A Brief History of Time
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A Brief History of Time: From the Big Bang to Black Holes is a popular-science book on cosmology (the study of the universe) by British physicist Stephen Hawking. It was first published in 1988. Hawking wrote the book for nonspecialist readers with no prior knowledge of scientific theories.
|Publisher||Bantam Dell Publishing Group|
|Media type||Print (Hardcover and Paperback)|
|LC Class||QB981 .H377 1998|
|Followed by||Black Holes and Baby Universes and Other Essays|
In A Brief History of Time, Hawking writes in non-technical terms about the structure, origin, development and eventual fate of the universe, which is the object of study of astronomy and modern physics. He talks about basic concepts like space and time, basic building blocks that make up the universe (such as quarks) and the fundamental forces that govern it (such as gravity). He writes about cosmological phenomena such as the Big Bang and black holes. He discusses two major theories, general relativity and quantum mechanics, that modern scientists use to describe the universe. Finally, he talks about the search for a unifying theory that describes everything in the universe in a coherent manner.
The book became a bestseller and sold more than 10 million copies in 20 years. It was also on the London Sunday Times bestseller list for more than five years and was translated into 35 languages by 2001.
Early in 1983, Hawking first approached Simon Mitton, the editor in charge of astronomy books at Cambridge University Press, with his ideas for a popular book on cosmology. Mitton was doubtful about all the equations in the draft manuscript, which he felt would put off the buyers in airport bookshops that Hawking wished to reach. With some difficulty, he persuaded Hawking to drop all but one equation. The author himself notes in the book's acknowledgements that he was warned that for every equation in the book, the readership would be halved, hence it includes only a single equation: E = mc2. The book does employ a number of complex models, diagrams, and other illustrations to detail some of the concepts it explores.
In A Brief History of Time, Stephen Hawking attempts to explain a range of subjects in cosmology, including the Big Bang, black holes and light cones, to the nonspecialist reader. His main goal is to give an overview of the subject, but he also attempts to explain some complex mathematics. In the 1996 edition of the book and subsequent editions, Hawking discusses the possibility of time travel and wormholes and explores the possibility of having a universe without a quantum singularity at the beginning of time.
Chapter 1: Our Picture of the UniverseEdit
In the first chapter, Hawking discusses the history of astronomical studies, including the ideas of Aristotle and Ptolemy. Aristotle, unlike many other people of his time, thought that the Earth was round. He came to this conclusion by observing lunar eclipses, which he thought were caused by the earth's round shadow, and also by observing an increase in altitude of the North Star from the perspective of observers situated further to the north. Aristotle also thought that the sun and stars went around the Earth in perfect circles, because of "mystical reasons". Second-century Greek astronomer Ptolemy also pondered the positions of the sun and stars in the universe and made a planetary model that described Aristotle's thinking in more detail.
Today, it is known that the opposite is true: the earth goes around the sun. The Aristotelian and Ptolemaic ideas about the position of the stars and sun were disproved in 1609. The first person to present a detailed arguments that the earth revolves around the sun was the Polish priest Nicholas Copernicus, in 1514. Nearly a century later, Galileo Galilei, an Italian scientist, and Johannes Kepler, a German scientist, studied how the moons of some planets moved in the sky, and used their observations to validate Copernicus's thinking.
To fit the observations, Kepler proposed an elliptical orbit model instead of a circular one. In his 1687 book on gravity, Principia Mathematica, Isaac Newton used complex mathematics to further support Copernicus's idea. Newton's model also meant that stars, like the sun, were not fixed but, rather, faraway moving objects. Nevertheless, Newton believed that the universe was made up of an infinite number of stars which were more or less static. Many of his contemporaries, including German philosopher Heinrich Olbers, disagreed.
The origin of the universe represented another great topic of study and debate over the centuries. Early philosophers like Aristotle thought that the universe has existed forever, while theologians such as St. Augustine believed it was created at a specific time. St. Augustine also believed that time was a concept that was born with the creation of the universe. More than 1000 years later, German philosopher Immanuel Kant thought that time goes back forever.
