Demons in Black Hole Thermodynamics: Bekenstein and Hawking
aa r X i v : . [ phy s i c s . h i s t - ph ] M a r Demons in Black Hole Thermodynamics:Bekenstein and Hawking
Galina Weinstein ∗ The Department of Philosophy, University of Haifa, Haifa, theInterdisciplinary Center (IDC), Herzliya, Israel.
March 3, 2021
Abstract
This paper comprehensively explores Stephen Hawking’s interactionwith Jacob Bekenstein. Both Hawking and Bekenstein benefited fromthe interaction with each other. It is shown that Hawking’s interactionwith Bekenstein drove him to solve the problems in Bekenstein’s blackhole thermodynamics in terms of a new thermal radiation mechanism.Hawking argued that Bekenstein’s thermodynamics of black holes wasriddled with problems. In trying to solve these problems, Hawking cutthe Gordian knot with a single stroke: black holes emit thermal radiation.Hawking derived the thermal radiation using a semi-classical approxima-tion in which the matter fields are treated quantum mechanically on aclassical space-time background. Hawking’s semi-classical approximationyielded a simple formula for the entropy of the black hole, which turnedout to be equivalent to Bekenstein’s equation for entropy.
This paper comprehensively explores Stephen Hawking’s interaction with JacobBekenstein. I show in this paper that both Hawking and Bekenstein benefitedfrom the interaction with each other.Bekenstein ascribed entropy to black holes and suggested that the entropyof black holes is proportional to the surface area of the event horizon of theblack hole (Section 6). Hawking proposed an area theorem, which suggested toBekenstein that the black hole entropy should tend to grow (Section 4).It is shown in this paper that Hawking’s interaction with Bekenstein drovethe former to solve the problems in Bekenstein’s black hole thermodynamics interms of a new thermal radiation mechanism (Section 14). ∗ This work is supported by ERC advanced grant number 834735.
In spring 1922, Albert Einstein gave lectures and discussion sessions at theColl`ege de France in Paris. In the second discussion session, Jacques Hadamard,a celestial mechanics professor at the Coll`ege de France, posed before Einsteina query: Since there is a singularity at r = 0 in the Schwarzschild solution[see equation (6) in Section 9], one may ask, what is the physical meaningof the Schwarzschild solution? We may ask whether r = 0 practically andphysically in our world? This question embarrassed Einstein. He said that ifthe radius term could really become zero somewhere in the universe, then itwould be an unimaginable disaster for his general theory of relativity. Einsteinconsidered that it would be a catastrophe, and jokingly called it the ”Hadamardcatastrophe” [35], pp. 155-156.Fast forward, Roger Penrose wrote: ”We may ask whether any connectionis to be expected between the existence of a trapped surface and the presenceof a physical space-time singularity such as that occurring at r = 0 in” theSchwarzschild solution? Penrose supplied the answer by a singularity theoremhe had proved together with Stephen Hawking: ”the presence of a trappedsurface always does imply the presence of some form of space-time singularity”.That is, a singularity must arise when a stellar-mass black hole is formed [38]pp. 1151-1152; [22], pp. 538-539.Brandon Carter and Werner Israel obtained uniqueness theorems. Israel pro-vided a theorem which states that the Schwarzschild and Reissner-Nordstr¨omsolutions are the only static solutions with an absolute event horizon [33], p.264. Carter established the family of Kerr solutions as the general pseudo-stationary, asymptotically flat, vacuum solutions with an absolute event horizon[18], p. 332. This is the Israel Carter Conjecture. The following picture thensuggested itself to Penrose. A certain body collapses to the size of the gravita-tional radius after which a trapped surface is found in the region surroundingthe matter. Outside the trapped surface there is a surface which is the absoluteevent horizon. ”Ultimately the field settles down to becoming a Kerr solution(in the vacuum case)” or a Kerr-Newman solution if a nonzero net charge istrapped in the black hole [38], pp. 1156-1157).The above theorem which Hawking called the Carter-Israel conjecture [23],p. 1344, states that ”if an absolute event horizon develops in an asymptoticallyflat space-time, then the solution exterior to this horizon approaches a Kerr-Newman solution asymptotically with time”. Moreover, the Kerr and Kerr-Newman solutions are explicit asymptotically flat stationary solutions of thefield equations of general relativity involving just three free parameters: mass,angular momentum and charge. That is, ”Only the mass, angular momentumand charge need survive as ultimate independent parameters”[38], p. 1158.The Carter-Israel conjecture was embellished by John Archibald Wheelerwho famously said: ”a black hole has no hair”. Hair refers to all the parameterswhich are hidden to outside observers. According to Wheeler, all the physicalproperties of the infalling matter into a black hole are eliminated and we cannotdistinguish between two black holes with the same mass, angular momentum(spin) and electrical charge [52], pp. 32-33. During 1971, Wheeler was pondering about things falling into a black hole andhaving everything disappear there. He thought to himself that there are someramifications that are not immediately obvious. He brooded about the secondlaw of thermodynamics that says that everything degenerates to a uniform tem-perature and imagined the following irreversible process: suppose you put a hottea cup next to a cold tea cup. They come into a common temperature. Byputting these two together, we have contributed to the degree of the entropy ofthe universe and its information loss.One day when he was still a graduate student at Princeton, Jacob Bekensteinentered Wheeler’s office and Wheeler told him: ”Jacob, if a black hole comesby, I can drop both tea cups into the black hole and conceal the evidence ofmy crime”. Wheeler’s demon, as Bekenstein called it, committed the perfectcrime against the second law of thermodynamics. That is because when thetea cups are dropped into the black hole, entropy is destroyed inside the blackhole. According to the no-hair theorem, no one outside the black hole wouldever notice any change in the black hole because all the physical properties ofthe infalling matter into the black hole are eliminated.After mulling over this problem, Bekenstein came back to Wheeler in afew months. In the meeting between the two, Bekenstein told Wheeler that3he demon had not avoided the entropy increase: ”you’ve just put the entropyincrease in another place. The black hole itself already has entropy and yousimply increase it” [8], pp. 24-25; [53].
In 1971 Hawking formulated an area theorem: the area of the black hole neverdecreases. It increases for all black holes. Hawking considered a situation inwhich there are initially two collapsed objects or black holes a considerabledistance apart. The black holes are assumed to have formed at some earliertime as a result of either gravitational collapses or, the amalgamation of smallerblack holes. Suppose now that the two black holes merge to form a single blackhole which settles down to a Kerr black hole with mass M BH and spin a . Thearea of the event horizon of the resulting black hole is greater than the sum ofthe areas of the event horizons around the original black holes: A K ≥ ( A + A ).In 1969 Penrose had written: ”I want to consider the question of whether it ispossible to extract energy out of a ’black hole’”. Penrose suggested mechanismsfor extracting energy from a black hole [38], p. 1159.Two years later Penrose and R. Floyd published a paper on the extractionof rotational energy from black holes which became to be known as the Penroseprocess. Penrose and Floyd discuss the question whether the mass-energy ofa rotating Kerr black hole could, under certain circumstances, be a source ofavailable energy. In order to answer this question they consider a process of anextraction of rotational energy from a Kerr black hole.Penrose and Floyd explain that there are two regions lying outside of thehorizon of a Kerr black hole: first, the stationary limit is a surface at which aparticle would have to travel with a velocity of light in order to appear stationaryto an observer at infinity. The stationary limit lies outside the event horizon andin between there is region called the ergosphere. A particle can enter and leavethe ergosphere but as viewed from infinity inside, it cannot remain stationary.The absolute event horizon is the effective boundary of the black hole insidewhich no information can escape to the outside world. The stationary limit andthe event horizon coincide in the case of zero angular momentum, i.e. for astationary Schwarzschild black hole.Penrose and Floyd consider the Killing vector K = ∂∂t . K becomes null atthe stationary limit and space-like within the ergosphere. Penrose and Floyduse K to define the energy of a particle with four-momentum (as measuredfrom infinity) p : E = − Kp . Since K is space-like within the ergosphere,it is possible for E to be negative there, even though the four-momentum istime-like.Suppose a particle with four-momentum p a enters the ergosphere. It thensplits into two. According to conservation of momentum: p a = p a + p a , andaccording to conservation of energy: E a = E a + E a . So that, the mass-energy4as measured from infinity) of the first particle is negative E a < E a >
0, escapes back to infinityand carries more mass-energy than the original particle possessed: E a > E a [41], pp. 177-178.According to the Carter-Israel no-hair theorem, the area of the event horizonof the newly formed Kerr-Newman black hole is given by: A = 8 πM BH r + , (1)where r + is the location of the surface of the horizon: r + = M BH + r M BH − JM BH − Q , (2)and J stands for the angular momentum. For a Kerr black hole Q = 0 [23], p.1345; [28], p. 35.For the particle that falls into the black hole Penrose and Floyd wrote aninequality equation: 2 M BH r + E a − ΩΛ ≥
0, where Ω is the angular velocity ofthe black hole, and Λ is the angular momentum of the first particle. The massof the black hole decreases: dM BH = − E a and the angular momentum of theblack hole also changes as a result of the rotational energy that is extracted: dJ = − Λ . Consequently, dA ≥ A ) increases even though itsmass can decrease. We thus retrieve the area theorem. Penrose and Floydconclude: “In fact, from general considerations one may infer that there shouldbe a natural tendency for the surface area of the event horizon of a black holeto increase with time whenever the situation is non-stationary. Thus the idealof maximum efficiency would appear to be achieved whenever this surface areaincrease is as small as possible” (i.e. a reversible process) [41], pp. 177-178.At the end of the paper the authors ”thank R. P. Geroch for discussions”.The paper was received in Dcember 1970. In December 1971 Robert Gerochwould present his thought experiment: the Geroch heat engine (see Section 9)[41], p. 179. In 1970 Demetrios Christodoulou, then a graduate student of Wheeler’s, intro-duced the concept of ”irreducible mass”, m ir , of the black hole and consideredtwo kinds of transformations: reversible and irreversible. He arrived at thefollowing formula: m = m ir + J m ir , (3)5here G = c = 1 and J is the angular momentum of the black hole. Thisequation is equivalent to E = m + p . Equation (3) says that the total mass-energy of the black hole is equal to m ir plus a rotational energy term, which can be changed by the extraction ofenergy from the black hole.In addition to writing equation (3), Christodoulou imposed four conditions:1) One can approach arbitrarily closely to reversible transformations thataugment or deplete the rotational contribution: J m ir to m .2) The attainable range of reversible transformation extends from: L = 0 , m = m ir to L = m , m = 2 m ir .
