kerr black holes as a carnot engine

7/27/2019 Kerr Black Holes as a Carnot Engine

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PHYSICAL REVIEW D VOLUME 43, NUMBER 2 15 JANUARY 1991Kerr black holes as a Carnot engine

Osamu Kaburaki*Theoretical Astrophysics, California Institute of Technology, Pasadena, California 91125

Isao OkamotoDivision of Theoretical Astrophysics, National Astronomical Observatory, Mizusatva, Itvate 023, Japan(Received 5 June 1990)It is shown through a series of thought experiments that rotating black holes immersed in a

strong magnetic field can act, in principle, as a Carnot engine. In contrast with the original Carnotengine, however, work is extracted in the form of electric power, so that a Kerr hole actually func-tions as an electric powerhouse which is driven by the energy of Hawking's thermal radiation. Thewhole cycle consists of (i) isothermal spin-down, (ii) adiabatic spin-up, (iii) isothermal spin-up and(iv) adiabatic spin-down processes of the hole. The maximum efficiency of this engine is attained ifstep (iii) can be performed in the extreme Kerr state.

I. INTRODUCTIONRotating black holes have attracted great attention inthe field of astrophysics as possible central engines of ac-tive galactic nuclei. ' The compactness of black holesmakes it possible to liberate a large amount of gravita-tional energy of accreting matter in the neighborhood oftheir event horizons. Moreover, it was pointed out byPenrose that the rotational energy of a Kerr hole is ex-tractable in principle. Although the process suggested byhim as an illustration of this factow called the Pen-rose processas believed to be astrophysically ir-relevant, a viable mechanism has been proposed later byBlandford and Znajek as an electromagnetic extractionmechanism. Thorne and his co-workers extensively dis-cussed the latter process and related problems from aviewpoint of "3+1"formalism of general relativity. ' Ithas also been proposed that the presence of an elec-tromagnetic field around Kerr holes can modify the origi-nal Penrose process significantly.According to the no-hair theorem of black holes, aKerr hole is completely specified only by its mass and an-gular momentum, M and J. Therefore all other quanti-ties associated with the hole can be expressed in terms ofthese two quantities. This fact is reminiscent of thermo-dynamics. The similarity in the behavior of black-holearea and of entropy in thermodynamics was first madeexplicit by Bekenstein. He also derived a temperaturewhich is proportional to the surface gravity. In order for

this analogy between black-hole mechanics and thermo-dynamics to be complete, however, black holes shouldemit thermal radiation. But this was thought to be inhib-ited by the nature of the black-hole event horizon.A striking breakthrough was achieved when Hawkingdiscovered that a black hole can emit thermal radiationquantum mechanically with a temperature given by&gH2~k

where gH is the surface gravity of the hole, and A and kare the Planck and Boltzmann constants, respectively.Corresponding to this relation, the entropy of the hole isdefined byk 3 (2 RG4 2 y P:

C2dM =TdS+Q dJ,with 0 and J being the angular velocity and the angularmomentum of a Kerr hole. Comparing this equationwith the first law of thermodynamics for a gas in a con-tainer, we can see the correspondence, c M+ E, A~p,and J~V, where E, p, and V are the internal energy, thepressure, and the volume of the gas. The second lawstates that

dS ~0 (4)for any change of isolated systems which consist of blackholes and normal matter.Pushing ahead the above correspondence between aKerr hole and a gas in a container, we show in the follow-ing sections that the Carnot cycles can be performed alsofor this rotating black hole and hence work can be ex-tracted in the course of the cycle. In contrast with thePenrose process, however, the consequence of this pro-cess is not an extraction of rotational energy from thehole. The hole returns to its initial state after a completecycle. Instead, a certain amount of thermal energy in theHawking radiation held in a thermal bath is convertedinto work. In the present paper, we restrict our attention

where 3 is the surface area of the hole, l~ is the Plancklength, G is the gravitational constant, and c is the lightvelocity. In terms of these quantities the first law ofblack-hole thermodynamics is expressed as

43 340 1991 The American Physical Society


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43 ERR BLACK HOLFSS AS A CARNOT ENGINE 341mainly to t"e "' - e iscus 'y the 1n-principle" discuss'e. e believe 111a 1 1-su%cient acad emic interest b' ve that this problope that it open y itself, but we ma 1may alsoP g Pysical applications.

