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Gibb's Free Energy of the Spinning Black Holes Physics & Astronomy International Journal Research Article Open Access Gibb’s free energy of the spinning black holes Abstract Volume 4 Issue 1 - 2020 The present gives a model for the change in Gibb’s free energy of spinning black holes 1 2 with corresponding change in the event horizon using the first law of black hole mechanics, Alok Ranjan, Dipo Mahto 1 mass-energy equivalence relation applied to the Gibb’s free energy of the Reissnner- Department of Physics, Research Scholar University, India 2 Nordstrom black hole and concludes that the magnitude of change in free energy with Associate Professor, Department of Physics, Marwari College, corresponding change in the event horizon is approximately similar to the magnitude of India change in temperature of spinning black holes with corresponding change in the mass in Correspondence: Dipo Mahto, Associate Professor, XRBs. Department of Physics, Marwari College, TMBU Bhagalpur, India, Keywords: free energy, surface gravity, XRBs Tel 9006187234, Email Received: January 31, 2020 | Published: February 12, 2020 Introduction cn=quantum gravity model dependent coefficients, sA =surface area of the black holes & Q=charge on the black holes. David Hochberg1 computed the O(h) corrections to the mass, thermal energy, entropy and free energy of the black hole due to In the case of black holes having charge (Q=0), then the equation the presence of hot conformal scalars, massless spinors and U(1) (1) takes its form gauge quantum fields in the vicinity of the black hole using the A recent solutions of the semi-classical back-reaction proble.1 You S = s (2) Gen Shen and Chang-Jun Gao calculated free energy and entropy of 4 diatomic black hole due to arbitrary spin fields using the membrane The equation (2) is known as standard Bekenstein entropy of black model based on the brick-wall model and showed that the energy of hole. For the spherically symmetric and stationary or Schwarzschild scalar field and the entropy of Fermionic field have similar formulas 2 black hole, its surface area is naturally given by the following containing only a numerical coefficient between them. David Kastor equation9,10 et al.3 analysed the free energy and also specific heat in the small and 2 large black hole limits and comment upon the Hawking-Page phase AR= 4π (3) bh transition for generic Ads-Lovelock black holes.3 Hugues Beauchesne Where the radius of event horizon for non-spinning and spinning and Ariel Edery has shown that the negative of the total Lagrangian black holes are given by equations (4) and (5) respectively. approaches the Helmholtz free energy of a Schwarzschild black hole at the time of collapse. He also computed the numerical value 2GM = of the interior Lagrangian to the expected analytical value of the Rbh 2 (4) interior Gibb’s free energy for different initial states.4 In the present c work, we have proposed a model for the change in free energy with GM = corresponding change in the event horizon using the first law of black And Rbh 2 (5) hole mechanics, mass-energy equivalence relation applied to the c Gibb’s free energy of the Reissnner-Nordstrom black hole. The entropy of black holes (S) can be obtained by putting (3) into Theoretical discussion eqn (2) as: 2 SR= π (6) Black holes are boost machines processing the high frequency bh input and deliver it as low frequency output, owing to the gravitational The above equation is differentiated, we have shift and also provide a glimpse of the world at very short distance 5 dS= 2π R dR (7) scales. This world consists of nothing, but vacuum fluctuations. bh bh Black hole is one of new physical phenomena predicted by General The Gibb’s free energy of the Reissnner-Nordstrom black hole is 6 relativity and defined as the solution of Einstein’s gravitational field given by the following equation3 equations in the absence of matter that describes the space-time 7 G= E − TS −Φ . Q (8) around a gravitationally collapsed star. The gravity of a black hole is H so abnormal that nothing can escape from it. The generalised form for Where E=ADM mass of Reissner–Nordström black hole which entropy of Reissnner-Nordstrom black holes in commutative space is 8 gives the total energy of a space-time as defined by an observer at given by the following equation spatial infinity, using the Hamiltonian formalism, for an asymptotically ∞ flat space-time and The ADM mass consists of two contributions: A 2 A 4 SQ=−++s π ln s cC black hole horizon and solitonic residue. It is always greater than the ∑ n (1) 44n=1 As Schwarzschild black holes. Where C=constant Submit Manuscript | http://medcraveonline.com Phys Astron Int J. 2020;4(1):23‒28. 