbaoaalaaakeaajugibiaadsgacaWGhbaakeaajugibiaadsga caWGsbqcfa4aaSbaaSqaaKqzadGaamOyaiaadIgaaSqabaaaaaGcca GLhWUaayjcSdqcLbsacqGH9aqpjuaGdaWcaaGcbaqcLbsacaaIWaGa aiOlaiaaicdacaaI3aGaaGynaiaaiIdacaaI4aaakeaajugibiaad2 eaaaaaaa@4C27@ . To get the nature of variation of free energy, we have plotted the graphs between the magnitude of change in free energy with corresponding change in the event horizon in XRBs as shown in the Figure 1.

Figure 1 The figure shows the graph plotted between the mass of spinning black holes and their corresponding value of free energy.

From the graph plotted between the magnitude of change in free energy with corresponding change in the event horizon in XRBs, it is obvious that the magnitude of change in free energy with corresponding change in the event horizon decreases gradually with increasing the mass of the spinning black holes as the magnitude of change in temperature with corresponding change in the mass in XRBs (Ref last). It should be noted that the mass and radius of the event horizon are the same thing in the case of spinning black holes using G=h=c=1 as clear from equation (39). Hence it can be concluded that the magnitude of change in free energy with corresponding change in the event horizon is approximately similar to the magnitude of change in temperature of spinning black holes with corresponding change in the mass in XRBs.17–19

Conclusion

In the present work, we can draw the following conclusions:

  1. The change in Gibb’s free energy with corresponding change in the event horizon depends on the surface gravity, mass, angular velocity and event horizon of spinning black holes
  2. The change in Gibb’s free energy of the spinning black holes of zero surface gravity is zero just like the change in entropy in the reversible process
  3. The change in free energy of spinning black holes of zero angular velocity is directly proportional to the corresponding change in the radius of event horizon
  4. The Gibb’s free energy of the spinning black holes of zero surface gravity is constant just like the entropy in the reversible process
  5. The change in the free energy of spinning black holes w.r.t. the event horizon existing in XRBs remains the same except for those spinning black holes having polar angle [0, 90] degree of lateral grid boundaries
  6. The magnitude of change in free energy of spinning black hole of angular velocity 0.5 per second with about maximum spin is inversely proportional to the mass
  7. The magnitude of change in free energy with corresponding change in the event horizon is approximately similar to the magnitude of change in temperature of spinning black holes with corresponding change in the mass in XRBs.

Acknowledgments

None.

Conflicts of interest

The author declares there is no conflict of interest.

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