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The thermodynamics of black holes: from Penrose process to Hawking radiation

Carla rodrigues almeida.

1 Institute for Advanced Studies in Humanities, Essen, Germany

2 Max Planck Research Group Historical Epistemology of the Final Theory Program, Berlin, Germany

Associated Data

This manuscript has associated data in a data repository. [Author’s comment: The manuscripts utilized in this research are available at the repository of the Max Planck Institute for the History of Science or Princeton University, except for Wheeler’s notebooks. These are publicly available, and there is a link to them at the Reference.]

In 1969, Roger Penrose proposed a mechanism to extract rotational energy from a Kerr black hole. With this, he inspired two lines of investigation in the years after. On the one side, the Penrose process, as it became known, allowed a comparison between black-hole mechanics and thermodynamics. On the other, it opened a path to a quantum description of those objects. This paper provides a novel take on the events that led to the rise of the thermodynamic theory of black holes, taking as a starting point the Penrose process. It studies the evolution of the research conducted independently by Western and Soviet physicists on the topic, culminating in Stephen Hawking’s groundbreaking discovery that black holes should radiate.

Introduction

The formulation of the thermodynamics of black holes is an important development that shaped our current understanding of those objects. Black holes are now understood to have the thermodynamic properties of entropy and temperature and to interact with their environments as a perfect black body, as defined in [ 40 , pp. 12–13]. The identification of black holes as perfect black bodies is, however, a theoretical one and, as such, should be considered critically [ 18 ]. A step toward this goal is to analyze the inception of this idea, retracing its origins and evolution to grasp how it was first conceived and the scientific methods employed in this hypothesis.

A black hole was hidden in the first solution of Einstein’s equation of general relativity, obtained in 1916. Schwarzschild’s Massenpunktes solution [ 41 ] described a black hole, but this conclusion was initially rejected. General relativity was in its infancy, and it was broadly misunderstood and underdeveloped [ 20 , p. 255]. The hypothesis that objects with the characteristics of a black hole could exist lacked observational evidence and was too counterintuitive [ 1 ]. Theoretical evidence of their existence became strong in the late thirties, indicating that it was a probable consequence of general relativity, but the astrophysics community was still reluctant to accept this result. After the renaissance of Einstein’s theory of gravitation in the mid-fifties [ 9 ] and the observation of super-dense bodies in the skies at the beginning of the following decade, the situation changed [ 11 ]. The rise of relativistic astrophysics and the possibility of identifying quasars as collapsed objects [ 22 , p. 27] renewed interest in the latter, and the sixties became a golden decade for black-hole physics. The concept was improved, new features were unveiled, and black holes were baptized with the name we know them by today.

By the end of the sixties, the main understanding was that black holes were dense astronomical objects with a strong gravitational field wrapped by a fictitious unidirectional membrane, the event horizon , which allowed the entrance but not the escape of matter and radiation from its interior and hid a singularity inside it. A black hole could be described entirely by three classical parameters: mass, charge, and angular momentum. A Schwarzschild black hole has no angular momentum and no charge. Rotating black holes are known as Kerr black holes or, if charged, Kerr–Newman. A rotating black hole would have an exterior region in which the reference frame rotates alongside the black hole, causing an effect called frame-dragging . This region was named ergosphere in 1971. 1

One influential person who contributed to the development of the concept of black holes during the sixties was Roger Penrose. Among many achievements, he showed that black holes could and would be formed as a consequence of the theory of general relativity [ 36 ], and it earned him a Nobel Prize in Physics in 2020 [ 26 ]. In 1969, Penrose added another layer to the theory of black holes when he proposed a mechanism to extract energy from a rotating black hole. The Penrose process would happen in the ergosphere, where a particle dropping from infinity would split into two components. One would fall into the black hole, while the other would escape with more mass energy than the original particle had, and thus, the rotational energy of the black hole would be transferred to the motion of this particle outside the ergosphere [ 38 , p. 45]. The energy extraction of a rotating black hole would slow it and decrease its mass, a process that would not be indefinite.

Penrose’s mechanism allowed a comparison between black-hole mechanics and thermodynamics—a correlation first accepted as a pure analogy. Penrose’s proposal was not the origin of this idea, but its pivotal moment—the missing piece in the puzzle of a consistent formulation of a thermodynamic theory of black holes. One student of John Archibald Wheeler had attempted to develop this theory but could not propose a robust formulation. Progress came after the understanding of Penrose’s mechanism and ingenious use of information theory. Meanwhile, the Soviet physicists were looking at the Penrose process through the lenses of quantum field theory, interpreting it as amplification of waves. A complete, semi-consistent description of the thermodynamics of black holes arose from a collaboration between Western and Soviet scientists, partially impaired by the Cold War.

This article puts Penrose’s proposal of a mechanism to extract energy from a rotating black hole into a historical context—which does not imply causation, but that had undeniable influence. The focus, however, will be on the ramifications of the formulation of the Penrose process. Starting from its presentation, then to the chain of reasoning that led first to an analogy between black-hole physics and thermodynamics and finally to the theoretical prediction of the Hawking radiation. This last phenomenon convinced the physics community that black holes might be indeed thermodynamical entities. The objective is to shed light on the motivation and methodology that led to this perception. For this, we shall revisit the original papers and analyze technical details to uncover the history they hide.

The history of the quantum phase of research on black holes is briefly outlined in [ 30 ], while [ 8 ] analyzes the use of information theory to assess the thermodynamical description of those objects. The friendly relationship between Western and Soviet physicists is described in the recollections of physicists who witnessed the events, like Kip Thorne [ 45 ] and Igor Novikov [ 34 ]. In [ 31 ], there is a historical review of the old Soviet academic practices and a glimpse at Yakob Zel’dovich’s early career.

The organization of this paper is as follows: Section  2 presents the historical details of the Penrose process, and Sect.  3 retraces the origins of the theory of black-hole thermodynamics. Section  4 focuses on the work of Jacob Bekenstein and his interpretation of this theory, while Sect.  5 explores his formulation of the three laws of black-hole thermodynamics. The criticism of his proposal is presented in Sect.  6 , contrasting with the acceptance of an analogy between black-hole mechanics and thermodynamics. Yakob Zel’dovich’s views of the Penrose process through the lenses of quantum field theory are outlined in Sect.  7 . Finally, Sect.  8 details the combination of Bekenstein’s and Zel’dovich’s works by Stephen Hawking, ending with his conclusion that black holes should evaporate.

The Penrose process

The inaugural congress of the European Physical Society—held between 8–12 of April 1969 in Florence, Italy—was a game-changer for the physics of black holes. Themed “The growth points of physics,” the package “astronomy, astrophysics, cosmology and relativity” was the first item of concern addressed there [ 32 ]. One lecturer in Florence was Roger Penrose, who did an extensive survey on this topic at the conference, exposing the characteristics and key properties known to that date [ 37 ]. More than that, he introduced a feature that set the course of research in the following years: a mechanism to extract rotational energy of a Kerr black hole, today known as the Penrose process.

The second half of the twentieth century was also a period for substantial technological progress. During the Cold War, energy consumption increased significantly, and the exploitation of nuclear [ 43 ] and fossil energy sources became a vital discussion in the political scenario [ 39 ]. Meanwhile, the space race was intensified when, in 1961, the Soviet cosmonaut Yuri Gagarin became the first human to leave the Earth. In 1969, Americans Neil Armstrong and Buzz Aldrin became the first men to step on the Moon [ 12 ]. In this context, the Dyson sphere became a symbol for extraterrestrial advanced energy exploitation. Freeman Dyson proposed in 1960 an alternative for scanning radio signals in the search for extraterrestrial intelligent life [ 19 ] in the form of detection of infrared radiation. He conceptualized an apparatus that an advanced civilization could use to harvest energy from stars and suggest searching for intelligent life by detecting its characteristic radiation. The Dyson sphere was a habitable thick spherical shell built around a star, with enough machinery to exploit the solar radiation falling onto it from inside.

By the end of his lecture in Florence, Penrose proposed a similar apparatus with a query about energy exploitation. “I want to consider the question of whether it is possible to extract energy out of a ‘black hole’ ” [ 37 , p. 270]. In a thought experiment, he imagined a convoluted system to demonstrate that it was feasible to harvest the energy from a rotating black hole. His first proposal of this apparatus was much like Dyson’s: two enormous structures around a black hole built by an advanced civilization, one moving in the direction of rotation, while the other remained still. Penrose gave an intuitive explanation of how it would work (Fig.  1 ), but he admitted the intricacy of his experiment: “Of course, [it] is hardly a practical method! Certain improvements may be possible, e.g. , using a ballistic method” [ 37 , p. 272].

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Extracted from [ 37 , p. 271]. “We imagine a civilization which has built some form of stabilized structure S surrounding the ‘black hole’. If they lower a mass slowly on a (light, inextensible, unbreakable) rope until it reaches L, they will be able to recover, at S, the entire energy content of the mass. If the mass is released as it reaches L then they will simply have bartered the mass for its energy content. (This is the highest-grade energy, however, namely wound-up springs!) But they can do better than this! They also build another structure S*, which rotates, to some extent, with the ‘black hole’. The lowering process is continued, using S*, to beyond L. Finally the mass is dropped through H, but in such a way that its energy content, as measured from S, is negative! Thus, the inhabitants of S are able, in effect, to lower masses into the ‘black hole’ in such a way that they obtain more than the energy content of the mass. Thus, they extract some of the energy content of the ‘black hole’ itself in the process. If we examine this in detail, however, we find that the angular momentum of the ‘black hole’ is also reduced. Thus, in a sense, we have found a way of extracting rotational energy from the ‘black hole’ ”

The mechanism of the Penrose process depends on the fact that inside the ergosphere, there exist orbits of negative energy when observed from infinity, while the orbits outside this region are of positive energy at infinity. This happens because the metric inside the ergosphere is such that Killing vectors inside it are space-like and those on the outside are time-like. Thus, by lowering an object’s orbit from outside to inside the ergosphere, to near the horizon, it can acquire negative energy. The energy it lost could, in theory, be retrieved by the ones who send the object.

By the time of publication in a special edition of the Rivista del Nuovo Cimento dedicated to that conference, Penrose had already worked out a ballistic description of the mechanism. The idea revolved around a particle p 0 dropping from infinity to inside the ergosphere—in Fig.  1 , it is the region from L to H , in between the absolute event horizon and the sphere of stationary solutions—and nearing the event horizon H , “at which point the particle splits into two particles p 1 and p 2 . The particle p 2 crosses H , but p 1 escapes ( . . . ) possessing more mass-energy content than p 0 !” [ 37 , p. 272] Throughout the years, Penrose refined the explanation of this phenomenon (Fig.  2 ).

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Extracted from [ 38 , p. 272]

More than speculation, Penrose aimed to strengthen the search for mechanisms to detect black holes in space by improving the existing theory. He added that its “real significance is to find out what can and what cannot be done in principle since this may have some indirect relevance to astrophysical situations” [ 37 , p. 272]. It indeed led to a theoretical development that Penrose himself could not have foreseen, which changed the perception of those objects. It was a necessary step toward a thermodynamic description of black holes.

Wheeler’s boys

John Archibald Wheeler’s interest in black holes started in the late fifties. First a critic of the idea, he grew to become one of the biggest enthusiasts of the concept and later popularized the term “black hole.” By the end of the sixties, a conjecture—first proposed and partially proved by physicist Werner Israel in 1967 [ 27 ]—intrigued Wheeler. The no-hair theorem, as Wheeler named it, says a black hole can be described by only three classical parameters. As framed in [ 33 , p. 876], “ the external gravitational and electromagnetic fields of a stationary black hole (a black hole that has settled down into its ‘final’ state) are determined uniquely by the hole’s mass M , charge Q , and intrinsic angular momentum S —i.e., the black hole can have no hair (no other independent characteristic).”

Not only was Wheeler a brilliant physicist, but he also became a prime supervisor and educator with a natural charisma for science communication. He used to meet with his research students once a month and regarded them as collaborators rather than apprentices. Wheeler often informally suggested various topics of research for them to explore and encouraged their creativity [ 8 ]. In 1969, on the project for the research group under his supervision, Wheeler exposed his concerns about black-hole physics [ 47 , p. 28]. As a reaction to the no-hair theorem, one of Wheeler’s specific research topics that year was to “[a]nalyze the meaning of ‘baryon conservation’ and ‘lepton conservation’ and the principle that ‘entropy always increases’ in the context of ‘black hole physics’, where complete collapse washes out all these familiar features of physics” [ 47 , p.30]. According to the no-hair theorem, a black hole has exactly three degrees of freedom. Thus, Wheeler asked how this constraint would fit within the known laws of conservation and thermodynamics.

Wheeler himself did not pursue this thread, however, delegating the task to his students. In his notebooks, he explained why he did not tackle issues like black-hole physics himself. “My reason? Does not reach to the heart of the mystery: pregeometry” [ 48 , p. 17]. At the time, he had a broader research agenda, working on a geometrical description of physics he called pregeometry . In his words, “we’ll accept Einstein’s general relativity, or ‘geometrodynamics’ in its standard 1915 form, translated of course into the appropriate quantum version. Second, we accept as tentative working hypothesis the picture of Clifford and Einstein that particles originate from geometry; that there is no such thing as a particle immersed in geometry, but only a particle built out of geometry” [ 49 , p. 32]. 2

The first of Wheeler’s students to touch on the subject was an undergraduate named Jeffrey M. Greif. In his 1969 Junior thesis Black Holes and Their Entropy , 3 he proposed a definition for the entropy of a black hole. He suggested it would be the limit of the entropy of a collapsing body right before reaching the singular radius. This definition had many issues. Most noticeable, it did not reconcile with the no-hair theorem. If the entropy depended on the state prior to collapse, the loss in degrees of freedom would be paradoxical. Jacob Bekenstein later noticed that “[Greif] examined the possibility of defining the entropy of a black hole, but lacking many of the recent results in black-hole physics, he did not make a concrete proposal” [ 5 , p. 2333].

One year later, Greek physicist Demetrios Christodoulou took advantage of the most recent tool available at the time to advance one step further in the direction pointed by Greif. Self-taught in physics, Christodoulou got a position as a graduate student in Princeton in 1968 under Wheeler’s guidance after finishing high school. In 1970, he published an important report, Reversible and Irreversible Transformations in Black Hole Physics , where he considered the loss of mass of a Kerr black hole in the occurrence of the Penrose process [ 14 ]. This extraction of rotational energy, Christodoulou argued, was not indefinite. It would end when the Kerr black hole turned into a Schwarzschild one, that is, when it stopped spinning. Christodoulou explained that losing angular momentum would lead to a decrease in mass and asked which would be the smallest Schwarzschild black hole derived from this process. He named its mass the irreducible mass of the black hole.

Conversely, given a Schwarzschild black hole, one could add rotation by adding massive particles carrying angular momentum. Christodoulou considered this case and the possibility of reversing this action by means of the Penrose process. A Schwarzschild–Kerr–Schwarzschild conversion that results in the original black hole (same mass, no angular momentum) would configure a reversible transformation . Otherwise, it would be an irreversible transformation . For the latter, Christodoulou analyzed that the final Schwarzschild black hole would always have more mass than the first (Fig.  3 ). Therefore, he concluded the irreducible mass remains constant for reversible transformation, increases for irreversible ones, and never decreases.

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Extracted from [ 15 ]

Christodoulou remarked that the energy harvesting from a black hole had limited efficacy. In fact, he pointed to the fact that 29% of the mass of an extreme Kerr black hole could be harvested as energy, “far more energy conversion than in nuclear fission or fusion” [ 15 , p. 66]. For a charged black hole, this value can reach up to 50% [ 3 , p. 104]. Christodoulou theorized that the rest would be released as relativistic energy highly ionized, like the relativistic jets escaping from the nuclei of some radio galaxies.

In 1971, Stephen Hawking, investigating the properties of the gravitational radiation generated by the collision of two black holes, discovered that the surface area of the event horizon of the resulting black hole is strictly greater than the sum of their individual areas before they merged [ 23 ]. This, he argued, can also be applied for the case of a single black-hole absorbing particles, and thus, he concluded that the surface area of a black hole never decreases. On this subject, Christodoulou remarked in his thesis, “We see, in Hawking’s theorem, a strong analogy between the surface area of a black hole and its irreducible mass. Both quantities cannot be reduced. What is the connection between the two quantities?” [ 15 , p. 95]. His answer:

where A is the surface area of the event horizon, and m ir is the irreducible mass of the black hole.

The third of Wheeler’s students to contribute to the subject was the young Mexican-Israeli physicist Jacob Bekenstein, who revolutionized the field, giving a new meaning to Christodoulou’s result.

A crazy-enough idea

The connection between black-holes physics and thermodynamics was a seed sprouting in Princeton by the end of the sixties. As mentioned before, Wheeler would often instigate his students to pursue this thread, and to Jacob Bekenstein, he posed an adapted version of an old thermodynamic thought experiment. “What would happen, Wheeler asked Bekenstein, if you mixed hot tea and cold water—thus maximizing the entropy—and poured it into a black hole? Would the entropy simply vanish, pumped out of physical reality? If so, the ‘crime’ of raising the total entropy would seem to have been pardoned” [ 8 , p. 74]. The beverage, in this case, represents a state of increased entropy in the universe: if it vanished into the event horizon, if the black hole did not have entropy, it would constitute a transgression of the second law of thermodynamics. To find the answer to Wheeler’s query, Bekenstein considered a similar contravention in a problem proposed by James Clerk Maxwell in 1867.

The total entropy of an isolated system always increases over time, and it remains constant only for reversible processes. This is the second law of thermodynamics, and Maxwell proposed a situation in which it would not hold. In the nineteenth century, instead of an advanced civilization, Maxwell considered a demon guarding the gate connecting two isolated compartments filled with gas. This demon could open the gate at the right moment to let the more agitated molecules of A enter B and the slower molecules of B enter A. This would cause A to warm up and B to cool down, decreasing its entropy. Heat would flow from the cooler environment to the warmer one, breaking the second law. This thought experiment became known as Maxwell’s Demon , and thus, Bekenstein named his supervisor’s problem Wheeler’s Demon .