3) An irreversible transformation is characterized by an increase in m ir .4) ”There exists no process which will decrease the irreducible mass”.Christodoulou examined the Penrose process and demonstrated that if con-dition 2) is fulfilled, then equation (3) applies to the particle that falls into theblack hole.In other words, the most efficient processes are those associated with re-versible changes of the black hole (reversible transformations). The irreversibletransformation which is characterized by an increase in m ir of the black holewas found to be less efficient [8], p. 25.Bekenstein and Christodoulou recognized that for a Schwarzschild black hole: m ir ∝ A/ . [see equation (14)] [19], pp. 1596-1597; [5], p. 2333; [8], p. 25.Moreover, in thermodynamics, reversible processes are the most efficientones for converting energy from one form to another. So, Bekenstein thought tohimself that black holes must be in harmony with thermodynamics [8], p. 25. Bekenstein had a sneaking suspicion that the area A of the black hole ”mightplay a role of the entropy of black holes” [8], p. 25.In 1972 Bekenstein ascribed entropy to black holes. In fact, Bekensteinhad already suggested in his Ph.D thesis that the entropy of black holes isproportional to the surface area of the event horizon of the black hole A : S BH = ηkAl p , (4)where η is a constant. On Wheeler’s suggestion Bekenstein took the propor-tionality constant of the order l p , where l p is the Planck length [see equation(37)] [4], p. 733; [13], p. 515; see explanation in Section 22.Bekenstein explained why he chose equation (4). Christodoulou’s researchin classical black hole physics and Hawking’s area theorem suggested to him6hat the central desirable property was that black hole entropy should tend togrow. Equation (4) complied with the area theorem because when two blackholes merge to form a single black hole which settles down to a Kerr black hole,the entropy of the resulting black hole will increase. The area of the blackhole (a geometric quantity) behaves very much like entropy (a thermodynamicproperty) [4], p. 733; [13], p. 515. According to Bekenstein, S BH can be interpreted as the information hiddeninside the black hole and the ignorance of an observer at infinity about thematter inside it.In 1948 Claude Shannon famously presented a definition of the entropy which”measures the amount of information generated by the source per symbol or persecond” [43], p. 396. I will not discuss Shannon’s ideas further. I will just addthat Shannon wrote: ”From our previous discussion of entropy as a measure ofuncertainty it seems reasonable to use the conditional entropy of the message,knowing the received signal, as a measure of this missing information” [43], p.407. In 1973 Bekenstein wrote that ”the entropy associated with the systemis given by Shannon’s formula”. And he considered the Shannon informationentropy of a system which measures our ignorance or lack of information aboutthe actual internal configuration of the system.L´eon Brillouin identified information with negative entropy [16], pp. IX,pp. 7-10. Bekenstein adopted Brillouin’s relationship between information andnegentropy and stressed that the more information entropy, the less informationwe have about the black hole [5], p. 2335.Bekenstein further explained that ”the entropy of an evolving thermody-namic system increases due to the gradual loss of information which is a conse-quence of the washing out of the effects of the initial conditions”. That is, theentropy of a thermodynamic system, i.e. a black hole, which is not in equilib-rium increases. That is because information about the internal configuration ofthe system is being lost during its evolution. This happens as a result of the wip-ing out of the initial conditions of the system. Bekenstein summoned Maxwell’sdemon, saying: ”It is possible for an exterior agent to cause a decrease in theentropy of a system by first acquiring information about the internal configu-ration of the system. The classic example of this is that of Maxwell’s demon”.But in acquiring information about the system, the demon inevitably causesan increase in the entropy of the rest of the universe. Thus, even though theentropy of the system decreases, the over-all entropy of the universe increasesin the process [5], p. 2336. In 1993, Bekenstein would raise a somewhat similarargument, see Section 26.As a black hole approaches equilibrium, it loses its hair. Only M , J , and Q are left as parameters of the black hole at later times. It is conjectured thatthe loss of information about the initial conditions of the black hole would bereflected in a gradual increase in S BH . According to Hawking’s area theorem,7 BH increases monotonically as the black hole evolves [5], p. 2336. Entropy ebbs and flows at will and Wheeler’s demon violates the second lawof thermodynamics. Entropy increases inside the black hole whilst there is adecrease of entropy outside the black hole (this is explained in detail in the nextsection). Hawking’s area theorem dS BH ≥ dS g , of the black hole dS BH and the matteroutside of the black hole dS M , must never decrease ( dS g ≥ dSg = dS BH + dS M ≥ . (5)He termed this, the Generalized Second Law of Thermodynamics (hereafterGSL).The area theorem requires only that A not decrease. But the GSL alsodemands that if S M is lost into black holes, A will increase sufficiently for theassociated increase in S BH to at least compensate for the decrease in S M [4], p.738; [5], p. 2334-2338; [7], pp. 3077; [9], p. 287. In December 1971, in a Colloquium at Princeton University, Geroch conceived aheat engine that employs a Schwarzschild black hole as an energy sink. Geroch’sthought experiment could violate Bekenstein’s GSL.Consider the Schwarzschild metric: ds = (cid:18) − R g r (cid:19) c dt − dr − R g r − r (cid:0) dθ sin θdφ (cid:1) , (6)where: R g = GM BH c (7)is the gravitational radius.Here are the steps of the process:1) We fill a box with radiation of energy E = (cid:0) M c (cid:1) , temperature T andhigh entropy S . We then slowly lower the box by a rope toward the horizon ofthe Schwarzschild black hole in its gravitational field. Let us suppose, for thesake of the argument that the box has no weight and the rope also has no mass.We want to extract work from the Geroch engine.2) Suppose the box is lowered as far as possible so that r → R g . r is theradius away from the black hole to which the box has been lowered.8he box never actually reaches 2 R g . The small box is then opened and theradiation is allowed to escape into the black hole. The gravitational field doeswork on the box in this process: W = R g r M c ,where the gravitational potential is: U = − R g r M c and the redshift factor is: α ≈ R g r . (8)3) The box is then pulled up back at expense of almost no work and can berefilled with thermal radiation from the reservoir.Since r → R g , the amount of work that can be extracted from the Gerochheat engine is: W = M c and the black hole would end up in the same statein which it began and all of the energy of the radiation could be converted towork in the laboratory from which one was doing the lowering [4], p. 148, pp.739-740.The Carnot efficiency of the Geroch heat engine is: η T H ≤ (cid:16) − T C T O (cid:17) where T O is the temperature at which the radiation entered the black hole and T C is the temperature of the black hole into which the box exhausts its radiation.If all the mass-energy is converted to work, then the efficiency η T H ≤ T C = 0. Consequently, Geroch stressed that black holes are systems at absolutezero temperature. And this would violate Bekenstein’s GSL because entropy inthe outer world would decrease dS M <
0, with no counter increase in entropyin the black hole dS BH = 0. Thus, dS g = dS M < S b.h. as demanded by the second law. [. . . ] It followsthat the primary particle is required to fall from far away to the horizon”.Bekenstein explained that in such a fall the particle will radiate into the blackhole gravitational waves with energy of the order of its rest mass. This radiationwill cause a non-negligible increase in the black hole area corresponding to theincrease in S BH demanded by the GSL. In this manner Bekenstein reconciledChristodoulou’s process with the second law [4], p. 150.He then tried to resolve the difficulty raised by the Geroch thought experi-ment. In an attempt to salvage his GSL, Bekenstein wrote: S BH = ηkAl p ¯ h , (9)where ¯ h = h π .This means that the entropy of a black hole would be enormous comparedwith the entropy of typical ordinary matter of comparable size and energy.9ekenstein thus assumed that the system is “dictated by quantum effects” andhe gave a quantum explanation for why the box would never reach 2 R g . Butin 1980 he admitted that these attempts to salvage his GSL “were forced andinelegant” [6], p. 3292, pp. 3295-3296, p. 3298; [8], pp. 27-28; [51], pp. 5-6;[47], p. 154.Hawking objected to Bekenstein’s entropy, equation (4). According to KipThorne, ”Bekenstein was not convinced. All the world’s black hole experts linedup on Hawking’s side – all, that is, except Bekenstein’s mentor, John Wheeler.’Your idea is just crazy enough that it might be right’, Wheeler told Bekenstein”.As said above, Bekenstein suggested that the black hole must have enormousamount of entropy [equation (9)] and he interpreted equation (4) in terms of an”internal configuration” of the black hole, see Section 7. ”Nonsense, respondedmost of the leading black-hole physicists, including Hawking and me. The hole’sinterior contains a singularity, not atoms or molecules”, so writes Thorne [45],pp. 425-426.
