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Properties of Expanding Universes

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Stephen Hawking's 1966 doctoral thesis made available for first time

Cambridge University says Properties of Expanding Universes is already most-requested item in open access repository

Anyone in the world can now download and read the doctoral thesis of a 24-year-old Cambridge postgraduate student, written in 1966; how many will fully understand Properties of Expanding Universes is another matter.

Stephen Hawking hopes that giving free access to his early work will inspire others, not just to think and learn but to share research. He said: “By making my PhD thesis open access, I hope to inspire people around the world to look up at the stars and not down at their feet; to wonder about our place in the universe and to try and make sense of the cosmos.

“Anyone, anywhere in the world should have free, unhindered access to not just my research, but to the research of every great and inquiring mind across the spectrum of human understanding.”

Cambridge University, which calls the thesis “historic and compelling”, says it is already the most-requested item in its open access repository, Apollo. “In just the past few months, the university has received hundreds of requests from readers wishing to download Prof Hawking’s thesis in full.”

The work considers implications and consequences of the expansion of the universe, and its conclusions include that galaxies cannot be formed through the growth of perturbations that were initially small.

The thesis is now freely available to all, to mark Open Access Week 2017 . Hawking said: “Each generation stands on the shoulders of those who have gone before them, just as I did as a young PhD student in Cambridge, inspired by the work of Isaac Newton, James Clerk Maxwell and Albert Einstein. It’s wonderful to hear how many people have already shown an interest in downloading my thesis – hopefully they won’t be disappointed now that they finally have access to it.”

Hawking’s thesis, with a typed dedication to his supervisor, and a handwritten “this dissertation is my original work – SW Hawking”, set him on the path to becoming one of the most famous scientists in the world. He remains a commanding figure decades after a form of motor neurone disease – diagnosed soon after his 21st birthday – left him confined to a wheelchair and dependent on a computerised voice system for speech.

His 1988 A Brief History of Time became one of the most successful popular science books ever published, with more than 10m copies sold worldwide in more than 40 languages.

At Oxford he studied physics though his father wanted him to take medicine, because his own original choice of mathematics wasn’t available. According to his website , “after three years and not very much work” he was awarded a first-class honours degree – he now holds 12 honorary degrees – and then moved to Cambridge to undertake research in cosmology. From 1979 to 2009 he was the Lucasian professor of mathematics at the university – a post once held by Isaac Newton – and he retains an office in the department.

The university hopes that Hawking’s thesis may encourage other scholars to make their work freely available online. From this October all PhD graduates will be required to deposit a digital copy of their theses, and will be urged to make them open access. Cambridge will also be trying to persuade all its former academics, who include 98 Nobel prize winners, to follow Hawking’s example.

The university library has 8m books and periodicals, and a million maps, on 125 miles of shelving expanding by two miles every year.

The treasures in the Apollo archive, which together have been downloaded a million times from all over the world in 2017, include 15,000 research articles, 10,000 images and 2,400 theses. However, the university says it is often a struggle to give free access to historic theses.

Arthur Smith, deputy head of scholarly communication, said more open-access research could lead to major breakthroughs. He said: “By eliminating the barriers between people and knowledge we can realise new breakthroughs in all areas of science, medicine and technology.

“It is especially important for disseminating the knowledge acquired during doctoral research studies. PhD theses contain a vast trove of untapped and unique information just waiting to be used, but which is often locked away from view and scrutiny.”

Hawking, whose original medical diagnosis more than half a century ago gave him two years to live, continues to work and and lecture internationally. Richard Branson has offered him a free place on his Virgin Galactic spacecraft , and his website says “he still hopes to make it into space one day”.

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Here's Stephen Hawking's 1966 Thesis That Crashed Cambridge's Website

Here's the full PDF.

In honor of Open Access Week, Cambridge University, Stephen Hawking’s alma mater, made his Ph.D. thesis, “Properties of Expanding Universes,” available for free on its website. Good luck accessing it, though. Ever since the document went live and media outlets began covering it, browser requests have been timing out. Demand for the thesis crashed the site .

