## Hawking

A long post is brewing (breaking my month-long silence), but as I was working on it, the sad news arrived that Stephen Hawking passed away. There’s little I can add to the tributes that poured in from around the world: like chocolate or pizza, Hawking was beloved everywhere *and actually deserved to be*. Like, probably, millions of other nerds of my generation, I read *A Brief History of Time* as a kid and was inspired by it (though I remember being confused back then about the operational meaning of imaginary time, and am *still* confused about it almost 30 years later). In terms of a scientist capturing the public imagination, through a combination of genuine conceptual breakthroughs, an enthralling personal story, an instantly recognizable countenance, and oracular pronouncements on issues of the day, the only one in the same league was Einstein. I didn’t agree with all of Hawking’s pronouncements, but the quibbles paled beside the enormous areas of agreement. Hawking was a force for good in the world, and for the values of science, reason, and Enlightenment (to anticipate the subject of my next post).

I’m sorry that I never really met Hawking, though I did participate in two conferences that he also attended, and got to watch him slowly form sentences on his computer. At one conference in 2011, he attended my talk—this one—and I was told by mutual acquaintances that he liked it. That meant more to me than it probably should have: who cares if some random commenters on YouTube dissed your talk, if the Hawk-Man himself approved?

As for Hawking’s talks—well, there’s a reason why they filled giant auditoriums all over the world. Any of us in the business of science popularization would do well to study them and take lessons.

If you want a *real* obituary of Hawking, by someone who knew him well—one that, moreover, actually explains his main scientific contributions (including the singularity theorems, Hawking radiation, and the no-boundary proposal)—you won’t do any better than this by Roger Penrose. Also don’t miss this remembrance in *Time* by Hawking’s friend and betting partner, and friend-of-the-blog, John Preskill. (**Added:** and this by Sean Carroll.)

Comment #1 March 16th, 2018 at 10:35 am

Funny. I spoke to Laura Green (Kelila’s mom) last night, who still is a travel agent, primarily running trips to Israel, and an avid AIPAC supporter. She remarked that Hawking was convinced, years back, by Palestinians not to go to some conference in Israel, but she explained that drugs developed in Israel kept him alive, and physicists in Israel actually were able to prove two of his theories.

Any truth to that?

Comment #2 March 16th, 2018 at 11:40 am

Saw that Penrose obit yesterday and learned something new: existence of Einstein-Cartan theory. Anyway, highly recommend Penrose’s obit as it is not hagiography, but quite good at explaining Hawking’s scientific achievements and human strengths and faults.

Comment #3 March 16th, 2018 at 12:23 pm

It’s sad to me that admiration for his scientific genius must unfortunately be tempered because of his sometimes abtuse moral sense — he was for boycotting Israel.

Comment #4 March 16th, 2018 at 12:42 pm

Arnie #1: I think there are some real facts that went through a game of “Telephone” before they got to you!

Hawking used to be a frequent visitor to Israel, and even shared the Wolf Prize (Israel’s most prestigious scientific prize) with Roger Penrose. But in 2013, yes, he did let the BDS activists convince him to boycott the country, a decision that deeply disappointed many of us who otherwise admired him. To me it seems obvious: if you disagree with Israeli government policy, say so and support those Israelis (including the majority of Israeli scientists and academics) who share your criticisms. Visit both Israel and Palestine, as Hawking did prior to his boycott. Don’t boycott the entire country,

unlessyou’re also prepared to boycott the US under Trump, China under Xi, and probably a hundred other countries for which at least as strong an argument could be made, and probably a stronger one.I hadn’t heard about drugs developed in Israel keeping Hawking alive—do you have a source for that? It’s true that, if you want to participate in the modern world, it’s hard to avoid microprocessors and other technologies that were at least partly developed in Israel, a practical problem that the BDS people learned about to their dismay.

I also have no idea what are the “two theories of Hawking’s that were proved by Israeli physicists”—again, do you have a source? What I know is that Jacob Bekenstein, an Orthodox Jewish Mexican-American physicist who emigrated to Israel and became a professor at the Hebrew University, played a central role in Hawking’s most important discovery, namely Hawking radiation and the thermodynamics of black holes. I knew Bekenstein reasonably well, and wrote a little obituary for him here (he died in 2015). But the situation was more that

Hawkingdid a calculation that provedBekenstein’searlier speculation.See this recent Haaretz article for more on the subject of Hawking’s relationship to Israel.

Comment #5 March 16th, 2018 at 4:38 pm

He was inspirational for so many people and I feel bad that one of my memories of him from Cambridge was horrible – I had a french girlfriend who was a literature student, and we were walking down a deserted Cambridge street when Hawking appeared in his wheelchair with a helper, I said to her “That’s Stephen Hawking”, and my girlfriend, to my horror, proceeded to laugh uncontrollably and very loudly.

I guess she thought I was making a sick joke, but it was really uncomfortable, the helper looked over to us, and I just tried to stop her laughing. She was mortified afterwards, when she understood what had happened, but imagine how much shit less famous disabled people go through. The “intelligent” university environments aren’t safe places either.