10 The Hawking, Bardeen and Carter four lawsof black hole mechanics
Now there was a little dispute between Hawking and Bekenstein lurking beneaththe surface. “I remember”, said Wheeler, “when Bekenstein’s paper came outthe whole argument seemed so implausible to Stephen Hawking and his friendthat they decided to write a paper to prove that it was wrong” [53]. Indeed,in 1973 Hawking, Bardeen and Carter, wrote: “The fact that the effective tem-perature of a black hole is zero means that one can in principle add entropy toa black hole without changing it in any way. In this sense a black hole can besaid to transcend the second law of thermodynamics”. That is, the second lawis made irrelevant [3], pp. 168-169.Bardeen, Carter and Hawking formulated the four laws of black hole me-chanics, which correspond to the ordinary three laws of thermodynamics andthe zeroth law of thermodynamics. Recall that according to the no-hair theo-rem, when an object collapses to create a black hole, it will result in a stationarystate that is characterized by only three parameters: M BH , J , and Q . Apartfrom these three properties the black hole apparently preserves no other infor-mation of the object that has collapsed. Now suppose that radiation falls ontothe black hole, these are the four laws of black hole mechanics [3] , pp. 167-168:0) The surface gravity κ is constant over the event horizon of a stationaryblack hole.1) The first law relates the change of the mass, angular momentum andcharge of the black hole to the change of its entropy: d (cid:0) M BH c (cid:1) = T BH dS BH − Ω dJ − Φ dQ, (10)where d (cid:0) M BH c (cid:1) is the corresponding change in the black hole energy, Ω standsfor the angular velocity of the black hole, J is the angular momentum, Φ rep-10esents the electric potential and Q is the electric charge. Ω dJ and Φ dQ areanalogous to P dV of the first law of thermodynamics. So, Ω dJ and Φ dQ rep-resent the work done on the black hole by adding to it angular momentum andcharge, respectively. As seen from equation (10), the first law is related to theno-hair theorem.2) Hawking’s 1971 area theorem: dA ≥ . (11)Hence, A can only increase and dS BH ≥ κ to zero in a finite numberof steps.Robert Wald writes: “The fact that black holes obey such laws was, in somesense, supportive of Bekenstein’s thermodynamic ideas” [51], p. 6.
11 Bekenstein’s black hole temperature
Bekenstein responded to Bardeen, Carter and Hawking’s paper by assigning tothe black hole a finite non-zero temperature that corresponds to the entropy.He wrote in 1973 equation (10) as follows: d ( M BH ) = κc πG dA − Ω dJ − Φ dQ, (12)where κ is given by equation (15) below. Hawking also wrote equation (12) in1973 [24], p. 270.According to equations (10) and (12), if G = c = 1, T BH dS BH = (cid:0) κ π (cid:1) dA. Using the above form of the first law of black hole mechanics and equation (4),Bekenstein wrote [5], p. 2335, p. 2338: T BH = κl p ηk . (13)Later Bekenstein explained that equation (4) ”incidentally, established theblack hole temperature [...] What this temperature meant operationally wasnot clear, though I did study the matter in detail” [13], p. 517.In 1975 Hawking stressed that Bekenstein ”was the first to suggest that somemultiple of κ should be regarded as representing in some sense the temperatureof black holes. He also pointed out that one has the relation” equation (12) [26],p. 192.
12 A black hole in a thermal radiation bath
Bekenstein’s GSL started to wobble slightly as it faced another thought exper-iment. Consider a Kerr black hole of mass M BH immersed in a black bodyradiation bath within a cavity of temperature T . We assume that the cavity11oes not rotate so that the radiation has zero angular momentum and thatthe process of accretion has reached a steady state. Then to radiation of en-ergy E (as measured from infinity) going down the black hole there corresponds dS BH = ET BH [see also equation (33) Section 21].Bekenstein explained that since the accretion onto the black hole takes placein steady state, then together with E , entropy S = ET flows into the black hole.And so, for T ≥ T BH we get dS g = dS BH − S ≥ T < T BH , however, the GSL is violated.In order to solve this problem, Bekenstein suggested that the wavelengthin the radiation ¯ hT is already larger than the size of the black hole (very longwavelength corresponding to low T ). The radiation no longer flows in as acontinuous fluid would, but rather most wavelengths in it must tunnel intothe black hole discretely and individually. Bekenstein argued that the rate oftunnelling is sensitive to the wavelength. The shorter the wavelength, the higherthe rate. Thus for T ≥ T BH the rate is high and for T < T BH , the rate is soslow that practically the radiation can hardly flow into the black hole.For T < T BH , Bekenstein further introduced an ad hoc hypothesis thatirreversible processes will go on in the cavity material (and perhaps in the radi-ation also), which will generate additional entropy. This extra entropy sufficesto make the GSL work in the regime T < T BH [6], p. 3298. This explanationseemed cumbersome and it did not comply with Occam’s razor.
13 Making order of the pandemonium
Bekenstein suggested on thermodynamic grounds that a black hole should havea finite entropy S BH that is proportional to the surface area of the event horizon A [see equation (4)]. He further proposed that the temperature of the black holeshould be regarded as proportional to the surface gravity κ , see equations (13)and (15) below [25] p. 31.Hawking reasoned that if the black hole has a finite entropy that is propor-tional to the area of the event horizon A : A = 4 π (2 R g ) = 16 π ( GM BH ) c , (14)it must necessarily have a finite temperature T BH that is proportional to thesurface gravity of the black hole κ : κ = GM BH (2 R g ) = c GM BH . (15) R g is defined by equation (7).Hawking finally argued that this in turn indicates that a black hole can bein equilibrium with thermal radiation at some temperature other than zero. Itis certainly true for bodies in classical physics, but it is not so for black holeswhich can devour matter and cannot emit anything [28], p. 37.12n 1973 Hawking, Bardeen and Carter, thought that Bekenstein’s resolutionto the paradox of the black hole-in-thermal-radiation-bath was far-fetched (seeSection 12). A black hole, said the three authors, cannot be in equilibrium withblack body radiation at a non-zero temperature. That is because no radiationcould be emitted from the black hole whereas some radiation would always crossthe horizon into the black hole. If the wavelength of the radiation were very long,as suggested by Bekenstein, corresponding to a low black body temperature, therate of absorption of radiation would be very slow. True equilibrium, however,would be possible only if there were no radiation present at all; that is, if theexternal black body radiation temperature were zero [3], p. 169.So, Hawking was baffled about Bekenstein’s efforts at salvaging his GSL.Which is why Hawking thought that no equilibrium of a black hole with thermalradiation at some temperature other than zero was possible because the blackhole would absorb any thermal radiation that fell onto it but would be unableto emit anything in return [28], p. 37.The problem Hawking was faced with was: either the temperature T BH isidentically zero, in which case S BH [equation (4)] is infinite and the concept ofblack hole entropy is meaningless, or black holes have to emit thermal radiationwith some finite nonzero temperature [27], p. 193.”The first case”, says Hawking, ”is what holds in purely classical theory, inwhich black holes can absorb but do not emit anything. Bekenstein ran intoinconsistencies because he tried to combine the hypothesis of finite entropy withclassical theory, but the hypothesis is viable only if one accepts the quantum-mechanical result that black holes emit thermal radiation” [27], p. 193.
14 Hawking radiation
At about 1974, Hawking solved the above paradox by suggesting that black holesemit particles at a steady rate. The outgoing particles have a spectrum that isprecisely thermal. The black hole creates and emits particles and radiation justas if it were an ordinary hot body with a temperature T that is proportional tothe surface gravity κ and inversely proportional to the mass M BH [see equations(16) and (17) below] [25]; [28], p. 37. This immediately solved all problems.Hawking set E = kT , where k is the Boltzmann constant. He showed thatblack holes emit precisely thermal radiation, like that of a black body, withtemperature: T = ¯ hκ πck . (16)From the point of view of an outside observer, a Schwarzschild black holeemits black body radiation with temperature T that is proportional to κ andinversely proportional to M BH : T = c ¯ h πGM BH k ≈ . × − Km , (17)13here m stands for the black hole mass expressed in terms of solar mass ≈ × kg : m = M BH M ⊙ . (18)Bekenstein said that to Hawking’s surprise, by his own research he foundthat black holes are hot [13], p. 518.
