The term black hole refers to an object so massive and so dense that nothing, not even light, can escape from its gravitational attraction. The concept was considered long before the name black hole was coined to replace more cumbersome terminology.

The concept had its origin in Albert Einstein's General Theory of Relativity which he propounded in 1915. Einstein developed a set of field equations usually expressed in the form

R_μν −½ρg_μν + λg_μν = (8πγ/c⁴)T_μν

where g_μν is the metric tensor of spacetime and R_μν is the Ricci curvature tensor of spacetime derived from it. On the RHS T_μν is the stress-energy tensor which represents the distribution mass and energy in spacetime. Spacetime is four dimensional and the indices μ and ν run from 0 to 3. The scalar quantities in the equation are defined as follows. The scalar curvature of spacetime is given as ρ, λ is a constant, called the cosmological constant of the Universe, γ is the gravitational constant and c is the speed of light in a vacuum.

The term tensor is generic. A vector, a matrix and even a scalar are tensors. Defining the Ricci matrix as R, the metric matrix as G and the stress-energy matrix as T the above equations may be expressed as

R + (λ −½ρ)G = (8πγ/c⁴)T

Note that the LHS of the equations involve only geometric quantities and the RHS only physical quantities. The physical quantities, in effect, force a warping of spacetime.

The matrices are 4×4 but they are symmetric so they have only 10 independent components instead of 16. Furthermore there are four Bianci identities that have to be satisfied so there only six degrees of freedom for the components of the equations. So the Einstein equations are a system of ten nonlinear, coupled partial differential equations.

Karl Schwarzschild

Soon after the Einstein equations were publicized physicists started trying to find solutions to simplified versions of the equations. One notable case of this was Karl Schwarzschild. In 1915 he was in the German army stationed in Russia. But when he received the journal issue in which Einstein's equations were published set about trying to find a simplified solution. He found one that had the possibility that for a star small enough and heavy enough there was a distance such anything getting within that distance could never get out. In effect, Schwarzschild had discovered the possible existence of what later came to be known as a black hole. Schwarzschild wrote up his results and sent them to Einstein who publicized them to the scientific community in 1916. Unfortunately Schwarzschild died of an autoimmune disease at his post in Russia. Einstein and other physicists made use of Schwarzschild's solution but were dubious of the existence of black holes.

Arthur Eddington

Arthur Eddington, a leading light in general relativity theory as well as astronomy, was one of those who doubted Schwarzschild's discovery. In 1926 Eddington published The Internal Constitution of the Stars which became the leading source in astrophysics. Eddington proposed that the source of energy for the radiation of stars was the fusion of hydrogen into helium. He also asserted that stars were prevented from collapsing under their own gravitation by the pressure due to the radiation emanating through them. Eddington's status and prestige in the field was such that Schwarzschild's discovery of the possibility of a black hole was forgotten.

Subramanyan Chandrasekhar

Subramanyan Chandrasekhar was a student at the University of Madras in South India when he became captivated by the excitement of the new developments in astrophysics. He read everything he could get, including Eddington's book The Internal Constitution of the Stars. By the time he was eighteen years of age he was able to write a paper that was published in the British journal Proceedings of the Royal Society. He decided to journey to England to pursue a Ph.D. at Cambridge University.

On the journey from India to England Chandrasekhar examined closely a paper written by Ralph Fowler, a protégé of Arthur Eddington. Fowler argued on the basis of the Uncertainty Principle and Pauli's Exclusion Principle that in white dwarf stars quantum effects would counterbalance the gravitational pull and allow white dwarf stars to continue to exist indefinitely. Chandrasekhar found that Fowler's analysis involved particles within a white dwarf star traveling at velocities close to the speed of light. He found that when Special Relativity was taken into account there is a maximum mass that can be sustained by a white dwarf star. If a white dwarf star exceeds that amount it will collapse into itself. Chandrasekhar wrote up his results and showed them to Eddington and Fowler when he got to Cambridge. They ignored Chandrasekhar and his analysis.

Chandrasekhar was able to present his analysis at a monthly meeting of the Royal Astronomical Society. Eddington made a presentation after Chandrasekhar and in his presentation dismissed Chandrasekhar's results as insignificant. Chandrasekhar was devastated and subsequently left England for an appointment to the staff of Yerkes Observatory in Chicago. He went on to write acclaimed books on stellar physics and was awarded the Nobel Prize in Physics for 1983.