“Professor Hawking is a phenomenon in his own right, but I don’t think they had expected quite so much interest,” a university spokesperson tells Inverse. “We’ve been overwhelmed in various ways by it.” According to Cambridge sources, around 428,000 people had viewed page by Tuesday morning, and 60,000 downloaded the thesis. There have been about 300,000 views for the digital library version.

With the servers clogged, we decided to host the entire 32-megabyte, 134-page PDF below. Enjoy.

If you liked this article, check out this video of Stephen Hawking ’s chilling prediction for the future of humanity.

Stephen Hawking’s PhD thesis is now available online

Fans have already crashed the website.

By Angela Chen

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Anyone can download and read Stephen Hawking’s 1966 doctoral thesis “ Properties of expanding universes” now that the scientist has given permission for it to be open access.

Hawking is a theoretical physicist famous for his research in relativity and black holes. The 134-page dissertation, written when Hawking was only 24, is the most-requested item at the Cambridge library, according to the university . About 200 requests have been made for it since May 2016; the next most requested thesis was requested 13 times. People had to pay $85 for a digital copy of Hawking’s thesis, or physically go to the library , but now it’s free for download at the University of Cambridge's Open Access repository Apollo .

Since Cambridge made the thesis available to everyone, more than 60,000 people have accessed the work, which has caused the website to occasionally crash .

In a statement, Hawking said that he hoped the thesis would “inspire people around the world to look up at the stars and not down at their feet,” and added that everyone should have free access to all research.

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Stephen Hawking: Everything you need to know about the thesis that 'broke the Internet'

Your cheat sheet into the mind of one of the world’s greatest physicists.

Scribbled in pencil on one of its early pages is "no copying without the author's consent". In October 2017, Stephen Hawking allowed his PhD thesis — Properties of Expanding Universes — to be made available online through the University of Cambridge's Apollo portal. The website crashed almost immediately under the sheer weight of traffic. It was downloaded almost 60,000 times in the first 24 hours alone.

Hawking was 24 years old when he received his PhD in 1966 and, despite being diagnosed with motor neurone disease at just 21, could still handwrite that "this dissertation is my original work." In a statement to accompany its release, the late physicist said: "By making my PhD thesis Open Access, I hope to inspire people around the world to look up at the stars and not down at their feet; to wonder about our place in the universe and to try and make sense of the cosmos." 

Here, we break it down, guiding you through the physics until we reach the conclusion that made Hawking a household name.

Step 1: What’s it about?

Hawking's PhD thesis relates to Albert Einstein's General Theory of Relativity — the more accurate theory of gravity that replaced Isaac Newton 's original ideas. Newton said gravity was a pull between two objects. Einstein said that gravity is the result of massive objects warping the fabric of space and time (space-time) around them. According to Einstein, Earth orbits the sun because we're caught in the depression our star makes in space-time.

Hawking applies the mathematics of general relativity to models of the birth of our universe ( cosmologies ). The earliest cosmologies had our universe as a static entity that had existed forever. This idea was so ingrained that when Einstein's original calculations suggested a static universe was unlikely, he added a "cosmological constant" to the math in order to keep the universe static. He would later reportedly call it his "greatest blunder".

Things began to change when Edwin Hubble made an important discovery. Hawking writes: "the discovery of the recession of the nebulae [galaxies] by Hubble led to the abandonment of static models in favor of ones in which we're expanding."

Step 2: Our expanding universe

Some astronomers seized the idea of an expanding universe to argue that the universe must have had a beginning — a moment of creation called the Big Bang . The name was coined by Fred Hoyle, an advocate of the alternative Steady State Model. This theory states that the universe has been around forever, and that new stars form in the gaps created as the universe expands. There was no initial creation event.

Hawking spends chapter one of his thesis taking down the premise, formally encapsulated in a model called Hoyle-Narlikar theory. Hawking laments that although the General Theory of Relativity is powerful, it allows for many different solutions to its equations. That means many different models can be consistent with it. He says that's "one of the weaknesses of the Einstein theory."