II. CARNOT CYCLE IN J-0 PLANEThe entro ay and temperaturepressed implicitli y in terms of its anure of a Kerr holee are ex-t b th e relations' n an-

I

A'S (fiGA S/7rkc2~k [IAGNI Slvrkc )]'c 0 +2(2vrkT/A) 2(2kT

4GQ,2~kT/A)[A +(2~kT/A)[0 + (2~kT/A ) ] ' i

The contoours of constant S andThe contours of con n ig. 1.qe origin with the tan p ane.g opes are

J' 0 h hn )0) exist only th

nstant-T curveb d hhII a ing oles) are distributed orom the up r left to th loh'1 ing tool g at tgo eyond

g th's fo b'dd pcurves leave ths and approach the 0 ax' e origin alongthey are viewed as fu s. ena w ich the hole's s ec' , eac as a maximum JSuch specific heat for conica pro erties es.

any pairurves a ways has one more

intersection besides ates at the orig . Thi.syp

p ane. Thereforese

thermal curves T & and T (T )Te ore, two pairs f 'z ' p'] 2 ave in general fo

curvessections. The state of aff'a' four nontrivial inter-Fi ~ 2.g. . If the change of a K

a e o affairs is expressed schematically inthe ro qo a Kerr hole is

' ', nbla er we consider onl gth in which the state ofyc e is the one i

e

Fi 2 Th h 1h 1 d o e cycle consists1n

own, adiabatic spin-u i , i.e., iso-d b ' '" d P O' Pn- own processes. n-up,Isothermal spin-down process

pAfter putting a K haw ing radiation oerr ole of tern era & n roof temperature T .S ' 't lt th b0 n angular velo 't, w ich is sli htl ci yig t y smaller than the

Sp

oW Schwarzschild HolesFIG. 1. The ccontours of constantE K h 1o es (T=O) ar0=( rt hild hol d es, w ile

tio b id hest eoneatth e origin. on y one in-

FIG. 2. A black-p a ic

t ss system during a 1yc e is AdJw ic can be extracted frompath ABCDA). (the

area enclosed b hthe


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342 OSAMU KABURAKI AND ISAO OKAMOTO 43hole's rotation 0, with their axes of rotation common toeach other [Fig. 3(a)].In this situation, the thermal particles evaporatingfrom the hole have the average angular velocity 0 andthe particles absorbed from the heat bath by the holehave 0( I). As a consequence, the hole is graduallyspun down within a period of the thermal relaxationtime. By keeping the rotation of the thermal bath alwaysslightly smaller than that of the hole, we can lead thestate of the hole from 3 to 8. During this process thehole absorbs a finite amount of entropy AS =SzS&.(ii) Adiabatic spin-up process (B~C). In this processthe black hole should be thermally disconnected from theoutside world in order not to emit nor absorb thermalparticles of any kind. This may be accomplished by set-

ting a spherical mirror somewhere in between the eventhorizon and the stretched horizon which is locatedslightly above the true horizon, covering the former corn-pletely. Then, an axisymmetric magnetic field around thehole's rotation axis is applied. We always adjust this fieldnot to be forced out from the hole; i.e., there is no po-loidal Aux which does not penetrate the hole. It is alsoassumed that the field is not so strong as to alter the ther-modynamical state of the hole, i.e., as to a8'ect the back-ground Kerr metric.Next, the hole is connected to an external batterywhich is placed sufficiently far from its horizon, and has avoltage Vz and a negligible internal resistance. In orderto lead the current to and from the hole's stretched hor-izon on which all the electromagnetic properties of the

Q(I-e}(zQ7~ ~

8 ~ . . ~

8 ~ ~

~ ' . ~Tl - ~

~ ~ ~

~ 8

. zs

(a)

Q(l g}T(~~3

T2

(c)FIG. 3. An illustration of the conceptual setups which are used in performing the four steps in a black-hole Carnot cycle. For thedetailed explanation of individual steps, see the text.