In 1929, astronomer Edwin Hubble discovered that galaxies are moving away from each other. Consequently, there was a time, between ten and twenty billion years ago, when they were all together in one singular extremely dense place. This discovery brought the concept of the beginning of the universe within the province of science. Today, scientists use two partial theories, Einstein's general theory of relativity and quantum mechanics, to describe the workings of the universe. Scientists are still looking for a complete unified theory that would describe everything in the universe. Hawking believes that the discovery of a complete unified theory may not aid the survival of our species, and may not even affect our life-style, but that humanity's deepest desire for knowledge is justification enough for our continuing quest,and that our goal is nothing less than a complete description of the universe we live in. 
Chapter 2: Space and TimeEdit
Stephen Hawking talks about how the Aristotle theory of absolute space came to an end by the Newton's theory that 'rest' and 'motion' can be the same state if an observer sees the event at rest or if he moves with the same speed as that of the event. So 'rest' can't be the standard position. Moreover, Galileo Galilei also disproves Aristotle theory that heavier body falls more quickly than the lighter one just because of its mass. He experimentally proves it by sliding objects of different weights, and even concludes that both these object would fall at same rate and would reach the bottom at the same time, unless external force acts on them.
Aristotle and Newton believed in absolute time. They believed that if an event is measured using two different clocks at different state of motion, they'll have to agree on the same time, if clocks used are synchronized, which by now we know it isn't. But the fact that the light travels with a finite speed was first explained by the Danish scientist Ole Rømer, by his observation of Jupiter and one of its moons Io. He observed that Io appeared sometimes quicker and sometimes later when it revolves around Jupiter, because the distance between Earth and Jupiter changes every time because of their orbital motion around the sun.
The actual propagation of light was published by James Clerk Maxwell who told that light travels with a fixed speed. Later, many argued that light must travel through a hypothetical fluid called aether, which was disproved by the Michelson–Morley experiment. Einstein and Poincaré later on argued that there's no need of aether provided one has to abandon absolute time. The Special Theory of Relativity is based on this, that light travels with a finite speed no matter what the speed of the observer is. Moreover, the speed of light is assumed to be the ultimate speed.
Mass and energy are also related by the famous equation E=mc^2, and so it would require infinite energy to get to the speed of light. A new way of defining a metre using speed of light is also developed. 'Events' can also be described by using the light cones, a space time graphical representation which restricts what all events are allowed and what are not based on the past and the future light cones. The new 4-dimensions is also described, how different the path is seen when one changes reference from 3D to 4D or 3D to 2D.
General Theory of Relativity explains about how path of light ray is affected by 'gravity' which according to Einstein is a mere illusion in contrast to Newton's views. It is space-time curvature where light moves in a straight path in 4D which is seen as a curve in 3D. These straight line paths are Geodesics. Twin paradox, a part theory of Relativity which explains that two twins can age differently if they move at relatively different speeds or even at different places where spacetime curvature is different. Special relativity is based upon arenas of space and time where events take place whereas General Relativity is dynamic where force could change spacetime curvature, which gives rise to the expanding universe. Hawking and Roger Penrose worked upon this and later proved using general Relativity that if the Universe had a beginning then it also must have an end.
Chapter 3: The Expanding UniverseEdit
In this chapter, Hawking first describes how physicists and astronomers calculated the relative distance of stars from the Earth. In the 18th century, Sir William Herschel confirmed the positions and distances of many stars in the night sky. In 1924, Edwin Hubble discovered a method to measure the distance using brightness of the stars. The luminosity, brightness and distance are related by a simple mathematical formula. Using all these, he fairly calculated distances of nine different galaxies. We live in a spiral galaxy just like other galaxies containing vast numbers of stars.
The stars are very far away from us, so we only observe their one characteristic feature, their light. When this light is passed through a prism, it gives rise to a spectrum. Every star has its own spectrum and since each element has its own unique spectra, we can know a star's composition. We use thermal spectra of the stars to know their temperature. In 1920, when scientists were examining spectra of different stars, they found that some of the characteristic lines of the star spectrum was shifted towards the red end of the spectrum. The implications of this phenomenon was given by the Doppler effect, and it was clear that some stars were moving away from us.
It was assumed that since some stars are red shifted, some stars would also be blue shifted. When found, none of them were blue shifted. Hubble found that the amount of redshift is directly proportional to relative distance. So, it was clear that the Universe is expanding. Despite this the concept of a static universe persisted until the 20th century. Einstein was so sure of a static universe that he developed 'Cosmological Constant' and introduced 'anti-gravity' forces to persist with the earlier claim. Moreover, many astronomers also tried to avoid the face value of General Relativity and stuck with their static universe except one Russian physicist Alexander Friedmann.
