Black Hole Theory Explanation – By Stephen Hawking

Black Hole Theory Explanation - By Stephen Hawking

Black Hole Theory Explanation – By Stephen Hawking

The term black hole is of very recent origin. It was coined in 1969 by American scientist John Wheeler as a graphic description of an idea that goes back at least two hundred years. At that time there were two theories about light. One was that it was made of particles; The second was that it was made of waves.

Now we know that in fact, both theories are true. By the wave/particle duality of quantum mechanics, light can be thought of as both a wave and a particle. Under the theory that light is made up of waves, it was unclear how it would react to gravity.

But if the light is made of particles, one might expect them to be affected by gravity in the same way that cannonballs, rockets, and planets are. On this assumption, Cambridge don, John Mitchell, wrote a paper in Philosophical Transactions of the Royal Society of London in 1783.

In it, he explained that a star that was sufficiently massive and compact would have a gravitational field so strong that light could not escape. Any light emitted from the star’s surface will be pulled back by the star’s gravitational attraction before it travels too far.

Mitchell suggested that there could be an enormous number of such stars. Although we will not be able to see them because the light from them will not reach us, we will still feel their gravitational attraction. There are objects that we now call Black Holes because they are what they are – the black voids in space.

Black Hole Image

A similar suggestion was made a few years later by the French scientist Marquis de Laplace, apparently independently of Michel. Interestingly, he included only the first and second editions of his book, The System of the World, and omitted it from later editions; Maybe he decided it was a crazy idea.

In fact, treating light like a cannonball in Newton’s theory of gravitation is not really consistent because the speed of light is fixed. A cannonball fired upwards from Earth would be slowed by gravity and would eventually stop and fall back. However, a photon must be moving upwards at a constant speed. So, how can Newton’s gravity affect light?

Related: Big Bang Theory Explanation – By Stephen Hawking

A coherent theory of how gravity affects light did not come about until Einstein proposed general relativity in 1915, and even then the implications of the theory of massive stars were worked out. To understand how black holes can form, we first need to understand the life cycle of a star.

A star is formed when a large amount of gas, mostly hydrogen, begins to collapse on itself due to its gravitational attraction. As it contracts, the gas atoms collide with each other more and more often and the gas heats up at a greater rate.

Eventually, the gas will be so hot that when hydrogen atoms collide they don’t bounce off each other but combine with each other to form helium atoms. The heat released in this reaction, which is like a controlled hydrogen bomb, makes stars shine.

This additional heat increases the pressure of the gas until it is sufficient to balance the gravitational attraction and the gas stops shrinking. It is like a balloon where there is a balance between the pressure of the air inside, which is trying to stretch the balloon, and the tension in the rubber, which is trying to make the balloon smaller.

Stars would be similarly stable for long periods of time, with heat from nuclear reactions balancing out the gravitational attraction. Eventually, however, the star will run out of its hydrogen and other nuclear fuel. And ironically, the more fuel a star starts out with, the quicker it ends.

This is because the more massive the star, the hotter it must be to balance its gravitational attraction. And the hotter it is, the faster it will use up its fuel. Our Sun has probably got enough fuel for the next five thousand million years, but more massive stars can use up their fuel in a hundred million years, much less than the age of the universe.

When the star’s fuel runs out, it will begin to cool and therefore shrink. What might happen to it was first understood only in the late 1920s. In 1928, an Indian graduate student named Subramanian Chandrasekhar left for England to study at Cambridge with the British astronomer Sir Arthur Eddington. Eddington was an expert in general relativity.

Black Hole Image

There is a story that a journalist told Eddington in the early 1920s that he heard that there are only three people in the world who understand general relativity. Eddington replied, “I’m trying to figure out who the third person is.” During his travels from India, Chandrasekhar worked out how massive a star could grow and tear itself apart against its own gravity even after using up all its fuel.

The idea was this: as the star gets smaller, the particles of matter become very close to each other. But the Pauli exclusion principle states that no two matter particles can have both the same position and the same velocity. Therefore the velocities of the particles of matter must vary greatly.

This moves them away from each other, and so the star has a tendency to expand. A star can therefore maintain itself at a constant radius by striking a balance between gravitational attraction and the repulsion arising from the exclusion principle as if earlier in its life gravity was balanced by heat.