15 The Bekenstein-Hawking entropy
Using equation (14) Hawking wrote the black hole entropy as follows [26], p.191, p. 197: S BH = kc G ¯ h A , (19)where S BH is called the Bekenstein-Hawking entropy .This equation is equivalent to Bekenstein’s equation (4) or equation (9).Hawking therefore fixed Bekenstein’s constant of proportionality η = . Thecalculation, therefore, has been satisfactorily concluded and Wheeler alludedthat Hawking ultimately came to the conclusion that Bekenstein was right afterall [53].In 1975 Hawking even wrote in his paper, ”black hole thermodynamics”:”A black hole of given mass, angular momentum, and charge can have a verylarge number of unobservable internal configurations which reflect the differentpossible configurations for the body that collapsed. If quantum effects were ne-glected, the number of different internal configurations would be infinite becauseone could form the black hole out of an indefinitely large number of indefinitelysmall mass particles. However, Bekenstein pointed out that the Compton wave-lengths of these particles might have to be restricted to be less than the radius ofthe black hole and that therefore the number of possible internal configurationsmight be finite though very large” [26], p. 192. Recall that according to Thorne,most physicists (including Hawking and Thorne) had responded ”Nonsense” tothis idea of Bekenstein [45], p. 426 (see Section 9).Delivering a lecture on his 60th birthday, Hawking outlined how he discov-ered Hawking radiation. At the end of 1973, Hawking’s work with Penrose onthe singularity theorems ([22], see Section 2) had shown to him that generalrelativity broke down at singularities. So he thought that the obvious next stepwould be to combine general relativity, the theory of the very large, with quan-tum theory, the theory of the very small. He had no background in quantumtheory so as a warm up exercise, he considered how particles and fields governedby quantum theory would behave near a black hole; he studied how quantumfields would scatter off black holes. ”But to my great surprise”, said Hawking,”I found there seemed to be emission from the black hole”. At first he thoughtthis must be a mistake in his calculation. But what persuaded him that it was14eal, was ”that the emission was exactly what was required to identify the areaof the horizon, with the entropy of the black hole [equation (19)]. S = Akc hG . I would like this simple formula to be on my tombstone”, said Hawking [31],pp. 112-113.In his 1975 paper, Hawking pointed out that Bekenstein’s entropy S BH [4] would have to be a function only of M , J , and Q with the following twoproperties: first, it always increased when matter or radiation fell into the blackhole. Second, when two black holes merged together, the entropy of the finalblack hole was bigger than the sum of the entropies of the initial black holes[26], p. 193. Bekenstein admitted that he very much guessed the form of thefunction in equation (4): S BH = f ( A ) [13], p. 515. In 1975 Hawking remarkedthat ”this is the only such quantity” [26], p. 193.I show in Section 18 that unlike Bekenstein who tried to unify thermody-namics and classical general relativity, Hawking endeavored to unify quantummechanics and classical general relativity. And using a semi-classical approxi-mation, Hawking arrived at equation (11).
16 The Bekenstein bound
Bekenstein’s mind still boggled at the Geroch paradox and the GSL, see Section9. That is because even if black holes do carry entropy S BH and we add ad-ditional mass dM to the black hole, the area of the horizon might not increaseenough to compensate for the loss of entropy. The GSL could in this case beviolated.Recall that the GSL is defined by equation (5) and we lower down the box onthe rope to the black hole and when it approaches 2 R g , we release the radiationof high entropy S . But the GSL is violated dS g = dS BH + S < dM = 0 and the area of the black hole would not change dS M = S = 0.So, in 1981, Bekenstein set himself to solve this problem from a thermody-namic perspective. He set G , c , k and ¯ h equal to 1. If a system of mass M ,energy E , effective radius R and entropy S is dropped into a black hole, themass increase of the black hole is: dM BH = ER M BH . (20)This equation takes into consideration the gravitational redshift factor α given by equation (8).For a Schwarzschild black hole: A = 16 πM BH [see equation (14)], so theincrease in the surface area of the horizon of the black hole can be expressed asfollows: dA = 32 πM BH dM BH . (21)15nserting equation (20) into equation (21) one obtains: dA = 32 πM BH ER M BH = 8 πER. (22)The GSL requires that dS g ≥ dS BH ≥ − dS M = S . That is, according toequation (19), dA ≥ S .So, dividing equation (22) by 4 yields: dA πER , (23)and the black hole entropy will increase by: 2 πER .Bekenstein stressed that if S does not exceed 2 πER , the GSL is not violated.So, Bekenstein decided to impose a limit on S saying that it cannot exceed whatwas later termed the Bekenstein bound: S ≤ πER. (24) S BH exactly saturates (satisfies) the Bekenstein bound [6], p. 3292; [9], p. 287;[17], p. 145; [51], p. 7; [2].
17 Buoyancy and Bekenstein’s bound
In response to Bekenstein’s paper, William Unruh and Wald published a paperin which they showed that equation (24) is not needed for the validity of theGSL [49].Wald explains that Bekenstein’s interpretation of equation (4) was thatquantum effects prevent Geroch’s box from reaching 2 R g . The quantum ef-fects eventually turned out to be Hawking radiation. If a black hole is placedin a radiation bath of temperature smaller than that given by equation (16),then Hawking radiation [with temperature given by equation (16) or (17) woulddominate over accretion and absorption. And Bekenstein’s GSL would not beviolated [51], p. 7.More specifically, Unruh and Wald argue that if we attempt to slowly lowerthe Geroch box containing rest energy E and entropy S into a black hole,there will be an effective buoyancy force on the box caused by the accelerationradiation felt by the box when it is suspended near the black hole. As a resultthere is a finite lower bound on the energy delivered to the black hole in thisprocess and a minimal area increase. This small area increase turns out to bejust sufficient to ensure that the GSL is satisfied. Consequently, less work isdelivered to infinity by the rope during the process of lowering the box andmore energy is transferred to the black hole in this process than would occurclassically. By the Archimedes principle of buoyancy this occurs when the energyof the box equals the energy of the displaced acceleration radiation [49], p. 942,p. 944; [47], p. 155; see Section 9. 16nruh and Wald wrote to Bekenstein to tell him about their finding. ButBekenstein had reservations about the buoyancy effect and he gave exampleswhere he showed that it was not sufficient to save the GSL. He stressed that hisentropy bound on matter, equation (24), would still be needed for the validityof the GSL. Unruh and Wald’s solution seemed to have obviated the need forBekenstein’s bound. Indeed, Bekenstein remarked that Unruh and Wald’s con-clusion was that bound equation (24) was not necessary for the GSL to hold.So, Bekenstein and Unruh and Wald had conflicting intuitions about the bound.Bekenstein, on his part, insisted that the fact remains that bound equation (24)”is supported by direct statistical arguments”. He further emphasized that ”thebound must apply to any object that can be lowered to the horizon of a blackhole” [51], p. 8; [10], p. 2262. [12], p. 1920.In 2002 Bekenstein explained that in the original derivation of equation (24)he considered the Geroch process and then applied the GSL to get the bound.This derivation was criticized for not taking into account the buoyancy of theGeroch box. ”A protracted controversy on this issue” led to the perception thata correction for buoyancy merely increases the 2 π coefficient in equation (24)”by a tiny amount” [14], pp. 4-5.
18 Hawking’s semi-classical approximation
Hawking discovered that black holes emit thermal radiation and gave a theoreti-cal basis for the thermal radiation emitted by black holes: quantum effects causethe black hole to radiate. He used the semi-classical approximation to arriveat equation (16). He wrote in 1978: ”The conclusion that black holes radiatethermally was derived using a semiclassical approximation in which the matterfields are treated quantum mechanically on a classical space-time background”[29], p. 24.I shall start with Hawking’s original derivation from 1974. In 1974 Hawkingdetermined the number of particles created and emitted to infinity in a gravita-tional collapse. He considered a simple example in which the collapse of a staris spherically symmetric. Outside the field is described by the Schwarzschild so-lution. The asymptotically flat space-time is divided into three regions: 1) Thefuture null infinity (light-like future infinity) I + : all null geodesics can reach thefuture null infinity. 2) The horizon null surface of the black hole. 3) Past nullinfinity (light-like past infinity) I − : no null geodesics can reach the future nullinfinity.Consider a wave which has a positive frequency on I + propagating backwardsthrough space-time with nothing crossing the event horizon. Part of this wavewill be scattered by the curvature of the static Schwarzschild metric outside theblack hole and will end up on I − with the same frequency. Another part of thewave will propagate backwards into the star, through the origin and out againonto I − .These waves can have a very large blue-shift and they can reach I − withasymptotic form of the wave function for time v < v where v stands for17he last advanced time that enables them to leave I − . These waves are thenabsorbed by the black hole. On the other hand, if v > v the waves can passthrough the origin and escape to I + . These waves are red-shifted.Hawking calculated (took Fourier transforms of the asymptotic form) thatthe total number of outgoing particles or wave packets created in a certainfrequency range (peaked at a certain frequency) is infinite. Hawking concludedthat this infinite number of particles corresponds to a steady-rate emission,which means that the black hole will emit thermal radiation until it completelyevaporates in an explosion (see the next section for black hole evaporation).Hawking compared the incoming waves that escaped to I + to the outgoingones that arrived at I − and was led to the following equation, which representsthe relation between absorption and emission (Hawking radiation): (cid:0) exp (cid:0) πωκ (cid:1) − (cid:1) − , that is, the number of particles emitted from the black holein a wave packet mode times the number of particles that would have beenabsorbed from a similar wave packet incident on the black hole from I − . ”Butthis is just the relation”, said Hawking, between absorption and emission thatone would expect from a black hole emitting thermal radiation with a tempera-ture given by equation (16): T = κ π . Hawking therefore justified the geometricreasoning made in black hole thermodynamics [25], pp. 30-31.In a paper he submitted in June 1975, ”Black Holes and Thermodynamics”,Hawking showed that the steady-rate emission turns out to have an exactlythermal spectrum and the following quantities arise from his 1974 semi-classicalapproximation: the area A of the event horizon [equation (14)], the surfacegravity κ [equation (15)], the angular frequency of rotation of the black holeΩ and the potential of the event horizon Φ (both are inversely proportional tothe area A [see equation (10) for the first law of black hole mechanics]). Andthe most important quantity that arises from the semi-classical approximationis the entropy of the black hole [equation (19)] [26], pp. 191-192.