The famous physicist then shows that a requirement of Hoyle-Narlikar theory appears to "exclude those models that seem to correspond to the actual universe." In short, the Steady State Model fails to match observation.

Step 3: Space: It looks the same everywhere

Hawking says that the assumptions of the Hoyle-Narlikar theory are in direct contradiction of the Robertson-Walker metric, named after American physicist Howard P. Robertson and British mathematician Arthur Walker. Today it is more widely called the Friedmann–Lemaître–Robertson–Walker (FLRW) metric. A metric is an exact solution to the equations of Einstein's General Theory of Relativity. Devised in the 1920s and 1930s, FLRW forms the basis of our modern model of the universe. Its key feature is that it assumes matter is evenly distributed in an expanding (or contracting) universe — a premise backed up by astronomical observations. 

Interestingly, Hawking offers Hoyle and Narlikar a ray of hope. "A possible way to save the Hoyle-Narlikar theory would be to allow masses of both positive and negative sign," he writes, before adding: "There does not seem to be any matter having these properties in our region of space." Today, we know that the expansion of the universe is accelerating, perhaps due to dark energy — a shadowy entity with an anti-gravitational property perhaps akin to particles with a negative mass.

Step 4: The problem with galaxies

Even geniuses get it wrong sometimes. Hawking's second chapter covers perturbations — small variations in the local curvature of space-time — and how they evolve as the universe expands. He says that a small perturbation “will not contract to form a galaxy." Later in the chapter he goes on to say: "We see that galaxies cannot form as the result of the growth of small perturbations."

That couldn't be further from our modern-day picture of how galaxies form. The key ingredient Hawking was missing is dark matter , an invisible substance thought to be spread throughout the universe, which provides a gravitational glue that holds galaxies together. Dark matter gathered around small space-time perturbations, eventually drawing in more and more material until early galaxies formed.

Our modern working cosmological picture is known as the ΛCDM model (pronounced Lambda CDM). Lambda is the Greek letter cosmologists use to denote the cosmological constant that Einstein originally introduced (albeit for the wrong reasons). CDM stands for cold dark matter. These two factors have been added to the FLRW model since Hawking wrote his thesis.

Step 5: Gravitational waves don’t disappear

Where Hawking was wrong on galaxies, he was very right on gravitational waves —ripples in the fabric of space-time that move outwards through the universe. They were predicted by Einstein when he first devised his Theory of General Relativity back in 1915, and in Hawking's time were also known as gravitational radiation.

Hawking uses Einstein's equations to show that gravitational waves aren't absorbed by matter in the universe as they travel through it, assuming the universe is largely made of dust. In fact, Hawking says that "gravitational radiation behaves in much the same way as other radiation fields [such as light]."

The physicist does note how esoteric the topic is in the 1960s. "This is slightly academic since gravitational radiation has not yet been detected, let alone investigated."

It would take physicists until September 2015 to detect gravitational waves for the first time using the Laser Interferometer Gravitational-Wave Observatory (LIGO). They were produced by the collision of two black holes — one 36- and the other 29-times the mass of the sun — about 1.3-billion-light-years away.

Related: Lab-grown black hole analog behaves just like Stephen Hawking said it would

Step 6: Are we living in an open, closed, or flat cosmos?

Hawking is heading for a groundbreaking conclusion, but first he sets himself up by introducing the idea of the overall shape of space. There are three general forms the curvature of space can take: open, closed, or flat.

A closed universe resembles Earth's surface — it has no boundary. You can keep traveling around the planet without coming to an edge. An open universe is shaped more like a saddle. A flat universe, as the name suggests, is like a sheet of paper. 

Imagine a triangle drawn onto the surface. We all learn at school that the angles inside a triangle sum to 180 degrees. However, that's only the case for triangles on flat surfaces, not open or closed ones. Draw a line from the Earth's North Pole down to the equator, before taking a 90-degree turn to travel along it. Then make another 90-degree turn back towards the North Pole. The angle between your path away from and towards the North Pole cannot be zero, so the angles inside that triangle must add up to more than 180 degrees.