Part of Bekenstein’s PhD thesis was devoted to assessing this question. Maxwell’s Demon was a well-known controversial problem in physics, and Bekenstein came across an intriguing answer to this paradox in a connection between thermodynamics and information theory. In 1950, Leon Brillouin proposed a solution to this dilemma using information theory [ 10 ], which inspired Bekenstein to address Wheeler’s Demon in a similar way. Brillouin suggested an increase in entropy relates to loss of information, adopting this relation as a definition for the concept. Bekenstein compared a black hole to a thermodynamic system in equilibrium and argued that, although from the outside only three classical parameters could be measured, from the inside, the other physical features of matter would be preserved. This inaccessibility of information to an exterior observer, this loss in degrees of freedom, would read as the entropy—as defined by Brillouin—of a black hole [ 5 , p. 2336].

According to the theorem by Hawking and Christodoulou, adding mass to a black hole would increase its surface area and decrease the amount of information an observer on the outside has of its interior. Bekenstein argued that the irreducible mass of a black hole would straightforwardly correlate to entropy, according to information theory. The terminology used by Christodoulou, reversible and irreversible transformations, hinted at a connection between his results and the second law of thermodynamics. Bekenstein formalized it: “The irreversibility of an increase in the area of a black hole is reminiscent of the irreversibility of an increase in the entropy of a closed thermodynamic system” [ 3 , p. 104].

Compared to Greif’s early attempt, Bekenstein proposed a more concrete definition of the entropy of a black hole. And he availed from the developments not available to Greif at the time, mainly the Penrose process, which led to the Christodoulou–Hawking theorem. Although the correlation between black-holes physics and thermodynamics was welcomed, Bekenstein’s approach also received plenty of criticism. He later remarked: “In those days in 1973 when I was often told I was headed the wrong way, I drew some comfort from Wheeler’s opinion that ‘black holes thermodynamics is crazy, perhaps crazy enough to work’ ” [ 7 , p. 28].

Bekenstein and the three laws of black-hole thermodynamics

Following up on Wheeler’s thought experiment, Jacob Bekenstein proposed the connection between thermodynamics and black-holes physics in his PhD thesis, Baryon Number, Entropy and Black Hole Physics . He argued this parallelism could help predict “new effects in black holes physics by analogy with familiar results in thermodynamics” [ 3 , p. 107] and exposed the equivalent of the three laws of thermodynamics for this case.

From Christodoulou’s equation ( 1 ), the entropy of a black hole is proportional to its irreducible mass or equivalently to the area of its event horizon. Thus, Christodoulou–Hawking theorem—the area of a black hole’s surface never decreases—reads as the second law.

The first law was not as evident as the second. It is a conservation law that says the energy of an isolated system is constant. As an equation, it translates as

where E , T , S , P , and V are the energy, temperature, entropy, pressure, and volume of the system, respectively. For a charged, rotating black hole with mass M , charge Q , and angular momentum J , 4 Bekenstein considered its rationalized area, 5

where a is the angular momentum per unit, that is, a = J / M . The differentiation of Eq. ( 3 ) results in,

Bekesntein concludes that Ω and Φ play the role of the rotational angular frequency and electric potential of the black hole, respectively.

Comparing this mass formulae ( 4 ) with Eq. ( 2 ), Bekenstein identified A / 4 π —the square of the irreducible mass m ir from Christodoulou’s formulæ  Eq. ( 1 )—as entropy and κ / 2 as temperature. 6 In Eq. ( 4 ), the term Ω · d J + Φ d Q represents the work done on the black hole by an external agent, increasing the angular momentum and charge by d J and dQ , respectively. Thus, it would correspond to ( - P d V ) , which is the work done on a thermodynamic system [ 5 , p. 2335]. With these identifications, the energy content falling into a black hole would be described by parameters depending only on the mass, charge, and angular momentum, in agreement with the no-hair theorem.

The third law came from the identification of κ / 2 as the temperature of a black hole. For a generalized Kerr black hole, the temperature is always nonnegative, and κ = 0 holds only if the black hole is extreme, that is, if for a given charge and angular momentum, it has minimal mass. Bekenstein considered the process of accretion of selected particles to transform an ordinary black hole into an extreme one. About this transformation, he concluded that “the nearer [ κ ] is to zero ( . . . ), the less efficient is further accretion in lowering [ κ ].” This, he argued, is equivalent to the third law of thermodynamics, which he phrased as “the nearer a system’s temperature T is to absolute zero, the less efficient is any process in lowering T further.” And “[t]hus we find an analog to for each of the three laws of thermodynamics in black hole physics!” [ 3 , pp. 106–107].

Coming from Wheeler’s thermodynamic thought experiment, Bekenstein questioned whether the correlation between irreducible mass and entropy could be regarded as more than comparison. “Is there a deeper connection between the two subjects? Clearly the concepts of energy and work have similar meanings in thermodynamics and black hole physics. But does the rationalized area of a black hole (the area divided by 16 π ) have anything to do with entropy?” [ 3 , p. 107]. Bekenstein argued yes, it does. He said, “In everyday physics entropy goes hand in hand with statistics, information, and degradation of energy,” via Boltzmann’s formulæ  for entropy, Brillouin’s informational explication, and the introduction of black-hole entropy as a measure of degradation of its energy [ 3 , pp. 129–130]. He proceeded to compare a black hole to a black body. Bekenstein formalized this view in 1973 when he said that “so far the analogies have been of a purely formal nature, primarily because entropy and area have different dimensions” [ 5 , p. 2335]. To obtain a formula for the entropy of a black hole with correct units, his strategy again relied on information theory, identifying information loss as an increase in entropy.

A good analogy

The correlation between thermodynamics and black-hole physics spiked the curiosity of John M. Bardeen, 7 Brandon Carter, and Stephen Hawking, who gathered to work on a formulation of their own at the Les Houches Summer School on black holes in 1972 [ 30 ] and resulted on a paper published in the following year [ 2 ]. In it, they proposed four laws of black-holes mechanics, comparing them with the four laws of thermodynamics. In a more mathematical approach, they explored the geometry of the Kerr solution to calculate the mass and angular momentum in terms of unique time translational and rotational Killing vectors, from where they derived the ensuing laws.

The zeroth law states that the proper acceleration experienced by a test particle near a stationary black hole due to its gravitational pull with a redshift correction, the surface gravity κ , is constant over the event horizon. This is the same constant κ which appears in Eq. ( 4 ), 8 but Bekenstein did not straightforwardly identify it this way, a fact Bardeen, Carter, and Hawking did not fail to notice. “We show that the quantities appearing in (Eq.  4 ) formulae have well-defined physical interpretation” [ 2 , p. 162].

The other three laws are similar to Bekenstein’s. The first law establishes that κ / 8 π is analogous to the temperature in the same way the area A is analogous to entropy, departing slightly from Bekenstein’s definition. The second law refers to the fact that the surface area of each black hole does not decrease with time. And finally, the third law, left unproven, 9 is the same as the one Bekenstein had already obtained, rewritten as “It is impossible by any procedure, no matter how idealized, to reduce κ to zero by a finite sequence of operations” [ 2 , p. 169].

Despite the agreement with Bekenstein’s results, Bardeen, Carter, and Hawking could not stress enough that this correlation was nothing but a good analogy. “It should however be emphasized that ( κ / 8 π ) and A are distinct from the temperature and entropy of a black hole,” otherwise, they argued, it should radiate. “In fact the effective temperature of a black hole is absolute zero. One way of seeing this is to note that a black hole cannot be in equilibrium with black body radiation at any non-zero temperature, because no radiation could be emitted from the hole whereas some radiation would always cross the horizon into the black hole” [ 2 , p. 168]. Their purely geometrical approach to the problem, analyzing a stationary axisymmetric solution of Einstein’s equation containing a black hole and surrounded by matter to derive the results, indicates the lack of a deeper connection with thermodynamics. At the same time, Bekenstein reasoned outside of a gravitational theory to understand these laws as more than an analogy. In a recollection told to Werner Israel, Hawking admitted that “in writing this paper [with Bardeen and Carter], I was motivated partly by my irritation with Bekenstein who, I felt, had misused my result about increase of the area of the event horizon” [ 28 , p. 264].

Bekenstein did not seem to consider the point raised by Bardeen et al. about the temperature a troublesome issue. In fact, he argued that a black hole could be regarded as a black body cavity with zero temperature. Therefore, Bekenstein agreed that the surface gravity was not to be regarded as the real temperature, calling it a characteristic temperature [ 6 , p. 3295]. The increase of the black hole’s entropy (its surface area), he argued, would compensate for the loss of thermal entropy [ 3 , p. 128]. If κ / 2 was not the actual temperature, there should not be any radiation associated with it.

A point Bekenstein perceived as the most serious problem with his view was the violation of the second law of thermodynamics. Ironically, it was the question he was trying to solve in the first place. The black hole’s entropy would increase with accretion of matter, causing the entropy of the visible universe to decrease. “It would seem that the second law of thermodynamics is transcendent here in the sense that an exterior observer can never verify by direct measurement that the total entropy of the whole universe does not decrease in the process” [ 5 , p. 2339]. This fault was avoided in Bardeen, Carter, and Hawking’s interpretation of this description as an analogy, as the surface area was not actually regarded as entropy. In a follow-up four-page report published in 1972, Black Holes and the Second Law , Bekenstein reformulated this law: “Common entropy plus black-hole entropy never decreases” [ 4 , p.738].

Physicist Werner Israel also positioned himself against Bekenstein’s views. “Black-hole dynamics has many points of resemblance with thermodynamics. ( . . . .) It is perhaps tempting to look for a deeper significance in this parallel. ( . . . .) If such a relationship holds it would have far-reaching consequences, for example in cosmology, where it would largely obviate the complications arising from having to treat a universe containing black holes as an open thermodynamical system” [ 28 , p. 267]. Israel argued that Bekenstein could not properly connect entropy influx with energy influx—something Bekenstein had done using information theory—and thus, an increase in the area would not equate to an increase in entropy. Israel proposed that the presence of an external potential energy could disrupt this connection. Bekenstein’s use of information theory was innovative and, as such, hard to grasp intuitively. Years later, Israel reflected on his own assessment, saying Bekenstein had “convoluted arguments which, at best, were just short of compelling and, in less fortunate instances, an easy target for criticism” [ 30 , p. 263].

Bekenstein further developed his formulation [ 6 ] in the following years. The description of thermodynamics as a statistical discipline and the interpretation of entropy as information loss guided his intuition. Although the proposal was over twenty years old at the time, the use of information theory to assess thermodynamics found one of its first application with this case [ 8 ], and this may explain why others were skeptical with the possibility of the thermodynamic description of black holes being more than an analogy.

A different angle

Since the rise of relativistic astrophysics in the sixties, the Soviet community had been influential in developing black-hole physics. The Cold War pushed a mostly friendly rivalry with Western physicists and highlighted the differences between their scientific backgrounds. In the West, the Penrose process allowed the combination of thermodynamics and black-hole mechanics. Meanwhile, the investigation of this mechanism to extract rotational energy from a Kerr black hole through the lenses of quantum field theory in the Soviet Union led to a wave interpretation of this phenomenon instead of Penrose’s ballistic approach.

Self-taught in science, Yakov Borisovich Zel’dovich rose quickly to Academician, the greatest academic position in Soviet Union [ 34 ]. He made major contributions to the fields of physical chemistry, nuclear physics, thermodynamics, and cosmology. In 1946, he became the head of the theoretical department of the military project Arzamas-16 and led the research of atomic weapons, working together with Andrei Sakharov on the conception of the Soviet hydrogen bomb [ 46 ]. In the sixties, Zel’dovich turned his attention to astrophysics, researching, among other things, the formation of black holes. He suggested, for example, the presence of an accretion disk around massive black holes [ 50 ].

As recalled by Kip Thorne [ 45 , p. 428], Zel’dovich learned about the Penrose process from discussions with Thorne himself, Charles Misner, and John Wheeler. In June 1971, Thorne visited Moscow, and he reported one particular interaction with Zel’dovich during that Summer, in which the latter proposed that rotating black holes should radiate. In Zel’dovich’s intuition, Thorne explained, the soviet physicist compared a black-hole system to a massive, rotating sphere inside a cylindrical surface. In his analogy, the sphere was the black hole and the interior of the cylinder, the ergosphere. The rotation of the spherical object would reflect and amplify an incident wave, using a fraction of its rotational energy to do so. This would perturb the vacuum fluctuation inside the cylinder and lead to spontaneous radiation.

Zel’dovich formalized this idea in a short paper published in a Soviet Journal in 1971. Although Thorne credited Zel’dovich’s inspiration to be the case of energy extraction of a black hole, the Academician mainly discussed the analogue experiment—the system of plane waves incident on a cylindrical shell with a rotating mass inside it. Zel’dovich concluded that not only the amplification but also the generation of waves would occur. “In the presence of an external reflector with small losses (resonator), the amplification following single scattering may turn into generation.” For a black hole described by a Kerr solution, he added that “in a quantum analysis of the wave field one should expect spontaneous radiation of energy and momentum by the rotating body. The effect, however, is negligibly small” [ 51 , pp. 180–181]. Zel’dovich also explained the phenomenon intuitively with a particle interpretation, arguing that the vacuum fluctuation would create a spontaneous pair of particle/anti-particle and the rotation would make this production unstable, and more particles would escape to infinity instead of falling into the horizon. Zel’dovich went beyond the Penrose process and, with a quantum perspective, predicted that rotating black holes should radiate.

The strategy of exploiting analogue experiments to infer properties of an equivalent system is to this date controversial [ 17 ], but Zel’dovich had done it before to study stellar configurations. He and Thorne reportedly discussed the physics of rotating starts using a system of a rotating rigid body as an analogue. In 1972, Brandon Carter published a report in favor of this strategy [ 13 ], arguing that when applied to the case of black holes, it replicated known features, like the Penrose process and Christodoulou’s reversible transformation. In this same year, Zel’dovich released an extended version of his 1971 paper with the title Amplification of cylindrical electromagnetic waves reflected from a rotating body [ 52 ]. 10 This time, he considered the analogy to black holes, but he did not mention, however, the spontaneous radiation again.

Zel’dovich’s papers did not generate attention. Even at that year’s event dedicated to black holes, the Les Houches Summer School of Theoretical Physics —which some of his associates attended, like Thorne and Soviet astrophysicist Igor Novikov—his results were not mentioned [ 45 , p. 433]. Both of Zel’dovich’s papers were published in Soviet journals, which probably contributed to this perceived disinterest. Another reason for this may have to do with Zel’dovich’s caution in exploring the phenomenon of energy extraction for the case of black holes, instead, focusing on the analogue systems, refraining from pursuing his bolder assumptions.

In 1973, Alexei Starobinskii, at the time a postdoctoral student of Zel’dovich, revisited his supervisor’s result with a more compelling title, Amplification of waves during reflection from a rotating “black hole.” In a different manner than Zel’dovich, Starobinskii dealt explicitly with black holes and the incidence of classical waves, properly identifying their amplification as the Penrose process [ 42 ]. In September that same year, Stephen Hawking visited Moscow for the first time. In his hotel room, with Zel’dovich and Thorne as witnesses, Starobinskii presented his and his supervisor’s results to Hawking, who was unaware of them. To Israel [ 28 , p. 264], Hawking recalled that he liked the idea but was not convinced by the calculations. He mentioned Starobinskii had told him about spontaneous radiation—a phenomenon which Starobinskii himself had not discussed in his paper. Hawking reportedly thought their arguments had physical ground, but he remained skeptical. Inspired by Zel’dovich’s and Starobinskii’s ideas, Hawking worked on this problem back home. His conclusion, he said, really annoyed him: all black holes must radiate.

The Hawking radiation

For the case of a Kerr black hole, Starobinskii repeatedly mentioned Zel’dovich’s analogue system but avoided his supervisor’s quantum approach. Instead, he considered a semiclassical scalar field satisfying a relativistic wave equation—the Klein–Gordon equation. As expected, the amplitude of the reflected wave was greater than of the incident one. It was a description of the Penrose process, Starobinskii argued, without the need to consider composite or unstable particles to work.

After the visit to Moscow, Stephen Hawking had Starobinskii calculations and Zel’dovich’s quantum analogue system in his mind. The Soviets convinced him that quantum effects should be considered when studying the evolution of black holes, and Hawking included this premise in a brief report he wrote to Nature [ 24 ]. He opened this article titled Black holes explosions? , acknowledging that those effects were usually ignored because near the event horizon of any black hole, the radius of space-time curvature is disproportionally large compared to Planck scale, and thus these contributions were locally small. Nevertheless, he reasoned that “they may still, however, add up to produce a significant effect over the lifetime of the Universe” [ 24 , p. 30].

Hawking adopted a strategy different than Starobinskii’s. Whereas the Soviets thought of amplification of waves, Hawking thought of creation and annihilation of particles. He calculated the rate of particle production near the horizon of a Schwarzschild black hole, using simple quantum field theory. He found the number of particles created and emitted to infinity was greater than those annihilated or absorbed, and this difference would amount with time to produce a steady rate of emission. He estimated the emission rate of bosonic particles to be ( exp ( 2 π ω / κ ) - 1 ) - 1 , the same expected for a body with temperature in geometrical units of κ / 2 π .

With this, not only rotating black holes would radiate, as suggested by Zel’dovich, but all black holes would. The prediction of the radiation was the last piece of theoretical evidence to convince Hawking that black holes are thermodynamical entities. “Bekenstein suggested on thermodynamic grounds that some multiple of κ should be regarded as the temperature of a black hole. He did not, however, suggested that all black hole could emit particles as well as absorb them. For this reason Bardeen, Carter and I considered that the thermodynamical similarity between κ and temperature was only an analogy. The present result seems to indicate, however, that there may be more to it than this” [ 24 , p. 31].

The change in status of the thermodynamical analogy of black holes is a key step for understanding its acceptance. This fact is nicely expressed in the words of Kip Thorne, “This [the Hawking radiation] (if Hawking was right) was incontrovertible proof that Bardeen-Carter-Hawking laws of black-hole mechanics are the laws of thermodynamics in disguise, and that, as Bekenstein had claimed two years later, a black hole has an entropy proportional to its surface” [ 45 , p. 436]. The use of analogues to assess different physical processes is a common practice but presents limitations to what kind of information one can extract about the original system. To derive the four laws of black holes as an analogue to thermodynamics was a valid option to study the mechanics of this inaccessible astrophysical system, although it had the constraint of being only an analogy. With the identification of black holes as black bodies, the problem of having to transfer the knowledge from a field to another was solved. 11

Zel’dovich intuitively predicted that rotating black holes would radiate, and Hawking extended his prediction to all black holes. This difference comes from their approaches to the problem. Zel’dovich’s wave description in his analogue system relied on the angular momentum of the massive sphere to destabilize the quantum vacuum fluctuation near the cylindrical shell, here representing the event horizon, causing the emission of particles. Hawking, on the other hand, considered the changing geometry of the space-time near the event horizon instead of the rotation of the black hole to calculate the absorption and emission rates. Hawking was not the first to use quantum field theory in a cosmological scenario. Years before, the American physicist Leonard Parker pioneered [ 35 ] investigating particle creation in an expanding universe, but Hawking did not cite him in his papers.