Friedmann made two very simple assumptions: the universe is identical in every direction, i.e. Homogenity and that this would be true wherever we look from, i.e. Isotropy. His results showed that the Universe is non-static. His assumptions were later proved when two physicists at Bell's laboratory, Arno Penzias and Robert Wilson found extra microwave radiation noise not only from the one particular part of the sky but from everywhere and by nearly the same amount. Then, Friedmann's first assumption was proved as true.
At around the same time, Robert H. Dicke and Jim Peebles were also working on microwave radiation. They argued that they should be able to see the glow of the early universe as microwave radiations. Wilson and Penzias had already done this, so they were awarded with the Noble Prize in 1978. In addition, our place in the Universe is not exceptional, so we should see the universe as the same from any other part of space, which proves Friedmann's second assumption. His work remained largely unknown until similar models were made by Howard Robertson and Arthur Walker.
Friedmann's model gave rise to three different types of model of the universe. First, the universe would expand for a given amount of time and if the expansion rate is less than the density of the universe (leading to gravitational attraction), it would ultimately lead to the collapse of the universe at a later stage. Secondly, the universe would expand and at sometime if the expansion rate and the density of the universe become equal, it would expand slowly and stop at infinite time, leading to a somewhat static universe. Thirdly, the universe would continue to expand forever if the density of the universe is less than the critical amount required to balance the expansion rate of the universe.
The first model depicts the space of universe to be curved inwards, a somewhat earth-like structure. In the second model, the space would lead to a flat structure, and the third model results in negative curvature, or saddle shaped. Even if we calculate, the current expansion rate is more than the critical density of the universe including the dark matter and all the stellar masses. The first model included the beginning of the universe as a big-bang from a space of infinite density and zero volume known as 'singularity', a point where General Theory of Relativity (Friedmann's solutions are based in it) also breaks down.
This concept of the beginning of time was against many religious beliefs, so a new theory was introduced. 'Steady state theory' by Hermann Bondi, Thomas Gold and Fred Hoyle to tackle the Big Bang theory. Its predictions also matched with the current Universe structure. But the fact that radiowave sources near us are far fewer than from the distant universe and there were numerous more radio sources than at present, resulted in failure of this theory and everybody finally supported the Big Bang theory.
Evgeny Lifshitz and Isaak Markovich Khalatnikov also tried to avoid the Big Bang theory but also failed. Roger Penrose used light cones and General Relativity to prove that a collapsing star could result in a region of zero size and infinite density and curvature called a Black Hole. Hawking and Penrose proved together that the universe should have arisen from a singularity, which Hawking himself disproved once Quantum effects are taken into account.
Chapter 4: The Uncertainty PrincipleEdit
The uncertainty principle says that the speed and the position of a particle cannot be found at the same time. To find where a particle is, scientists shine light at the particle. If a high frequency light is used, the light can find the position more accurately but the particle's speed will be unknown (because the light will change the speed of the particle). If a lower frequency light is used, the light can find the speed more accurately but the particle's position will be unknown. The uncertainty principle disproved the idea of a theory that was deterministic, or something that would predict everything in the future.
How light behaves is also talked more about in this chapter. Some theories say that light acts like particles even though it really is made of waves; one theory that says this is Planck's quantum hypothesis. A different theory also says that light waves also act like particles; a theory that says this is Heisenberg's uncertainty principle.
Light waves have crests and troughs. The highest point of a wave is the crest, and the lowest part of the wave is a trough. Sometimes more than one of these waves can interfere with each other - the crests and the troughs line up. This is called light interference. When light waves interfere with each other, this can make many colors. An example of this is the colors in soap bubbles.
Chapter 5: Elementary Particles and Forces of NatureEdit
Quarks are very small things that make up everything we see (matter). There are six different "flavors" of quarks: the up quark, down quark, strange quark, charmed quark, bottom quark, and top quark. Quarks also have three "colors": red, green, and blue. There are also anti-quarks, which are the opposite of the regular quarks. In total, there are 18 different types of regular quarks, and 18 different types of anti quarks. Quarks are known as the "building blocks of matter" because they are the smallest thing that make up all the matter in the universe.