However, Chandrasekhar realized that there is a limit to the repulsion that the exclusion principle can provide. The principle of relativity limits the maximum difference in the velocity of particles of matter in a star to the speed of light.

This meant that when the star became sufficiently dense, the repulsion due to the exclusion principle would be less than the gravitational attraction. Chandrasekhar calculated that a cold star with more than one and a half times the mass of the Sun would not be able to support itself against its own gravity.

This mass is now known as the Chandrasekhar limit. This had serious implications for the eventual fate of the massive stars. If a star’s mass is below the Chandrasekhar limit, it may eventually stop shrinking and settle into a possible final state as a White Dwarf with a radius of a few thousand miles and a density of hundreds of tons per cubic inch.

Related: How Friedman discovered the Expanding Universe before Edwin Hubble’s Publication?

A white dwarf is supported by the exclusion principle repulsion between electrons in its matter. We see a significant number of these white dwarf stars. The first to be discovered is a star orbiting Sirius, the brightest star in the night sky.

It was also realized that there was also another possible final state for a star whose mass was about one or two times the mass of the Sun, but much smaller than a white dwarf. These stars would be supported by the exclusion principle repulsion between neutrons and protons rather than electrons. That’s why they were called neutron stars. 

They would have had a radius of only ten miles or more and a density of hundreds of millions of tons per cubic inch. At the time they were first predicted, there was no way neutron stars could be observed, and they were not detected until much later.

On the other hand, stars with masses greater than the Chandrasekhar limit have a major problem when they come to the end of their fuel. In some cases, they may explode or manage to eject enough matter to reduce their mass beyond the limit, but it was hard to believe that this always happened, no matter how massive the star was.

How will she know if she needs to lose weight? And even if every star managed to lose enough mass, what if you added more mass to take a white dwarf or neutron star over the limit? Will it collapse to infinite density?

Eddington was surprised by its effects and refused to believe Chandrasekhar’s result. He thought it was not possible for a star to fall to a point. This was the opinion of most scientists.

Einstein himself wrote a paper in which he claimed that stars would not shrink to zero sizes. The hostility of other scientists, notably Eddington, his former teacher and leading authority on the structure of stars, persuaded Chandrasekhar to abandon this line of work and turn to other problems in astronomy.

However, when he was awarded the Nobel Prize in 1983, it was, at least in part, for his early work on the finite mass of cold stars. Chandrasekhar had shown that the exclusion principle could not prevent the collapse of a star with a mass greater than the Chandrasekhar limit.

But the problem of understanding what would happen to such a star, according to general relativity, was not solved until 1939 by a young American, Robert Oppenheimer. However, their results suggested that there would be no observational results that could be detected by the telescopes of that day.

Then the war intervened, and Oppenheimer himself became involved in the atomic bomb project. And after the war, the problem of the gravitational collapse was largely forgotten because most scientists were then interested in what happens at the scale of the atom and its nucleus.

In the 1960s, however, interest in the large-scale problems of astronomy and cosmology was revived by the enormous increase in the number and range of astronomical observations brought about by the application of modern technology. Oppenheimer’s work was rediscovered and expanded upon by many.

The picture we now have from Oppenheimer’s work is as follows: the gravitational field of a star causes the paths of light rays to diverge in space-time they would not have existed.

The light cones, which indicate paths followed by flashes of light emanating from their ends in space and time, are curved slightly inward near the surface of the star.

This can be seen in the twisting of light from distant stars as seen during an eclipse of the Sun. As the star shrinks, the gravitational field on its surface becomes stronger and the light cone bends more inward. This makes it more difficult for light to escape from the star, and the light appears dim and red to the distant viewer.

Eventually, when the star shrinks to a certain critical radius, the gravitational field at the surface becomes so strong that the light cone bends inward enough that light can no longer exist. 

According to the theory of relativity, no object can travel faster than light. Thus, if light cannot escape, nothing else. Everything is pulled back by the gravitational field. So one has a set of events, a region of space-time, from which it is not possible to escape to reach a distant observer.

We now call this region a Black Hole. Its boundary is called the Event Horizon. This corresponds to the paths of light rays that fail to exit the black hole. To understand what you would see if a star collapsed to form a black hole, it has to be remembered that there is no absolute time in the theory of relativity.