Step 7: The universe is flat!

Hawking then links the idea of open and closed universes to Cauchy surfaces, named after the French mathematician and physicist Augustin-Louis Cauchy (1789—1857). A Cauchy surface is a slice through space-time, the equivalent of an instant of time. All points on the surface are connected in time. A path along a Cauchy surface cannot see you revisit a previous moment. In Hawking's own words: “A Cauchy surface will be taken to mean a complete, connected space-like surface that intersects every time-like and null line once and once only.”

He then says that closed universes are known as "compact" Cauchy surfaces, and open universes as “non-compact” ones. The former example is said to have "positive" curvature, the latter "negative" curvature. 

A flat universe has zero curvature. He moves on to set up the landmark assertions he's about to make about singularities by saying they are “applicable to models... with surfaces... which have negative or zero curvature.” Modern astronomers believe the universe is flat, meaning its zero curvature satisfies Hawking's conditions. 

Step 8: Hawking drops a bombshell

Most of the early chapters of Hawking's thesis are unremarkable — they don't offer anything particularly revolutionary, and he even gets a few things wrong. However, in his final chapter the physicist drops a bombshell that will make his name and ignite a stellar career, during which he will become one of the most famous scientists on the planet.

He says that space-time can begin and end at a singularity , and what's more he can prove it. A singularity is an infinitely small and infinitely dense point. It literally has zero size, and space and time both end (or begin) at a singularity. They had been predicted for decades, particularly when physicists started to apply Einstein's General Theory of Relativity to the picture of an expanding universe. 

If the universe is expanding today then it was smaller yesterday. Keep working back, and you find all matter in the universe condensed into a tiny, hot point — the moment of creation, a Big Bang. But just how do you prove that you can indeed get singularities in space-time?

Step 9: Hawking’s proof that the Big Bang happened

Hawking's proof leans on a very old method for proving a mathematical theory: Proof by contradiction. First you assume the thing you are trying to prove isn't true, then show that the resulting conclusions are demonstrably false. In fact, Hawking's most important section begins with the words "assume that space-time is singularity-free." There then follows some very complicated maths to show that such a universe would be simultaneously both open and closed — compact and non-compact at the same time. "This is a contradiction," Hawking said. "Thus the assumption that space-time is non-singular must be false." 

In one swoop, Hawking had proven that it is possible for space-time to begin as a singularity — that space and time in our universe could have had an origin. The Big Bang theory had just received a significant shot in the arm. Hawking started to write his PhD in October 1965, just 17 months after the discovery of the Cosmic Microwave Background — the leftover energy from the Big Bang. Together, these discoveries buried the Steady State Model for good.

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Stephen Hawking’s PhD thesis crashed its host website – here’s what it says in simple terms

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The PhD thesis of perhaps the world’s most famous living scientist, Professor Stephen Hawking, was recently made publicly available online . It has proved so popular that the demand to read it reportedly crashed its host website when it was initially uploaded.

But given the complexity of the topic – “Properties of Expanding Universes” – and the fact that Hawking’s book A Brief History of Time is also known as the most unread book of all time, you might benefit from a summary of its main result.

The thesis covers several topics, including recently discovered gravitational radiation , but the final chapter is the part that many physicists consider the most significant. It deals with the birth of the universe itself, and is simply titled “Singularities”.

Creation theories

The major achievement of Hawking’s thesis was to effectively show that the Big Bang theory of how the universe began from a single point was physically possible. It wasn’t just a mathematical nuisance that sprang out of the equations physicists had developed to describe the possible evolution of the cosmos.

The concept that the universe started a finite time ago in a Big Bang is now an accepted scientific fact, and yet it remains an astounding idea. Imagine: all the matter in your body was once – in one form or another – compressed into the same tiny volume as the most distant galaxy and everything in between. About 14 billion years ago , this point rapidly expanded to create space and time. It continues to expand today.