Hawking published an extended version of his report a year later, entitled Particle Creation by Black Holes [ 25 ], extending the ideas proposed in the first paper. He reinforced that black holes should radiate with the temperature given by a factor of the surface gravity, and its mass would decrease in the process as the emission rate increased. This, Hawking concluded, meant that black holes not only radiate, but they can also evaporate. He explained, “For a black hole of solar mass ( 10 33 g) this temperature is much lower than the 3 K temperature of the cosmic microwave background. Thus black holes of this size would be absorbing radiation faster than they emitted it and would be increasing in mass. However, in addition to black holes formed by stellar collapse, there might also be much smaller black holes which were formed by density fluctuations in the early universe. These small black holes, being at a higher temperature, would radiate more than they absorbed. ( . . . ) When the temperature got up to about 10 12 K or when the mass got down to about 10 14 g the number of different species of particles being emitted might be so great that the black hole radiated away all its remaining rest mass on a strong interaction time scale of the order of 10 23 s” [ 25 , p. 202].

Despite providing a necessary argument for black holes to be understood as black bodies, using Bekenstein’s thermodynamical description and Zel’dovich’s quantum approach, Hawking did not solve every problem with this identification. In fact, he created new ones. For example, if a black hole evaporated, its irreducible mass and surface area would decrease, contradicting the theorem that carries his and Christodoulou’s name. Hawking explained this by saying there is a constant influx of negative energy into the black hole Christodoulou had not considered, which would constitute a violation of the weak energy condition, 12 caused by the indeterminacy of particle number and energy density in curved space-time [ 25 , p. 203].

The ad hoc merge between general relativity and quantum field theory is also a point of contention, but Hawking’s calculations were solid, and this issue was considered something to be explored instead of being a fundamental crack in the method. Decades later, a debate about the fate of information inside a black hole gained notoriety and exposed another weak spot of the theory. If black holes evaporated, what would happen to the information in their interior? If it also evaporated, it would contradict the postulate that information cannot be lost. This question remains unresolved to this date, and the controversy that ensues it is known as the Black Hole War , thanks to physicists Leonard Susskind and Gerard ’t Hooft [ 44 ].

The history of black-holes physics has many pivot points and eventualities that may be a key to understand the scientific practices of highly theoretical areas of knowledge. The development of a thermodynamical theory of black holes, in particular, changed not only our perception of those bodies but also the fields of astrophysics, information theory, and quantum gravity. This paper focused on this first change, outlining the origin of the black-holes thermodynamics idea until the event that solidified this theory as more than a ploy to explore the mechanics of those objects. It identifies the Penrose process as the critical result that allowed the formulation of this thesis and the Hawking radiation as the step to solidify it. This analysis led us full circle around the globe, starting in Europe, with Penrose’s lecture and proposal of a mechanism to extract rotational energy of a black hole; then, to the USA, with John Archibald Wheeler’s exceptional students; to the Soviet Union, with Zel’dovich’s ingenious thinking and interpretation of the Penrose Process; and finally back to Europe, with Hawking’s prediction that black holes should radiate and evaporate.

In the sixties, the concept of black holes gained shape with the discovery of more properties, and candidates in the sky made their existence highly probable. In 1969, Roger Penrose compiled this knowledge—that he actively helped to build and for which he won the Nobel Prize in Physics—in a lecture he presented at the inaugural conference of the European Physical Society. In it, he revised and extended the theory of black holes, when he proposed an additional feature, a mechanism to collect rotational energy from a Kerr black hole. He hoped this would lead to new advances in astrophysics and urged the community to search for a black hole in space. The Penrose process proved to be an essential step toward a thermodynamical description of black holes, developed in the following years.

The idea to investigate thermodynamical properties of black holes was a sparkle in John A. Wheeler’s mind at that point, and he encouraged his students to pursue it. The first to try was Jeffery M. Greif, who could not make a sound proposal. It was only after the suggestion of the Penrose process that a more robust proposition was possible, and with it, another of Wheeler’s students, Demetrios Christodoulou, hinted at a correlation between thermodynamics and black-holes physics. It was Jacob Bekenstein, however, who formalized this connection.

Bekenstein’s use of information theory to assess the problem was pioneering. He observed a parallel between the three laws of thermodynamics and some properties of black-holes physics, the second law tied to the Penrose process via Christodoulou’s reversible and irreversible transformations. Although Bekenstein’s idea was welcomed as a good analogy, he understood it as a genuine connection. He suggested that black holes have a well-defined entropy and are, therefore, thermodynamical entities. This interpretation received many criticisms. Among the critics were John M. Bardeen, Brandon Carter, and Stephen Hawking, who noticed that the temperature in Bekenstein’s theory remained ill-defined and that for this connection to be true black holes would have to radiate.

Meanwhile, in the Soviet Union, Yakov B. Zel’dovich, upon learning about the Penrose process, reinterpreted it in a more familiar way to him, using quantum theory. He proposed an analogue experiment that would replicate Penrose’s mechanism in a system composed of a massive sphere rotating inside a resonant cylindrical shell. Zel’dovich predicted that Kerr black holes would radiate, although he did not pursue this thread. Alexei Starobinskii extended his supervisor’s suggestion focusing primarily on black holes and presented it to Stephen Hawking in 1973. Hawking was intrigued and worked out his own calculations on the subject. To his surprise, he concluded that not only Kerr but all black holes should radiate. More than that, they should eventually evaporate.

The Hawking radiation is the starting point of a new chapter in the history of black-holes physics, the quantum phase of research, as Werner Israel put it [ 30 ]. The thermodynamical properties of black holes remain theoretical, but it opened venues and possibilities yet to be explored, hopefully in the near future.

Acknowledgements

I wrote this article while I was working at the Max Planck Research Group Historical Epistemology of the Final Theory Program during the COVID-19 pandemic, and I am grateful for the support and encouragement they gave me during this stressful time, in particular the guidance of Alexander Blum and the patience of Kseniia Mohelsky. I edited this paper during my first months as an International Fellow at the Kulturwissenschafliches Institut (KWI), and I appreciate the support of Julika Griem, Sabine Voßkamp, Sebastian Hartwig, and Britta Acksel. I heartily thank Alexander Blum for helping me in every step of research; Stephan Furlan, Rocco Gaudenzi, Gabriela Radulesco for the insightful discussions; and Ricarda Menn for assisting me with proofreading. Also, I offer my deepest gratitude for the suggestions and corrections by the unknown referees. Thank you!

Data Availability Statement

1 For a thorough characterization of black holes by the end of the sixties, see [ 33 ].

2 Wheeler’s approach to black holes in the sixties is detailed in [ 21 ].

3 Unfortunately, due to limitations caused by the COVID-19 pandemic, this Princeton manuscript was unavailable for external researchers by the time this article was written. The information presented here comes from Jacob Bekenstein’s description of Greif’s work [ 3 ]. They were colleagues working under Wheeler during an overlapping period.

4 Bekenstein uses L for the angular momentum. For the sake of clarity, here and throughout the text, a unified notation will be used, overriding the original one.

5 Equation (1) of [ 23 ] adapted by Bekenstein to the case of a charged black hole.

6 In the original notation, κ / 2 is actually denoted by Θ , and A / 4 π = α .

7 Son of two-times Nobel laureate John Bardeen.

8 And the reason why this notation was adopted in this text, instead of Bekenstein’s Θ .

9 Werner Israel published a proof of this law in 1986 [ 29 ].

10 In June 2020, Marion Cromb and collaborators claimed to have shown the effect predicted by Zel’dovich in the analogue system [ 16 ].

11 The history of the use of analogue systems in physics is part of the work of Rocco Gaudenzi. I am thankful for our enlightening discussions on this topic.

12 The energy conditions are common assumptions about a generic matter content. In simpler words, the weak energy condition states that the local energy density for any observer is always nonnegative.

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General Relativity and Quantum Cosmology

Title: singularities, black holes, and cosmic censorship: a tribute to roger penrose.

Abstract: In the light of his recent (and fully deserved) Nobel Prize, this pedagogical paper draws attention to a fundamental tension that drove Penrose's work on general relativity. His 1965 singularity theorem (for which he got the prize) does not in fact imply the existence of black holes (even if its assumptions are met). Similarly, his versatile definition of a singular space-time does not match the generally accepted definition of a black hole (derived from his concept of null infinity). To overcome this, Penrose launched his cosmic censorship conjecture(s), whose evolution we discuss. In particular, we review both his own (mature) formulation and its later, inequivalent reformulation in the PDE literature. As a compromise, one might say that in "generic" or "physically reasonable" space-times, weak cosmic censorship postulates the appearance and stability of event horizons, whereas strong cosmic censorship asks for the instability and ensuing disappearance of Cauchy horizons. As an encore, an appendix by Erik Curiel reviews the early history of the definition of a black hole.

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Roger Penrose wins 2020 Nobel Prize in Physics for discovery about black holes

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Professor Sir Roger Penrose

Professor Sir Roger Penrose, Honorary Fellow and alumnus of St John’s College Cambridge and honorary doctor of the University, has jointly won the 2020 Nobel Prize in Physics for the discovery that black hole formation is a robust prediction of the general theory of relativity.

His ground-breaking proof of the formation of black holes is a landmark contribution Heather Hancock

Penrose is an emeritus professor at the Mathematical Institute, University of Oxford. He becomes the 110th affiliate of the University of Cambridge to be awarded a Nobel Prize.

The Royal Swedish Academy of Sciences made the announcement this morning (6th October).

According to the Nobel Prize website : “Penrose used ingenious mathematical methods in his proof that black holes are a direct consequence of Albert Einstein’s general theory of relativity.” 

Einstein himself did not believe that black holes really existed. But in January 1965, ten years after Einstein’s death, Penrose proved that black holes really can form and described them in detail. His ground-breaking article, published in January 1965, continues to be viewed as the most important contribution to the general theory of relativity since Einstein.

David Haviland, chair of the Nobel Committee for Physics, said: “The discoveries of this year’s Laureates have broken new ground in the study of compact and supermassive objects. But these exotic objects still pose many questions that beg for answers and motivate future research. Not only questions about their inner structure, but also questions about how to test our theory of gravity under the extreme conditions in the immediate vicinity of a black hole”.

Penrose arrived at St John’s in 1952 as a graduate student and completed his PhD thesis on tensor methods in algebraic geometry in 1957. He remained at the College as a Research Fellow until 1960 and was elected as an Honorary Fellow in 1987. Penrose is the College's sixth Nobel prize-winner in Physics and tenth Nobel laureate overall . Heather Hancock, current Master of St John’s, said: “We are delighted to see Sir Roger Penrose receive the recognition and accolade of the Nobel Prize for his outstanding contribution to physics. His ground-breaking proof of the formation of black holes is a landmark contribution to the application of Einstein’s general theory of relativity. We offer our warmest congratulations to Roger.”

In the 1970s, Penrose collaborated with Cambridge’s Stephen Hawking and in 1988, they shared the Wolf Foundation Prize for Physics for the Penrose–Hawking singularity theorems. 

Prof Martin Rees, Astronomer Royal and Fellow of Trinity College, University of Cambridge, said: “Penrose is amazingly original and inventive, and has contributed creative insights for more than 60 years. There would, I think, be a consensus that Penrose and Hawking are the two individuals who have done more than anyone else since Einstein to deepen our knowledge of gravity. (Other key figures would include Israel, Carter, Kerr, and numerous others.) Sadly, this award was too much delayed to allow Hawking to share the credit with Penrose.

“It was Penrose, more than anyone else, who triggered the renaissance in relativity in the 1960s through his introduction of new mathematical techniques. He introduced the concept of a 'trapped surface’. On the basis of this concept, he and Hawking (more than a decade younger) together showed that the development of a singularity - where the density 'goes infinite' - was inevitable once a threshold of compactness had been crossed (even in a generic situation with no special symmetry). This crucial discovery firmed up the evidence for a big bang, and led to a quantitative description of black holes.”

Penrose shares the 2020 Physics Nobel with Reinhard Genzel and Andrea Ghez who developed methods to see through the huge clouds of interstellar gas and dust to the centre of the Milky Way. Stretching the limits of technology, they refined new techniques to compensate for distortions caused by the Earth’s atmosphere, building unique instruments and committing themselves to long-term research. Their work has provided the most convincing evidence yet of a supermassive black hole at the centre of the Milky Way.

Professor Penrose was awarded an honorary doctorate by the University of Cambridge in 2020.

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roger penrose phd thesis pdf

The thermodynamics of black holes: from Penrose process to Hawking radiation

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  • Published: 05 August 2021
  • volume  46 , Article number:  20 ( 2021 )
  • Carla Rodrigues Almeida 1 , 2  

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In 1969, Roger Penrose proposed a mechanism to extract rotational energy from a Kerr black hole. With this, he inspired two lines of investigation in the years after. On the one side, the Penrose process, as it became known, allowed a comparison between black-hole mechanics and thermodynamics. On the other, it opened a path to a quantum description of those objects. This paper provides a novel take on the events that led to the rise of the thermodynamic theory of black holes, taking as a starting point the Penrose process. It studies the evolution of the research conducted independently by Western and Soviet physicists on the topic, culminating in Stephen Hawking’s groundbreaking discovery that black holes should radiate.

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1 Introduction

The formulation of the thermodynamics of black holes is an important development that shaped our current understanding of those objects. Black holes are now understood to have the thermodynamic properties of entropy and temperature and to interact with their environments as a perfect black body, as defined in [ 40 , pp. 12–13]. The identification of black holes as perfect black bodies is, however, a theoretical one and, as such, should be considered critically [ 18 ]. A step toward this goal is to analyze the inception of this idea, retracing its origins and evolution to grasp how it was first conceived and the scientific methods employed in this hypothesis.

A black hole was hidden in the first solution of Einstein’s equation of general relativity, obtained in 1916. Schwarzschild’s Massenpunktes solution [ 41 ] described a black hole, but this conclusion was initially rejected. General relativity was in its infancy, and it was broadly misunderstood and underdeveloped [ 20 , p. 255]. The hypothesis that objects with the characteristics of a black hole could exist lacked observational evidence and was too counterintuitive [ 1 ]. Theoretical evidence of their existence became strong in the late thirties, indicating that it was a probable consequence of general relativity, but the astrophysics community was still reluctant to accept this result. After the renaissance of Einstein’s theory of gravitation in the mid-fifties [ 9 ] and the observation of super-dense bodies in the skies at the beginning of the following decade, the situation changed [ 11 ]. The rise of relativistic astrophysics and the possibility of identifying quasars as collapsed objects [ 22 , p. 27] renewed interest in the latter, and the sixties became a golden decade for black-hole physics. The concept was improved, new features were unveiled, and black holes were baptized with the name we know them by today.

By the end of the sixties, the main understanding was that black holes were dense astronomical objects with a strong gravitational field wrapped by a fictitious unidirectional membrane, the event horizon , which allowed the entrance but not the escape of matter and radiation from its interior and hid a singularity inside it. A black hole could be described entirely by three classical parameters: mass, charge, and angular momentum. A Schwarzschild black hole has no angular momentum and no charge. Rotating black holes are known as Kerr black holes or, if charged, Kerr–Newman. A rotating black hole would have an exterior region in which the reference frame rotates alongside the black hole, causing an effect called frame-dragging . This region was named ergosphere in 1971. Footnote 1

One influential person who contributed to the development of the concept of black holes during the sixties was Roger Penrose. Among many achievements, he showed that black holes could and would be formed as a consequence of the theory of general relativity [ 36 ], and it earned him a Nobel Prize in Physics in 2020 [ 26 ]. In 1969, Penrose added another layer to the theory of black holes when he proposed a mechanism to extract energy from a rotating black hole. The Penrose process would happen in the ergosphere, where a particle dropping from infinity would split into two components. One would fall into the black hole, while the other would escape with more mass energy than the original particle had, and thus, the rotational energy of the black hole would be transferred to the motion of this particle outside the ergosphere [ 38 , p. 45]. The energy extraction of a rotating black hole would slow it and decrease its mass, a process that would not be indefinite.

Penrose’s mechanism allowed a comparison between black-hole mechanics and thermodynamics—a correlation first accepted as a pure analogy. Penrose’s proposal was not the origin of this idea, but its pivotal moment—the missing piece in the puzzle of a consistent formulation of a thermodynamic theory of black holes. One student of John Archibald Wheeler had attempted to develop this theory but could not propose a robust formulation. Progress came after the understanding of Penrose’s mechanism and ingenious use of information theory. Meanwhile, the Soviet physicists were looking at the Penrose process through the lenses of quantum field theory, interpreting it as amplification of waves. A complete, semi-consistent description of the thermodynamics of black holes arose from a collaboration between Western and Soviet scientists, partially impaired by the Cold War.

This article puts Penrose’s proposal of a mechanism to extract energy from a rotating black hole into a historical context—which does not imply causation, but that had undeniable influence. The focus, however, will be on the ramifications of the formulation of the Penrose process. Starting from its presentation, then to the chain of reasoning that led first to an analogy between black-hole physics and thermodynamics and finally to the theoretical prediction of the Hawking radiation. This last phenomenon convinced the physics community that black holes might be indeed thermodynamical entities. The objective is to shed light on the motivation and methodology that led to this perception. For this, we shall revisit the original papers and analyze technical details to uncover the history they hide.

The history of the quantum phase of research on black holes is briefly outlined in [ 30 ], while [ 8 ] analyzes the use of information theory to assess the thermodynamical description of those objects. The friendly relationship between Western and Soviet physicists is described in the recollections of physicists who witnessed the events, like Kip Thorne [ 45 ] and Igor Novikov [ 34 ]. In [ 31 ], there is a historical review of the old Soviet academic practices and a glimpse at Yakob Zel’dovich’s early career.