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KERR BLACK HOLES AS A CARNOT ENGINE 343hole are endowed, two devices are attached to the hole.One is a north-polar pipe whose surface coincides withone of the poloidal magnetic (lux tube [see Fig. 3(b)]. Theother is an equatorial thin disk. Both are made of per-fectly conducting material and divided into a large num-ber of continuously placed, infinitely narrow rings. Theyare, therefore, able to rotate differentially keeping theelectric contact between any pair of adjacent rings. Wearrange the pipe and the disk to be at rest in the zero an-gular momentum observer's (ZAMO's) frame. ' '" Theleading wires which connect the battery to the outeredges of the pipe and the disk are also infinitely conduct-ing and are placed sufficiently far from the hole.The Kerr hole in a magnetic field acts as a unipolar in-ductor and generates an electromotive force (EMF)

Q ~y2VH = B,coHp~d 0,c oowhere all quantities are evaluated at the surface of thestretched horizon, and B is the normal component ofthe magnetic field, co=g &&, and p=g & in Boyer-Lindquist coordinates. The lower limit of the integra-tion, 6j0, denotes the colatitude of the polar pipe's surfacetouching to the stretched horizon. The nonvanishing ra-dius of the pipe guarantees the finiteness of the hole's to-tal resistance:

If a current flows in this circuit, the Lorentz force at thestretched horizon exerts a torque on the hole. However,there is no torque on the polar pipe since the currentAows along the poloidal field, and on the equatorial disksince there is no magnetic Aux. Although there is atorque on the portion of the leading wire which is im-mersed in the magnetic field, the torque cannot do workon the wire since it is fixed. In order for the torque at thehorizon to spin up the hole, the external battery shoulddrive a current against the hole's EMF. Therefore thevoltage of the battery must be larger than that of thehole, i.e., ~ V~~ & ~ V~ ~. In this situation, we can calculatethe total current from Kirchhoff's second law as

VH+ VxRH

where the sign of the current is defined as positive in thedirection of the hole's EMF.The entropy generation on the hole's surface throughthe Joule dissipation of the current can be kept negligibleas far as this spin-up process proceeds sufficiently slowly.This point will be discussed in the next section. Anotherremark to be made is that the external battery is specialin the following sense. Namely, it should drive, say, elec-trons and positrons in the opposite directions from itsnegative and positive terminals, respectively. This is dueto the fact that current-carrying electrons cannot comeout from the hole's horizon to close the electric current.(iii) Isothermal spin-up process (C~D). The globalsetup is similar to that in the isothermal spin-down case.The differences from that case are in the points that the

temperature of the thermal bath is lower, i.e., T2 & Ti,and that the angular velocity of the bath is always keptslightly faster than that of the hole, i.e., Q(1+5). As aresult of emission and absorption of the Hawking radia-tion in a quasiequilibrium state, the hole is spun up grad-ually and the excess entropy is transferred to the thermalbath. Therefore, the force acting on the bath to rotate itwith a required angular velocity should do positive work.

(iv) Adiabatic spin-down process (D~A). The globalsetup is similar to that in the adiabatic spin-up case. Theonly difference is in the point that an electric load withresistance Rz is inserted in the external circuit instead ofa battery, so that the current is driven by the hole's EMF.The load may actually be a special type of electric motorwhich can be operated by the special type of currentmentioned in (ii) above. The total current Aowing in thiscircuit is

The area in the J-0 plane enclosed by the closed pathABCDA rejects this work. It is the work done by themotor minus the work by the external battery and by theforce keeping the thermal bath's angular velocity at thevalue near $1 in the processes (i) and (iii). Analogously tothe case of ordinary Carnot engines, the extracted energycomes from the hotter bath. The efficiency of our cycle isalso given by the same expression as in the usual case:w Qi Q2Q) Q) (12)

where Q denotes the net energy exchanged between thehole and the thermal baths. It is evident from this ex-pression that the maximum efficiency q= 1 is attainedwhen the lower-temperature state corresponds to an ex-treme Kerr hole. Actually, however, the maximumefficiency would not be attained even in principle for twomain reasons. One is that it seems impossible to producean extreme Kerr hole. ' The other is that even if such ahole is produced the adiabatic processes considered abovecannot be operated because the magnetic Aux threadingthe hole becomes zero when the hole is maximally rotat-ing.

'III. DISCUSSIDN QF EACH PROCESS

In both isothermal and adiabatic processes introducedin the preceding section, there are a few problems whichshould be considered more carefully. We discuss suchproblems here.A quasiequilibrium between the hole and the heat bathhas been assumed in steps (i) and (iii). In order for thisassumption to hold with good accuracy, the characteris-tic time for the changes in these processes should be

VaI= RH+R~After completing a cycle the hole returns to its initialstate. Nevertheless, net work is extracted from the sys-tem, which amounts to


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OSAMU KABURAKI AND ISAO OKAMOTO 43much longer than the thermal relaxation time of the sys-tem. The latter may be characterized by the evaporationtime' of a Schwarzchild hole in a vacuum:

I e-.g .27Tct)H (19)8.7X 10 M sec for M ))10' g,H 4.8X10 M sec for 10' g))M))5X10' g.

