From Singularities to Black Hole Mechanics: Hawking’s Scientific Endeavour

The most recognizable scientist of our times has, sadly, left us this week. Stephen Hawking ushered in fundamental contributions in physics and cosmology and greatly revolutionized our understanding of black holes. It is no doubt that he left a remarkable legacy of scientific inquiry and his passing is a huge loss to the scientific community and to the world.

A Brief History of Black Holes 

Hawking’s first significant contribution to the world of physics was his theorem of singularity. In the early 1960s, the existence of black holes was a point of contention. While theoretical predictions pointed to their existence, the conception was not widely received. The very idea of a black hole itself, however, was, by no means, a novel one. The concept had already resurfaced a number of times in the late 1770s and early 1800s before disappearing into oblivion. The earliest promoters of such an idea included the natural philosophers John Michell and Pierre-Simon Laplace. They both provided the reasoning that a spherical mass could become so compact and their escape velocity would exceed the velocity of light. These ideas, however, of critical circumference were only advanced  in the context of Newtonian physics and the corpuscular theory of light and after the wave theory of light gained mainstream acceptance at the beginning of the 19th Century, Michell’s and Laplace’s ideas were, thereafter, abandoned. It was hard to apply Newton’s laws of gravity to the wavelike behaviour of light and it would take more than a century before interest in such an idea would, again, be revived.

It wasn’t until 1915 when Albert Einstein formulated his general theory of relativity that our conception of gravity as distortions in spacetime took over the reins. Only a few months later, the German astronomer Karl Schwarzchild formulated his solutions to Einstein’s field equations. He described the spacetime geometry, known as Schwarzchild geometry, around a spherically symmetric compact centre, for instance a star, and showed that the structure of this geometry depends on the total mass of the object. It follows that a massive enough object concentrated into a small enough space would result in an infinite curvature of space-time. Space-time, essentially, would curve back on itself such that light leaving would be bent inwards so much that it would never escape. This boundary of no return is what is more commonly known as an event horizon. The radius of this event horizon became known as the Schwarzchild radius. And, it would later be apparent that the Schwarzchild radius, in the context of the General Theory of Relativity, is just Michell’s and Laplace’s idea of a critical circumference. The major difference is that Michell’s and Laplace’s model treated light as particles that cannot escape a small and dense object.

BlackholeTimeSingularity-536x321
A black hole singularity is a point of infinite curvature of space-time. All laws of physics break down at the singularity.

These developments were all essential steps that helped initiate the concept of a black hole but they were, still, not sufficient enough to produce a theory of black holes yet. Crucially however, further interest in the study of dense gravitational systems was stimulated after the physicist Subrahmanyan Chandrasekhar’s work on white dwarfs where he showed that white dwarfs cannot exist if their mass exceeds about 1.4 Solar masses, a limit above which they cannot be supported by electron degeneracy. In 1939, Oppenheimer and Snyder’s work, which provided a mathematical description of the collapse of a neutron star into a black hole, elevated black-hole theory into centre-stage prominence in the astrophysics community. In particular, it showed the inevitability of a collapse of sufficiently massive stars to form a black hole in the context of General Relativity, when there is nothing to prevent the condensation of matter. Such a collapse asymptotes towards the Schwarzchild singularity, a point of infinite density, infinite pressure, and infinite curvature of space-time. Unfortunately, however, by that time, the war had greatly intervened and much of the intellectual prowess that initially went into the study and pursuit of these phenomena was diverted for defence research and constructing things such as the atomic bomb. Nobody would give this notion of black holes any more thought until the 1950s and Oppenheimer and Snyder’s work was largely forgotten.

While much of the theoretical background behind the existence of black holes was laid down, “black hole” would not enter the astrophysics lexicon until 1967 when John Wheeler unveiled the term in a talk he gave to the American Association for the Advancement of Science in New York City. Wheeler revived Oppenheimer and Snyder’s work and after having initially opposed the Oppenheimer-Snyder model, he later became convinced of the existence of black holes. By that time as well, advancements in technology ensued and radio telescopes pointed to the sky discovered objects such as pulsars, which were rotating neutron stars. Observational evidence began to emerge. X-ray sources appeared to have invisible companions whose masses were too high to be neutron stars. By the end of the 1970s, it was commonly accepted that black holes were an inevitable consequence of Einstein’s theory of relativity.