23 ©2020 Ranjan et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and build upon your work non-commercially. Copyright: Gibb’s free energy of the spinning black holes ©2020 Ranjan et al. 24 T=Temperature of black hole. Putting (7) in the above equation, we have S=Entropy of black hole. δM= κδ RR +Ω δ J (18) bh bh Q=Charge of the black hole. For maximum spin of black hole (a*=1), the angular momentum of the black hole is given by.12 Φ H =Electrostatic potential at the outer horizon of black hole. 2 JM= (19) Actually the astronomical black hole is not likely to have any significant charge, because it will usually neutralised by surrounding This condition corresponds to spinning black holes 11 plasma. Hence the charge Q=0, the equation (8) becomes Or δδJ= 2 MM (20) = − G E TS (9) Putting the above value in eqn (18), we have The product of temperature (T) and entropy (S) for the Reissnner- δM= κδ R R +Ω2 MM δ (21) Nordstrom black hole is given by.3 bh bh (12−ΩMM)δ = κδ RR 1 22 bh bh TS= M − Q (10) 2 κ δδM= RRbh bh (22) For Q=0, (12−ΩM ) M TS = (11) Putting the above value in equation (15), we have 2 κ According to Einstein well-known mass-energy equivalence δδG= RRbh bh (23) relation, we know that 21( −Ω 2 M ) 2 E= Mc (12) The above equation gives the change in Gibb’s free energy with corresponding change in the event horizon in terms of surface gravity, Putting (11) and (12) into equation (9), we have mass, angular velocity and event horizon of spinning black holes. In 2 M the case of spinning black holes, the surface gravity of a black hole is G= Mc − (13) given by the Kerr solution.12,13 2 4 2 1/2 Putting c=1 throughout our research work, the equation becomes ()MJ− κ = H 2 4 2 1/2 (24) M 2{MM+− ( M J )} GM= − H 2 Each black hole is characterised by just three numbers: mass M, M spin parameter a* defined such that the angular momentum of the G = (14) black hole is a*GM2/c.11,12 2 Hence we have The change in free energy of the Reissnner-Nordstrom black 2 hole due to change in the mass of black hole can be obtained by J= a*/ GM c (25) H differentiating the above equation Using G=c=h=1, we have 1 2 J= aM* (26) dG= dM (15) H 2 The radius is smaller in the case of spinning black holes, tending to The first law of black hole mechanics is simply an identity relating GM/c2 as a* tends to 111 and in the case of spinning black holes having the change in mass M, angular momentum J, horizon area A and spin parameter (a*=1), then we have, charge Q, of a black hole. The first order variations of these quantities 12,13 2 in the vacuum satisfy. where JMH = (27) k using the relation (27) into (24), we have δM= δ A +Ω δ JQ − υδ (16) k = 0 (28) 8π Where Ω =Angular velocity of the horizon. υ =difference in the The equation (23) becomes electrostatic potential between infinity and horizon. dG = 0 (29) For Q=0, dQ=0 and using the relation (2), the equation (16) This equation shows that the change in free energy is zero for the becomes spinning black holes spinning at max. Spin. The above equation can κ be written as: δM= δδ SJ +Ω (17) 2π G= constant (30) 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 Copyright: Gibb’s free energy of the spinning black holes ©2020 Ranjan et al. 25 The above equation shows that the total Gibb’s free energy of Using above equation, eqn (23) becomes spinning black holes spinning at max. Rate has constant free energy. dG .1517Rbh Wang, D has shown that the angular velocity ( ΩH ) evolves in a = dR2 M (1−Ω 2 M ) non-monotonous way in the case of thin disk-pure-accretion attaining bh a maximum at a*=0.994 and turns out to depend on the radial gradient Ω 14 of p near the BH horizon. One black hole at the heart of galaxy dG .07588R NGC1365 is turning at 84% the speed light. It has reached the = bh (34) cosmic speed limit and cannot spin any faster without revealing its dR M(1−Ω 2 M ) bh singularity.15 The equation (34) gives the change in Gibb’s free energy with For convenience, let us assume a*=0.9 corresponding change in the event horizon in terms of mass, angular 2 9 2481 = or = 2 or JM= and hence, velocity and event horizon of spinning black holes.
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