Lev Landau

In the Soviet Union the brilliant physicist Lev Landau attempted to explain the source of energy for stars. Eddington had already proposed the correct explanation of the fusion of hydrogen into helium but this was not universally accepted. Landau proposed that stars had internal cores made up entirely of neutrons. This was definitely wrong but at the time Landau wrote it was a plausible possibility. At the same time he was working out his ideas on astrophysics he was involved in producing some anti-Stalin propaganda and was arrested and spent a year in Lubyanka Prison as a consequence.

Landau's article was published in the premier British science journal Nature so it was well known to Western physicists. Robert Oppenheimer at the University of California in Berkeley decided to refute Landau's analysis. First Oppenheimer, in collaboration with one of his students, published a paper that refuted the notion that stars could have neutron cores. Then Oppenheimer, again with another student, showed that a neutron core would be strongly unstable. And finally with yet another student, Hartland Snyder, Opperheimer published an article about the collapse of a star into a black hole, although at that time such an entity was not called a black hole. This was not a primary interest of Oppenheimer and he went on to other things.

At the Institute for Advanced Study where Einstein lived after he came to America there was another brilliant individual he enjoyed discussing things with. That individual was the mathematician Kurt Gödel. Gödel, as a result of his contact with Einstein, decided to apply his intellect to cosmology. He formulated a model in which the Universe rotates. It had some interesting properties, such as travel forward or backward in time, but was rejected by everyone except Gödel.

A young physicist from New Zealand, Roy Kerr, joined a group of scientists at the University of Texas in Austin who were researching General Relativity. He developed an analysis that gave a more general solution to the Einstein equation than the solution of Schwarzschild.

Roger Penrose

In the Soviet Union under Stalin most of the top physicists had been engaged in the successful program of the development of nuclear weapons. After the death of Stalin some of those were freed to pursue other fields. One of them was a Belorussian named Yakov Zel'dovich. He organized a research group to study questions relevant to General Relativity. Two members of that group, Isaac Khalatnikov and Evgeny Lifshitz, produced a paper arguing that the irregularities and asymmetries of a star would prevent it from collapsing into a singularity of a black hole. Their paper was published in the journal Soviet Physics and then presented at a conference in London in 1965 on General Relativity and Cosmology. There was in the audience at that conference the young British physicist, Roger Penrose. Penrose had already proved a theorem that singularities would always occur. Isaac Khalatnikov and Evgeny Lifshitz, working with one of their students, Vadimir Belinski, found an error in their analysis and with that error corrected their analysis proved that singularities always occur for a star sufficiently large. That limit is now concerned to be 1.4 times the mass of the Sun. With Penrose's Singularity Theorem the theoretical issue of whether black holes could form was settled. The next issue was to search for evidence that black holes actually exist in physical reality. Up until 1967 these objects were called completely collapsed gravitational objects. This was too cumbersome, especially for lectures. During a lecture by John Wheeler the term black body was proposed and Wheeler adopted it and soon the rest of the physics community soon did also.

John Wheeler

The Search for Evidence of Black Holes

At first thought there would appear to be no way to observe a black hole at the astronomical distances which are involved. However astronomers were finding puzzling sources of astronomical radiation.

In Britain the project to develop radar drew in the top talent in science in the same the Manhattan Project in the U.S. did. After the war the British scientists decided to apply the knowledge they had acquired during the war to research involving radio wars. In the days when radio was the major source of entertainment people were quite conscious of the interference experienced. This interference, called static, fluctuated but was almost always there. Some scientists decided to investigate that interference. They built directable antenna which came to be called radio telescope. They sought to identify the sources of the static. At first the measurements were quite crude and not accurate enough to pinpoint the sources. But over time the techniques improved by orders of magnitude in accuracy and the radio astronomers were able to give the light telescope astronomers the locations of the radio sources. Initially it was not certain whether the sources were in the Mil.ky Way or whether they were extra-galactic.

There ensued a bitter debate within Cambridge University over this question. On one side there were the theoreticians; Fred Hoyle and his associates, Hermann Bondi and Thomas Gold. Fred Hoyle rose to public acclaim when, by chance, he was allowed by the BBC to give a series of lectures on cosmology. He used these lectures to promote his personal theory of the Universe being in a steady state contrary to the expanding Universe model held by astronomers in general.