In the 1970s, Roger Penrose and Stephen Hawking laid down some of the most fundamental developments in our understanding of gravitational collapse. Their work provided precise mathematical descriptions of singularities at the heart of black holes, birthed new methods for studying spacetime, and attempted to pinpoint where in spacetime singularities occur. In particular, Penrose’s work showed that every black hole must contain a singularity and introduced the concept of the closed trapped surface, a two-dimensional space-like surface whereupon a singularity will develop.  When the gravitational field is so strong, the ingoing and outgoing wavefronts of light from a trapped surface converge and the areas of the wavefronts decrease. Ultimately, no light emitted by the star escapes through the event horizon. Penrose also introduced the cosmic censorship hypothesis, which claimed that all singularities are hidden in event horizons and must remain hidden from distant observers who are protected from the consequences of a singularity provided they do not fall into it. Further work by Hawking extended the singularity theorems in order to explain that, in the context of classical theory of relativity, the beginning of time was a spacetime singularity. This however applies within certain conditions, namely the strong energy condition, and such a singularity may not, in fact, exist, especially when viewed within the light of the notion that the classical theory of relativity may not be valid for high energy densities of matter

Hawking Radiation 

In the 1970s, using complex calculations, Stephen Hawking shattered the notion of black holes being entirely black by discovering the phenomenon of Hawking radiation. It was commonly thought that nothing can escape a black hole. Hawking’s calculations, however, showed that large black holes could actually emit radiation by virtue of quantum fluctuations that occur at the event horizon of the black hole and that the classical restriction of nothing escaping out of a black hole does not apply when taken within the context of quantum mechanics. Effectively, the black hole radiates until it eventually dissolves and evaporates.

This notion derives from the idea that the vacuum of spacetime around a black hole is not in fact empty, but rather, filled with virtual pairs of particles and anti-particles that appear by virtue of quantum fluctuations, as required by the Heisenberg’s Uncertainty Principle, which denies us the possibility of ascribing a precise location and motion to an object. The particle pairs pop out of nothingness and materialize in empty space for a fraction of a second before annihilating together. While short, this time is, nonetheless, long enough for the particle pair to separate before annihilating. One of the pair of particles that appeared near the event horizon would be captured, while the other half would escape. To an observer, it would seem that the black hole had emitted thermal radiation.

In order to balance out the radiation energy release that has been engendered by the particle that has tunneled out, there results a decrease in the total mass of the black hole. As the black hole decreases in mass, the process of dissipation accelerates at an ever-increasing rate, until to an observer, it appears to have eventually dissipated all its energy. Hawking’s work showed that it was possible to use the mass of a black hole to calculate its temperature and also that it was possible to calculate the intensity of this so-called Hawking radiation. It follows that, counter-intuitively, the temperature of a black hole is inversely proportional to its mass. The black hole, essentially, emits radiation as that of a black body. The smaller the black hole gets, the higher the temperature it produces as it decays quickly and thus the higher the rate of radiation. Primordial black holes, of masses less than 1015 grams are thought to have evaporated by now. Verifiable observational evidence for any such tiny black hole events, however, has, thus far, been non-existent. The largest black holes in the universe would take up to 10100 years to evaporate that changes in their mass would not be perceptible.

Building on the work of Jacob Bekenstein, Hawking also showed that there is a relation between the surface area of a black hole and its entropy. In a black hole merger, the total surface area of the event horizon is greater than the sum of the the event horizons (surfaces) of the two individual black holes, in a property that is analogous to the Second Law of Thermodynamics. A black hole that radiates a real particle loses entropy. That, however, does not violate the law of thermodynamics since the combined entropy of the black hole and the radiated particle is greater than the individual entropy of the individual black hole.

Whether the information that is absorbed by a black is destroyed and lost forever, or whether it might be recovered from the phenomenon of Hawking radiation, is a very fundamental question in modern physics.

 

Bibliography 

Bekenstein JD. 1973. Black holes and entropy. Physical Review D. 7(8): 2333–2346.

Hawking SW. 1988. A Brief History of Time. Bantam Press, London

Hawking SW. 1974. Black hole explosions?. Nature. 248(5443): 30–31.

Hawking SW and Ellis GFR. 1973. The Large Scale Structure of Space–Time. Cambridge University Press, Cambridge.

Hawking SW and Penrose R. 1970. The Singularities of Gravitational Collapse and Cosmology. Proceedings of the Royal Society A. 314(1519): 529–548.

Thorne KS. 1994. Black Holes and Time Warps. Norton, W. W. & Company, New York.

 

 

 

 

 

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