All particles (for example, the quarks) have something called spin. The spin of a particle shows us what a particle looks like from different directions. For example, a particle of spin 0 looks the same from every direction. A particle of spin 1 looks different in every direction, unless the particle is spun completely around (360 degrees). Hawking's example of a particle of spin 1 is an arrow. A particle of spin two needs to be turned around halfway (or 180 degrees) to look the same.
The example given in the book is of a double-headed arrow. There are two groups of particles in the universe: particles with a spin of 1/2, and particles with a spin of 0, 1, or 2. All of these particles follow the Pauli exclusion principle. Pauli's exclusion principle says that particles cannot be in the same place or have the same speed. If Pauli's exclusion principle did not exist, then everything in the universe would look the same, like a roughly uniform and dense "soup".
Particles with a spin of 0, 1, or 2 move force from one particle to another. Some examples of these particles are virtual gravitons and virtual photons. Virtual gravitons have a spin of 2 and they represent the force of gravity. This means that when gravity affects two things, gravitons move to and from the two things. Virtual photons have a spin of 1 and represent electromagnetic forces (or the force that holds atoms together).
Besides the force of gravity and the electromagnetic forces, there are weak and strong nuclear forces. Weak nuclear forces are the forces that cause radioactivity, or when matter emits energy. Weak nuclear force works on particles with a spin of 1/2. Strong nuclear forces are the forces that keep the quarks in a neutron and a proton together, and keeps the protons and neutrons together in an atom. The particle that carries the strong nuclear force is thought to be a gluon. The gluon is a particle with a spin of 1. The gluon holds together quarks to form protons and neutrons. However, the gluon only holds together quarks that are three different colors. This makes the end product have no color. This is called confinement.
Some scientists have tried to make a theory that combines the electromagnetic force, the weak nuclear force, and the strong nuclear force. This theory is called a grand unified theory (or a GUT). This theory tries to explain these forces in one big unified way or theory.
Chapter 6: Black HolesEdit
Black holes are talked about in this chapter. Black holes are stars that have collapsed into one very small point. This small point is called a singularity. Black holes suck things into their center because they have very strong gravity. Some of the things it can suck in are light and stars. Only very large stars, called super-giants, are big enough to become a black hole.
The star must be one and a half times the mass of the sun or larger to turn into a black hole. This number is called the Chandrasekhar limit. If the mass of a star is less than the Chandrasekhar limit, it will not turn into a black hole; instead, it will turn into a different, smaller type of star. The boundary of the black hole is called the event horizon. If something is in the event horizon, it will never get out of the black hole.
Black holes can be shaped differently. Some black holes are perfectly spherical - like a ball. Other black holes bulge in the middle. Black holes will be spherical if they do not rotate. Black holes will bulge in the middle if they rotate.
Black holes are difficult to find because they do not let out any light. They can be found when black holes suck in other stars. When black holes suck in other stars, the black hole lets out X-rays, which can be seen by telescopes.
In this chapter, Hawking talks about his bet with another scientist, Kip Thorne. Hawking bet that black holes did not exist, because he did not want his work on black holes to be wasted. He lost the bet.
Chapter 7: Black Holes Ain't So BlackEdit
This chapter explains more about black holes.
Hawking realized that the event horizon of a black hole could only get bigger, not smaller. The area of the event horizon of a black hole gets bigger whenever something falls into the black hole. He also realized that when two black holes combine, the size of the new event horizon is greater than or equal to the sum of the event horizons of the two original black holes. This means that a black hole's event horizon can never get smaller.
Disorder, also known as entropy, is related to black holes. There is a scientific law that has to do with entropy. This law is called the second law of thermodynamics, and it says that entropy (or disorder) will always increase in an isolated system (for example, the universe). The relation between the amount of entropy in a black hole and the size of the black hole's event horizon was first thought of by a research student (Jacob Bekenstein) and proven by Hawking, whose calculations said that black holes emit radiation. This was strange, because it was already said that nothing can escape from a black hole's event horizon.
This problem was solved when the idea of pairs of "virtual particles" was thought of. One of the pair of particles would fall into the black hole, and the other would escape. This would look like the black hole was emitting particles. This idea seemed strange at first, but many people accepted it after a while.
Chapter 8: The Origin and Fate of the UniverseEdit
How the universe started and how it might end is discussed in this chapter.