The organization of this paper is as follows: Section  2 presents the historical details of the Penrose process, and Sect.  3 retraces the origins of the theory of black-hole thermodynamics. Section  4 focuses on the work of Jacob Bekenstein and his interpretation of this theory, while Sect.  5 explores his formulation of the three laws of black-hole thermodynamics. The criticism of his proposal is presented in Sect.  6 , contrasting with the acceptance of an analogy between black-hole mechanics and thermodynamics. Yakob Zel’dovich’s views of the Penrose process through the lenses of quantum field theory are outlined in Sect.  7 . Finally, Sect.  8 details the combination of Bekenstein’s and Zel’dovich’s works by Stephen Hawking, ending with his conclusion that black holes should evaporate.

2 The Penrose process

The inaugural congress of the European Physical Society—held between 8–12 of April 1969 in Florence, Italy—was a game-changer for the physics of black holes. Themed “The growth points of physics,” the package “astronomy, astrophysics, cosmology and relativity” was the first item of concern addressed there [ 32 ]. One lecturer in Florence was Roger Penrose, who did an extensive survey on this topic at the conference, exposing the characteristics and key properties known to that date [ 37 ]. More than that, he introduced a feature that set the course of research in the following years: a mechanism to extract rotational energy of a Kerr black hole, today known as the Penrose process.

The second half of the twentieth century was also a period for substantial technological progress. During the Cold War, energy consumption increased significantly, and the exploitation of nuclear [ 43 ] and fossil energy sources became a vital discussion in the political scenario [ 39 ]. Meanwhile, the space race was intensified when, in 1961, the Soviet cosmonaut Yuri Gagarin became the first human to leave the Earth. In 1969, Americans Neil Armstrong and Buzz Aldrin became the first men to step on the Moon [ 12 ]. In this context, the Dyson sphere became a symbol for extraterrestrial advanced energy exploitation. Freeman Dyson proposed in 1960 an alternative for scanning radio signals in the search for extraterrestrial intelligent life [ 19 ] in the form of detection of infrared radiation. He conceptualized an apparatus that an advanced civilization could use to harvest energy from stars and suggest searching for intelligent life by detecting its characteristic radiation. The Dyson sphere was a habitable thick spherical shell built around a star, with enough machinery to exploit the solar radiation falling onto it from inside.

By the end of his lecture in Florence, Penrose proposed a similar apparatus with a query about energy exploitation. “I want to consider the question of whether it is possible to extract energy out of a ‘black hole’ ” [ 37 , p. 270]. In a thought experiment, he imagined a convoluted system to demonstrate that it was feasible to harvest the energy from a rotating black hole. His first proposal of this apparatus was much like Dyson’s: two enormous structures around a black hole built by an advanced civilization, one moving in the direction of rotation, while the other remained still. Penrose gave an intuitive explanation of how it would work (Fig.  1 ), but he admitted the intricacy of his experiment: “Of course, [it] is hardly a practical method! Certain improvements may be possible, e.g. , using a ballistic method” [ 37 , p. 272].

figure 1

Extracted from [ 37 , p. 271]. “We imagine a civilization which has built some form of stabilized structure S surrounding the ‘black hole’. If they lower a mass slowly on a (light, inextensible, unbreakable) rope until it reaches L, they will be able to recover, at S, the entire energy content of the mass. If the mass is released as it reaches L then they will simply have bartered the mass for its energy content. (This is the highest-grade energy, however, namely wound-up springs!) But they can do better than this! They also build another structure S*, which rotates, to some extent, with the ‘black hole’. The lowering process is continued, using S*, to beyond L. Finally the mass is dropped through H, but in such a way that its energy content, as measured from S, is negative! Thus, the inhabitants of S are able, in effect, to lower masses into the ‘black hole’ in such a way that they obtain more than the energy content of the mass. Thus, they extract some of the energy content of the ‘black hole’ itself in the process. If we examine this in detail, however, we find that the angular momentum of the ‘black hole’ is also reduced. Thus, in a sense, we have found a way of extracting rotational energy from the ‘black hole’ ”

The mechanism of the Penrose process depends on the fact that inside the ergosphere, there exist orbits of negative energy when observed from infinity, while the orbits outside this region are of positive energy at infinity. This happens because the metric inside the ergosphere is such that Killing vectors inside it are space-like and those on the outside are time-like. Thus, by lowering an object’s orbit from outside to inside the ergosphere, to near the horizon, it can acquire negative energy. The energy it lost could, in theory, be retrieved by the ones who send the object.

By the time of publication in a special edition of the Rivista del Nuovo Cimento dedicated to that conference, Penrose had already worked out a ballistic description of the mechanism. The idea revolved around a particle \(p_0\) dropping from infinity to inside the ergosphere—in Fig.  1 , it is the region from L to H , in between the absolute event horizon and the sphere of stationary solutions—and nearing the event horizon H , “at which point the particle splits into two particles \(p_1\) and \(p_2\) . The particle \(p_2\) crosses H , but \(p_1\) escapes ( . . . ) possessing more mass-energy content than \(p_0\) !” [ 37 , p. 272] Throughout the years, Penrose refined the explanation of this phenomenon (Fig.  2 ).

figure 2

Extracted from [ 38 , p. 272]

More than speculation, Penrose aimed to strengthen the search for mechanisms to detect black holes in space by improving the existing theory. He added that its “real significance is to find out what can and what cannot be done in principle since this may have some indirect relevance to astrophysical situations” [ 37 , p. 272]. It indeed led to a theoretical development that Penrose himself could not have foreseen, which changed the perception of those objects. It was a necessary step toward a thermodynamic description of black holes.

3 Wheeler’s boys

John Archibald Wheeler’s interest in black holes started in the late fifties. First a critic of the idea, he grew to become one of the biggest enthusiasts of the concept and later popularized the term “black hole.” By the end of the sixties, a conjecture—first proposed and partially proved by physicist Werner Israel in 1967 [ 27 ]—intrigued Wheeler. The no-hair theorem, as Wheeler named it, says a black hole can be described by only three classical parameters. As framed in [ 33 , p. 876], “ the external gravitational and electromagnetic fields of a stationary black hole (a black hole that has settled down into its ‘final’ state) are determined uniquely by the hole’s mass M , charge Q , and intrinsic angular momentum S —i.e., the black hole can have no hair (no other independent characteristic).”

Not only was Wheeler a brilliant physicist, but he also became a prime supervisor and educator with a natural charisma for science communication. He used to meet with his research students once a month and regarded them as collaborators rather than apprentices. Wheeler often informally suggested various topics of research for them to explore and encouraged their creativity [ 8 ]. In 1969, on the project for the research group under his supervision, Wheeler exposed his concerns about black-hole physics [ 47 , p. 28]. As a reaction to the no-hair theorem, one of Wheeler’s specific research topics that year was to “[a]nalyze the meaning of ‘baryon conservation’ and ‘lepton conservation’ and the principle that ‘entropy always increases’ in the context of ‘black hole physics’, where complete collapse washes out all these familiar features of physics” [ 47 , p.30]. According to the no-hair theorem, a black hole has exactly three degrees of freedom. Thus, Wheeler asked how this constraint would fit within the known laws of conservation and thermodynamics.

Wheeler himself did not pursue this thread, however, delegating the task to his students. In his notebooks, he explained why he did not tackle issues like black-hole physics himself. “My reason? Does not reach to the heart of the mystery: pregeometry” [ 48 , p. 17]. At the time, he had a broader research agenda, working on a geometrical description of physics he called pregeometry . In his words, “we’ll accept Einstein’s general relativity, or ‘geometrodynamics’ in its standard 1915 form, translated of course into the appropriate quantum version. Second, we accept as tentative working hypothesis the picture of Clifford and Einstein that particles originate from geometry; that there is no such thing as a particle immersed in geometry, but only a particle built out of geometry” [ 49 , p. 32]. Footnote 2

The first of Wheeler’s students to touch on the subject was an undergraduate named Jeffrey M. Greif. In his 1969 Junior thesis Black Holes and Their Entropy , Footnote 3 he proposed a definition for the entropy of a black hole. He suggested it would be the limit of the entropy of a collapsing body right before reaching the singular radius. This definition had many issues. Most noticeable, it did not reconcile with the no-hair theorem. If the entropy depended on the state prior to collapse, the loss in degrees of freedom would be paradoxical. Jacob Bekenstein later noticed that “[Greif] examined the possibility of defining the entropy of a black hole, but lacking many of the recent results in black-hole physics, he did not make a concrete proposal” [ 5 , p. 2333].

One year later, Greek physicist Demetrios Christodoulou took advantage of the most recent tool available at the time to advance one step further in the direction pointed by Greif. Self-taught in physics, Christodoulou got a position as a graduate student in Princeton in 1968 under Wheeler’s guidance after finishing high school. In 1970, he published an important report, Reversible and Irreversible Transformations in Black Hole Physics , where he considered the loss of mass of a Kerr black hole in the occurrence of the Penrose process [ 14 ]. This extraction of rotational energy, Christodoulou argued, was not indefinite. It would end when the Kerr black hole turned into a Schwarzschild one, that is, when it stopped spinning. Christodoulou explained that losing angular momentum would lead to a decrease in mass and asked which would be the smallest Schwarzschild black hole derived from this process. He named its mass the irreducible mass of the black hole.

figure 3

Extracted from [ 15 ]

Conversely, given a Schwarzschild black hole, one could add rotation by adding massive particles carrying angular momentum. Christodoulou considered this case and the possibility of reversing this action by means of the Penrose process. A Schwarzschild–Kerr–Schwarzschild conversion that results in the original black hole (same mass, no angular momentum) would configure a reversible transformation . Otherwise, it would be an irreversible transformation . For the latter, Christodoulou analyzed that the final Schwarzschild black hole would always have more mass than the first (Fig.  3 ). Therefore, he concluded the irreducible mass remains constant for reversible transformation, increases for irreversible ones, and never decreases.

Christodoulou remarked that the energy harvesting from a black hole had limited efficacy. In fact, he pointed to the fact that 29% of the mass of an extreme Kerr black hole could be harvested as energy, “far more energy conversion than in nuclear fission or fusion” [ 15 , p. 66]. For a charged black hole, this value can reach up to 50% [ 3 , p. 104]. Christodoulou theorized that the rest would be released as relativistic energy highly ionized, like the relativistic jets escaping from the nuclei of some radio galaxies.

In 1971, Stephen Hawking, investigating the properties of the gravitational radiation generated by the collision of two black holes, discovered that the surface area of the event horizon of the resulting black hole is strictly greater than the sum of their individual areas before they merged [ 23 ]. This, he argued, can also be applied for the case of a single black-hole absorbing particles, and thus, he concluded that the surface area of a black hole never decreases. On this subject, Christodoulou remarked in his thesis, “We see, in Hawking’s theorem, a strong analogy between the surface area of a black hole and its irreducible mass. Both quantities cannot be reduced. What is the connection between the two quantities?” [ 15 , p. 95]. His answer:

where A is the surface area of the event horizon, and \(m_{\text {ir}}\) is the irreducible mass of the black hole.

The third of Wheeler’s students to contribute to the subject was the young Mexican-Israeli physicist Jacob Bekenstein, who revolutionized the field, giving a new meaning to Christodoulou’s result.

4 A crazy-enough idea

The connection between black-holes physics and thermodynamics was a seed sprouting in Princeton by the end of the sixties. As mentioned before, Wheeler would often instigate his students to pursue this thread, and to Jacob Bekenstein, he posed an adapted version of an old thermodynamic thought experiment. “What would happen, Wheeler asked Bekenstein, if you mixed hot tea and cold water—thus maximizing the entropy—and poured it into a black hole? Would the entropy simply vanish, pumped out of physical reality? If so, the ‘crime’ of raising the total entropy would seem to have been pardoned” [ 8 , p. 74]. The beverage, in this case, represents a state of increased entropy in the universe: if it vanished into the event horizon, if the black hole did not have entropy, it would constitute a transgression of the second law of thermodynamics. To find the answer to Wheeler’s query, Bekenstein considered a similar contravention in a problem proposed by James Clerk Maxwell in 1867.

The total entropy of an isolated system always increases over time, and it remains constant only for reversible processes. This is the second law of thermodynamics, and Maxwell proposed a situation in which it would not hold. In the nineteenth century, instead of an advanced civilization, Maxwell considered a demon guarding the gate connecting two isolated compartments filled with gas. This demon could open the gate at the right moment to let the more agitated molecules of A enter B and the slower molecules of B enter A. This would cause A to warm up and B to cool down, decreasing its entropy. Heat would flow from the cooler environment to the warmer one, breaking the second law. This thought experiment became known as Maxwell’s Demon , and thus, Bekenstein named his supervisor’s problem Wheeler’s Demon .

Part of Bekenstein’s PhD thesis was devoted to assessing this question. Maxwell’s Demon was a well-known controversial problem in physics, and Bekenstein came across an intriguing answer to this paradox in a connection between thermodynamics and information theory. In 1950, Leon Brillouin proposed a solution to this dilemma using information theory [ 10 ], which inspired Bekenstein to address Wheeler’s Demon in a similar way. Brillouin suggested an increase in entropy relates to loss of information, adopting this relation as a definition for the concept. Bekenstein compared a black hole to a thermodynamic system in equilibrium and argued that, although from the outside only three classical parameters could be measured, from the inside, the other physical features of matter would be preserved. This inaccessibility of information to an exterior observer, this loss in degrees of freedom, would read as the entropy—as defined by Brillouin—of a black hole [ 5 , p. 2336].

According to the theorem by Hawking and Christodoulou, adding mass to a black hole would increase its surface area and decrease the amount of information an observer on the outside has of its interior. Bekenstein argued that the irreducible mass of a black hole would straightforwardly correlate to entropy, according to information theory. The terminology used by Christodoulou, reversible and irreversible transformations, hinted at a connection between his results and the second law of thermodynamics. Bekenstein formalized it: “The irreversibility of an increase in the area of a black hole is reminiscent of the irreversibility of an increase in the entropy of a closed thermodynamic system” [ 3 , p. 104].

Compared to Greif’s early attempt, Bekenstein proposed a more concrete definition of the entropy of a black hole. And he availed from the developments not available to Greif at the time, mainly the Penrose process, which led to the Christodoulou–Hawking theorem. Although the correlation between black-holes physics and thermodynamics was welcomed, Bekenstein’s approach also received plenty of criticism. He later remarked: “In those days in 1973 when I was often told I was headed the wrong way, I drew some comfort from Wheeler’s opinion that ‘black holes thermodynamics is crazy, perhaps crazy enough to work’ ” [ 7 , p. 28].

5 Bekenstein and the three laws of black-hole thermodynamics

Following up on Wheeler’s thought experiment, Jacob Bekenstein proposed the connection between thermodynamics and black-holes physics in his PhD thesis, Baryon Number, Entropy and Black Hole Physics . He argued this parallelism could help predict “new effects in black holes physics by analogy with familiar results in thermodynamics” [ 3 , p. 107] and exposed the equivalent of the three laws of thermodynamics for this case.

From Christodoulou’s equation ( 1 ), the entropy of a black hole is proportional to its irreducible mass or equivalently to the area of its event horizon. Thus, Christodoulou–Hawking theorem—the area of a black hole’s surface never decreases—reads as the second law.

The first law was not as evident as the second. It is a conservation law that says the energy of an isolated system is constant. As an equation, it translates as

where E , T , S , P , and V are the energy, temperature, entropy, pressure, and volume of the system, respectively. For a charged, rotating black hole with mass M , charge Q , and angular momentum \(\mathbf {J}\) , Footnote 4 Bekenstein considered its rationalized area, Footnote 5

where a is the angular momentum per unit, that is, \(\mathbf {a} = \mathbf {J}/M\) . The differentiation of Eq. ( 3 ) results in,

Bekesntein concludes that \(\mathbf {\Omega }\) and \(\Phi \) play the role of the rotational angular frequency and electric potential of the black hole, respectively.

Comparing this mass formulae ( 4 ) with Eq. ( 2 ), Bekenstein identified \(A/4\pi \) —the square of the irreducible mass \(m_{ir}\) from Christodoulou’s formulæ  Eq. ( 1 )—as entropy and \(\kappa /2\) as temperature. Footnote 6 In Eq. ( 4 ), the term \(\mathbf {\Omega } \cdot d\mathbf {J} + \Phi dQ\) represents the work done on the black hole by an external agent, increasing the angular momentum and charge by \(d\mathbf {J}\) and dQ , respectively. Thus, it would correspond to \((-PdV)\) , which is the work done on a thermodynamic system [ 5 , p. 2335]. With these identifications, the energy content falling into a black hole would be described by parameters depending only on the mass, charge, and angular momentum, in agreement with the no-hair theorem.

The third law came from the identification of \(\kappa /2\) as the temperature of a black hole. For a generalized Kerr black hole, the temperature is always nonnegative, and \(\kappa = 0\) holds only if the black hole is extreme, that is, if for a given charge and angular momentum, it has minimal mass. Bekenstein considered the process of accretion of selected particles to transform an ordinary black hole into an extreme one. About this transformation, he concluded that “the nearer [ \(\kappa \) ] is to zero ( . . . ), the less efficient is further accretion in lowering [ \(\kappa \) ].” This, he argued, is equivalent to the third law of thermodynamics, which he phrased as “the nearer a system’s temperature T is to absolute zero, the less efficient is any process in lowering T further.” And “[t]hus we find an analog to for each of the three laws of thermodynamics in black hole physics!” [ 3 , pp. 106–107].

Coming from Wheeler’s thermodynamic thought experiment, Bekenstein questioned whether the correlation between irreducible mass and entropy could be regarded as more than comparison. “Is there a deeper connection between the two subjects? Clearly the concepts of energy and work have similar meanings in thermodynamics and black hole physics. But does the rationalized area of a black hole (the area divided by \(16\pi \) ) have anything to do with entropy?” [ 3 , p. 107]. Bekenstein argued yes, it does. He said, “In everyday physics entropy goes hand in hand with statistics, information, and degradation of energy,” via Boltzmann’s formulæ  for entropy, Brillouin’s informational explication, and the introduction of black-hole entropy as a measure of degradation of its energy [ 3 , pp. 129–130]. He proceeded to compare a black hole to a black body. Bekenstein formalized this view in 1973 when he said that “so far the analogies have been of a purely formal nature, primarily because entropy and area have different dimensions” [ 5 , p. 2335]. To obtain a formula for the entropy of a black hole with correct units, his strategy again relied on information theory, identifying information loss as an increase in entropy.