Comment #6 March 16th, 2018 at 9:41 pm

Read an interesting tidbit about how better AI programs meant Hawking’s speech selection programs were getting faster and faster to the point where he didn’t have to type much because the program could anticipate what he wanted to say:

“By processing Hawking’s books, articles and lecture scripts, the system got so good that he did not even have to type the term people most associate with him, “the black hole.” When he selected “the,” “black” would automatically be suggested to follow it, and “black” would prompt “hole” onto the screen.”

https://theconversation.com/stephen-hawking-warned-about-the-perils-of-artificial-intelligence-yet-ai-gave-him-a-voice-93416

I’m just as perplexed as you about imaginary time Scott. I note though that the use of the complex plane lets one treat time as a vector with 2 components (real and imaginary).

I did find a 4-part series of articles on John Baez’s blog where he was trying to formulate an analogy between classical thermodynamics and quantum mechanics and it involved the use of imaginary quantities here:

https://johncarlosbaez.wordpress.com/2011/12/22/quantropy/

Finally, it sounds like your next blog post is going to be on the topic of sociology/politics/history. Here’s my wiki-book on that:

https://en.wikipedia.org/wiki/User:Zarzuelazen/Books/Reality_Theory:_Sociology%26Politics

(434 wikipedia entries on key topics)

More modularity and I further sub-divided into 3 core areas:

History&Futurology (104 entries)

Law&Politics (169 entries)

Memes (161 entries)

Any way, the upshot is I identified ~100 key events that shaped history and are probably needed to grasp the origins of enlightenment values and the scientific revolution.

Comment #7 March 17th, 2018 at 11:16 am

@mjgeddes #6

It is called Wick rotation and it is a well known tool of quantum theory, transforming your problem into an exercise in statistical mechanics.

It is not so clear how it works for quantum gravity, but in special cases it can be done (e.g. by Hartle&Hawking).

Comment #8 March 17th, 2018 at 3:23 pm

Hawkings appeared on ST-TNG, Futurama, The Simpsons, and more also The Big Bang Theory (7 times!). On TBBT they also mention him often. This of course makes sense since TBBT is about a photon of physicists (little known fact: Photon is the plural word for physicists).

He always played himself.

I suspect these appearances made viewers more aware of science. I wonder if, just as Scott and Lance (see Complexity Blog) were influenced (positively!) by Hawking’s books, if others may be influenced (positively!) by his TV appearances.

Comment #9 March 17th, 2018 at 6:25 pm

Extending functions into the complex plane is of course common, already Sundman was doing that when he solved the 3-body problem. I take it that Scott was not asking about that basic idea, but about the implications in this case.

Comment #10 March 17th, 2018 at 6:32 pm

Perhaps I should add that Hawking’s scientific achievements and his appetite for life were both admirable. The world is a poorer place without him.

Comment #11 March 18th, 2018 at 6:49 am

Jr #9: IIRC, he was presenting imaginary time not just as a useful mathematical trick, but as an important component of reality.

But it was one of his trademarks to make brief, bold, and enigmatic proclamations of that kind, and leave it for others to figure out what they meant. That’s something that he could get away with and most of us only wish we could. 🙂

Comment #12 March 18th, 2018 at 8:26 am

On “imaginary time”, I found that learning about the method of steepest descent inspired in me a striking mental picture of what is going on. I emphasize that I’m not familiar with Hawking’s work and so I don’t know how accurate this picture actually is.

These quantum mechanical questions are about oscillatory integrals over the infinite-dimensional space of field configurations. Instead, I’ll consider the easier question of oscillatory integrals on a one-dimensional space. A classical example is the Airy function, which up to normalization is defined by Ai(x) = int exp(i(xt+t^3/3))dt. When x<0 this function can be approximated using the stationary phase method, which is also what’s used to calculate quantum mechanical wavefunctions that closely approximate a classical trajectory. The idea is that for most values of t the expression exp(i(xt+t^3/3)) oscillates rapidly, and so the nearby values of t cancel each other out and the integral is small. However, near the critical values t=+-sqrt(-x), which are the points where (d/dt)(xt+t^3/3)=0, the integral concentrates around a particular phase for a longer interval, and the dominant contribution to the integral comes from around these points. It can be calculated that the width of these regions nonoscillatory regions is around (-x)^(-1/4), and so the value of Ai(x) is also around (-x)^(-1/4) for x<0, times some phase factor. These calculations be refined to give very precise estimates of Ai(x) as x goes to negative infinity.

What about when x is greater than 0? Then there are no critical points; the function xt+t^3/3 has positive derivatives everywhere. Since there are no critical points, the estimate given by the stationary phase method as above is zero. More precisely, it can be shown this way that Ai(x) goes to zero faster than x^(-N) for any N as x goes to positive infinity. However, Ai(x) is not exactly zero, and we need a new method to determine exactly how small it gets.