Each observer has his own time measurement. The time for someone on a star will be different for someone at a distance due to the gravitational field of that star.

This effect has been measured in an experiment with clocks at the top and bottom of a water tower on Earth. 

Suppose an intrepid astronaut on the surface of a collapsing star sent a signal to his spacecraft every second, according to his clock, as it orbited the star. At some point in his clock, let’s say at eleven o’clock, the star would shrink below the critical radius at which the gravitational field had become so strong that the signal would no longer reach the spacecraft.

His companions watching from the spacecraft will find the interval between consecutive signals from the astronauts growing closer to eleven o’clock and getting longer. However, the effect will be much less before 10:59:59.

They would have to wait for only a little over a second between the astronaut’s 10:59:58 signal and his clock reading 10:59:59, but they would have to wait forever until eleven o’clock for the signal.

The light waves emanating from the star’s surface between 10:59:59 and eleven o’clock on the astronaut’s watch would have stretched for eternity, as seen from the spacecraft.

The time interval between successive waves arriving at the spacecraft would get longer and longer, and so the light from the star would appear redder and redder and fainter and fainter. Eventually, the star will become so dim that it can no longer be seen by spacecraft.

All that would be left would be a black hole in space. However, the star will continue to exert the same gravitational force on the spacecraft. This is because the star is still visible to spacecraft, at least in theory. It’s just that light from the surface is so red-shifted by the star’s gravitational field that it cannot be seen.

However, the redshift itself does not affect the gravitational field of the star. Thus, the spacecraft will continue to orbit the black hole.

The work that Roger Penrose and I(Stephen Hawking) did between 1965 and 1970 showed that, according to general relativity, there must be a singularity of infinite density within a black hole.

It’s like the big bang at the beginning of time, only it will be the end of time for the collapsing body and the astronauts.

At the singularity, the laws of science and our ability to predict the future will be broken. However, any observer who lives outside a black hole will not be affected by this failure of predictability, as neither light nor any other signal from the singularity can reach them.

Black Hole Image

This remarkable fact prompted Roger Penrose to propose the cosmic censorship hypothesis, which can be understood as “God hates a naked singularity”. In other words, singularities resulting from gravitational collapse only occur in places like black holes, where they are decently hidden from outside view by an event horizon.

Strictly speaking, this is known as the weak cosmic censorship hypothesis: protect observers who live outside the black hole from the consequences of a predicted breakdown at the singularity. But it does nothing for the poor unfortunate astronaut who falls into the hole. Should not even God protect his own shame?

There are some solutions to the equations of general relativity in which it is possible for our astronauts to see a naked singularity. He may be able to avoid colliding with the singularity and instead fall through a “wormhole” and come out into another region of the universe.

This would offer great possibilities for travel in space and time, but unfortunately, it seems that all solutions may be highly unstable. Minimal turbulence, such as the presence of an astronaut, can alter them so that the astronaut cannot see the singularity until it hits it and its time is up.

In other words, the singularity is always rooted in its future and never in its past. The stronger version of the cosmic censorship hypothesis states that in a realistic solution, the singularity always occurs either entirely in the future, like the gravitational collapse singularity, or entirely in the past, like the Big Bang.

It is to be hoped that there is some version of the censorship hypothesis, as it may be possible to travel in the past close to naked singularities. While this would be fine for science fiction writers, it would mean that one’s life would never be safe. Someone can go past and kill your father or mother before you conceive. 

In a gravitational collapse to form a black hole, the movements would be disrupted by the emission of gravitational waves.

So one would expect that it wouldn’t take long for a black hole to settle into a steady state. It was generally believed that this final steady-state would depend on the details of the body that collapsed to form the black hole.

A black hole can have any shape or size, and its shape may not be fixed but can be pulsating.

However, in 1967, the study of black holes was revolutionized by a paper in Dublin by Werner Israel. Israel showed that any black hole that is not rotating must be perfectly round or spherical. Moreover, its size will depend only on its mass.

It can, in fact, be described by a special solution to Einstein’s equations that had been known since 1917, when it was discovered by Karl Schwarzschild shortly after the discovery of general relativity.