6 A good analogy

The correlation between thermodynamics and black-hole physics spiked the curiosity of John M. Bardeen, Footnote 7 Brandon Carter, and Stephen Hawking, who gathered to work on a formulation of their own at the Les Houches Summer School on black holes in 1972 [ 30 ] and resulted on a paper published in the following year [ 2 ]. In it, they proposed four laws of black-holes mechanics, comparing them with the four laws of thermodynamics. In a more mathematical approach, they explored the geometry of the Kerr solution to calculate the mass and angular momentum in terms of unique time translational and rotational Killing vectors, from where they derived the ensuing laws.

The zeroth law states that the proper acceleration experienced by a test particle near a stationary black hole due to its gravitational pull with a redshift correction, the surface gravity \(\kappa \) , is constant over the event horizon. This is the same constant \(\kappa \) which appears in Eq. ( 4 ), Footnote 8 but Bekenstein did not straightforwardly identify it this way, a fact Bardeen, Carter, and Hawking did not fail to notice. “We show that the quantities appearing in (Eq.  4 ) formulae have well-defined physical interpretation” [ 2 , p. 162].

The other three laws are similar to Bekenstein’s. The first law establishes that \(\kappa /8 \pi \) is analogous to the temperature in the same way the area A is analogous to entropy, departing slightly from Bekenstein’s definition. The second law refers to the fact that the surface area of each black hole does not decrease with time. And finally, the third law, left unproven, Footnote 9 is the same as the one Bekenstein had already obtained, rewritten as “It is impossible by any procedure, no matter how idealized, to reduce \(\kappa \) to zero by a finite sequence of operations” [ 2 , p. 169].

Despite the agreement with Bekenstein’s results, Bardeen, Carter, and Hawking could not stress enough that this correlation was nothing but a good analogy. “It should however be emphasized that \((\kappa /8\pi )\) and A are distinct from the temperature and entropy of a black hole,” otherwise, they argued, it should radiate. “In fact the effective temperature of a black hole is absolute zero. One way of seeing this is to note that a black hole cannot be in equilibrium with black body radiation at any non-zero temperature, because no radiation could be emitted from the hole whereas some radiation would always cross the horizon into the black hole” [ 2 , p. 168]. Their purely geometrical approach to the problem, analyzing a stationary axisymmetric solution of Einstein’s equation containing a black hole and surrounded by matter to derive the results, indicates the lack of a deeper connection with thermodynamics. At the same time, Bekenstein reasoned outside of a gravitational theory to understand these laws as more than an analogy. In a recollection told to Werner Israel, Hawking admitted that “in writing this paper [with Bardeen and Carter], I was motivated partly by my irritation with Bekenstein who, I felt, had misused my result about increase of the area of the event horizon” [ 28 , p. 264].

Bekenstein did not seem to consider the point raised by Bardeen et al. about the temperature a troublesome issue. In fact, he argued that a black hole could be regarded as a black body cavity with zero temperature. Therefore, Bekenstein agreed that the surface gravity was not to be regarded as the real temperature, calling it a characteristic temperature [ 6 , p. 3295]. The increase of the black hole’s entropy (its surface area), he argued, would compensate for the loss of thermal entropy [ 3 , p. 128]. If \(\kappa /2\) was not the actual temperature, there should not be any radiation associated with it.

A point Bekenstein perceived as the most serious problem with his view was the violation of the second law of thermodynamics. Ironically, it was the question he was trying to solve in the first place. The black hole’s entropy would increase with accretion of matter, causing the entropy of the visible universe to decrease. “It would seem that the second law of thermodynamics is transcendent here in the sense that an exterior observer can never verify by direct measurement that the total entropy of the whole universe does not decrease in the process” [ 5 , p. 2339]. This fault was avoided in Bardeen, Carter, and Hawking’s interpretation of this description as an analogy, as the surface area was not actually regarded as entropy. In a follow-up four-page report published in 1972, Black Holes and the Second Law , Bekenstein reformulated this law: “Common entropy plus black-hole entropy never decreases” [ 4 , p.738].

Physicist Werner Israel also positioned himself against Bekenstein’s views. “Black-hole dynamics has many points of resemblance with thermodynamics. ( . . . .) It is perhaps tempting to look for a deeper significance in this parallel. ( . . . .) If such a relationship holds it would have far-reaching consequences, for example in cosmology, where it would largely obviate the complications arising from having to treat a universe containing black holes as an open thermodynamical system” [ 28 , p. 267]. Israel argued that Bekenstein could not properly connect entropy influx with energy influx—something Bekenstein had done using information theory—and thus, an increase in the area would not equate to an increase in entropy. Israel proposed that the presence of an external potential energy could disrupt this connection. Bekenstein’s use of information theory was innovative and, as such, hard to grasp intuitively. Years later, Israel reflected on his own assessment, saying Bekenstein had “convoluted arguments which, at best, were just short of compelling and, in less fortunate instances, an easy target for criticism” [ 30 , p. 263].

Bekenstein further developed his formulation [ 6 ] in the following years. The description of thermodynamics as a statistical discipline and the interpretation of entropy as information loss guided his intuition. Although the proposal was over twenty years old at the time, the use of information theory to assess thermodynamics found one of its first application with this case [ 8 ], and this may explain why others were skeptical with the possibility of the thermodynamic description of black holes being more than an analogy.

7 A different angle

Since the rise of relativistic astrophysics in the sixties, the Soviet community had been influential in developing black-hole physics. The Cold War pushed a mostly friendly rivalry with Western physicists and highlighted the differences between their scientific backgrounds. In the West, the Penrose process allowed the combination of thermodynamics and black-hole mechanics. Meanwhile, the investigation of this mechanism to extract rotational energy from a Kerr black hole through the lenses of quantum field theory in the Soviet Union led to a wave interpretation of this phenomenon instead of Penrose’s ballistic approach.

Self-taught in science, Yakov Borisovich Zel’dovich rose quickly to Academician, the greatest academic position in Soviet Union [ 34 ]. He made major contributions to the fields of physical chemistry, nuclear physics, thermodynamics, and cosmology. In 1946, he became the head of the theoretical department of the military project Arzamas-16 and led the research of atomic weapons, working together with Andrei Sakharov on the conception of the Soviet hydrogen bomb [ 46 ]. In the sixties, Zel’dovich turned his attention to astrophysics, researching, among other things, the formation of black holes. He suggested, for example, the presence of an accretion disk around massive black holes [ 50 ].

As recalled by Kip Thorne [ 45 , p. 428], Zel’dovich learned about the Penrose process from discussions with Thorne himself, Charles Misner, and John Wheeler. In June 1971, Thorne visited Moscow, and he reported one particular interaction with Zel’dovich during that Summer, in which the latter proposed that rotating black holes should radiate. In Zel’dovich’s intuition, Thorne explained, the soviet physicist compared a black-hole system to a massive, rotating sphere inside a cylindrical surface. In his analogy, the sphere was the black hole and the interior of the cylinder, the ergosphere. The rotation of the spherical object would reflect and amplify an incident wave, using a fraction of its rotational energy to do so. This would perturb the vacuum fluctuation inside the cylinder and lead to spontaneous radiation.

Zel’dovich formalized this idea in a short paper published in a Soviet Journal in 1971. Although Thorne credited Zel’dovich’s inspiration to be the case of energy extraction of a black hole, the Academician mainly discussed the analogue experiment—the system of plane waves incident on a cylindrical shell with a rotating mass inside it. Zel’dovich concluded that not only the amplification but also the generation of waves would occur. “In the presence of an external reflector with small losses (resonator), the amplification following single scattering may turn into generation.” For a black hole described by a Kerr solution, he added that “in a quantum analysis of the wave field one should expect spontaneous radiation of energy and momentum by the rotating body. The effect, however, is negligibly small” [ 51 , pp. 180–181]. Zel’dovich also explained the phenomenon intuitively with a particle interpretation, arguing that the vacuum fluctuation would create a spontaneous pair of particle/anti-particle and the rotation would make this production unstable, and more particles would escape to infinity instead of falling into the horizon. Zel’dovich went beyond the Penrose process and, with a quantum perspective, predicted that rotating black holes should radiate.

The strategy of exploiting analogue experiments to infer properties of an equivalent system is to this date controversial [ 17 ], but Zel’dovich had done it before to study stellar configurations. He and Thorne reportedly discussed the physics of rotating starts using a system of a rotating rigid body as an analogue. In 1972, Brandon Carter published a report in favor of this strategy [ 13 ], arguing that when applied to the case of black holes, it replicated known features, like the Penrose process and Christodoulou’s reversible transformation. In this same year, Zel’dovich released an extended version of his 1971 paper with the title Amplification of cylindrical electromagnetic waves reflected from a rotating body [ 52 ]. Footnote 10 This time, he considered the analogy to black holes, but he did not mention, however, the spontaneous radiation again.

Zel’dovich’s papers did not generate attention. Even at that year’s event dedicated to black holes, the Les Houches Summer School of Theoretical Physics —which some of his associates attended, like Thorne and Soviet astrophysicist Igor Novikov—his results were not mentioned [ 45 , p. 433]. Both of Zel’dovich’s papers were published in Soviet journals, which probably contributed to this perceived disinterest. Another reason for this may have to do with Zel’dovich’s caution in exploring the phenomenon of energy extraction for the case of black holes, instead, focusing on the analogue systems, refraining from pursuing his bolder assumptions.

In 1973, Alexei Starobinskii, at the time a postdoctoral student of Zel’dovich, revisited his supervisor’s result with a more compelling title, Amplification of waves during reflection from a rotating “black hole.” In a different manner than Zel’dovich, Starobinskii dealt explicitly with black holes and the incidence of classical waves, properly identifying their amplification as the Penrose process [ 42 ]. In September that same year, Stephen Hawking visited Moscow for the first time. In his hotel room, with Zel’dovich and Thorne as witnesses, Starobinskii presented his and his supervisor’s results to Hawking, who was unaware of them. To Israel [ 28 , p. 264], Hawking recalled that he liked the idea but was not convinced by the calculations. He mentioned Starobinskii had told him about spontaneous radiation—a phenomenon which Starobinskii himself had not discussed in his paper. Hawking reportedly thought their arguments had physical ground, but he remained skeptical. Inspired by Zel’dovich’s and Starobinskii’s ideas, Hawking worked on this problem back home. His conclusion, he said, really annoyed him: all black holes must radiate.

8 The Hawking radiation

For the case of a Kerr black hole, Starobinskii repeatedly mentioned Zel’dovich’s analogue system but avoided his supervisor’s quantum approach. Instead, he considered a semiclassical scalar field satisfying a relativistic wave equation—the Klein–Gordon equation. As expected, the amplitude of the reflected wave was greater than of the incident one. It was a description of the Penrose process, Starobinskii argued, without the need to consider composite or unstable particles to work.

After the visit to Moscow, Stephen Hawking had Starobinskii calculations and Zel’dovich’s quantum analogue system in his mind. The Soviets convinced him that quantum effects should be considered when studying the evolution of black holes, and Hawking included this premise in a brief report he wrote to Nature [ 24 ]. He opened this article titled Black holes explosions? , acknowledging that those effects were usually ignored because near the event horizon of any black hole, the radius of space-time curvature is disproportionally large compared to Planck scale, and thus these contributions were locally small. Nevertheless, he reasoned that “they may still, however, add up to produce a significant effect over the lifetime of the Universe” [ 24 , p. 30].

Hawking adopted a strategy different than Starobinskii’s. Whereas the Soviets thought of amplification of waves, Hawking thought of creation and annihilation of particles. He calculated the rate of particle production near the horizon of a Schwarzschild black hole, using simple quantum field theory. He found the number of particles created and emitted to infinity was greater than those annihilated or absorbed, and this difference would amount with time to produce a steady rate of emission. He estimated the emission rate of bosonic particles to be \(( \exp (2 \pi \omega / \kappa ) - 1 )^{-1}\) , the same expected for a body with temperature in geometrical units of \(\kappa /2 \pi \) .

With this, not only rotating black holes would radiate, as suggested by Zel’dovich, but all black holes would. The prediction of the radiation was the last piece of theoretical evidence to convince Hawking that black holes are thermodynamical entities. “Bekenstein suggested on thermodynamic grounds that some multiple of \(\kappa \) should be regarded as the temperature of a black hole. He did not, however, suggested that all black hole could emit particles as well as absorb them. For this reason Bardeen, Carter and I considered that the thermodynamical similarity between \(\kappa \) and temperature was only an analogy. The present result seems to indicate, however, that there may be more to it than this” [ 24 , p. 31].

The change in status of the thermodynamical analogy of black holes is a key step for understanding its acceptance. This fact is nicely expressed in the words of Kip Thorne, “This [the Hawking radiation] (if Hawking was right) was incontrovertible proof that Bardeen-Carter-Hawking laws of black-hole mechanics are the laws of thermodynamics in disguise, and that, as Bekenstein had claimed two years later, a black hole has an entropy proportional to its surface” [ 45 , p. 436]. The use of analogues to assess different physical processes is a common practice but presents limitations to what kind of information one can extract about the original system. To derive the four laws of black holes as an analogue to thermodynamics was a valid option to study the mechanics of this inaccessible astrophysical system, although it had the constraint of being only an analogy. With the identification of black holes as black bodies, the problem of having to transfer the knowledge from a field to another was solved. Footnote 11

Zel’dovich intuitively predicted that rotating black holes would radiate, and Hawking extended his prediction to all black holes. This difference comes from their approaches to the problem. Zel’dovich’s wave description in his analogue system relied on the angular momentum of the massive sphere to destabilize the quantum vacuum fluctuation near the cylindrical shell, here representing the event horizon, causing the emission of particles. Hawking, on the other hand, considered the changing geometry of the space-time near the event horizon instead of the rotation of the black hole to calculate the absorption and emission rates. Hawking was not the first to use quantum field theory in a cosmological scenario. Years before, the American physicist Leonard Parker pioneered [ 35 ] investigating particle creation in an expanding universe, but Hawking did not cite him in his papers.

Hawking published an extended version of his report a year later, entitled Particle Creation by Black Holes [ 25 ], extending the ideas proposed in the first paper. He reinforced that black holes should radiate with the temperature given by a factor of the surface gravity, and its mass would decrease in the process as the emission rate increased. This, Hawking concluded, meant that black holes not only radiate, but they can also evaporate. He explained, “For a black hole of solar mass ( \(10^{33}\) g) this temperature is much lower than the \(3\,\) K temperature of the cosmic microwave background. Thus black holes of this size would be absorbing radiation faster than they emitted it and would be increasing in mass. However, in addition to black holes formed by stellar collapse, there might also be much smaller black holes which were formed by density fluctuations in the early universe. These small black holes, being at a higher temperature, would radiate more than they absorbed. ( . . . ) When the temperature got up to about \(10^{12}\,\) K or when the mass got down to about \(10^{14}\) g the number of different species of particles being emitted might be so great that the black hole radiated away all its remaining rest mass on a strong interaction time scale of the order of \(10^{23}\) s” [ 25 , p. 202].

Despite providing a necessary argument for black holes to be understood as black bodies, using Bekenstein’s thermodynamical description and Zel’dovich’s quantum approach, Hawking did not solve every problem with this identification. In fact, he created new ones. For example, if a black hole evaporated, its irreducible mass and surface area would decrease, contradicting the theorem that carries his and Christodoulou’s name. Hawking explained this by saying there is a constant influx of negative energy into the black hole Christodoulou had not considered, which would constitute a violation of the weak energy condition, Footnote 12 caused by the indeterminacy of particle number and energy density in curved space-time [ 25 , p. 203].

The ad hoc merge between general relativity and quantum field theory is also a point of contention, but Hawking’s calculations were solid, and this issue was considered something to be explored instead of being a fundamental crack in the method. Decades later, a debate about the fate of information inside a black hole gained notoriety and exposed another weak spot of the theory. If black holes evaporated, what would happen to the information in their interior? If it also evaporated, it would contradict the postulate that information cannot be lost. This question remains unresolved to this date, and the controversy that ensues it is known as the Black Hole War , thanks to physicists Leonard Susskind and Gerard ’t Hooft [ 44 ].

9 Conclusion

The history of black-holes physics has many pivot points and eventualities that may be a key to understand the scientific practices of highly theoretical areas of knowledge. The development of a thermodynamical theory of black holes, in particular, changed not only our perception of those bodies but also the fields of astrophysics, information theory, and quantum gravity. This paper focused on this first change, outlining the origin of the black-holes thermodynamics idea until the event that solidified this theory as more than a ploy to explore the mechanics of those objects. It identifies the Penrose process as the critical result that allowed the formulation of this thesis and the Hawking radiation as the step to solidify it. This analysis led us full circle around the globe, starting in Europe, with Penrose’s lecture and proposal of a mechanism to extract rotational energy of a black hole; then, to the USA, with John Archibald Wheeler’s exceptional students; to the Soviet Union, with Zel’dovich’s ingenious thinking and interpretation of the Penrose Process; and finally back to Europe, with Hawking’s prediction that black holes should radiate and evaporate.

In the sixties, the concept of black holes gained shape with the discovery of more properties, and candidates in the sky made their existence highly probable. In 1969, Roger Penrose compiled this knowledge—that he actively helped to build and for which he won the Nobel Prize in Physics—in a lecture he presented at the inaugural conference of the European Physical Society. In it, he revised and extended the theory of black holes, when he proposed an additional feature, a mechanism to collect rotational energy from a Kerr black hole. He hoped this would lead to new advances in astrophysics and urged the community to search for a black hole in space. The Penrose process proved to be an essential step toward a thermodynamical description of black holes, developed in the following years.

The idea to investigate thermodynamical properties of black holes was a sparkle in John A. Wheeler’s mind at that point, and he encouraged his students to pursue it. The first to try was Jeffery M. Greif, who could not make a sound proposal. It was only after the suggestion of the Penrose process that a more robust proposition was possible, and with it, another of Wheeler’s students, Demetrios Christodoulou, hinted at a correlation between thermodynamics and black-holes physics. It was Jacob Bekenstein, however, who formalized this connection.

Bekenstein’s use of information theory to assess the problem was pioneering. He observed a parallel between the three laws of thermodynamics and some properties of black-holes physics, the second law tied to the Penrose process via Christodoulou’s reversible and irreversible transformations. Although Bekenstein’s idea was welcomed as a good analogy, he understood it as a genuine connection. He suggested that black holes have a well-defined entropy and are, therefore, thermodynamical entities. This interpretation received many criticisms. Among the critics were John M. Bardeen, Brandon Carter, and Stephen Hawking, who noticed that the temperature in Bekenstein’s theory remained ill-defined and that for this connection to be true black holes would have to radiate.

