## Black Hole Explosions? Offprint from: Nature, Vol. 248, No. 5443, 1 March 1974.

[London: Macmillan, 1974].

First edition, extremely rare offprint issue, and Hawking’s own file copy, of his most important article, predicting that black holes emit radiation, now known as ‘Hawking radiation’. This would cause relatively low mass black holes (less than 10^{12} kg) to ‘evaporate’ completely in a time less than the age of the universe. The discovery that black holes radiate was a complete shock to the scientific community since it had been assumed that nothing, not even light, could escape from a black hole (that was, of course, why they were called black holes!). Hawking’s Royal Society obituary stated: “It is fair to say that Stephen’s discovery [of Hawking radiation] ranks as one of the most important results ever in fundamental physics” (Carr, et al.). John Archibald Wheeler once said that just talking about Hawking radiation was like “rolling candy on the tongue” (*ibid*.). Hawking’s doctoral advisor, Dennis Sciama, said of Hawking radiation, “That’s a work of genius, in my opinion, that discovery” (Sciama, Oral History). At the heart of Hawking’s theory is an equation (see below) relating the entropy of a black hole, or equivalently the amount of information it contains, to its surface area (the two are proportional). The Harvard physicist Andrew Strominger, who collaborated with Hawking on one of his last works, wrote: “The [area/entropy] equation stands alongside the Einstein equation and the Schrödinger equation among the most important equations of 20^{th}-century physics. However, unlike the other equations, which describe distinct areas of physics, the area/information law unites disparate areas, as indicated by the striking appearance [in the equation] of nearly all the most basic constants of nature” (Page et al.). This formula, which Hawking wanted to have engraved on his tombstone, suggests a profound and hitherto unsuspected relation between general relativity, quantum theory, and thermodynamics. The formula gives a glimpse of a future theory of quantum gravity, the holy grail of theoretical physics. “The challenge of uniting quantum theory and general relativity in a successful theory of quantum gravity has arguably been the greatest challenge facing theoretical physics for the past eighty years. One avenue that has seemed particularly promising is the attempt to apply quantum theory to black holes. This is in part because, as purely gravitational entities, black holes present an apparently simple but physically important case for the study of the quantization of gravity. Further, because the gravitational force grows without bound as one nears a standard black-hole singularity, one would expect quantum gravitational effects (which should come into play at extremely high energies) to manifest themselves in the interior of black holes. In the event, studies of quantum mechanical systems in black hole spacetimes have revealed several surprises that threaten to overturn the views of space, time, and matter that general relativity and quantum field theory each on their own suggests or relies on … it has become increasingly clear that there are profound connections among general relativity, quantum field theory, and thermodynamics … In particular, a remarkable parallel between the laws of black holes and the laws of thermodynamics indicates that gravity and thermodynamics may be linked in a fundamental (and previously unimagined) way. This linkage strongly suggests, among many things, that our fundamental ideas of entropy and the nature of the Second Law of thermodynamics must be reconsidered, and that the standard form of quantum evolution itself may need to be modified” (*Stanford Encyclopedia of Philosophy*). Nobel Laureate Kip Thorne wrote: “Few, if any, of Einstein’s successors have done more [than Hawking] to deepen our insights into gravity, space and time’, and Roger Penrose stated ‘We remember Newton for answers. We remember Hawking for questions. And Hawking’s questions themselves keep on giving, generating breakthroughs decades later. When ultimately we master the quantum gravity laws, and fully comprehend the birth of our universe, it will be by standing on the shoulders of Hawking’ (Carr et al.). We are not aware of any other copy of this offprint having appeared on the market.

*Provenance*: the estate of Professor Stephen Hawking (1942-2018).

“The simplest picture of a black hole (the term was coined by Wheeler in 1967) is that of a system whose gravity is so strong that nothing, not even light, can escape from it. Systems of this type are already possible in the familiar Newtonian theory of gravity. The escape velocity of a body is the velocity at which an object would have to begin to travel to escape the gravitational pull of the body and continue flying out to infinity, without further acceleration. Because the escape velocity is measured from the surface of an object, it becomes higher if a body contracts and becomes more dense. If the object were to become sufficiently dense, the escape velocity could therefore exceed the speed of light, and light itself would be unable to escape … Taking relativistic considerations into account, however, we find that black holes are far more exotic entities” (*SEP*).

The existence of black holes in the context of general relativity was predicted by the German astronomer Karl Schwarzschild who, in 1915, found the first exact solution to Einstein’s equations, which Einstein had perfected just one month earlier. This solution described the gravitational field, or equivalently the nature of the spacetime, outside a spherically symmetric mass. In Schwarzschild’s solution there was a sphere concentric with the mass, later called the ‘event horizon’, such that nothing, not even light, could escape from inside the event horizon to the outside. There was a singularity at the centre of the black hole at which the notions of space and time broke down. Later, solutions of Einstein’s equations were found which described rotating black holes, and black holes with an electric charge. It is important to emphasize that, in general relativity, a black hole is a purely gravitational object: a black hole is not a thing *in* spacetime; it is instead a feature of spacetime itself.