At first, Israel’s result was interpreted by many, including Israel himself, as evidence that black holes would only form from the collapse of objects that were perfectly round or spherical. Since no real body will be perfectly spherical, this means that in general, the collapse of gravity will produce naked singularities.

However, Israel’s result had a different interpretation, advocated in particular by Roger Penrose and John Wheeler. It was that a black hole should behave like a ball of fluid. Although a body may start out in a non-spherical state as it collapsed to form a black hole, it will settle into a spherical state due to the emission of gravitational waves.

Further calculations supported this view and it came to be generally adopted. Israel’s result had only dealt with the case of black holes composed of non-rotating bodies. On the analogy of a ball of liquid, one would expect that a black hole formed by the collapse of a rotating body would not be perfectly round.

This would be a bulge around the equator due to the effect of rotation. We see such a small bulge in the sun, which is caused by its rotation once every twenty-five days. In 1963, Roy Kerr, a New Zealander, had found a set of black-hole solutions to the equations of general relativity comparable to that of the Schwarzschild solutions.

These “Kerr” black holes rotate at a constant rate, their size, and shape, depending only on their mass and rate of rotation. If the rotation was zero, then the black hole was perfectly round and the solution was similar to the Schwarzschild solution. But if the rotation was not zero, the black hole would have emerged outward near its equator.

It was therefore natural to speculate that a rotating body would end up in the state described by the Kerr solution in order to form a black hole. In 1970, a colleague and fellow research student of mine, Brandon Carter, took the first steps toward proving this conjecture. They showed that in a stationary rotating black hole having an axis of symmetry, such as a spinning top, its size and shape would depend only on its mass and rate of rotation.

Then, in 1971, I proved that any stationary rotating black hole would indeed have such an axis of symmetry. Finally, in 1973, David Robinson at King’s College, London, used Carter and my results to show that the conjecture was correct: such a black hole did indeed have to be the Kerr solution.

So after the gravitational collapse, a black hole must settle into a state in which it can spin, but cannot pulsate. Furthermore, its size and shape would depend only on its mass and rate of rotation, not on the nature of the body that collapsed to form it.

This result is known by the proverb “A black hole has no hair.” This means that a huge amount of information about the body’s collapse must be lost when the black hole is formed because the latter, all we can possibly measure about the body, is its mass and rate of rotation.

Its importance will be seen in the next lecture. The no-hair theorem also has great practical significance as it greatly limits the possible types of black holes. So one can build detailed models of objects that may contain black holes, and compare the model’s predictions with the observations.

Black holes are one of only a very few cases in the history of science where a theory was developed into a mathematical model in great detail before there was any evidence from observations that it was correct. Indeed, this used to be the main argument of opponents of black holes. How can one believe objects for which calculations based on the dubious theory of general relativity were the only evidence?

In 1963, however, Maarten Schmidt, an astronomer at Mount Palomar Observatory in California, found a faint, star-like object in the direction of the source of the radio waves named 3C273, source number 273 in the Third Cambridge Catalog of Radio Sources.

When they measured the object’s redshift, they found it was too large to be caused by a gravitational field: if it were a gravitational redshift, the object would have to be so massive and near us enough that it would disturb the orbits of planets in the Solar System.

This suggested that the redshift was instead caused by the expansion of the universe, which meant that the object was far away. And to be visible at such a great distance, the object must be very bright and emit a large amount of energy.

The only mechanism people could think of that would produce such a huge amount of energy appears to be the gravitational collapse of not just a single star but the entire central region of a galaxy.

Several other similar “quasistellar objects” or quasars have been discovered, all with large redshifts. But they are all too far away, and too hard to observe to provide conclusive evidence of a black hole.

Further impetus for the existence of black holes came in 1967 with the discovery of certain objects in the sky by Jocelyn Bell, a research student at Cambridge, that was emitting regular pulses of radio waves. 

At first, Jocelyn and her supervisor, Anthony Hewish, thought that perhaps they had made contact with an alien civilization in the Milky Way.

In fact, at the symposium at which he announced his discovery, I remember him calling LGM 1–4, the first four sources to be found LGM, standing for “Little Green Men”. In the end, however, he and everyone else came to the less romantic conclusion that these objects, which were named pulsars, were actually just rotating neutron stars.

