# Stephen Hawking’s PhD thesis can now be downloaded—here’s what it says

The paper, on the “big bang” theory of the universe, offers a glimpse into the scope and ambition of Hawking’s mind from his formative years

The Hawking effect strikes again. All those unread copies of *A Brief History of Time* are seemingly being joined by electronic copies of the world’s most famous living scientist’s PhD thesis from 1966. Recently put online by the University of Cambridge, it created such demand that the university’s open-access repository Apollo crashed.

It would be easy to scoff about how few of those readers will understand a word of “Properties of Expanding Universe.” But there’s something rather charming about the enthusiasm and curiosity for this unlikely scientific item. Among other things, the scrawled statement at the beginning that “This dissertation is my original work—SW Hawking” in the messy hand of a schoolboy turning in his homework creates a certain frisson, not least because the wobbly writing reveals the early onset of the motor neuron disease that soon left Hawking confined to a wheelchair and thereafter inexorably eroded his ability to talk, move or do pretty much anything for himself other than think.

To any doctoral graduate of a certain age, the thesis also holds some charm as a reminder of the days when it all had to be typed in carbon copy, with equations hand-written in the spaces left blank. The 119-page thesis describes the work that Hawking did at Cambridge from 1962 to 1965. What does it say?

The central motivation is to understand what the expansion of the universe means for its origin. That expansion was observed in the 1920s and 30s by astronomers, in particular the American Edwin Hubble. He deduced from the so-called redshift of distant galaxies—the fact that their light is at longer wavelengths than it should be, due to the Doppler effect that alters waves broadcast between objects in relative motion—that those galaxies are receding from us. The further they are, the faster the recession.

“This dissertation is my original work—SW Hawking”

Hawking points out that this creates three possibilities. In “the so-called ‘big-bang’ model” (a term allegedly coined with derogatory connotations by the astrophysicist Fred Hoyle) we are seeing the aftermath of a gigantic “explosion” from an initially very small, dense state. But some cosmologists wondered if this state might have been preceded by an endless series of expansions and contractions: the “bouncing” or oscillating universe, in which a contraction generates a new “bang” once it reaches a certain density. Or maybe—this was Hoyle’s preference—the expansion had been going on forever, with fresh matter continually forming in the gaps so that the universe doesn’t get diluted away. That was the “steady-state” model.

Before it was observed, the expansion of the universe was predicted by Einstein’s theory of general relativity, unveiled in 1916. Although because that then ran counter to the prevailing view, Einstein didn’t believe it and modified his equations to avoid it. However, general relativity, which furnished a theory of gravitation and spacetime, was seen as the appropriate theoretical framework for constructing a theory of the cosmos, and Hawking writes down its central equation at the foot of the third page of his introduction. Even for a non-mathematician it looks deceptively simple:

R_{ab} – ½ g_{ab}R = T_{ab}

It is, however, formidably difficult to solve these equations in most circumstances. General relativity had, until the 1960s, been explored mostly by mathematicians rather than physicists; the latter didn’t really know how to apply it to real-world situations. The 1960s saw a renaissance in studies of general relativity. In particular, some physicists found ways to solve the equations for the case of burnt-out stars collapsing under their own gravity—which led to the prediction of black holes, in which all the matter collapses to a “singularity,” a point of zero size and infinite density. Singularities were rather embarrassing, even affronting, to physics, since they didn’t seem to refer to anything that could be real. Yet Einstein’s equations seemed to insist on them.

And here then is Hawking’s key point, stated at the end of his Abstract. If you apply general relativity to the whole universe, “a singularity is inevitable provided that certain very general conditions are satisfied.” Hawking’s proposal was that the Big Bang could be considered a kind of cosmic black hole in reverse: as you track back in time, the universe contracts until it becomes a singularity, with all matter focused into a single, infinitely dense point.

That was bad news for Hoyle’s steady-state cosmological theory. In his first chapter, Hawking shows that the theory that Hoyle developed with his student Jayant Narlikar can indeed produce a steady-state picture from Einstein’s equations, but only by making assumptions that can’t be true in the real universe. Anyone who has seen *The Theory of Everything* or the earlier 2004 TV biopic *Hawking* (with Benedict Cumberbatch in the title role) will know of the prickly relationship between Hoyle and this brash young student. At one point in the latter movie, based on a real incident in 1964, Hawking stands up in a lecture at the Royal Society given by Hoyle to point out an error. “Would you like to tell us how you know this, young man?” Hoyle demands. “I worked it out,” says Hawking. Hawking had originally hoped that Hoyle would be his PhD supervisor at Cambridge; God knows how that would have worked out.

In chapter two, Hawking considers how the equations for describing this expanding universe are affected by “lumpiness”—the fact that matter isn’t dispersed perfectly evenly throughout it. That matters, if the whole point is that the motion of stuff in the universe depends on its mutual gravitational pull. Here and in chapter three Hawking looks at the behaviour of gravitational waves, the ripples in spacetime caused by extreme changes in the way matter is distributed (such as the collision of black holes) that Einstein’s theory predicted. Experimental confirmation of these waves was found only in late 2015, and won this year’s Nobel prize in physics.

“Many feared that Hawking’s thesis might be all we’d get before his debilitating disease claimed his life”

The main conclusion appears in chapter four, where Hawking shows that obtaining a singularity in the equations of general relativity, from which the expanding universe might have begun, doesn’t (as previously thought) require an extrapolation backwards in time from a perfectly symmetrical universe—which would seem an unrealistic view of our own. The conditions for a singularity are less stringent than that. In other words, getting to a situation primed to “bang” from a cosmic singularity is not as hard as it seemed.

It’s a bold and brilliant thesis, which established the young Hawking at the centre of the new enthusiasm for general relativity. What it doesn’t hint at, however, is Hawking’s genius for bringing quantum theory into the equation. General relativity seems a flawless theory for understanding the cosmos at grand scales, but we know it can’t be the whole truth because it assumes that space and time are perfectly smooth, without the “granular” character on which quantum mechanics insists at the smallest scales. In particular, mathematical singularities that give objects of zero size an infinite density are not possible in quantum theory, since below a certain size scale (called the Planck scale, which is far, far smaller than the size of an atomic nucleus) it’s simply not possible to speak of space as smooth.

When Hawking began to introduce quantum theory into this picture of singularities, big bangs and black holes, the picture became richer. In the 1970s he showed that quantum effects mean black holes are not really black, but emit radiation from close to their “event horizon” (beyond which light is trapped). This carries away energy, meaning that ultimately the black hole evaporates. Hawking has also attempted to develop an understanding of the initial moments of the Big Bang that is consistent with a quantum way of describing the absurdly dense, tiny object that became the universe, leading him to the conclusion that it’s not possible to meaningfully assign a “starting point” for the event.

That Hawking’s thesis looks eye-wateringly complicated is no big deal: you’d find that in most theses coming out of departments of physics, both then and now. It’s important too to recognise that Hawking is not alone in exploring these issues—his work, like his thesis, has always drawn heavily on that of others. His dissertation doesn’t have the elegance, concision, originality and penetrating vision evident in the groundbreaking papers by Einstein in 1905 on special relativity or quantum theory. But pretty much nothing else in physics does either.

What you do get is a glimpse of the scope and ambition of Hawking’s mind from his formative years. And it’s surely uplifting that, while many of his peers feared that the thesis might be all we’d get from him before his debilitating disease claimed his life, he’s still here theorising today. It seems safe to conclude that he’s not now going to solve the deepest of the mysteries he confronted—but perhaps that tantalising prospect played a big part in keeping Hawking going for such an astonishing length of time.

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