19 Hawking’s ansatz of pair production
Two months later, Hawking submitted a paper in which he was more explicitsaying: ”quantum mechanics allows particles to tunnel on spacelike or past-directed world lines. It is therefore possible for a particle to tunnel out ofthe black hole through the event horizon and escape to future infinity. One caninterpret such a happening as being the spontaneous creation in the gravitationalfield of the black hole of a pair of particles, one with negative and one withpositive energy with respect to infinity. The particle with negative energy wouldfall into the black hole [...]. The particles with positive energy can escape toinfinity where they constitute the recently predicted thermal emission from blackholes” [27], p. 2462.The paper was published in 1976 and a year later, Hawking explained thatquantum field theory implies that particle-antiparticle virtual pairs (an electronand a positron) are continually produced and recombined in the vacuum state.18awking showed that when such a particle-antiparticle pair is produced near theevent horizon of a black hole, one particle may fall into the black hole, leavingthe other particle outside without a partner with which to annihilate. “The for-saken particle or antiparticle may fall into the black hole after its partner”, saysHawking, “but it may also escape to infinity, where it appears to be radiationemitted by the black hole.Hawking wrote in 1978 that ”If a black hole is present, one member of a pairmay fall into the hole leaving the other member without a partner with whomto annihilate” [29], p. 23.Another way of looking at this process, says Hawking, is to regard the par-ticle that fell into the black hole (the antiparticle) as being a particle that wastraveling backwards in time. It would come from the singularity and wouldtravel backwards in time out of the black hole to the point where the particle-antiparticle pair first appeared. There it would be scattered by the gravitationalfield into a particle traveling forwards in time. Hawking reasoned that this wayone can think of the radiation from a black hole as having come from the sin-gularity and having quantum mechanically tunnelled out of the black hole [29],p. 32; [28], p. 37. In 1979 Hawking elaborated this picture by suggesting aradiation from white holes (see Section 22).This way of looking at the pair-production process is reminiscent of theRichard Feynman-Ernst Stueckelberg interpretation of the positron, which ismoving forward in time but appearing to us to be an electron traveling back-ward in time: a photon spontaneously creates an electron-positron pair, withan electron flying off to a distant region and the positron meeting an additionalelectron, resulting in mutual annihilation. This description involves three par-ticles: electron, positron, and another electron. Feynman suggested describingthat same process using just one electron moving first forward in time, thenbackward in time and finally again forward in time [21] p. 749.Hawking concludes that quantum mechanics allows a particle to escape frominside a black hole, something that is not allowed in classical mechanics [28], p.38. Simply put, from the point of view of an observer at infinity: the black holecreates particles in pairs, with one particle always falling into the black hole andthe other escaping to infinity [27], p. 2460.This pair production picture rings a familiar bell. Recall that Penroseand Floyd had suggested a process in which a particle with four-momentumis dropped into the ergosphere. It then splits into two particles, so that themass-energy of the first particle is negative and it is swallowed by the black holewhilst the second particle escapes back to infinity and carries more mass-energyto infinity. Penrose and Floyd showed that the area theorem is valid. See Sec-tion 4. On the other hand, as I show in the next section, Hawking radiationviolates the area theorem. Both processes, however, do not violate the GSL.More on the GSL in Section 21.Unruh has recently said that one of the problems with Hawking’s derivationis that although it is mathematically correct, it really makes no physical sense.There is nothing wrong with the mathematics but physically it simply makes nosense. Why is it problematic? The particles (Hawking radiation) cannot come19ithin the black hole because nothing can come out of the black hole. They areactually created outside of the black hole but an observer at infinity sees theblack hole as if it is emitting a Hawking radiation [48].
20 Hawking’s evaporation time
Now let us obtain the power P emitted by a Schwarzschild black hole of mass M BH . But when obtaining P there is a snag. As Hawking himself put it in1974: as the black hole emits the thermal radiation with temperature ≈ − [see equations (16) and (17)] “one would expect it to lose mass. This in turnwould increase the surface gravity and so increase the rate of emission. Theblack hole would therefore have a finite life of the order of 10 (cid:0) M ⊙ M (cid:1) − s . For ablack hole of solar mass this is much longer than the age of the Universe” [25],p. 30.The Stefan-Boltzmann law of black body radiation states that the radiatedpower P per unit area of a black body is given by: P = σǫAT , (25)where σT represents the flux at which a black body emits, ǫ = 1 because a blackhole is a perfect black body, and σ stands for the Stefan-Boltzmann constant: σ = π k c ¯ h . (26)We combine equations (25) and (26) and equations (14) and (17) to obtainthe rate at which a black hole emits energy across a surface with an area of theblack hole horizon, from the point of view of an outside observer: P = − dEdt = ¯ hc πG M BH = 9 × − m ergs , (27)where m is given by equation (18). This is terribly low. Inserting: E = M BH c , (28)into equation (27), the left-hand side can be written as follows: − dEdT = − c dM BH dt . (29)Rearranging the terms in equation (27) yields: − dM BH dt = ¯ hc πG M BH , (30)from which we obtain the evaporation time t for the black hole, from the pointof view of an outside observer: 20 = 5120 πG M BH ¯ hc ≈ m × . × s, (31)where m is given by equation (18). This is a tremendous amount of time.When particles escape from the black hole, the black hole loses a smallamount of its energy and consequently, according to equation (28), its massdecreases. Its temperature T increases as a result of emitting Hawking radiation.It therefore radiates more Hawking radiation, loses mass M BH , becomes againhotter and radiates even faster until it finally evaporates. The smaller the blackhole gets, the higher its temperature T goes and in turn the loss of mass M BH accelerates. By the time the black hole reaches the size of a micro black holewith a Planck mass: m p = r ¯ hcG = 2 . × − kg, (32)it almost evaporates completely and its temperature T is so high that it willexplode. Which means that Hawking’s area theorem (the second law of blackhole mechanics) is violated because the area of the black hole gets smaller andsmaller as the black hole radiates Hawking radiation.