The solution is to extend the function exp(i(xt+t^3/3)) to complex values of t and interpret Ai(x) as a contour integral. Then it is possible to shift the contour to get a more tractable integral. Specifically, shifting the contour for short distances in the positive imaginary direction makes the amplitude of this function exponentially smaller. This continues until you reach the

complexcritical point t_0=i*sqrt(x). Shifting the contour any higher makes the amplitudes increase again. It is possible to shift the contour in such a way so that it reaches a peak value at t=t_0 and rapidly decreases away from this point. It follows that Ai(x)~exp(i(x*t_0+t_0^3/3)) =exp(-2/3*x^(3/2)).My mental picture for imaginary time, which I repeat is only an educated guess, is that something similar is happening. Hawking is calculating the probability amplitude of a certain hypersurface as a path integral over all field configurations of a four-dimensional manifold with this hypersurface as a boundary. There is no classical trajectory which leads to this hypersurface without a singular boundary, and this leads to the stationary phase method failing to give an answer. However, by complexifying the space of field configurations it is possible to find a complexified trajectory which is a critical point of the Lagrangian and shift the contour of the path integral so that configurations around that trajectory are the dominant contributors to the integral. I believe that what people are calling imaginary time is related to the imaginary component of this trajectory in the complexified space of field configurations.

I would appreciate it if somebody who knows more about Hawking’s cosmological model would corroborate or refute the picture I just gave.

Comment #13 March 18th, 2018 at 8:27 am

‘Imaginary time’ apparently has a physical interpretation as related to ‘inverse temperature’.

From the wikipedia article:

https://en.wikipedia.org/wiki/Wick_rotation

“Wick rotation connects statistical mechanics to quantum mechanics by replacing inverse temperature with imaginary time”

Here are some quotes from an expert on Quora (published in Forbes):

https://www.forbes.com/sites/quora/2016/11/15/einstein-and-hawking-had-different-ideas-about-the-concept-of-imaginary-time/#381e3ab83e0e

—

The “imaginary time” associated with Hawking comes … from Feynman’s path integral formulation of quantum mechanics.

In basic terms, the path integral approach gives you the probability with which any state will transition from state a to state b, by evaluating the integral of all possible paths between them.

Analyzing this problem is hard, and the integral of all possible paths blows up in your face. But there is a trick that (seemed) to simplify it, called “Wick rotation”. This trick crops up across physics, where it’s useful in classical and statistical mechanics problems, as well as in QM. What a Wick rotation does is to transform a problem posed in Euclidean space into a slightly different one posed in Minkowski space (and vice versa). Often, the analogous problem in the other space turns out to be easier, so you solve it there, and then transform back again. And so it proves here: the integrals turn out to be much easier once they are Wick rotated. The key substitution is similar to Minkowski’s original (give or take few constants), it’s usually written as t=−it′.

Stephen Hawking and James Hartle built this trick into an approach called “Euclidean Quantum Gravity”, and at the time that Hawking wrote A Brief History of Time, it was top of Hawking’s mind as a promising research direction, and so he wrote a popular account of it in the book. However, the approach has not developed as Hartle and Hawking hoped, and Euclidean Quantum Gravity is now very much a minority interest.

—

Comment #14 March 18th, 2018 at 12:53 pm

Thank you for the link to the obituary by Roger Penrose.

Comment #15 March 18th, 2018 at 7:13 pm

Wishing a speedy recovery to Dick Lipton, who is recovering from heart surgery according to the most recent post on his blog:

https://rjlipton.wordpress.com/

Comment #16 March 19th, 2018 at 4:05 pm

mjgeddes #6

“Read an interesting tidbit about how better AI programs meant Hawking’s speech selection programs were getting faster and faster to the point where he didn’t have to type much because the program could anticipate what he wanted to say”

Hmm… Once the matching probability is 99.9999% can we claim that the mind has been effectively “uploaded” into the machine?

Comment #17 March 20th, 2018 at 10:59 pm

Well that Wick rotation stuff is very interesting indeed.

The way I interpret it then is this: classical thermodynamics can be generalized to obtain analogous notions in quantum physics. So it looks like the Schrodinger equation is analogous to the ‘heat equation’ and ‘action’ (in the quantum physics sense) becomes analogous to kinetic energy.

Then imaginary time is a sort of generalized notion of ‘temperature’. If you imagine temperature as related to thermal oscillations of matter caused by kinetic energy, then ‘imaginary time’ is referring to the oscillations of quantum paths (in Feynman’s path integral formulation). The magnitude of these oscillations is given by the ‘action’.

Quantum states can be interpreted as taking ‘random walks’ in a ‘space of possibility’ (quantum Brownian motion so to speak)! There’s a ‘space of possibility’ (imaginary time), which, unlike real time, has the characteristic that you can move back just as easily as you can move forward. Movements through ‘possibility space’ correspond to Feynman’s sum over histories – states of matter exploring quantum possibilities at random.