(13)Then, the mass of a black hole which evaporates withinthe present age of the Universe is in the range 5X 10'g ~M& ~ 7X 10' g. The result changes only by a factorof -2 even for extreme Kerr holes. Therefore, the idealisothermal processes can be performed in the realUniverse only for mini-black-holes whose masses aremuch smaller than Mz.Another point to be mentioned for these processes isthe stability of the thermal equilibrium between the holeand the bath. As is well known, the specific heat for con-stant 0 is always negative for any Kerr holes. ' ' Thismeans that the equilibrium is unstable for a given 0, asfar as the energy in the thermal bath is infinite. However,the instability can be avoided' by restricting the volumeof the bath such that

220 4 gG (14)55 p 5

The vectors es and e& are the unit vectors in the 8 and Pdirections. It can be seen from (17) that the quasistaticcondition requires I to be of order ~ since 0 and VHare quantities of order unity. Then, Eq. (18) tells us thatthe entropy production rate is a quantity of order ~and hence is negligible compared with dJ/dt. Thus wehave

2 dM dS dJ dJdt dt dt dt (20)

VX=VH(1+ e), (21)where e is a positive quantity of order ~ '. Then we havefrom (9) that I=e( VH/RH )=0(r ') (0, and thiscurrent really spins up the hole as is confirmed from (17).On the contrary, the current should have a positive value,i.e. , I=O(r ))0, in the adiabatic spin-down process.This can be realized in Eq. (10) by setting the resistanceof the external load infinitely large:

for any change in steps (ii) and (iv).In the adiabatic spin-up process, the voltage of theexternal battery should be set at a value

where U is the total energy of the hole and the bath, o. isdefined by R~=O(r) . (22)2 k4 (lib + 1lf+ ll )c'A'

(16)noting that the rate dJ/dt can generally change sign evenin a process of monotonic change in A. Any ideal quasi-static process attains finite AJ by taking an infinitely longtime ~ with the rate of change infinitely small, i.e.,Idj/dtl=0(r ').In our current circuit, the rates of change in the rota-tional and irreducible masses are given, respectively, by

dJ =0f H(cFH XB).SHe~&A =IVH,dt (17)

and n&, n&, and n, are the numbers of bosons withnonzero spin, of fermions, and of scalar particles, respec-tively. Of course, the thermal energy of the heat bathshould not be so large as to aff'ect the background Kerimetric. This requires that AS c M/T.Turning our attention to the adiabatic processes, wefirst discuss the entropy production due to the currentdissipation on the surface of the hole. We define thefinite change in the hole's angular momentum in an adia-batic process by b, (c M)))26(c m, ), (23)where

b, (c'M)= f 'lIl V ,2b(c m, )= j lIdr, (24)(25)

and e and m, are the charge and mass of an electron, re-spectively. We have assumed for simplicity that the posi-tive component of the current-carrying particles is posi-trons.The inequality (23) is reduced to a criterion for thestrength of the surface magnetic field,B. B (26)

There is another factor in the above-mentioned electro-dynamic processes which contributes to the increase ofthe hole's entropy. It is the infall of current-carrying par-ticles into the hole. This time, however, we cannotreduce the total increase of entropy by preforming theprocess infinitely slowly, since the total number of parti-cles swallowed by the hole does not depend on ~. In or-der for this effect to be negligible, the increase in the irre-ducible mass due to the particle infall should be muchsmaller than the change in the hole's total mass duringthe electrodynamical process:

T = fHEH cPHdA =IVH I RH, sdt (18) by using expression (7) for VH and that for the gravita-tional radius,where dA is the infinitesimal element of the hole's surfacearea, VH=IRH is the voltage drop across the hole, and8H is the surface current on the hole which is given by 2GM 1~H c2 ]+h (27)


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43 KERR BLACK HOLES AS A CARNOT ENGINE 345where h is a nondimensional parameter which measuresthe rapidness of the hole's rotation