21 Saving the GSL from refutation
Hawking radiation unveiled an unexpected possibility. The black hole is emit-ting Hawking radiation, is losing mass and entropy in the course of time. Con-sequently, Hawking radiation (and evaporation) slowly drains the black holemass-energy and the black hole entropy decreases. That is because accordingto equation (19), S BH ∝ M BH . This posed a problem for the GSL.In 1975 Hawking set himself to solve this problem. First, given the entropyof a system as a function of the energy E of the system, the temperature canbe defined as: 1 T = ∂S∂E . (33)Thus, the temperature of a black hole can be defined as:1 T h = ∂S h ∂M B H , (34)where S h is the entropy of the black hole and T h is the temperature of the blackhole.Second, says Hawking, the GSL is then equivalent to the requirement thatheat should not run from a cooler system to a warmer one.So, Hawking returned to the thought experiment of a black hole which issurrounded by black body radiation at some temperature T M (see Section 12).For any nonzero T M there will be some rate of accretion of this radiation intothe black hole. 21ow suppose the black hole is in a hotter radiation bath, T M > T h , it followsfrom equation (34) that dS M < dS h . dS M is caused by the accreting radiation.In this case, the GSL holds. The black hole can absorb matter but does notemit anything. Hawking remarked that this case holds in purely classical theory[26], p. 193.On the other hand, if the black hole is in a colder radiation bath, T M < T h ,the GSL is violated. In classical theory, one sets T h = 0 in equation (34), andso S h is infinite. In this case, equation (4) is meaningless. One inevitably runsinto inconsistencies. Hawking concluded that the black hole has to emit thermalradiation with some finite nonzero temperature.So, if T M > T h , the accretion onto the black hole is greater than the Hawkingradiation and dS h > dS M . But if T M < T h , the emission of Hawking radiationis greater than the accretion onto the black hole and dS M > dS h . Hawkingthen explained that the fact that the black hole emits Hawking radiation witha temperature [equation (16)] enables one to establish the GSL and prove thatthe entropy S h [equation (19)] is finite. Hawking integrated equation (16). Thisallows one to integrate the first law of black hole mechanics, equation (12) anddeduce that: S h = 14 A + const. (35)If one makes the reasonable assumption that the entropy tends to zero as M BH →
0, the constant in equation (35) must be zero. Thus, the fact that theblack hole emits quantum radiation with a temperature given by equation (16),enables one to prove the GSL and establish that S h , equation (19), is finite [26],p. 193.In 1975 Bekenstein reanalyzed the same thought experiment of the black holeimmersed in a black body radiation bath within a cavity of temperature T M ,sufficiently low in relation to T BH . S M is the entropy of radiation and accretingmatter outside the black hole. Bekenstein reasoned that the flow of radiationfrom the cavity into the black hole (lower to higher temperature) will violatethe GSL unless some process generates entropy outside the black hole. In 1974he had suggested a cumbersome and illogical mechanism to accomplish this, seeSection 12. But in 1975 he recognized that Hawking radiation was a simplerresolution of the difficulty. So, Bekenstein showed that the GSL in averagedform is respected: after the radiation emitted by the black hole is assimilatedinto the ambient bath, S BH plus S M will be larger than before. AlthoughHawking radiation causes the surface area A to steadily decrease, eventuallythe mean dSg = dS BH + dS M > T and vanishes onlywhen T M = T BH [7], p. 3079.Bekenstein realized that the only way the GSL could hold was for the Hawk-ing radiation to carry enough entropy to overcompensate for the loss [13], p.518. During the radiation process, the area of the Kerr black hole can decreasebut the radiation entropy increases [see equations (1) and (2) for the area of aKerr black hole]. Bekenstein gives the analogy between a black hole and a hot22ody: just as for the hot body, one can expect that the black hole entropy plusthe radiation entropy would most likely increase [7], pp. 3077-3078.Bekenstein explained that the GSL was designed to replace the ordinarysecond law of thermodynamics. Accordingly, the GSL makes a stronger state-ment than Hawking’s area theorem. The area theorem requires only that A notdecrease (see Section 8). By contrast the GSL, ”being an intrinsically quantumlaw could be expected to fare better. And indeed in the astonishing quantumprocess of spontaneous radiation by a Kerr black hole discovered by Hawking,the area theorem is flagrantly violated, but the increase in exterior entropy dueto the radiation is expected to suffice to uphold the GSL”. The GSL is notexactly ”an intrinsically quantum law”. But Bekenstein considered Hawking’sproof as a ”confirmation of the validity of the GSL for a process not even dreamtof at the time of its inception” and stressed that this is a ”striking evidence ofthe versatility” of the GSL and of ”the physical meaningfulness of the conceptof black-hole entropy which underpins it” [7], p. 3077.In his 1975 paper, Hawking did not forget Bekenstein when he wrote thatthe latter ran into inconsistencies because he tried to combine the hypothesis offinite entropy with classical theory [26], p. 193. Bekenstein though understoodthings quite differently. He stressed that his argument (from 1975) is implicitin Hawking’s suggestion that the GSL is respected in the black hole radiationprocess. Of course, Bekenstein emphasized that it was Hawking who ”gave asimple proof” that dSg = dS BH + dS M ”is larger than before”, i.e. that equation(5) holds. In other words, Hawking gave a resolution to the difficulty posed bythe case T M < T BH and by that salvaged the GSL from refutation (see Section15). Bekenstein was pleased as ever. He concluded that Hawking has ”started asa vocal critic of” the GSL but ”became the person who made it fully consistentwith” the thought experiments [8], p. 29.
22 Mini black holes and experimental verifica-tion
Hawking radiation is widely accepted but we still lack observational evidenceconfirming Hawking’s predictions. According to equation (17), the tempera-ture of Hawking radiation for a stellar mass black hole is formidably law andabout six to eight orders of magnitude below that of the cosmic microwave back-ground radiation. This means that we cannot detect Hawking radiation fromstellar-mass black holes, let alone from supermassive black holes. Put differ-ently, as Hawking himself was well aware in 1974, according to equation (31),the evaporation time is proportional to M BH and a black hole of one solarmass M ⊙ ≈ × g will have evaporation time of more than t ≈ × y ,see Section 20. This is much longer than the age of the universe ≈ y . Theminimum mass of a stellar-mass black hole is 3 . M ⊙ and for the supermassiveblack hole at the center of our galaxy the evaporation time is approximately10 y . The evaporation time is in fact longer because black holes are absorbing23atter whilst gradually shrinking to awfully small sizes.So, maybe we can detect Hawking radiation from micro black holes? A blackhole mass cannot be below a Planck mass, see equation (32). That is becauseif it were, the black hole would be smaller than its own Compton length: λ c = ¯ hm p c , (36)which is equal to its Schwarzschild radius 2 l p [2 R g , see equation (7)] and is ofthe same order as l p : l p = r G ¯ hc ≈ . × − m. (37)Consequently, the black hole would not exhibit the black hole hallmark, theevent horizon [15], p. 32.The Planck time is: G m p ¯ hc = t p = r G ¯ hc ≈ . × − s, (38)and according to equation (31), the evaporation time t for a Planck-size blackhole is: t = 5120 πG m p ¯ hc = 5120 πt p ≈ × − s. (39)So, an extremely small black hole radiates all its mass-energy in a tinyamount of time.In 1974 Hawking wondered whether there might ”be much smaller blackholes which were formed by fluctuation in the early Universe. Any such blackhole of mass less than 10 g would have evaporated by now. Near the end ofits life the rate of emission would be very high” (see explanation in Section 20).Hawking conjectured that in the last 0 . s the black hole’s rate of emission wouldbe 10 erg . Although by astronomical standards, says Hawking, the explosion ofthis black hole would be fairly small, it is equivalent to about a million hydrogenbombs of one megaton each [25], p. 30.Hawking therefore suggests that we detect Hawking radiation from hypo-thetical primordial mini black holes (about the size of a mountain on Earth)with M BH ≤ g . A hypothetical 10 g primordial black hole, which mighthave been formed in the early universe, would radiate its mass over the age ofthe universe and would now be entering its final stages of evaporation. Theevaporation time for such a low-mass black hole is t ≈ . × y . If such tinyblack holes in fact exist, then we could see them evaporate in an explosion.In 1975 Hawking suggested that the mini black holes are equivalent to thehypothetical white holes. The time reverse of black holes, says Hawking, a whitehole, must also occur. While black holes emit Hawking radiation and their eventhorizons swallow matter and radiation, white holes emit matter and radiationand nothing can enter into their event horizons. Hawking explains that white24oles emit radiation at the same rate as black holes. That is, ”if one makes thereasonable assumption that the emission of a white hole is independent of itssurroundings, it follows that it is the same as the rate of emission of a blackhole of the same mass, angular momentum, and charge”. Hawking concludsthat black and white holes are identical to external observers. In 1979 Hawkingsaid in an interview: ”I think there’s only one entity. There are only holes.They appear black when they are big and white when they are small” [36], p.107; [26], p. 196.
23 Unruh’s acoustic black holes
The inability to detect Hawking radiation led Unruh in 1981 to suggest thatone could detect Hawking radiation in the laboratory. Unruh proposed thefollowing idea: let us create a small artificial black hole in the laboratory whosephysics and mathematics is analogous to that of the astrophysical black hole.The artificial black hole is based on the idea of sound waves propagating in amoving fluid. The waves are dragged by the fluid. Co-propagating waves movefaster and counter-propagating waves are slowed down. At some point the fluidexceeds the speed of the waves and beyond this point, no counter-propagatingwave would be able to propagate upstream. The point where the moving fluidmatches the wave velocity is an event horizon [46], p. 1351, p. 1353; [34], pp.2852-2853.But the acoustic black hole is hobbled by a serious problem. It cannotreproduce S BH , the Bekenstein-Hawking entropy, equation (19). The reason isthat Hawking radiation is a purely kinematic effect that depends only on theexistence of a certain metric and some sort of horizon. In contrast, S BH isassociated with dynamical effects, i.e. the area of the event horizon. Accordingto the first law of black hole mechanics, entropy equals one quarter the areaonly if the Einstein field equations are valid [50], p. 651.