At the time of Hawking’s PhD in the 1960s, scientists were still arguing over the idea. A popular alternative to the Big Bang was the Steady State model . Proponents of the Steady State model were uncomfortable with a universe of finite age that began in this way. In fact, the moniker “Big Bang” was coined as a derisive term by Steady State champion Fred Hoyle . To understand how Hawking showed it really was possible, we need some background physics.

Spacetime and singularities

In the early 20th century, Albert Einstein revolutionised our understanding of gravity through his general theory of relativity . Einstein showed that we could think of gravity as the curvature of spacetime, caused by the presence of mass or energy.

Spacetime is a way of thinking about the framework of the universe that combines three-dimensional space and one-dimensional time. All objects exist and all events happen somewhere in spacetime. But it’s hard for most people to imagine because, although we can move freely in three-dimensional space, we cannot travel where we like through time. It’s a bit like being an insect trapped on the surface of a pond. It can only move in two dimensions, despite there being another spatial dimension to explore.

General relativity expresses how space and time are linked. In his theory, Einstein elegantly described how the curvature of spacetime is related to the density of mass and energy in his “field equations”.

After these equations were published, other scientists used them to explore what happens to spacetime in different physical situations. In the case of objects where all the matter is concentrated into a single point, the field equations predict something unusual: the curvature of spacetime becomes so extreme that even light cannot escape. Today we know these objects actually exist as black holes, and we’ve since found evidence for them in space. These situations where the solutions to the equations become infinite are called “singularities”. Hawking’s final thesis chapter explored this idea of singularities, not for the spacetime around black holes, but for the entire universe.

From black holes to the Big Bang

In cosmology, a central tenet is that space must, on average, be homogeneous and isotropic . In other words, over a large scale, the contents of the universe must be pretty evenly distributed and look the same in every direction.

The simplest solution to Einstein’s field equations that satisfies these conditions is called the “ Robertson-Walker metric ”, named after the scientists involved in its development. The metric is simply the term we use for describing the interval between two events in spacetime.

Importantly, the Robertson-Walker solution allows the spatial part of the metric to change with time. That means it can describe a universe in which space itself is expanding. Edwin Hubble found evidence that the universe really is expanding in the 1920s by showing that other galaxies are moving away from us.

The Robertson-Walker metric and field equations allow us to describe this expansion in terms of what cosmologists call the “scale factor”, describing how much space has expanded or contracted between a particular point in time and the present day.

If the universe is expanding, it should have been smaller and denser in the past. Run the clock back far enough and the scale factor should go to zero. All the matter and energy in the universe must have been contained in a single point with infinite density: a cosmological singularity. This is the basis of the Big Bang model, a bit like a black hole in reverse.

Scrapping the Steady State

The Steady State model tried to eliminate the cosmological singularity, which many argued wasn’t plausible. Singularities were seen as shortcomings of the predictions of general relativity and not in line with the known laws of physics.

In the Steady State model, the universe is eternal and doesn’t have a beginning at all. Its apparent expansion can be explained by adding a “creation field” or C-field to Einstein’s equations that would mean matter is continuously created in the space between galaxies as they move apart.

But in the final chapter of his PhD thesis, Hawking argued that the idea of a C-field came with its own set of problems and that the right model involved the Robertson-Walker solution describing an initial singularity.

What he did next was what many consider groundbreaking. Building on the work of fellow British physicist Roger Penrose , Hawking mathematically proved that singularities were not a shortcoming of theory but expected features of nature. He effectively demonstrated that general relativity allowed for a universe that began in a singularity.

Half a century later, the observational evidence for the Big Bang creation scenario is overwhelming and the Steady State model has long been abandoned. Hawking has gone on to make further monumental contributions to cosmology and theoretical physics. Reading Hawking’s thesis is an insight into an exceptionally creative mind – and the first steps of discovery in what has been a remarkable scientific journey.

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