They were emitting pulses of radio waves due to a complex signal between their magnetic field and the surrounding matter. This was bad news for space Western writers, but there was great hope for the small number of us who believed in black holes at the time.

This was the first positive evidence that neutron stars existed. A neutron star has a radius of about ten miles, which is only a few times the critical radius at which a star becomes a black hole. If a star could collapse to such a small size, it was not unreasonable to expect that other stars could collapse to an even smaller size and become a black hole.

How can we expect to detect a black hole, because by its definition it emits no light? It might sound like a black cat looking for a coal cellar. Fortunately, there is a way, as John Mitchell pointed out in his pioneering paper in 1783, that a black hole still exerts a gravitational force on nearby objects.

Astronomers have observed several systems in which two stars orbit around each other, attracted to each other by gravity. They also observed systems in which there is only one visible star orbiting an unseen companion.

Of course, one cannot immediately conclude that the companion is a black hole. It may just be a star that is very weak to see. However, some of these systems, such as Cygnus X-I, are also strong sources of X rays.

The best explanation for this phenomenon is that the X-rays originate from matter that has been blown off the surface of the visible star. As it falls toward the invisible companion, it develops a spiral motion—as if water is flowing out of a bath—and it becomes very hot, releasing X rays.

For this mechanism to work, the invisible object would have to be very small, such as a white dwarf, neutron star, or black hole. Now, from the observed motion of the visible star, one can determine the minimum possible mass of the invisible object. In the case of Cygnus X-I, it is about six times the mass of the Sun.

According to Chandrasekhar’s result, this is too much for the unseen object to be a white dwarf. It is too large to be a neutron star. Therefore, it seems that it must be a black hole. There are other models to explain Cygnus X-I that do not involve black holes, but they are all far-fetched.

A black hole appears to be the only real natural explanation of the observations. Despite this, my bet with Kip Thorne of the California Institute of Technology is that Cygnus X-I doesn’t actually have a black hole.

It is a form of insurance policy for me. I’ve done a lot of work on black holes, and it would all be doomed if it turned out that black holes don’t exist. But in that case, I would have the consolation of winning my bet, which would have earned me four years of Private Eye magazine.

If black holes do exist, then Kip would only get one year of the penthouse, because when we bet in 1975, we were 80 percent certain that Cygnus was a black hole. So far I would say we are almost 95 percent sure, but the stakes are yet to be settled.

There is evidence of black holes in many other systems in our galaxy and very large black holes in the centers of other galaxies and quasars. One can also consider the possibility that there could be black holes with masses much less than the mass of the Sun. Such black holes could not be formed by gravitational collapse, because their mass is less than the mass limit of Chandrasekhar.

Stars of this low mass can support themselves against the force of gravity even when their nuclear fuel is exhausted. Therefore, low-mass black holes can only form when the matter is compressed to enormous densities by very large external pressures.

Such situations can happen in a very large hydrogen bomb. Physicist John Wheeler once calculated that if one carried heavy water into all of the world’s oceans, one could create a hydrogen bomb that would compress the matter at the center so much that a black hole would form. Unfortunately, though, no one will be left to see it.

A more plausible possibility is that such low-mass black holes would have formed under the high temperatures and pressures of the very early universe. Black holes could have formed if the early universe was not perfectly smooth and uniform because then a smaller region that was denser than average could have compressed in such a way as to form a black hole.

Black Hole Image

But we do know that there must have been some irregularities, because otherwise the matter in the universe would still be perfectly evenly distributed in the present age, rather than clump together into stars and galaxies. Whether or not a significant number of these primitive black holes would have formed due to the irregularities required for stars and galaxies depends on the details of the conditions in the early universe.

So if we could determine how many early black holes there are now, we would learn a lot about the early stages of the universe. Primordial black holes with masses over a thousand million tons – the mass of a large mountain – can only be detected by their gravitational influence on other visible matter or on the expansion of the universe.

However, as we’ll learn in the next lecture, black holes aren’t actually black: they glow like a hot body, and the smaller they are, the brighter they get. So, paradoxically, smaller black holes may actually be easier to detect than larger black holes.


Related: How Edwin Hubble Discovered the Expanding Universe?

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Information Source: Stephen Hawking Book – Theory of Everything

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