24 Penrose’s Hawking points
Penrose has raised the following suggestion. We know that the Hawking temper-ature of black holes of several solar masses is extremely cold and the Hawkingradiation they emit is ridiculously tiny. We will therefore not see any signif-icant effect in these black holes. Penrose has asked: how can we measure ordetect Hawking radiation? He suggests that we have to wait until the universeexpands and expands and the ambient temperature goes down and down andultimately gets less than the temperature of the black hole. Then the black holestarts to lose its energy and, therefore, it loses its mass and ultimately it wouldevaporate. When will this happen? Penrose answers that this will probablyhappen in years. Of course, this seems like an awfully long time for stellar-massblack holes and Penrose gathers that with regards to supermassive black holesthat we now see, they have a much longer time than that must be expected.25ut he proposes a cyclic picture in which we actually do not need to wait thisimmensely long time because we can observe the Hawking radiation in the formof what he calls
Hawking points in our present universe. Let us Pause for amoment on Penrose’s hypothesis.Penrose claims that if we are waiting far way-way into the future, as theuniverse expands, clusters of galaxies tend to cling together, and every now andagain the supermassive black holes will run into each other and merge. Forinstance, our galaxy is on a collision course with the Andromeda galaxy. It willnot be for three thousand million years from now so do not worry, says Penrose.But when the galaxies will eventually collide, the black holes will spiral aroundinto each other and there will be a great explosion ending up with a huge blackhole, much bigger than the one at the center of the Andromeda galaxy.The majority of material in the galaxies will ultimately be swallowed by thehuge black holes sitting around the clusters of galaxies and gravitational waveswill carry the energy. As the universe expands and expands, in the very-very farfuture, in googolplex years or so, says Penrose, what will eventually remain arebig black holes that will ultimately disappear. They will disappear with whatPenrose calls a “pop” (an explosion) and almost the entire mass-energy of theblack holes goes in the form of Hawking radiation of photons.Now, this is a thing, says Penrose, that used to worry him because whatwe have got is a universe full of photons. On the whole as far as particlesnumbers are concerned, in the remote future, the universe would completely bedominated by cold photons, which are conformally invariant. Ultimately themass asymptotically goes to zero.In the big bang, on the other hand, you have the opposite situation becausewe have very high temperatures. When those temperatures get higher andhigher, the energy of individual particles is completely dominated by the motionand the mass becomes utterly unimportant. Hence, in this limit again, youwould have massless things.Penrose put forth the following hypothesis called the conformal picture: asyou approach the boundary at both ends – the big bang and eternity, the ar-gument is that the cosmic conformal cyclic cosmology allows a crossover fromone “aeon” (universe) to the next, to be described in terms of the conformalphysics. Penrose argues that we in our universe are looking here through thebig bang of the previous universe, aeon, which is due to Einstein’s cosmologicalconstant.How can we confirm that very hypothesis of Penrose? Penrose suggests thathis hypothesis may be confirmed by the Hawking points. Before explaining whatare these points, Penrose says there are lots of mathematical details which needto be worked out and he has not yet done so. Hence, his explanation is verygeneral and his theory is not fully baked.Penrose explains that rather than thinking about going into the future, wethink about going into the past, the big bang. We know that according tothe second law of thermodynamics, the entropy increases in time. The samestatement of the second law says that as we go back and back in time into thepast, the entropy is supposed to go down and down. The best evidence for26he big bang that we have is the microwave cosmic background radiation whichcomes from all directions. What we see at this early stage, says Penrose, ismaximum entropy and this is quite evident as the main feature of this radiationand its spectrum of black body radiation. So, Penrose says he thought therewas something fishy about that because at the early universe, as far as gravity isconcerned, the universe expanded and expanded, matter clumped into galaxiesand black holes were created and black holes represent maximum entropy. Onthe other hand, the universe started with very low entropy at the beginning.Penrose’s conjecture consists of the following main idea: we have a hugeexplosion of the supermassive black hole, one burst which causes the evapo-ration of the supermassive black hole from the previous universe, aeon. Theevaporation basically leads to photons that spread out. But where can we findthese photons? When you think of how far does this radiation spread out, saysPenrose, the answer is, almost nothing. So, all the Hawking radiation from theevaporation is eventually concentrated into one point. This entire Hawking ra-diation over years is simply squashed down into one tiny little point, a Hawkingpoint, probably smaller than the Planck scale. The Hawking points are createdjust when inflation is turned off, as we go back again to the big bang. Accord-ingly, we have a burst of radiation coming through at that little point. So, it isa matter of a cyclic model of stretching and squashing, stretching and squashingof aeons.Hawking once argued that the black holes eventually evaporate away andinformation must be lost in the process, see Sections 26 and 27. Though overthe years he backed off this claim, Penrose has adopted it. According to theprinciple of unitarity of quantum mechanics, you cannot lose information. But,says Penrose, unitarity is violated by the way we use quantum mechanics whenwe make measurements, see Section 27. Penrose says he can make very cogentreasons why the information is lost and this is related to his hypothesis becausewhen black holes disappear the effective entropy comes down and it means thatwe arrive at the beginning of the universe, the big bang.Penrose admitted that he would often “talk about these things throughlectures and I thought well, it’s perfectly safe, I could talk about conformalcyclic cosmology until the end of time because nobody will ever be able todisprove me. Then I began to think, is that true?” Maybe we can have a testof conformal cyclic cosmology? Maybe one can detect Hawking radiation in theform of spots that came about and spread from the big bang and three hundredand eighty thousand years afterwards? It means that what Penrose calls theHawking points, the Hawking radiation concentrated in those spots at 4 degreesKelvin, spread out from the big bang to a much more uniform distribution towhat we actually see in the microwave cosmic background radiation. If youhappen to ask Penrose: “Why haven’t people seen these before?” His answer isplain and simple: “Because they haven’t been looking!” [37], [39], [1] and [40].Suppose we detected hot (4K) spots in the cosmic microwave backgroundradiation. We cannot say that those spots are a conclusive evidence for Penrose’scyclic universe, namely, clear-cut evidence for Hawking radiation from a universebefore the big bang. Those spots could be explained by as many as possible27odels. This means that one major evidence cannot support an entire theory.
25 Evaporation and the censorship hypothesis
In 1969 Penrose pondered whether there exists a “cosmic censor” who forbidsthe appearance of naked singularities, clothing each one in an event horizon.Penrose answered that it is not known whether naked singularities would everarise in a collapse which starts off from a non-singular initial state [38], p. 1160.In 1975 Hawking formulated “The ‘cosmic censorship’ hypothesis: Natureabhors a naked singularity” , namely, any singularities which are developed fromgravitational collapse will be hidden from the view of observers at infinity byan event horizon. Hawking pointed out that evaporation violates the classicalcensorship hypothesis. He explained that if one tries to describe the processof a black hole losing mass and eventually disappearing and evaporating by aclassical space-time metric, then “there is inevitably a naked singularity whenthe black hole disappears. Even if the black hole does not evaporate completelyone can regard the emitted particles as having come from the singularity insidethe black hole and having tunneled out through the event horizon on spaceliketrajectories” [27], p. 2461.So, Penrose said that if “the singularity is visible, in all its nakedness, to theoutside world!” [38], p. 1160, God forbid, Hawking answered that “an observerat infinity cannot avoid seeing what happens at the singularity” [27], p. 2461.
26 Bekenstein summons Maxwell’s demon
According to the classical no-hair theorem and general relativity, informationthat enters the black hole is lost forever. Since the black hole evaporates, ¯ h → T BH → S BH →
0. At the end thereis only Hawking radiation with no black hole. The question we need to be askingis: what happens to the information about the actual internal configuration ofthe back hole?Hawking thus had been wracking his brain about a new problem that poppedout: a black hole of M BH , J , and Q can have a large number of differentunobservable internal configurations which reflect the possible different initialconfigurations of the matter which collapsed to produce the black hole. Thelogarithm of this number can be regarded as S BH . S BH is a measure of theamount of information about the initial state which was lost in the formationof the black hole [26], pp. 191.In fact Bekenstein was the first to recognize the relationship between infor-mation and black hole entropy. Unlike Hawking, Bekenstein defined entropyas follows (see Section 7): S BH can be regarded as our ignorance or lack ofinformation about the actual internal configuration of the back hole.Bekenstein was troubled by Hawking radiation, which seemed to him to haveled to a deep problem. A black hole is prepared from matter in pure state but28hen it evaporates and radiates away its mass in the form of thermal Hawkingradiation. Bekenstein found two contradictions: The first contradiction : One is left with a high-entropy mixed state of Hawk-ing radiation (see Section 21). Bekenstein said that he found a contradiction inHawking’s attempt to provide an interpretation of the black hole entropy as themeasure of the information hidden in the black hole, i.e. the information aboutthe ways the black hole might have been formed. But this was Bekenstein’sinterpretation, not Hawking’s! Bekenstein went on to say that the final state ofthe black hole has a lot of entropy and therefore we are faced with a large intrin-sic loss of information. The way Bekenstein saw it was that a thermal radiationis incapable of conveying detailed information about its source. Consequently,hidden information remains sequestered as the black hole radiates, and whenthe black hole finally evaporates away, the information must be lost forever [11],p. 3680.In order to resolve this paradox Bekenstein showed that it would be necessaryto summon Maxwell’s demon. Bekenstein first exorcised Wheeler’s demon (seeSection 3) and then he summoned Maxwell’s demon. This was the second timethat Bekenstein raised Maxwell’s demon (see Section 7).Leo Szilard wrote in 1929: ”A measurement procedure underlies the entropydecrease effected by the intervention of intelligent beings”, [44], p. 304. In 1993Bekenstein explained that ”just as in Szilard’s famous discussion of Maxwell’sdemon where acquisition of information about the location of the molecule in thebox was tantamount to the ’gas’ having less entropy than expected, so here, thegradual information outflux is tantamount to the black hole entropy becominggradually less and less than originally expected”. Bekenstein concluded thatindeed, the black hole entropy [equation (19)] decreases. But Hawking provedin 1975 that this was not the case (see Section 21).In 1976, Hawking suggested “a quantum version of the ‘no hair’ theorems”which “implies that an observer at infinity cannot predict the internal stateof the black hole apart from its mass, angular momentum, and charge: If theblack hole emitted some configuration of particles with greater probability thanothers, the observer would have some a priori information about the internalstate” [27], p. 2462. In 1978 Hawking argued that a large amount of informationis irretrievably lost in the formation of a black hole. The number of bits ofinformation lost can be identified with the entropy of the black hole. Becausepart of the information about the quantum state of the system is lost down theblack hole, the final situation after the black hole has evaporated is describedby a mixed state rather than a pure quantum state [29], pp. 24.