Physicists, notably including Einstein himself, were initially sceptical that Schwarzschild’s solution described anything that could exist in the real universe. Indeed, in 1939 Einstein wrote a paper purporting to show that black holes could not exist. But at almost the same time, J. Robert Oppenheimer and his student Hartland Snyder published a paper which gave a precise mechanism whereby a collapsing star could become a black hole.

It was only in the 1960s that important results on the nature of black holes began to appear. One of the first was the ‘no-hair’ theorem (so called by Wheeler), which states that black holes are completely characterised by just three externally observable quantities: their mass, electric charge and angular momentum. All other information (for which ‘hair’ is a metaphor) about the matter that formed a black hole or is falling into it ‘disappears’ behind the event horizon and is therefore permanently inaccessible to external observers. The no-hair theorem was followed by the discovery of the four ‘laws’ of black holes:

- The surface gravity of a stationary black hole is constant over its entire surface.
- A change in the total mass of the black hole is determined in a fixed way by changes in its area, angular momentum, and electric charge, so that the total quantity is conserved.
- (The Area Theorem) The total surface area of a black hole never decreases.
- No physical process can reduce the surface gravity of a black hole to zero.

There is an obvious formal analogy between these laws and the laws of classical thermodynamics: the 0^{th} law of thermodynamics states that a body in thermal equilibrium has constant temperature throughout; the 1^{st} law is a statement of the conservation of energy; the 2^{nd} law states that entropy never decreases; and the 3^{rd} law states that it is impossible to reduce the temperature of a system to zero by any physical process. Accordingly, if in the laws for black holes one takes ‘stationary’ to stand for ‘thermal equilibrium’, ‘surface gravity’ to stand for ‘temperature’, ‘mass’ to stand for ‘energy’, and ‘area’ to stand for ‘entropy’, then the formal analogy is perfect. In the early 1970s, Jacob Bekenstein suggested that this was more than an analogy: he argued that a black hole should be assigned an entropy proportional to its area.

“Stephen was highly sceptical [of Beckenstein’s suggestion] because black holes would then have a non-zero temperature. Any object with a temperature emits black body radiation, but black holes were thought to be unable to emit anything. Stephen began to study how particles and fields governed by quantum theory would [interact with] a black hole. Earlier studies [by the Soviet astrophysicists Yakov Zeldovich and Alexei Starobinsky] had shown that an incoming wave could be amplified by scattering off a spinning black hole and had even indicated, as an extension, that a spinning black hole could … caus[e] spontaneous particle emission that spins the black hole down.

“At the Krakow meeting on Cosmology in September 1973, Stephen met Zeldovich and Starobinsky, who pointed out to him this spontaneous emission. Stimulated by these discussions, Stephen discovered a few months later, much to everyone’s surprise, that all black holes, even those that do not spin and thus have no spin energy to extract, emit particles at a steady rate. Initially, he thought there must be a mistake in his calculation. That black holes could emit particles went against the long-held belief that black holes could only be absorbers and never emitters of anything. The result refused to go away. What finally convinced Stephen was that the outgoing particles had precisely a thermal spectrum—the black hole created and emitted particles and radiation just like any hot body … This phenomenon came to be known as Hawking radiation.

“For a non-rotating black hole [of mass *M*], the temperature is given by Stephen’s most famous equation,

*T = hc*^{3}/16π^{2}*GkM*,

where *h* is Planck’s constant, *c* is the speed of light, *G* is Newton’s gravitational constant, and *k *is Boltzmann’s constant. [This implies that] the entropy *S* of the black hole can be identified as

*S* = *πAkc*^{3}/2*Gh*

[where *A* is its surface area]” …

“Consequently, an isolated black hole emits thermal radiation and so loses mass. Since the temperature is inversely proportional to the mass, the temperature increases, leading to a runaway process and eventually, it is presumed, to the disappearance of the black hole. The process takes an inordinately long time for a solar-mass black hole, about 10^{67} years. But for small black holes, which could have been formed in the early universe, their lifetimes could be much shorter. For example, a black hole of mass 10^{12} kg will last about 10^{10} years, approximately the current age of the universe” (Carr et al.).

“Although careful to be discreet until he was ready to make an official announcement, [Hawking] could not resist telling Dennis Sciama, the man who had helped shape his early career. Afterward, the shocking news spread through Hawking’s inner circle. Martin Rees, a colleague at Cambridge [and future President of the Royal Society and Astronomer Royal] welcomed a visit from Sciama with ‘Have you heard? Have you heard what Stephen had discovered? Everything is different, everything has changed!’” (Larsen, p. 42).

“Stephen first announced this result at a meeting on quantum gravity at the Rutherford Laboratory on 15-16 February 1974, and it was published in *Nature* shortly afterwards [the offered paper]. The prediction has now been derived in several different ways and it is such a beautiful result—unifying quantum theory, general relativity and thermodynamics—that most experts accept that it is correct” (Carr et al.). “Hawking wrote a longer, more complete description of his discovery and submitted it to *Communications in Mathematical Physics* in March, 1974. He did not receive any acknowledgement, and a year later inquired as to the status of his paper. He was told that the journal had lost his manuscript and was asked to resubmit it. When it was finally published, the submission date was erroneously listed as April 1975” (Larsen, p. 43).