Meanwhile, in the Soviet Union, Yakov B. Zel’dovich, upon learning about the Penrose process, reinterpreted it in a more familiar way to him, using quantum theory. He proposed an analogue experiment that would replicate Penrose’s mechanism in a system composed of a massive sphere rotating inside a resonant cylindrical shell. Zel’dovich predicted that Kerr black holes would radiate, although he did not pursue this thread. Alexei Starobinskii extended his supervisor’s suggestion focusing primarily on black holes and presented it to Stephen Hawking in 1973. Hawking was intrigued and worked out his own calculations on the subject. To his surprise, he concluded that not only Kerr but all black holes should radiate. More than that, they should eventually evaporate.

The Hawking radiation is the starting point of a new chapter in the history of black-holes physics, the quantum phase of research, as Werner Israel put it [ 30 ]. The thermodynamical properties of black holes remain theoretical, but it opened venues and possibilities yet to be explored, hopefully in the near future.

Data Availability Statement

This manuscript has associated data in a data repository. [Author’s comment: The manuscripts utilized in this research are available at the repository of the Max Planck Institute for the History of Science or Princeton University, except for Wheeler’s notebooks. These are publicly available, and there is a link to them at the Reference.]

For a thorough characterization of black holes by the end of the sixties, see [ 33 ].

Wheeler’s approach to black holes in the sixties is detailed in [ 21 ].

Unfortunately, due to limitations caused by the COVID-19 pandemic, this Princeton manuscript was unavailable for external researchers by the time this article was written. The information presented here comes from Jacob Bekenstein’s description of Greif’s work [ 3 ]. They were colleagues working under Wheeler during an overlapping period.

Bekenstein uses L for the angular momentum. For the sake of clarity, here and throughout the text, a unified notation will be used, overriding the original one.

Equation (1) of [ 23 ] adapted by Bekenstein to the case of a charged black hole.

In the original notation, \(\kappa /2\) is actually denoted by \(\Theta \) , and \(A/4\pi = \alpha \) .

Son of two-times Nobel laureate John Bardeen.

And the reason why this notation was adopted in this text, instead of Bekenstein’s \(\Theta \) .

Werner Israel published a proof of this law in 1986 [ 29 ].

In June 2020, Marion Cromb and collaborators claimed to have shown the effect predicted by Zel’dovich in the analogue system [ 16 ].

The history of the use of analogue systems in physics is part of the work of Rocco Gaudenzi. I am thankful for our enlightening discussions on this topic.

The energy conditions are common assumptions about a generic matter content. In simpler words, the weak energy condition states that the local energy density for any observer is always nonnegative.

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Acknowledgements

I wrote this article while I was working at the Max Planck Research Group Historical Epistemology of the Final Theory Program during the COVID-19 pandemic, and I am grateful for the support and encouragement they gave me during this stressful time, in particular the guidance of Alexander Blum and the patience of Kseniia Mohelsky. I edited this paper during my first months as an International Fellow at the Kulturwissenschafliches Institut (KWI), and I appreciate the support of Julika Griem, Sabine Voßkamp, Sebastian Hartwig, and Britta Acksel. I heartily thank Alexander Blum for helping me in every step of research; Stephan Furlan, Rocco Gaudenzi, Gabriela Radulesco for the insightful discussions; and Ricarda Menn for assisting me with proofreading. Also, I offer my deepest gratitude for the suggestions and corrections by the unknown referees. Thank you!

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Sir Roger Penrose, OM, FRS, of the University of Oxford was presented with a Clay Award for Dissemination of Mathematical Knowledge in recognition of his outstanding contributions to geometry, relativity, and other branches of mathematics, and of his tireless work in explaining mathematical ideas to the public through popular books, public lectures, and broadcasts.

Roger Penrose has made extraordinary and wide-ranging contributions to mathematics and its applications, often making novel and inspiring connections across disciplinary boundaries. He started out in algebraic geometry under W V D Hodge and J A Todd at Cambridge, but within a few years of completing his PhD thesis on  Tensor Methods in Algebraic Geometry  he had laid the foundations of the modern theory of black holes with his celebrated paper on gravitational collapse. For this work he was awarded the Wolf Prize in Physics, with Stephen Hawking. His exploration of foundational questions in relativistic quantum field theory and quantum gravity, based on his twistor theory, had a huge impact on differential geometry. In other areas, a paper that he wrote as a student introduced what is now known as the Moore-Penrose inverse. His discovery of ‘Penrose tilings’ spawned a new field in geometry, as well as having an impact in crystallography. It was Penrose who first suggested that Church’s lambda calculus could be a powerful tool in exploring programming language semantics. In his collaboration with M C Escher, he crossed over into the visual arts.

His work in public engagement goes beyond the simple exposition of mathematics.  In his popular books, he has set out, with considerable success, to engage the public with the development of his ideas and to draw them into an interconnected world of intellectual endeavour which is the common property of all, not just of ‘specialists’. So his books on artificial intelligence aim not just to inform his readers about deep mathematical ideas of relevance to the subject but also to enable and encourage participation of the public in discussion of the nature of consciousness.

He is an energetic public lecturer, travelling widely and attracting large audiences because he speaks in direct and engaging terms about mathematical ideas that he thinks are worth communicating and that his audiences think are worth trying to understand. Many of his lectures have been recorded and made available online, along with recordings of television appearances.  They have been viewed by hundreds of thousands of people.

The Award was presented following a Public Lecture by Roger Penrose on Monday, September 24, 2018, during CMI’s 20th Anniversary Conference.

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Dennis Sciama (1926–99)

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In the years after the Second World War, general relativity theory and cosmology were transformed. Previously seen as only of philosophical interest, they have become central strands in physics, and the new subject of relativistic astrophysics has come into being. Dennis Sciama, who died on 18 December last year, was one of the far-sighted physicists involved in this transition.

Sciama was a student of Paul Dirac, and like him became fascinated with Mach's principle — loosely, the idea that the nature of local physical laws is affected by the state of the whole Universe. Sciama developed a theory of gravity that incorporated this principle, his PhD thesis on the subject being widely quoted. Benefiting from many discussions with Hermann Bondi, Thomas Gold and Fred Hoyle, he developed the broad theme of how local physics relates to the Universe.

Sciama became a passionate believer in the Bondi–Gold–Hoyle steady-state theory (in which the Universe always expands at a constant rate), helping to develop observational tests of that theory. As he emphasized, the theory had the virtue of being disprovable. When it became untenable because of the radio-source and quasar number counts (there are more such sources at large distances than that theory allows), he reluctantly abandoned it. Sciama then became one of the pioneers investigating astrophysical interactions in the evolving and expanding Universe — in particular in studying the way radiation and matter have interacted and how the burgeoning fields of radio and X-ray astronomy could provide information on the thermal history of the Universe.

Here, the transforming event was the discovery in 1965 of the 3 K cosmic blackbody radiation — the relic radiation of the hot big bang — by Arno Penzias and Robert Wilson. This was then tied in to the theory of synthesis of light elements in the early Universe from protons and neutrons, and the theory of growth of large-scale structures (galaxies and clusters of galaxies) in the era after decoupling of matter and the background radiation. The resulting inhomogeneities caused anisotropies in the cosmic background radiation intensity measured today. With Martin Rees, Sciama helped develop the theory of these anisotropies. His considerable understanding of the interlocking set of astrophysical interactions in the early Universe was summarized in Modern Cosmology (Cambridge Univ. Press, 1971).

Sciama's career moves took in the universities of Cambridge, Oxford and Texas (Austin), and then the International School of Advanced Studies (SISSA) in Trieste, his interests becoming increasingly broad. They encompassed the structure of radio sources and quasars, X-ray astronomy, the physics of the interstellar and intergalactic medium, astroparticle physics, and the nature of the mysterious dark matter that pervades the Universe, as well as the thermodynamics of black holes and the nature of the vacuum in quantum theory. He also worked on gravitational theory. In particular, he investigated the inclusion of torsion in the theory (allowing for a kind of intrinsic spin in the geometry) and developed an intriguing integral formulation of the Einstein field equations.

More recently, Sciama worked hard on a scheme in which the dark matter in the Universe is taken to consist mainly of massive neutrinos which decay with a long half-life. These neutrinos produce photons, which could then be responsible for a range of observable astrophysical phenomena such as ionization of intergalactic gas. He formulated this theory so that it too was eventually falsifiable; it was a sadness for him when observations failed to support it.

The real impact of Sciama's career lay, however, not in his own contributions, impressive as they are, but in his creation of an enormously influential school of students. Sciama devoted his life to finding high-quality, dedicated students and helping them in their careers. He supervised over 70 PhD students, among them Stephen Hawking, Brandon Carter (formulator of the Anthropic Principle in cosmology), Sir Martin Rees, Philip Candelas, John Barrow and David Deutsch (originator of quantum computing). Students of students number well over 180, including such luminaries as Roger Blandford, Craig Hogan, Nick Kaiser and Peter Coles. A ‘family tree’ of this impressive group of relativists and astrophysicists is given in the Sciama Festschrift, The Renaissance of General Relativity and Cosmology , edited by myself and colleagues (Cambridge Univ. Press, 1993).

Sciama inculcated in his students the importance of physical and astrophysical understanding, the significance of rigorous mathematical analysis whenever this is possible, and the power of combining the two in a way leading to testable predictions. He encouraged them to be both adventurous and rigorous, and to make their thinking relevant to the physical problem at hand.

And as well as having his own students, he was a proponent of relativistic astrophysics and cosmology wherever he went. Indeed it was Sciama who interested Roger Penrose in gravitation theory when they were together at Cambridge, thereby opening the way to the systematic study of the causal properties of space–times and the famous Penrose–Hawking singularity theorems. These showed that (according to classical general relativity theory) there is indeed a beginning to the Universe, as well as an end to space–time when a massive object collapses to a black hole.

Sciama was a warm person with a love for the civilized things in life. He enjoyed good company, music, and opera, in the later part of his life often retreating to his flat in Venice where, with his wife Lydia, he enjoyed the Venetian lifestyle. When at work he always displayed an intellectual determination to get to grips with the issues concerned, bringing a formidable knowledge of physics and mathematics to bear in this endeavour. His forte was working on these problems with his students and colleagues. He was like a father to many of them, and will be missed both personally and as an inspirational figure in the field of study he loved so much.

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Works by Roger Penrose

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I was a student of Stephen Hawking’s – here’s what he taught me

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roger penrose phd thesis pdf

Like many students of my generation, Stephen Hawking had already had enormous influence on me long before we ever met. When I was hesitating about my A-level choices, it was his book A Brief History of Time that convinced me to continue with physical sciences. In 1994, Hawking and mathematical physicist Roger Penrose gave a series of inspiring lectures about cosmology in Cambridge. As a direct result, I chose courses on black holes and relativity for my fourth year of study at the University of Cambridge.

I first saw Hawking when I was an undergraduate. At that time he was living in an apartment building just behind my student house. He was already so famous that friends would come to my room just to watch him leaving and entering his apartment. But as an undergraduate I never tried to talk with him, feeling much too junior and intimidated.

After I finished my fourth year, I was invited in to talk to Hawking, who was already using a speech synthesizer, about options for my PhD. I was quite nervous when I first met him, but he jumped straight into physics and soon we were discussing black holes. I became a student at the time of the “ Second String Theory Revolution ” in theoretical physics. Hawking had not worked actively in string theory, but he was very keen to understand the new ideas.

Following that meeting, he sent me off to read all the papers that Edward Witten , a famous string theorist, had written that year. My task was to come back and summarise them for him – the student teaching the master. It’s difficult to describe how hard this task actually was: Hawking expected me to jump straight to the frontier of string theory as a starting graduate student. He also chose the title for my PhD thesis: “Problems in M-theory”, which I worked on from 1995 to 1998.

I can only hope that my explanations of string theory were helpful. Hawking went back and forth on his views on M-theory, but eventually ended up thinking that it may be our best bet for a theory of everything.

No hand-holding

PhD students were enormously important to Hawking. In the early phase of his illness, his students helped take care of him. By the time I became his student he needed round-the-clock nursing. At this point, his students were no longer involved in his physical care, but remained essential to his research. Theoretical physics begins with ideas and concepts, but these then evolve into explicit detailed calculations. Hawking had a remarkable ability to do complex calculations in his head, but he still relied on collaborators to develop and complete his research projects.

Theoretical physicists typically give early PhD students “safe” research projects, and guide them through the calculations required. As the students develop, the projects become more ambitious and risky and students are expected to work independently. However, PhD students working with Hawking did not have the luxury of this gentle introduction – he needed us to work on his own high-risk, high-gain projects.

Hawking’s communication via his speech synthesizer was necessarily concise and he simply could not provide detailed guidance about calculations, making it extremely challenging to work with him. But it was also stimulating, forcing students to be creative and independent. He did give praise when he thought it was due. He once sent me away with a very hard problem – finding exact rotating black hole solutions of Einstein’s equations with a cosmological constant – and was stunned when I came back a few days later with the solution. I can’t even remember exactly what he said but I will never forget his enormous smile.

Hawking was a determined and stubborn person. On many occasions he got through serious medical issues with sheer determination . This same determination could make him very difficult to work. But it could also push research projects forward: Hawking would refuse to give up on seemingly unsolvable problems.

In fact, never giving up is the main thing Hawking has taught me – to keep attacking problems from different directions, to reach for the hardest problems and find a way to solve them. It’s immensely important as a scientist, but also in other aspects of life.

Pithy one-liners

Hawking was devoted to his family. His eyes would light up when one of his children came to visit or when he proudly showed us pictures of his first grandchild. In many respects, Hawking treated his PhD students and collaborators as a second family. However busy he was, he always made time for us, often making dignitaries wait outside his office while he talked physics with a student. He would eat lunch with us several times per week, and funded a weekly lunch for the wider group to bring everyone together.

roger penrose phd thesis pdf

There were many occasions when physics discussions merged seamlessly into social activities: going to the pub, eating dinner at one of his favourite Cambridge restaurants, and so on. Hawking had a wonderful sense of humour. He turned his communication difficulties into an advantage, composing pithy one-liners. For instance when changing his mind about what happens to information in a black hole, he announced it in the pub by turning the volume up on his synthesizer, saying simply: “I’m coming out.” He would discuss anything and everything in a social setting: politics, movies, other branches of science, music.

As we worked in closely related fields, we saw each other regularly even after I finished my PhD. In 2017, I attended a conference in Cambridge celebrating his 75th birthday. The list of participants illustrates Hawking’s influence on academia and beyond. Many of his former students and collaborators have gone on to become leaders in research in cosmology, gravitational waves, black holes and string theory. Others have had huge impact outside academia, such as Nathan Myhrvold at Microsoft.

There is currently pressure on academics to demonstrate the immediate impact of their research on society. It is perhaps worth reflecting that impact is not easily measurable on short time scales. Hawking’s was truly blue-sky research – and yet it has fascinated millions, attracting many into scientific careers. His academic legacy is not just the remarkable science he produced, but the generations of minds he shaped.

There’s no doubt Hawking’s death is a huge loss to physics. But personally, what I will miss most is his humour and the general feeling of inspiration I got from being around him.

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Quantum Consciousness in the Penrose-Hameroff Model

by Anthony P Bermanseder

2019, Quantum Consciousness in the Penrose-Hameroff Model

The number of microtubules in the Hameroff-Penrose model for physicalized consciousness can be related to the actual scale of the classical electron radius and the cosmological spacetime matrix. In this article, the author explores cosmological relevance of microtubules as conformal wormhole quantization in quantum geometry. Using conformal mapping from the Quantum Big Bang 'singularity' from the electric charge in brane bulk space as a magnetic charge onto the classical spacetime of Minkowski and from the Planck parameters onto the atomic-nuclear diameters in 2R e c 2 = e* from the Planck length conformally maps the Planck scale onto the classical electron scale. A conformal scale of 2.5 fm, which is close to the classical electron radius of about 2.8 fermi and as defined in the alpha electromagnetic fine structure and the related mass-charge definition for the eigen energy of the electron in m e c 2 =ke 2 /R e. Also in a model of quantum relativity (QR), there is a quantization of exactly 10 10 wormhole 'singularity-bounce' radii defining the radian-trigonometric Pi ratio as R wormhole /R electron = 360/2π.10 10 or 10 10 = {360/2π}{R e /r wormhole }

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Free Related PDFs

Anthony P Bermanseder

The number of micro tubules in the Hameroff-Penrose model for physicalized consciousness can be related to the actual scale of the classical electron radius and the cosmological spacetime matrix. Using this conformal mapping from the Quantum Big Bang 'singularity' from the electric charge in brane bulk space as a magnetic charge onto the classical spacetime of Minkowski and from the Planck parameters onto the atomic-nuclear diameters in 2Rec 2 = e* from the Planck length conformally maps the Planck scale onto the classical electron scale. A conformal scale of 2.5 fm, which is close to the classical electron radius of about 2.8 fermi and as defined in the alpha electromagnetic fine structure and the related mass-charge definition for the eigen energy of the electron in mec 2 = ke 2 /Re.