27 Black hole information loss and noise
Szilard wrote: ”A measurement procedure”. Does this hold for black holes?Bekenstein was led to:
The second contradiction : It seemed to Bekenstein that”a pure state could be converted into a mixed one through the catalysing in-fluence of a black hole!” [15], p. 35. This contradicts the quantum principle29f unitarity that says that a pure state will always remain pure under unitaryevolution. ”Hawking was led by this conclusion to assert that gravity violatesthe unitarity principle of quantum theory”. ”To be sure”, said Bekenstein,”Hawking’s inference has remained controversial: whereas general relativity in-vestigators have tended to accept this conclusion, particle physicists have stoodby the unitarity principle and orthodox quantum theory” [15], p. 35.Let us leave aside the mathematics for the moment. According to quan-tum mechanics, a pure quantum state is a solution of the deterministic time-dependent Schr¨odinger equation. The complete information of a state at onepoint of time allows us to determine the state at all other times. And so, theunitary evolution is reversible. Hence, no information is lost and we must beable to invert things and retrieve all information that went into the black hole,so long as no measurement is made on the system. But that is exactly the point;or maybe not. The thing is that the initial pure state that formed the blackhole cannot be reconstructed. That is because the pure state has evolved intoa mixed state. And we measure a mixed state. But what if we did not exist?Carlo Rovelli pointed out that according to Shannon’s definition of information(the one adopted by Bekenstein, see Section 7), ”We do not need a human being,a cat, or a computer to make use of this notion of information” [42], p. 1641.Dozens of papers have been published in scientific journals on the informationloss paradox and possible solutions to this paradox.Back to Bekenstein. ”The two contradictions”, concluded Bekenstein, ”arefacets of the black hole information loss paradox” [11], p. 3680.From Hawking’s perspective, the information paradox concerned the follow-ing contradiction: “This is the information paradox: How does the informationof the quantum state of the infalling particles re-emerge in the outgoing radia-tion? This has been an outstanding problem in theoretical physics for the lastforty years” [32] p. 2.In 2015 Hawking tried to propose the following solution: “The informationabout the ingoing particles is returned, but in a highly scrambled, chaotic anduseless form. This resolves the information paradox. For all practical purposes,however, the information is lost” [32], p. 3. Hawking had already suggested thissolution to the information loss paradox in 1978: ”The loss of information isequivalent to the acquisition of new random information or ’noise’”[29]. p. 24.
28 Discussion: Bekenstein and Hawking
In response to Wheeler’s tea cups thought experiment, Bekenstein suggested onthermodynamic grounds that a black hole should have a finite entropy that isproportional to the surface area of the event horizon. Bekenstein formulatedthe GSL and inaugurated the field of black hole thermodynamics.However, Bekenstein then discovered a problem. Geroch invented a thoughtexperiment which could violate Bekenstein’s GSL. Suppose a massless box isfilled with radiation and then slowly lowered by a massless rope toward thehorizon of a Schwarzschild black hole in its gravitational field. It is opened at30 → R g and the radiation is allowed to escape into the black hole. The amountof work that can be extracted from the Geroch heat engine is M c and the blackhole would end up in the same state in which it began and all of the energyof the radiation could be converted to work in the laboratory from which onewas doing the lowering. If all the mass-energy is converted to work, then blackholes are systems at absolute zero temperature. In an attempt to salvage hisGSL, Bekenstein assumed that the black hole entropy would be enormous andhe gave a contrived quantum explanation for why the box would never reach2 R g .This argument seemed implausible to Hawking because if the effective tem-perature of a black hole is zero, then this means that one can in principle addentropy to a black hole without changing it in any way. Bekenstein respondedto this claim by assigning to the black hole a finite non-zero temperature thatcorresponds to the finite black hole entropy. Bekenstein’s temperature of theblack hole is proportional to the surface gravity.Hawking on his part maintained that this in turn indicated that a black holecould be in equilibrium with thermal radiation at some temperature other thanzero. But the problem is that black holes which can devour matter do not emitanything. So Bekenstein tried to solve this paradox in terms of his GSL and thesecond thought experiment was invented.Imagine a black hole which is surrounded by thermal radiation at sometemperature. If the black hole is in a hotter radiation bath, the GSL holds. Theblack hole can absorb matter but does not emit anything. On the other hand,if the black hole is in a colder radiation bath, the GSL is violated. Bekensteingave a cumbersome explanation for why his GSL was not violated in this case.Hawking thought that this resolution was far-fetched because no equilibrium ofa black hole with thermal radiation at some temperature other than zero waspossible. The black hole would absorb any thermal radiation that fell onto itbut would be unable to emit anything in return.Hawking was now faced with the following problem: either the temperatureof the black hole is identically zero, in which case its entropy is infinite and theconcept of black hole entropy is meaningless, or black holes have to emit thermalradiation with some finite nonzero temperature. Hawking realized that the firstcase is what holds in purely classical theory, in which black holes can absorb butdo not emit anything. Hawking argued that Bekenstein ran into inconsistenciesbecause he tried to combine the hypothesis of finite entropy with classical theory.Accordingly, in 1974 Hawking offered the following resolution: the paradoxesare solved only if one accepts the quantum mechanical result that black holesemit thermal radiation. Hawking gave an explanation for ”how the thermalradiation arises”. He used the semi-classical approximation to arrive at equation(16). That is, he combined quantum field theory and classical general relativityand obtained equation (16) [25], p. 30.In June 1975, Hawking showed that the steady-rate emission of Hawking ra-diation turns out to have an exactly thermal spectrum and the following quan-tities arise from his 1974 semi-classical approximation: the area of the eventhorizon, the surface gravity of the black hole, the angular frequency of rota-31ion of the black hole and the potential of the event horizon (both are inverselyproportional to the the area, as expressed by the first law of black hole me-chanics). And the most important quantity that arises from the semi-classicalapproximation is the entropy of the black hole [26], pp. 191-192. Hawking there-fore used the semi-classical approximation to obtain black hole thermodynamicquantities.In August 1975, Hawking submitted a paper in which he gave a thoroughtheoretical explanation for Hawking radiation based on the preliminary expla-nation he had given in 1974 [27].Underlying this story is a sense of dispute between two opposing worldviews:that of Bekenstein guided by black hole thermodynamics, and that of Hawking,led by quantum field theory.In 1974 Hawking wrote: “Bekenstein suggested on thermodynamic groundthat some multiple of κ should be regarded as the temperature of the blackhole. He did not, however, suggest that a black hole could emit particles as wellas absorb them” [25], p. 31. When Hawking did the calculation, as he wrotein his well-known book, A Brief History of Time , “I found, to my surpriseand annoyance, that even nonrotating black holes should apparently create andemit particles at a steady state. At first I thought that this emission indicatedthat one of the [semi-classical] approximations I had used was not valid. Iwas afraid that if Bekenstein found out about it, he would use it as a furtherargument to support his ideas about the entropy of black holes, which I still didnot like. However, the more I thought about it, the more it seemed that theapproximations really ought to hold” [30] p. 105.Bekenstein, on his part, maintained that ”Hawking had been a leader of thevociferous opposition to black hole thermodynamics”. He believed that Hawkingwas trying to discredit his black hole thermodynamics. Bekenstein said that inhis paper with Bardeen and Carter, Hawking ”argued against ” Bekenstein’s en-tropy and ”Ironically, many uninformed authors still cite” the Hawking, Bardeenand Carter paper ”as one of the sources of black hole thermodynamics!” [13],p. 518.Bekenstein praised Hawking’s achievement but said that Hawking “providedthe missing pieces of black hole thermodynamics” [8], p. 29. Which is notexactly what Hawking intended to do. But in 1980 Hawking’s fears came truewhen Bekenstein stated: “Having started as a vocal critic of the generalizedsecond law, Hawking became the person who made it fully consistent with thegedanken experiments” [8], p. 29.A suggestion has been made that Hawking radiation should more appropri-ately be called Bekenstein-Hawking radiation, but Bekenstein himself rejectedthis. He said: “The entropy of a black hole is called Bekenstein-Hawking en-tropy, which I think is fine. I wrote it down first, Hawking found the numericalvalue of the constant, so together we found the formula as it is today. Theradiation was really Hawking’s work. I had no idea how a black hole could radi-ate. Hawking brought that out very clearly. So that should be called Hawkingradiation” [20]. 32 cknowledgement
I still remember the inspiring conversations I had in 2005 with Sir Roger Penrosein Israel on Einstein, general relativity and black holes. I would like to thankJuan Maldacena for the historical comments he gave me during our exchangeof emails on black holes. I am grateful to Gil Kalai for discussions and helpfulcomments. Finally, I would like to thank Yaron Sheffer for pointing out an errorin an equation in the first version of this paper.
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