Most scientists agree that the universe started in an expansion called the Big Bang. The model for this is called the "hot big bang model". When the universe starts getting bigger, the things inside of it also begin to get cooler. When the universe was first beginning, it was infinitely hot. The temperature of the universe cooled and the things inside the universe began to clump together.
Hawking also discusses how the universe could have been. For example, if the universe formed and then collapsed quickly, there would not be enough time for life to form. Another example would be a universe that expanded too quickly. If a universe expanded too quickly, it would become almost empty. The idea of many universes is called the many-worlds interpretation.
Inflationary models and the idea of a theory that unifies quantum mechanics and gravity also are discussed in this chapter.
Each particle has many histories. This idea is known as Feynman's theory of sum over histories. A theory that unifies quantum mechanics and gravity should have Feynman's theory in it. To find the chance that a particle will pass through a point, the waves of each particle needs to be added up. These waves happen in imaginary time. Imaginary numbers, when multiplied by themselves, make a negative number. For example, 2i X 2i = -4.
Chapter 9: The Arrow of TimeEdit
In this chapter Hawking talks about why "real time" as humans observe and experience it (in contrast to the "imaginary time" in the laws of science) seems to have a certain direction, notably from the past towards the future. The things that give time this property are the arrows of time.
Firstly, there is the thermodynamic arrow of time. According to this, starting from any higher order organized state, the overall disorderliness in the world always increases as time passes. This is why we never see the broken pieces of a cup gather themselves together to form a whole cup. Even though human civilizations have tried to make things more orderly, the energy dissipated in this process has created more overall disorder in the universe.
The second arrow is the psychological arrow of time. Our subjective sense of time seems to flow in one direction, which is why we remember the past and not the future. Hawking claims that our brain measures time in a way where disorder increases in the direction of time. We never observe it working in the opposite direction. In other words, the psychological arrow of time is intertwined with the thermodynamic arrow of time.
Thirdly there is the cosmological arrow of time, the direction of time in which our universe is expanding and not contracting. Hawking believes that in order for us to observe and experience the first two arrows of time, the universe would have to begin in a very smooth and orderly state. And then as it expanded, it became more disorderly. So the thermodynamic arrow agrees with the cosmological arrow.
Because of the "no boundary" proposal for the universe, after a period of expansion, the universe will probably start to contract. It will probably not go backwards in time to a more smooth, orderly state. The thermodynamic arrow in the contracting phase will not be as strong.
As for why humans experience these three arrows of time going in the same direction, Hawking postulates that humans have been living in the expanding phase of the universe. He thinks that intelligent life couldn't exist in the contracting phase of the universe. Only the expanding phase of the universe is suitable for intelligent beings like humans to exist, because it contains a strong thermodynamic arrow. Hawking calls this the "weak anthropic principle".
Chapter 10: The Unification of PhysicsEdit
Physicists have come up with partial theories to describe a limited range of things, but a complete, unified and consistent theory which can take into account all of these partial theories remain unknown. Hawking is cautiously optimistic that such a unified theory of the universe may be found soon. Such a theory must combine the classical theory of gravity with the uncertainty principle found in quantum mechanics. Attempts to do that have led to the occurrence of absurd infinitely massed particles or an infinitely small universe. In 1976, the theory of "supergravity" was suggested as a solution. But the calculations to verify the theory was deemed time-consuming and thus abandoned.
In 1984, another set of theories called the "string theories", where basic objects are not particles but two-dimensional strings, became popular among physicists. They were claimed to explain the existence of certain particles better than supergravity and other theories. However, according to string theories, instead of the usual four space-time dimensions, the universe could have dozens of them. It is imagined that humans do not experience the other dimensions because these are too tightly curled up.
This is due to the "weak anthropic principle", according to which intelligent beings like humans cannot exist in any other way. String theories appear to allow this situation for certain regions of the universe, but there may be other regions of the universe where more than four dimensions are prominent. Furthermore, supergravity, p-brane and string theories all describe different situations with similar results, as if using different approximations of the same theory.
Hawking thus proposes three possibilities: 1) there exists a complete unified theory that we will eventually find; 2) there are an infinite number of theories that overlap and describe the universe more and more accurately and 3) there is no ultimate theory. The third possibility has been sidestepped by acknowledging the limits set by the uncertainty principle. The second possibility describes what has been happening in physical sciences so far, with increasingly accurate partial theories.