Quantum Consciousness in the Penrose-Hameroff Model Cosmological relevance of microtubules to the size of the electron as conformal wormhole quantization in quantum geometry

Andrei Lucian Drăgoi (Dragoi)

(QGR - v1.0 - 30.01.2020 - 11 A4 pages [without References]) (toy-model) Sketching a new Quantum General Relativity (QGR) variant mainly based on the redefinition of leptons (as quantum micro black holes composed from highly compressed single triquarks under a very strong quantum gravitational field [QGF]), a dual electro-gravitational interpretation of the fine structure constant (FSC), Planck wormholes (“Planck tubes”) and a reinterpretation of Planck units in the “spirit” of Einstein’s GR. #DONATIONS. Anyone can donate for dr. Dragoi’s independent research and original music at: https://www.paypal.com/donate/?hosted_button_id=AQYGGDVDR7KH2

(QGR - v1.0 - 30.01.2020 - 11 A4 pages [without References]) A new Quantum General Relativity variant mainly based on the redefinition of leptons (as quantum micro black holes composed from highly compressed single triquarks), Planck wormholes etc

Erasmo Recami

Micro--universes and``strong black holes'': a purely geometric approach to elementary particles

Abhay Ashtekar

2005, Proceedings of Symposia in Pure Mathematics

Quantum geometry in action: Big bang and black holes

2023, Research Gate

This paper proposes a toy-model of a Kerr-Newman quantum black-hole electron (with finite non-zero volume of about 10^23 Planck volumes and density of about 10^50kg) governed by very strong gravity (of about 10^30G), showing that matter is actually spacetime curvature at Planck scales and explaining quantum spin. Furthermore, besides the nature’s principle “no electromagnetic charge without a non-zero rest mass” (as if charge sine-qua-nonly needs mass to be “stored” on, practically implying “no electromagnetic field without a gravitational field”), we launch another analogue nature’s hypothetical principle: “no rest mass without a finite non-infinitesimal non-zero 3D volume”, as if non-zero rest mass sine-qua-nonly needs 3D volume to be “stored” on. This paper continues (from an alternative angle of view) the work of other past articles/preprints of the same author in physics (cited in anti-chronological order, from the latest to the oldest). #DONATIONS. Anyone can donate for dr. Dragoi’s independent research and original music at: https://www.paypal.com/donate/?hosted_button_id=AQYGGDVDR7KH2

(QBH-electron - RG preprint - version 1.1 - 8.03.2023 - 2 A4 pages without ref.) A toy-model of a Kerr-Newman quantum black-hole electron governed by very strong gravity, showing that matter is actually spacetime curvature at Planck scales and explaining quantum spin

Octavio Obregon

1999, Modern Physics Letters A

We review the anomaly induced effective action for dilaton coupled spinors and scalars in large-N and s-wave approximation. It may be applied to study the following fundamental problems: construction of quantum corrected black holes (BHs), inducing of primordial wormholes in the early Universe (this effect is confirmed) and the solution of initial singularity problem. The recently discovered anti-evaporation of multiple horizon BHs is discussed. The existence of such primordial BHs may be interpreted as SUSY manifestation. Quantum corrections to BHs thermodynamics may also be discussed within such scheme.

Unified Approach to Study Quantum Properties of Primordial Black Holes, Wormholes and of Quantum Cosmology

Christian Wuthrich

Approaching the Planck scale from a generally relativistic point of view: A philosophical appraisal of loop quantum gravity

Angel Garces Doz

A.Garcés Doz

In this paper shown; on the one hand, as the tensor-scalar polarization modes B ratio, it is derived from the initial properties of the vacuum due to the unification of gravitation and electromagnetism. This ratio suggest that it is 2/Pi ^ 2 (0.20262423673), as an upper bound. Secondly; demonstrates that it is not necessary to introduce any inflaton scalar field or similar ( ad-hoc fields ); If on the other hand, is the same structure of the vacuum and the quantization of gravity which perfectly explains this initial exponential expansion of the universe. In some respects exponential vacuum emptying has certain similarities with the emission of radiation of a black hole. This quantization of gravity and its unification with electromagnetic field, shown in previous work; It allows deriving complete accurately the exponential factor of inflation; and therefore calculate accurately the Hubble constant, mass of the universe, matter density, the value of the vacuum energy density, the GUT mass scale ( bosons X,Y ), the gravitino mass and more. The method of quantize gravity used in this work; It is based on dimensionless constants that must be enforced in accordance with general relativity. We demonstrate the existence of quantum wormholes as the basic units of space-time energy, as an inseparable system.These quantum wormholes explain the instantaneous speed of propagation of entangled particles. Or what is the same: an infinite speed, with the condition of zero net energy. Another consequence of the dimensionless quantization of gravity; is the existence of a constant gravitational acceleration that permeates all space. Its nature is quantum mechanical, and inseparable from the Hubble constant. This work is not mere speculation; since applying this vacuum gravitational acceleration; first allows us to explain and accurately calculate the anomaly of the orbital eccentricity of the Moon. This anomaly was detected and accurately measured with the laser ranging experiment. This same constant acceleration in vacuum (in all coordinate of space), which interacts with the masses; explains the almost constant rotation curves of galaxies and clusters of galaxies. Therefore there is no dark matter. Current interpretation of quantum mechanics is completely erroneous. We explain that; as the de Broglie–Bohm theory, also known as the pilot-wave theory; is a much more realistic and correct interpretation of quantum mechanics. The current assumption that there is reality no defined; until the act of observation does not occur; is an aberrant, illogical assertion false and derived from the obsolete current interpretation of quantum mechanics. The age of the universe derived from the Hubble constant is a wrong estimate; due to absolute ignorance of the true nature of this constant and its physical implications. The universe acquired its current size in the very short period of time, a unit of Planck time. We understand that this work is dense and completely revolutionary consequences. Experiments reflection of lasers; type of the laser ranging experiment; undoubtedly will confirm one of the main results: the existence of an intrinsic acceleration of vacuum of gravitational quantum mechanical nature, which explains the rotation curves of galaxies and clusters of galaxies; and which thus makes unnecessary the existence of dark matter.

The BICEP2 Experiment And The Inflationary Model: Dimensionless Quantization of Gravity. Predictive Theory of Quantum Strings.Quantum Wormholes and Nonlocality of QM.The Absence Of Dark Matter

Ignazio Licata

2019, Entropy

Einstein’s equations of general relativity (GR) can describe the connection between events within a given hypervolume of size L larger than the Planck length L P in terms of wormhole connections where metric fluctuations give rise to an indetermination relationship that involves the Riemann curvature tensor. At low energies (when L ≫ L P ), these connections behave like an exchange of a virtual graviton with wavelength λ G = L as if gravitation were an emergent physical property. Down to Planck scales, wormholes avoid the gravitational collapse and any superposition of events or space–times become indistinguishable. These properties of Einstein’s equations can find connections with the novel picture of quantum gravity (QG) known as the “Einstein–Rosen (ER) = Einstein–Podolski–Rosen (EPR)” (ER = EPR) conjecture proposed by Susskind and Maldacena in Anti-de-Sitter (AdS) space–times in their equivalence with conformal field theories (CFTs). In this scenario, non-traversable wormhole co...

General Relativistic Wormhole Connections from Planck-Scales and the ER = EPR Conjecture

2003, Annales Henri Poincaré

Addressing Challenges of Quantum Gravity Through Quantum Geometry: Black holes and Big-bang

Russell Kramer

Abstract The fundamental relation between space and time is motion expressed as the ratio of space over time for motion in space and the ratio of time over space for motion in time. This indicates that space and time are co-existent reciprocal aspects of motion. While inseparable and interdependent both space and time have distinct geometric properties. There are two fundamental quantum holographic interference patterns which most closely exemplify these structural properties. These are separately identified and defined consistent with the space-time reciprocal relationship. Quantum time potentials and space time networks are defined. The first network consists of two interacting quantum time potentials forming a space-time network whereby space is an emergent feature; there being an inverse structure with inverse properties. The phenomenon of mass and force are emergent features from the various permutations of interconnections between nodes within this space-time network. The resulting structure implies the existence of a coordinate system where each node represents coordinates defined by the rays from each pole. The coordinates form an information field and indicate that space and time ARE information, The connections between the nodes are determined by pre-mathematical connection algorithms indicating the underlying mechanism of creation. Further properties of the space-time network are identified and reveal underlying mechanisms to account for elusive and anomalous physical phenomenon including non-locality, quantum entanglement and quantum gravity.

A New Look at Space-Time Towards a Unified Quantum Geometry

Marek-Lars Kruusen

2023, New discoveries in Einstein’s physics

Quantum gravity is a very important research topic in theoretical physics because it is believed to bridge quantum mechanics and Einstein's theory of general relativity. This may be true, but quantum gravity is important to basic science primarily because it would show how gravity works on an extremely small scale of spacetime, the quantum level. This work presents many new discoveries that reveal the nature of gravity at the quantum level. Albert Einstein described how gravity works on the macro- and mega-level (i.e. the level of planets and galaxies), but not on the quantum level. In order to understand gravity at the quantum level, it is first necessary to know Einstein's theory of general relativity, with which this work begins. In this paper, Albert Einstein's theory of general relativity has been presented and derived as simply as possible, which means that all irrelevant aspects of general relativity have been omitted. Modern theoretical physics tries to simplify existing theories as much as possible, eliminating all irrelevant information. General relativity can be mathematically extremely complex and a very bulky physical theory, but in this work it is presented and also derived in the most direct and simple way that has not been done anywhere and ever before. This is necessary in particular for understanding quantum gravity. Another comprehensive topic is the physics of wormholes. The physical and mathematical interpretation of wormholes has actually been known for about a hundred years, but their technical creation has not been possible until now. However, in this work, apart from the nature of the wormhole, its technical feasibility is also shown, which has not been presented before. In particular, one specific part of the technology for creating wormholes (the outer part of the machine) is described, which is currently being processed by the United States Patent and Trademark Office ( in 2023 – 2024 ). Thus, the inventor of such technology is licensed by the USPTO. One of the biggest obstacles to the technical creation of wormholes was the production of the necessary energy, but the new discoveries presented in this work show that this is not necessary. The key to the creation of wormholes lies in the changes or creation of energy fields that occur exactly at the speed of light. Physical states that change or arise at the speed of light are also accompanied by the emergence of abstract geometric surfaces with time and space curved to infinity. The actual creation of wormholes is crucial to humanity's plans for the future. They would be practically indispensable for, for example, space travel to great distances, and wormholes would also allow people to travel in time to the past and the future. Time travel could be used in the study of human history and would be a very good tool for accurate weather forecasting, for example. For a long time, wormholes were considered the realm of science fiction and fantasy literature, and in some cases even pseudoscience. However, new discoveries concerning the possibility of their creation, which are thoroughly presented in this work, make the existence of wormholes a tangible reality. This work is primarily aimed at an academic audience, such as students, graduate students, lecturers, researchers, but also interested parties, industrialists and even managers of technology companies and organizations. This work requires the readers to have university-level knowledge of theoretical physics and to a lesser extent of engineering. This work is a part of a larger research work, the content of which is the development of the physical theory of time travel and its technology. References concerning it are also presented in this work.

New discoveries in Einstein's physics: Quantum gravity and the creation of wormholes

Mauricio Bellini

The study of geometrodynamics was introduced by Wheeler in the 50’s decade in order to describe particle as geometrical topological defects in a relativistic framework[1], and, in the last years has becoming a very intensive subject of research[2]. In the last decades Loop Quantum Gravity (LQG) have provided a picture of the quantum geometry of space, thanks in part to the theory of spin networks[3]. The concept of spin foam is intended to serve as a similar picture for the quantum geometry of spacetime. LQG is a theory that attempts to describe the quantum properties of the universe and gravity. In LQG the space can be viewed as an extremely fine favric of finite loops. These networks of loops are called spin networks. The evolution of a spin network over time is called a spin foam. The more traditional approach to LQG is the canonical LQG, and there is a newer approach called covariant LQG, more commonly called spin foam theory. However, at the present time, it is not possible to ...

O ct 2 01 5 Towards relativistic quantum geometry

Daniel Arteaga

2008, Classical and Quantum Gravity

Peyresq Physics Workshops 11 and 12—'Micro and Macro Structure of Spacetime', Peyresq, Alpes de Haute Provence, France (17–23 June 2006 and 16–22 June 2007)

Arnab Kundu

The European Physical Journal C

Wormholes are intriguing classical solutions in General Relativity, that have fascinated theoretical physicists for decades. In recent years, especially in Holography, gravitational Wormhole geometries have found a new life in many theoretical ideas related to quantum aspects of gravity. These ideas primarily revolve around aspects of quantum entanglement and quantum information in (semi-classical) gravity. This is an introductory and pedagogical review of Wormholes and their recent applications in Gauge-Gravity duality and related ideas.

Wormholes and holography: an introduction

Reginald T Cahill

The new dynamical theory of space is further confirmed by showing that the black hole masses M_BH in 19 spherical star systems, from globular clusters to galaxies with masses M, satisfy the prediction that M_BH=(alpha/2)M, where alpha is the fine structure constant. As well the necessary and unique generalisations of the Schrodinger and Dirac equations permit the first derivation of gravity from a deeper theory, showing that gravity is a quantum effect of quantum matter interacting with the dynamical space. As well the necessary generalisation of Maxwell's equations displays the observed light bending effects. Finally it is shown from the generalised Dirac equation where the spacetime mathematical formalism, and the accompanying geodesic prescription for matter trajectories, comes from. The new theory of space is non-local and we see many parallels between this and quantum theory, in addition to the fine structure constant manifesting in both, so supporting the argument that sp...

Black Holes and Quantum Theory: The Fine Structure Constant Connection

Thomas Görnitz

Quantum-theoretic considerations for the ground state of a black hole result in a change of its interior solution. lt is shown hat the interior of a Schwarzschild black hole can be modelled by an ur-theoretically described Robertson-Walker space-time. Thereby the Schwarzschild singularity is changed into a Friedman singularity. 1. INTRODUCTON One expects that an appropriate unification of quantum theory and gravitation theory should lead to an explanation of the observed smallness of the cosmological constant and to a better understanding of the space-time singularities of classical general relativity. We do not think that one should try to avoid or even remove space-time singularities in quantized gravity; we rather take singularities as precious hints to look for a new type of unification. The usual attempts to construct a union of quantum theory and gravity are applications of quantization procedures to gravitation theory retaining the space-time continuum even at very small distances. In this paper we do not presuppose a space-time continuum first but start with abstract quantum theory, i.e., the quantum theory of binary alternatives (Drieschner et al., 1987; Görnitz, 1988 a, b). Space-time is introduced via the invariance group of the "ur," the quantized binary alternative. This invariance group turns out to be U(2).

Connections between Abstract Quantum Theory and Space-Time-Structure. III. Vacuum Structure and Black Holes

Flavio Gimenes Alvarenga

1999, General Relativity and Gravitation

Wormholes, Classical Limit and Dynamical Vacuum in Quantum Cosmology

Amira Baker

2016, NeuroQuantology

The Unified Spacememory Network: from Cosmogenesis to Consciousness

2019, The Monopolar Quantum Relativistic Electron: An Extension of the Standard Model & Quantum Field Theory (Part 3)

In this paper, a particular attempt for unification shall be indicated in the proposal of a third kind of relativity in a geometric form of quantum relativity, which utilizes the string modular duality of a higher dimensional energy spectrum based on a physics of wormholes directly related to a cosmogony preceding the cosmologies of the thermodynamic universe from inflaton to instanton. In this way, the quantum theory of the microcosm of the outer and inner atom becomes subject to conformal transformations to and from the instanton of a quantum big bang or qbb and therefore enabling a description of the macrocosm of general relativity in terms of the modular T-duality of 11-dimensional supermembrane theory and so incorporating quantum gravity as a geometrical effect of energy transformations at the wormhole scale. Part 3 of this article series includes: A Mapping of the Atomic Nucleus onto the Thermodynamic Universe of the Hyperspheres; The Higgsian Scalar-Neutrino; & The Wave Matter of de Broglie. We consider the universe's thermodynamic expansion to proceed at an initializing time tps=fss at lightspeed for a light path x=ct to describe the hypersphere radii as the volume of the inflaton made manifest by the instanton as a lower dimensional subspace and consisting of a summation of a single spacetime quantum with a quantized toroidal volume 2π²rweyl and where rweyl=rps is the characteristic wormhole radius for this basic building unit for a quantized universe (say in string parameters given in the Planck scale and its transformations). At a time tG, say so 18.85 minutes later, the count of space time quanta can be said to be 9.677x10 102 for a universal 'total hypersphere radius' of about rG=3.391558005x10 11 meters and for a G-Hypersphere volume of so 7.69x10 35 cubic meters from N{2π 2 .rps 3 } = Volume = 2π2.RHk3.

The Monopolar Quantum Relativistic Electron: An Extension of the Standard Model & Quantum Field Theory (Part 3)

Stuart Hameroff

Consciousness in the Universe: Neuroscience, Quantum Space-Time Geometry and Orch OR Theory

In this paper, a particular attempt for unification shall be indicated in the proposal of a third kind of relativity in a geometric form of quantum relativity, which utilizes the string modular duality of a higher dimensional energy spectrum based on a physics of wormholes directly related to a cosmogony preceding the cosmologies of the thermodynamic universe from inflaton to instanton. In this way, the quantum theory of the microcosm of the outer and inner atom becomes subject to conformal transformations to and from the instanton of a quantum big bang or qbb and therefore enabling a description of the macrocosm of general relativity in terms of the modular T-duality of 11-dimensional supermembrane theory and so incorporating quantum gravity as a geometrical effect of energy transformations at the wormhole scale.

The Monopolar Quantum Relativistic Electron: An Extension of the Standard Model & Quantum Field Theory (Part 4

2014, Springer Handbook of Spacetime

Gravity, Geometry, and the Quantum

Richard Amoroso

Gravitation and cosmology : from the Hubble radius to the Planck scale

In this paper, starting from vortices we are finally lead to a treatment of Fermions as Kerr-Newman type Black Holes wherein we identify the horizon at the particle's Compton wavelength periphery. A naked singularity is avoided and the singular processes inside the horizon of the Black Hole are identified with Quantum Mechanical effects within the Compton wavelength. Inertial mass, gravitation,

Quantum Mechanical Black Holes: Towards a Unification of Quantum Mechanics and General Relativity

Joseph J . Jean-Claude

Table of Content of Vol I - Revision I (2017) of this Title A vast program of axiomatization of physics in direct response to David Hilbert's 6th Problem with tangible and verifiable numeric results. A resolution of long-time search for theoretical derivation of the fundamental physical constants and origin of the primordial mathematical-physics symmetries subtending the quantum realm and the cosmological arena.