On the other side was an engineering professional at Cambridge named Martin Ryle. Ryle compiled catalogues of radio sources. He completed three catalogues of increasing accuracy. In the debate with the Hoyle group Ryle tended to overstate the accuracy of his measurements and announce prematurely the discrediting of Hoyle's steady state theory of the Universe.

The debate resulted in improvements in the cosmological analysis as well as the accuracy of radio astronomy and made the general public aware of quasi-stellar radio sources, subsequently known as quasars.

The investigation of the energy outputs of quasars revealed that one of them had an energy output equal to the light energy outputs of two galaxies. Theorists suggested that quasars might be black holes at the center of galaxies. Logically the radio waves would not have come from the black holes themselves but from the destruction of material in the accretion disks that would surround the black holes in the same way a ring surrounds Saturn.

Theorists then found mechanisms that could result in the black holes themselves radiating energy. They also developed what amounted to the thermodynamics of black holes.

The Laws of Black Hole Mechanics

The name and field of black hole mechanics derive from an article in 1973 by James Bardeen, Brandon Carter and Stephen Hawking. The article in Communications in Mathematical Physics 31 (2): pp. 161–170, was entitled, . "The four laws of black hole mechanics" The laws have a rough correspondence to the laws of thermodynamics. But a bit of terminology must be covered.

There is spherical zone of radius R_H such that anything entering that zone cannot escape, This is also referred to as the event horizon. The material in a black hole would be contained in a far smaller zone but the relevant scale of the black hole is based on its event horizon. The surface area of the black hole is the area of the spherical zone at the event horizon; i.e., 4πR_H². The radius R of the event horizon is given by

R_H = 2γm

where γ is the gravitational constant and m is the mass of the black hole.

The gravitational attraction on a unit mass at distance R from the center of the black hole is

g = γm/R² = ½R_H/R²

The value of g at the event horizon is denoted as κ and its value is

κ = ½/R_H = 1/(2γm)

Here is a schematic depiction of a black hole.

The gray sphere represents the event horizon of the black hole. It should be black but it is made transparent to show the mass of the black holes as a smaller sphere. The smaller white sphere represents the mass of the black hole. It could be radiating but the radiation can never pass the event horizon and so any radiation from the mass is returned to the mass.

Stephen Hawking and his associates take the mass of a black hole to be concentrated as a point!

• The Zeroeth Law:
  A stationary black hole has constant surface gravity at the event horizon.

  The zeroeth law of thermodynamics can be construed to say that any physical system in thermal equilibrium has a constant temperature throughout.

  This means that κ, the magnitude of the gravitational field intensity at the event horizon, corresponds to thermodynamic temperature. Specifically the temperature of black hole T_H (called the Hawking temperature) is equal to κ/(2π). .

• The First Law:
  All aspects of a stationary black hole are determined by it event horizon area A, its angular momentum J and its charge Q. The change of energy is related to the changes of area, angular momentum, and electric charge by:

  dE = (κ/8π)dA + ΩdJ + ΦdQ,

  where E is the energy, κ is the surface gravity, Ω is the angular velocity, and Φ is the electrostatic potential.

  The absence of irregularities and asymmetries of black holes is referred to as "Black holes have no hair." There is a theorem to that effect.

• The Second Law:
  The area A at the event horizon does not decrease,

  (dA/dt) ≥ 0

  This means that A corresponds to thermodynamic entropy. Specifically the blackhole entropy S_BH is equal to A/(4l_P², where l_P is Planck's distance.

• The Third Law:
  It is not possible to form a black hole with surface gravity equal to zero.

  This corresponds to not being able to reduce a physical body's temperature to absolute zero.

The analysis of the mechanics of black holes can be said to have been initiated by J. Beckenstein in his Ph. D. dissertation at Princeton University in 1972. Stephen Hawking disagreed with Beckenstein's analysis and commenced his own analysis of the topic culminating in his demonstration that through quantum effects black holes radiate thermally for a temperature κ/(2π). For a presentation of the analysis of the mechanics of black holes see Mechanics.

(To be continued.)

Sources:

Ferreira, Pedro G., The Perfect Theory, Houghton Mifflin Harcourt, New York, 2014.

McMahon, David, String Theory Demystified, McGraw-Hill, New York, 2009.