Hawking believes that such refinement has a limit and that by studying the very early stages of the universe in a laboratory setting, it is possible to finally find a complete unified theory in the 21st century. Such a theory might not be proven but would be mathematically consistent. The predictions of such a basic set of laws would match our observations. However, given the complicated nature of realistic situations, it would only be a first step to a complete understanding of the events around us.
Chapter 11: ConclusionEdit
Humans have always wanted to make sense of the universe and their place in it. At first, events were considered random and controlled by human-like emotional spirits. But in astronomy and in some other situations, regularities were observed. With the advancement of the human civilization in the modern age, more regularities and laws were discovered. Laplace suggested at the beginning of the nineteenth century that the universe's structure and evolution could eventually be precisely explained by a set of laws. However, the origin of these laws was left in God's domain. In the twentieth century, quantum theory introduced the uncertainty principle, which set limits to the predictive accuracy of laws to be discovered.
The big bang implied by the general theory of relativity indicates that a creator of the universe or God has the freedom to choose the origin and the laws of the universe. When one combines theory of relativity with quantum mechanics, however, a unified and completely self-contained theory may emerge, in which God has little or no role to play. So the search of a unified theory may shed light on the nature of God. However, most scientists today are working on the theories themselves rather than asking such philosophical questions. On the other hand, these physical theories are so mathematical and technical that philosophers are not discussing them like they used to do, let alone ordinary people. Hawking would like to see that eventually everybody would one day talk about these theories in order to understand the true origin and nature of the universe, accomplishing the ultimate triumph of human reasoning.
- 1988: The first edition included an introduction by Carl Sagan that tells the following story: Sagan was in London for a scientific conference in 1974, and between sessions he wandered into a different room, where a larger meeting was taking place. "I realized that I was watching an ancient ceremony: the investiture of new fellows into the Royal Society, one of the most ancient scholarly organizations on the planet. In the front row, a young man in a wheelchair was, very slowly, signing his name in a book that bore on its earliest pages the signature of Isaac Newton... Stephen Hawking was a legend even then." In his introduction, Sagan goes on to add that Hawking is the "worthy successor" to Newton and Paul Dirac, both former Lucasian Professors of Mathematics.
The introduction was removed after the first edition, as it was copyrighted by Sagan, rather than by Hawking or the publisher, and the publisher did not have the right to reprint it in perpetuity. Hawking wrote his own introduction for later editions.
- 1994, A brief history of time – An interactive adventure. A CD-Rom with interactive video material created by S. W. Hawking, Jim Mervis, and Robit Hairman (available for Windows 95, Windows 98, Windows ME, and Windows XP).
- 1996, Illustrated, updated and expanded edition: This hardcover edition contained full-color illustrations and photographs to help further explain the text, as well as the addition of topics that were not included in the original book.
- 1998, Tenth-anniversary edition: It features the same text as the one published in 1996, but was also released in paperback and has only a few diagrams included. ISBN 0553109537
The New York's Metropolitan Opera had commissioned an opera to premiere in 2015–16 based on Hawking's book. It was to be composed by Osvaldo Golijov with a libretto by Alberto Manguel in a production by Robert Lepage. The planned opera was changed to be about a different subject and eventually canceled completely.
- A Brief History of Time is based on the scientific paper J. B. Hartle; S. W. Hawking (1983). "Wave function of the Universe". Physical Review D. 28 (12): 2960. Bibcode:1983PhRvD..28.2960H. doi:10.1103/PhysRevD.28.2960.
- Paris, Natalie (26 April 2007). "Hawking to experience zero gravity". The Daily Telegraph. London.
- "Hawking's briefer history of time". news.bbc.co.uk. 15 October 2001. Retrieved 2008-08-06.
- White, Michael and John Gribbin (1992). Stephen Hawking: a life in science. Viking Press. ISBN 978-0670840137.
- Hawking, Stephen (1988). A Brief History of Time. Bantam Books. ISBN 978-0-553-38016-3.
- A brief history of time – An interactive adventure
- A new Robert Lepage at the Met ‹See Tfd›(in French)
- Cooper, Michael (29 November 2016). "Osvaldo Golijov's New Opera for the Met is Called Off". The New York Times.