QUANTO-GEOMETRY: Overture of Cosmic Consciousness or Universal Knowledge for All - Volume I – Revision I

Paulo Moniz

2017, Physics of the Dark Universe

Quantum deformation of quantum cosmology: A framework to discuss the cosmological constant problem

Ali Yousif Hassan Edriss

This book is written after watched lectures from Prof. Carlo Rovelli( ) about the interpretation of Loop Quantum Gravity for the time and space at tiny amounts of them to extent of the quantization, and I concluded that space and time have certain lower values to be and then they will expand refusing any more compressing, this lower values are 〖1×10〗^(-27) meter and 〖1×10〗^(-35) second which memorizing us to Einstein cosmology constant and the concept of the dark energy density. Beneath these values the space will expand and time will dilate. The interception between time and distance the mass is laying. So from the above concept I tried to put many conditions to differentiate the mass from the energy on contrary to Einstein famous concept that “mass and energy are two face of the same coin and they could interchange to one to other”, these conditions are: Mass makes time dilates on its surface than corresponding empty space that confined to the same dimensions, and energy couldn’t do that. Also mass makes space expands (finitely)than the empty space by making U – V two dimensions space shrinks, and so does the energy but by expands U – V see the last chapter. Mass changes space characteristics especially light speed and enlarge the electrical permittivityε_0 and magnetic permeability μ_0 and that through lessening the speed of light( ) and energy does the opposite by diluting the space through enlarging the light speed and lessening ε_0 and μ_0. Mass is energy condensate. the energy has depression propriety and mass has aggregate propriety. So in this book I tracked the increasing of the mass and the effect of that on the space through recognizing time flow rate and distance expanding (or you can say space flow rate) and then the effect of the interactions of the space and the mass on the mass itself. So the first chapter speaks about studying the space expansion on empty space beginning from planks dimensions 〖10〗^(-35) meter to the entire universe through: Electron dimensions, proton’s, atom’s, earth’s, galaxy’s, then the entire observable universe and not observable universe when inflationary era been taken on account. All these systems have been studied numerically. The second chapter deals with time dilation in empty space just like the first with the same manner, the third deals with the both time and space but in case of matter existence. The fourth deals with the mass itself and the effect of the interaction with space on it by manipulating it in empty and occupied space. The last chapter deals with the energy and its interactions with matter. All that in order to find the answer of the title. Then the last topics are the derivation of the equations that I used in this book, after that is the references. Loop Quantum Gravity dealt with the same manner through all chapters and I concluded that the space-time has its own way to protect itself from the very huge masses from tearing it, so at very small space the time is dilates at huge speed and space is expands exponentially. I found the answer of the title through the third chapter and I am glad to inform the whole world that from equation that define a black hole we can derivate all aspects of matter with its shapes, and the BHs in the center of the galaxies have grateful power to keep our habitable planets healthy for life, not BHs alone but the sun also. I hope this book inspire the other scientists to make huge steps on humanity knowledge. My greetings: #Ali_Yousif_Hassan_Edriss

Cosmos’ Structure 34th Episode: Why Elementary Particles take their Specific Shapes and Masses? Loop Quantum Gravity May Answer (Part 1, Mass Considerations)

Alejandro Corichi

2007, Classical and Quantum Gravity

Quantum black holes within the loop quantum gravity (LQG) framework are considered. The number of microscopic states that are consistent with a black hole of a given horizon area $A_0$ are counted and the statistical entropy, as a function of the area, is obtained for $A_0$ up to $550 l^2_{\rm Pl}$. The results are consistent with an asymptotic linear relation and a logarithmic correction with a coefficient equal to -1/2. The Barbero-Immirzi parameter that yields the asymptotic linear relation compatible with the Bekenstein-Hawking entropy is shown to coincide with a value close to $\gamma=0.274$, which has been previously obtained analytically. However, a new and oscillatory functional form for the entropy is found for small, Planck size, black holes that calls for a physical interpretation.

Quantum geometry and microscopic black hole entropy

Victor Berezin

1997, arXiv: General Relativity and Quantum Cosmology

Towards the mass spectrum of quantum black holes and wormholes

2023, The Fibonacci Torus and the Quantum of Consciousness

This paper explains a link between the Fibonacci-Lucas 'Golden Mean 'sacred geometry' in the context of its relationship to the alpha electromagnetic finestructure constant. The Fibonacci Torus narrative by Mark Sims is subsequently supported and detailed in a number of referenced published science based pdf's and books. A multi-dimensional quantum cosmology is proposed to derive from an extraterrestrial hypothesis information entangled with a time shifted evolution of the human planetary civilization paralleled by interstellar and intergalactic starhuman observers, mainly based on ex-human time travellers. The purpose for this human-starhuman ‘soul-consciousness’ agenda is found in the planetary history of Earth; as well as the origin and ontology of the universe emerging from a primal self-state of collective universal consciousness existing in pre-spacetime then self-simulating itself in spacetime in a creation event – the birth of the universe. The birth of space and time then allows the self-simulation of the creator-creation monadic dyad to distribute itself in parts of individuated physicalized consciousness in spacetimed wormhole volumars evolving in energy parameters to become data collectors and information processors for the self-simulator. The questions regarding the extraterrestrial hypothesis and including UFO-UAP-USO phenomena and their planetary presence and interactions are linked to a number of videos, constructed from available data sources. In particular, the question of “Why Are They Here?” and for “How Long Have They Been Here?” is addressed in detail to show the presence of the human ancestors in the form of a multi-tiered and interwoven multi-dimensional universally applicable cosmology of universal intelligent conscience.

The Fibonacci Torus and the Quantum of Consciousness

Yasunori NOMURA

Modern Physics Letters A

We portray the structure of quantum gravity emerging from recent progress in understanding the quantum mechanics of an evaporating black hole. Quantum gravity admits two different descriptions, based on Euclidean gravitational path integral and a unitarily evolving holographic quantum system, which appear to present vastly different pictures under the existence of a black hole. Nevertheless, these two descriptions are physically equivalent. Various issues of black hole physics — including the existence of the interior, unitarity of the evolution, the puzzle of too large interior volume, and the ensemble nature seen in certain calculations — are addressed very differently in the two descriptions, still leading to the same physical conclusions. The perspective of quantum gravity developed here is expected to have broader implications beyond black hole physics, especially for the cosmology of the eternally inflating multiverse.

From the black hole conundrum to the structure of quantum gravity

2019, monopolar electron

In this paper, a particular attempt for unification shall be indicated in the proposal of a third kind of relativity in a geometric form of quantum relativity, which utilizes the string modular duality of a higher dimensional energy spectrum based on a physics of wormholes directly related to a cosmogony preceding the cosmologies of the thermodynamic universe from inflaton to instanton. In this way, the quantum theory of the microcosm of the outer and inner atom becomes subject to conformal transformations to and from the instanton of a quantum big bang or qbb and therefore enabling a description of the macrocosm of general relativity in terms of the modular T-duality of 11-dimensional supermembrane theory and so incorporating quantum gravity as a geometrical effect of energy transformations at the wormhole scale. Part 1 of this article series includes: Introduction; The Electromagnetic Mass Energy and the [v/c] 2 Velocity Ratio Distribution; The Extension of Newton's Law in Relativistic Momentum & Energy and the Magnetopolar Self-Interaction of the Electron; and Frequency permutation states in the monopolar velocity distribution.

The Monopolar Quantum Relativistic Electron: An Extension of the Standard Model & Quantum Field Theory (Part 1)

Soo-jong Rey

1989, Nuclear Physics B

The collective dynamics and the correlations of wormholes in quantum gravity

2005, Space-Time Structure: Einstein and Beyond

Quantum Geometry and Its Ramifications

Andrew Beckwith

Linkage of New Formalism for Cosmological “constant” with the Physics of Wormholes

Jan Govaerts

2002, Contemporary Problems in Mathematical Physics

The Quantum Geometer's Universe: Particles, Interactions and Topology

Paola Zizzi

2005, Modern Physics Letters A

We argue that the model of a quantum computer with N qubits on a quantum space background, which is a fuzzy sphere with n = 2N elementary cells, can be viewed as the minimal model for quantum gravity. In fact, it is discrete, has no free parameters, is Lorentz-invariant, naturally realizes the holographic principle, and defines a subset of punctures of spin networks' edges of loop quantum gravity labelled by spins j = 2N-1-½. In this model, the discrete area spectrum of the cells, which is not equally spaced, is given in units of the minimal area of loop quantum gravity (for j = 1/2), and provides a discrete emission spectrum for quantum black holes. When the black hole emits one string of N bits encoded in one of the n cells, its horizon area decreases of an amount equal to the area of one cell.

A Minimal Model for Quantum Gravity

Octavian Micu

2016, International Journal of Modern Physics D

Horizon quantum mechanics: A hitchhiker’s guide to quantum black holes

Some time ago, when I first inquired as to ‘what quantum cosmology is about’, I did approach the hall with a combination of caution as well as eagerness [...]

Editorial to the Special Issue “Quantum Cosmology”

Da-Ming Chen

1999, International Journal of Theoretical Physics

Quantum cosmology and the quantum wormhole witha conformal complex scalar field are discussed, thecorresponding Wheeler-DeWitt equations are obtained, andthe cosmological wave functions and wormhole wave functions are calculated, respectively,with different boundary conditions. From thecosmological wave function it is found that theprobability density of the universe is zero at a = 0,while at the ground state the most probable radius is aboutthe Planck scale. It is also shown that there exist twodifferent types of universes, which can be connected bythe quantum tunneling effect, transiting from one region to another. It follows from thewormhole wave function that the most probable radius ofthe wormholes is about the Planck scale, which impliesthat the wormhole is steady due to the quantumeffect.

Cosmological Wave Function and Wormhole Wave Function with a Conformal Complex Scalar Field

Janko Kokošar

2017, viXra

This paper was sent to the futuristic Physics Today contest: Physics in 2116. It suggests which theory developments and experiments to do, that we will come closer to explanations of quantum gravity and consciousness. This is partially physics, partially futuristics, and partially science fiction.

Quantum Gravity and Consciousness, the Most Fundamental Physical Unknowns

Arthur Pletcher

NeuroQuantology

This article proposes a novel analogy between the escalating polarization which occurs between rational agents, and the emergence of spacetime in quantum decoherence. The former occurs within a framework of reinforcement loops of recursive information exchange between agents, and escalates to higher scales (local, regional, national and international) of polarization. The latter occurs within the split consciousness of the single observer, and thus a particle evolution becomes determined, as the particle becomes entangled with the polarized state of the observer and the observer’s local environment. A modified Stern Gerlach experimental is proposed as a proof. The premise of this analogy is mathematically supported. A radical concept of gravity is a corollary

Quantum Cosmology and the Role of Consciousness

TEJINDER P SINGH

2017, International Journal of Modern Physics D

We show why and how Compton wavelength and Schwarzschild radius should be combined into one single new length scale, which we call the Compton–Schwarzschild length. Doing so offers a resolution of the black hole information loss paradox, and suggests Planck mass remnant black holes as candidates for dark matter. It also compels us to introduce torsion, and identify the Dirac field with a complex torsion field. Dirac equation and Einstein equations, are shown to be mutually dual limiting cases of an underlying gravitation theory which involves the Compton–Schwarzschild length scale, and includes a complex torsion field.

A new length scale for quantum gravity: A resolution of the black hole information loss paradox

2013, Journal of Physics: Conference Series

Two-dimensional quantum dilaton gravity and the quantum cosmological constant

José García Raya

1997, Physical Review D

Circular strings, wormholes, and minimum size

2010, Classical and Quantum Gravity

Shape in an atom of space: exploring quantum geometry phenomenology

2002, General Relativity and Gravitation - Proceedings of the 16th International Conference

Quantum Geometry and Gravity: Recent Advances

Tensor methods in algebraic geometry

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COMMENTS

  1. PDF Theoretical Foundation for Black Holes and The Supermassive Compact

    Roger Penrose . for the discovery that black hole formation is a robust prediction of the general theory of relativity . and the other half jointly to . ... The theoretical work by Penrose in the other half of this year's Prize was essential for the study of black hole physics, and a great motivation for astronomers in their search for good ...

  2. Roger Penrose's research works

    Roger Penrose's 100 research works with 8,706 citations and 46,751 reads, including: First results of the LARES 2 space experiment to test the general theory of relativity

  3. The thermodynamics of black holes: from Penrose process to Hawking

    In 1969, Roger Penrose proposed a mechanism to extract rotational energy from a Kerr black hole. With this, he inspired two lines of investigation in the years after. On the one side, the Penrose process, as it became known, allowed a comparison between black-hole mechanics and thermodynamics. On the other, it opened a path to a quantum ...

  4. (PDF) Roger Penrose, Laureate of the Physics Nobel Prize 2020 Swiss

    PDF | On Feb 1, 2021, Norbert Straumann published Roger Penrose, Laureate of the Physics Nobel Prize 2020 Swiss Physical Society, Communications, Nr. 63, p.8 | Find, read and cite all the research ...

  5. Singularities, black holes, and cosmic censorship: A tribute to Roger

    Download a PDF of the paper titled Singularities, black holes, and cosmic censorship: A tribute to Roger Penrose, by Klaas Landsman. Download PDF Abstract: In the light of his recent (and fully deserved) Nobel Prize, this pedagogical paper draws attention to a fundamental tension that drove Penrose's work on general relativity. His 1965 ...

  6. Roger Penrose wins 2020 Nobel Prize in Physics for discovery about

    Penrose arrived at St John's in 1952 as a graduate student and completed his PhD thesis on tensor methods in algebraic geometry in 1957. He remained at the College as a Research Fellow until 1960 and was elected as an Honorary Fellow in 1987. Penrose is the College's sixth Nobel prize-winner in Physics and tenth Nobel laureate overall ...

  7. 2020 Physics Nobel laureate Roger Penrose and the Penrose ...

    Three physicists shared the Nobel Prize in Physics for 2020. Roger Penrose (1931, Fig. 1) received half of the prize "for the discovery that black hole formation is a robust prediction of the general theory of relativity."The other half was divided between Reinhard Genzel (1952) and Andrea Ghez (1965) "for the discovery of a supermassive compact object at the center of our galaxy."

  8. PDF A Little Homage to Roger Penrose

    A Little Homage to Roger Penrose Michele Emmer Sir Roger Penrose was awarded the 2020 Physics Nobel Prize at the age of 90 (he was born on August 8, 1931) for his research on the black holes of the universe, which he had initiated many decades earlier. He worked for many years with Stephen Hawking, who died in 2018. In Hawking's life filmThe ...

  9. The thermodynamics of black holes: from Penrose process to ...

    In 1969, Roger Penrose proposed a mechanism to extract rotational energy from a Kerr black hole. With this, he inspired two lines of investigation in the years after. On the one side, the Penrose process, as it became known, allowed a comparison between black-hole mechanics and thermodynamics. On the other, it opened a path to a quantum description of those objects. This paper provides a novel ...

  10. Roger Penrose

    Roger Penrose has made extraordinary and wide-ranging contributions to mathematics and its applications, often making novel and inspiring connections across disciplinary boundaries. He started out in algebraic geometry under W V D Hodge and J A Todd at Cambridge, but within a few years of completing his PhD thesis on Tensor Methods in Algebraic ...

  11. Roger Penrose

    Sir Roger Penrose OM FRS HonFInstP (born 8 August 1931) [1] is an English mathematician, mathematical physicist, philosopher of science and Nobel Laureate in Physics. [2] He is Emeritus Rouse Ball Professor of Mathematics in the University of Oxford, an emeritus fellow of Wadham College, Oxford, and an honorary fellow of St John's College ...

  12. (PDF) Roger Penrose Research Papers

    Sir Roger Penrose (born 8 August 1931) is an English mathematical physicist, mathematician and philosopher of science. He is Emeritus Rouse Ball Professor of Mathematics at the University of Oxford and an emeritus fellow of Wadham College, Oxford. Penrose has made contributions to the mathematical physics of general relativity and cosmology.

  13. PDF 2. The Penrose-Hameroff Approach.

    The Penrose-Hameroff Approach. Perhaps the most ambitious attempt to create a quantum theory of consciousness is the one of Roger Penrose and Stuart Hameroff. Their proposal has three parts: The Gödel Part, The Gravity Part, and ... 1994) explaining and defending this thesis. However, the Harvard philosopher Hillary Putnam challenged Penrose ...

  14. PDF Consciousness in the Universe: Neuroscience, Quantum Space-Time

    In the 1980s Penrose and Hameroff (separately) began to address these issues, each against the grain of mainstream views. 3. Microtubules as Biomolecular Computers Hameroff had been intrigued by seemingly intelligent, organized activities inside cells, accomplished by protein polymers called microtubules (Hameroff and Watt, 1982; Hameroff, 1987).

  15. Roger Penrose: Collected Works: Volume 1: 1953-1967

    Abstract. Professor Sir Roger Penrose's work, spanning fifty years of science, with over five thousand pages and more than three hundred papers, has been collected together for the first time and arranged chronologically over six volumes, each with an introduction from the author. Where relevant, individual papers also come with specific ...

  16. Dennis Sciama (1926-99)

    Dennis Sciama, who died on 18 December last year, was one of the far-sighted physicists involved in this transition. Sciama was a student of Paul Dirac, and like him became fascinated with Mach's ...

  17. Works by Roger Penrose

    Roger Penrose - 2014 - Foundations of Physics 44 (5):557-575. This paper argues that the case for "gravitizing" quantum theory is at least as strong as that for quantizing gravity. Accordingly, the principles of general relativity must influence, and actually change, the very formalism of quantum mechanics.

  18. I was a student of Stephen Hawking's

    PhD students were enormously important to Hawking. In the early phase of his illness, his students helped take care of him. By the time I became his student he needed round-the-clock nursing. At ...

  19. Roger Penrose

    Sir Roger Penrose, mathematical physicist, mathematician and philosopher of science, is the Emeritus Rouse Ball Professor of Mathematics at the Mathematical Institute of the University of Oxford, as well as an Emeritus Fellow of Wadham College. Penrose is known for his work in mathematical physics, in particular for his contributions to general relativity theory and cosmology, most notably for ...

  20. INSPIRE

    Sir Roger Penrose . astro-ph; PhD Advisor: Todd, John Arthur; edit. Updated on Sep 13, 2023. Research works (1,583,096) Cited By . Seminars. Date of paper. 1687 2024. ... pdf DOI cite claim. reference search 0 citations. Exploring the texture structure of quark and Lepton mass matrices #5.

  21. Quantum Consciousness in the Penrose-Hameroff Model

    2019, Quantum Consciousness in the Penrose-Hameroff Model. The number of microtubules in the Hameroff-Penrose model for physicalized consciousness can be related to the actual scale of the classical electron radius and the cosmological spacetime matrix. In this article, the author explores cosmological relevance of microtubules as conformal ...

  22. (PDF) The 1965 Penrose singularity theorem

    Abstract and Figures. We review the first modern singularity theorem, published by Penrose in 1965. This is the first genuine post-Einstenian result in General Relativity, where the fundamental ...

  23. INSPIRE

    Tensor methods in algebraic geometry. Roger Penrose. St John's Coll., Cambridge)