I was fourteen years old when I first met Stephen Hawking, in 1995. Not the flesh and bones Hawking, of course, but something more interesting: his brain, in the form of his ‘A Brief History of Time’.

I wish I could say the book inspired me to pursue Physics further, but the best I can say is that it was funny and insightful at many points. I felt myself a failure at understanding many of the ideas there discussed. Today, I know it was not entirely my fault, but back then I finished the book a bit frustrated. While Hawking was quite an example of life and the power of mind over matter, I can’t say I much admired him as a science writer. My real inspiration to do Physics end up being another brain’s creation, that of Mr. Albert Einstein. I was back then trying to understand Relativity and its many ‘paradoxes’, so thrilled by the whole thing I needed no further inspiration: it was a done deal I could do nothing else but Physics, to try to get to the bottom of it.

And I am glad that I did, because only afterward I could appreciate the greater insights Hawking’s brain had to offer me - not as a science writer, but as a scientist. It is about that Hawking I want to talk about.

Back in the 60’s, very many things we now take for granted weren’t well understood. ‘Black holes’ were not branded yet, and its most distinguishing feature, the event horizon wherefrom nothing can escape, was a source of great confusion. Due to a 1939 work of Openheimer et al (the same one from Manhattan project fame), people initially thought the collapse of a dying star would get ‘frozen’ at the Schwarzschild (or event horizon) radius, which we nowadays identify as the ‘point of no return’ from a black hole. It was a very strange result, because no one could figure out why the attractive gravitational force would somehow ‘stop’ working to make the collapse go all the way down.

It took almost 20 years for Relativists to realize, in a work by David Finkelstein in 1958, that Openheimer’s results were a mirage caused by the coordinate system he used. If you addressed that mistake, you end up concluding that every dying star, above a certain threshold of mass, will end up as a black hole: a most simple solution of Einstein’s Equations, where all matter ends up dying in a central ‘singularity’ surrounded by pure vacuum, all hidden behind that ‘point of no return’ event horizon.

For many people, such a simple universal result sounded too good to be true, or pretty weird. ‘Too good’ because it would render all big stars, which could differ in so many ways, all exactly alike in their deaths (aren’t we all?). And ‘weird’ because that central singularity was, well, singular - a point where Einstein’s equation breaks down and nothing else in the theory can be calculated meaningfully.

So it is not surprising some physicists, in particular the Russian school represented by Landau & Lifschitz, got second thoughts about those results. The initial calculations were all based in simplifying hypotheses, like perfect spherical symmetry of the collapsing star for example, that could be easily broken in any real life star. They launched themselves into the task of showing that, in any more realistic setting, stars could well end up in other configurations, maybe without a singularity and preserving some of their initial complexity, as opposed to being a mere point of mass ‘M’ in spacetime. At some point in the 60’s, they published their final theorems in prestigious scientific journal, supposedly showing the end state of stars would generically be free of those nagging singularities.

And that’s the point where Mr. Hawking comes again to our narrative. Starting his PhD studies in the begin of the 60’s, his advisor gave him the problem of studying how general would be the presence of singularities in the begin of the Universe, what we call nowadays the ‘big bang’. The Russian school results looked to imply that, maybe, the universe would be born out of a well behaved solution, instead of an initial all-encompassing singularity.

As it happens, geniuses rarely operate in a vacuum. They often are immersed in a battle of great ideas, provoking and being provoked by other great minds. And here comes to our story the great mind of Roger Penrose, the British mathematician who first showed, right in the begin of Hawking studies, that for the case of black holes, their singularity was indeed a universal phenomena: that, no matter how the stars started (spherical or squared, you choose), their end state would indeed be that of the singular black hole.

Penrose’s mathematical methods were new and alien to much of the community of Physicists, and to the credit of Hawking, he not only learned well and fast that new mathematics, but went on to apply it to what is, in a way, the inverse problem: matter being born out of a singularity, as opposed to it being the graveyard of a star. Adapting many of Penrose’s methods and hypotheses, Hawking was able to show that cosmological singularities are also a generic feature of cosmological spacetimes (actually, there are ways to dodge that theorem too, by changing its initial hypotheses, but let me play fast and loose here). The Russians ought to have made a mistake after all, it was the inexorable conclusion. They did (and they didn’t, it is much about which definition of singularity you take, but let me keep playing fast and loose).

Such results were of great impact in the initial days of black hole physics, giving the young Hawking a position in Cambridge right after his PhD. To our luck, he was only beginning. His contributions in the following decades were many, but I will fast forward to what is, perhaps, the greatest of all.

By the begin of the 70’s, a young graduate student of John Archibald Wheeler in Princeton, one Mr. Jacob Bekenstein (a jew born in Mexico - apparently and contra D. Trump, the country did send people other than rapists to America), made a most ingenious argument. By gedankenexperimenting what should happen to a box full of an ideal gas approaching a black hole, Bekenstein showed that, were we allow for a black hole to keep being defined as a void labelled only by its mass, we would run into big trouble. You could continually unload the gas of boxes brought into the vicinity of black holes, in such a way to make its entropy to disappear out of our universe.

So what? Well, by doing so you would promptly violate our cherished Second Law of Thermodynamics - which is like opening the Pandora box, for once you violate the 2nd, you can violate them all, virtually engendering any sort of perpetual motor giving you infinite energy out of nothing.

Physicists love free lunches at faculty meetings, but they abhor them in our theories. The solution was simple though, a black hole ought to have entropy, and one proportional to its horizon area, postulated Bekenstein in order to save our well ordered world.

Hawking, who already (sort of unseriously) played with the idea of black holes mimicking our First Law of Thermodynamics (in the way they absorb energy and grow their horizon areas), was not happy at all. As every physicist know, macroscopic entropy translates to internal degrees of freedom, quantifying how they may all sum up to give us a well defined macroscopic state. But much of Hawking’s career so far was built upon showing that no such internal degrees of freedom exist: stars die into (and universes are born out of) faceless simple singular states!

So, trying to prove Bekenstein wrong -- while coping with his final loss of walking capacity, in the middle of absolute immersion into his calculations -- Hawking ends up proving him right, and incidentally gives us one of the most beautiful results of last half-century Physics: black holes do have entropy, an enormous one, maybe the greatest possible to achieve in Nature. Furthermore, they are not so black at all: they emit wavelight in one of the most precious ways Physicists can think of, the Planck-like (or blackbody-like) one, which is the same kind of radiation a red hot stove often emits, and was also the key through which Max Planck first discovered quantum physics in 1900.

In other words, a self gravitating collection of bodies like stars and black holes, while being described by Einstein’s Equations of general relativity, end up obeying the very same laws of Thermodynamics we discovered while building steam machines in XVIII to XIX-century Europe. Being the physics of relativistic stars and universes so different from our classical mechanics of old, who ordered that Thermodynamics should transcend it all in such a way? Our Lord is subtle indeed, as Einstein once reminded us.

It is thus easy to understand why Hawking, playing a final joke upon us physicists, wanted for his tomb to display his cherished formula for the black hole entropy, mimicking the famous entropy formula in the grave of Ludwig Boltzmann, the man who created the concept of entropy as we know it. After all, “life would be tragic if it weren’t funny”.

## 3 comments:

Excellent. It gave me a couple of things to think about.

I hope you teach as well as do research?

I teach 2 or 3 courses per semester, Bret.

Often basic courses for those 4 first semesters every engineer student does, plus some more advanced course for students majoring in Physics.

As for research, this is my last submitted paper.

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