Hawking’s first picture to understand his radiation was in terms of ‘virtual particles.’ “According to Einstein’s special theory of relativity, mass can be converted into energy and energy can be converted into mass [namely, a particle-antiparticle pair]. According to … the Heisenberg Uncertainty Principle, it is possible to create energy out of nothing so long as the energy debt is paid back in a very short period of time … The energy is, in a sense, borrowed from the very fabric of the universe – from the vacuum, as it is called. This happens randomly every second of every day, with the borrowed energy creating a particle-antiparticle pair (called a virtual pair, since they only exist temporarily). After the passage of a minute amount of time, the pair rejoins and annihilates, repaying the energy debt to the universe. As strange as this may sound, it has been verified experimentally. Parallel metal plates in a vacuum set very close together will feel a tiny attractive force due to the presence of the virtual pairs – this is called the ‘Casimir effect.’

“Suppose the creation of the particle-antiparticle pair happened just outside the event horizon of a black hole. If one of the members of the pair fell into the black hole, the remaining particle could not annihilate with its partner when the time was up, and the energy debt appears to be unpaid. A real particle or antiparticle has been created, seemingly from nothing, or, rather, it looks to an outside observer as if it leaked from the black hole! But the energy needed to create the particle had to come from somewhere – namely from the black hole itself. In this way, the energy of the black hole, or, better yet, its mass (since mass and energy are convertible, according to Einstein) decreases, and the black hole radiates away” (Larsen, p. 41).

The existence of Hawking radiation creates two deep problems for physics: ‘naked’ singularities and the ‘information loss paradox’. Work by Hawking, the Oxford physicist Roger Penrose and others in the late 1960s established several singularity theorems which showed that, if certain physically reasonable premises were satisfied, then in certain circumstances singularities could not be avoided. These theorems indicate that our universe began with an initial singularity (the Big Bang), and that in certain circumstances collapsing matter must form a black hole with a central singularity. While spacetime singularities are frequently viewed with fear and suspicion, physicists offer the reassurance that, even if they are real, we expect them to be hidden away behind the event horizons of black holes. They could not therefore affect the universe outside the event horizon. A naked singularity, on the other hand – one that is not hidden behind an event horizon – seems to imply a profound breakdown in the fundamental structure of spacetime: even determinism itself would collapse entirely near a naked singularity. The existence of Hawking radiation raises the possibility that naked singularities could actually be formed in our universe: as a black hole emits Hawking radiation, its event horizon shrinks, and eventually (in a finite time) disappears altogether, apparently leaving the singularity at the centre of the black hole exposed.

We noted above that when a black hole is formed, by the collapse of a star, for example, all information about the star (e.g. the elements of which is composed) is lost, except for the three quantities mass, electric charge, and angular momentum. This was not regarded as a problem, as it could be assumed that this information was somehow stored inside the event horizon of the black hole, even though it was inaccessible. But if the black hole eventually disappears as a result of emitting Hawking radiation, it seems that the information about everything that fell into the black hole will be irretrievably lost – the Hawking radiation is random (it has no relation to the original information). This loss of information is a serious problem because it seems to imply that black holes cannot be described by quantum mechanics. In standard quantum mechanics, a knowledge of the quantum state of a physical system at one time is sufficient to determine it, via Schrödinger’s equation, at any later time: this implies that quantum evolution “preserves information”. Although Hawking agonized about the paradox on and off over the next 40 years, it was only in one of his very last works that a solution began to appear. In a 2016 joint paper with Strominger and the Cambridge physicist Malcolm Perry, Hawking suggested that the vacuum of general relativity might provide a ‘memory matrix’ that preserves the information in the universe, even after a black hole has evaporated.

Carr, Ellis, Gibbons, Hartle, Hertog, Penrose, Perry, & Thorne, ‘Stephen William Hawking CH CBE. 8 January 1942 – 14 March 2018,’ *Bibliographical Memoirs of Fellows of the Royal Society*, Vol. 66, June 2019 (royalsocietypublishing.org/doi/10.1098/rsbm.2019.0001). Page, Taylor, Preskill, Hertog, Ellis, Guth, Unruh, Strominger, Rocek, Galfard, Berry, Carr & Hartle, ‘Stephen Hawking (1942-2018),’ *Physics Today*, 14 March 2018 (physicstoday.scitation.org/do/10.1063/PT.6.4.20180314a/full/). Hawking, Perry & Strominger, ‘Soft hair on black holes,’ *Physical Review Letters*, vol. 116, 231301 (2016). Larsen, *Stephen Hawking: A Biography*, 2005. Sciama, Oral History Interview, 14 April 1978, American Institute of Physics (aip.org/history-programs/niels-bohr-library/oral-histories/4871).

Large 8vo (280 x 212mm), pp. 30-31. Single leaf (punch holes in inner margin, not affecting text).

Item #5076

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Price:
$35,000.00
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