2GM 03C

The critical field is given by

(29)where the numerical value has been calculated for thecase of extreme Kerr holes (h= I). The physical meaningof (26) is evident. We have to reduce the total current forsuppressing the total number of infalling particles. In or-der to keep the Lorentz force unchanged, however, weneed a strong magnetic field in compensation.The required field strength is reasonably weak for ablack hole of a solar mass but becomes as large as 10'G for M-5X10' g. Therefore, the critical field is un-comfortably large for a mini-black-hole of mass M ((Mzwhich can realize an ideal isothermal process. Undersuch strong fields various quantum effects may make the

situation complex, and moreover the background metricmay also be affected by the fields. At any rate, it seemsvery difficult to perform an ideal cycle in the realUniverse. However, these circumstances are not sodifferent from the case of usual thermodynamics. Also inthe latter case, an ideal process requires an infinitely longtime and there is no ideal Carnot engine working in theworld.The adiabatic spin-down process described in step (iv)of the black-hole Carnot cycle is an idealized limit(Rz~ ~) of the Blandford-Znajek process. In this sense,step (ii) of the same cycle may be called the inverseBlandford-Znajek process. It may also be an interestingtask to find an astrophysical counterpart of this processin black-hole-accretion disk systems, but this belongs to afuture investigation.

ACKNOWLEDGMENTSOne of the authors (O.K.) would like to thank K.S.

Thorne for his discussions, suggestions, and encourage-ment. He is also grateful to R.D. Blandford and E.S.Phinney for their discussions and interest in this work.

*Present address: Astronomical Institute, Faculty of Science,Tohoku University, Sendai, Miyagi 980, Japan.For a general review, see M. C. Begelman, R. D. Blandford,and M. J.Rees, Rev. Mod. Phys. 56, 255 (1984).2R. Penrose, Riv. Nuovo Cirnento 1, 252 (1969);R. Penrose andR.M. Floyd, Nature Phys. Sci. 229, 177 (1971).3R. D. Blandford and R. L. Znajek, Mon. N. R. Astron. Soc.179, 433 (1977); also, see E. S. Phinney, Ph.D. thesis, Univer-sity of Cambridge, 1983.4K. S. Thorne and D. A. Macdonald, Mon. Not. R. Astron. Soc.198, 339 (1982); D. A. Macdonald and K. S. Thorne, ibid.198, 345 (1982); D. A. Macdonald and W. -M. Suen, Phys.Rev. D. 32, 848 (1985).5K. S. Thorne, R. H. Price, and D. A. Macdonald, Black Holes:The Membrane Paradigm (Yale University Press, NewHaven, 1986).See, for example, S.M.Wagh and N. Dadhich, Phys. Rep. 183,137 (1989).7See, for example, B. Carter, in General Relativity: an EinsteinCentenary Survey, edited by S. W. Hawking and W. Israel(Cambridge University Press, Cambridge, England, 1979), p.294.

8J. D. Bekenstein, Phys. Rev. D 7, 233 (1973); 9, 3292 (1974);also see J.M. Bardeen, B. Carter, and S.W. Hawking, Corn-mun. Math. Phys. 31, 161 (1973).S. W. Hawking, Nature (London) 248, 30 (1974); Commun.Math. Phys. 25, 152 (1975);Phys. Rev. D 13, 191 (1976).I. Okamoto and O. Kaburaki, Mon. Not. R. Astron. Soc. 247,244 (1990)."Originally this idea appeared in J. M. Bardeen, W. H. Press,and S.A. Teukolsky, Astrophys. J. 178, 347 (1972).See, for example, R. Penrose, in General Relativity: an Ein-stein Centenary Survey, edited by S.W. Hawking and W. Isra-el (Cambridge University Press, Cambridge, England, 1979),p. 581.J. Bicak and L. Dvorak, Phys. Rev. D 22, 2933 (1980); J.Bica.k and V. Janis, Mon. Not. R. Astron. Soc. 212, 899(1985).D. N. Page, Phys. Rev. D 13, 198 (1976); 14, 3260 (1976); 16,2402 (1977).~~P. C.W. Davies, Rep. Prog. Phys. 41, 1313 (1978).6S. W. Hawking, Phys. Rev. D 13, 191 (1976);G. W. Gibbonsand M. J. Perry, Proc. R. Soc. London A358, 467 (1978).

kerr black holes as a carnot engine

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