If people know anything about physics, it’s the guy in a wheelchair who speaks with a computer. Google “most famous scientist alive” and the answer is “Stephen Hawking.” But if you ask a physicist, what exactly is he famous for?

Hawking became “officially famous” with his 1988 book “A Brief History of Time.” Among physicists, however, he’s more renowned for the singularity theorems. In his 1960s work together with Roger Penrose, Hawking proved that singularities form under quite general conditions in General Relativity, and they developed a mathematical framework to determine when these conditions are met.

Before Hawking and Penrose’s work, physicists had hoped that the singularities which appeared in certain solutions to General Relativity were mathematical curiosities of little relevance for physical reality. But the two showed that this was not so, that, to the very contrary, it’s hard to avoid singularities in General Relativity.

Since this work, the singularities in General Relativity are understood to signal the breakdown of the theory in regions of high energy-densities. In 1973, together with George Ellis, Hawking published the book “The Large Scale Structure of Space-Time” in which this mathematical treatment is laid out in detail. Still today it’s one of the most relevant references in the field.

Only a year later, in 1974, Hawking published a seminal paper in which he demonstrates that black holes give off thermal radiation, now referred to as “Hawking radiation.” This evaporation of black holes results in the black hole information loss paradox which is still unsolved today. Hawking’s work demonstrated clearly that the combination of General Relativity with the quantum field theories of the standard model spells trouble. Like the singularity theorems, it’s a result that doesn’t merely indicate, but prove that we need a theory of quantum gravity in order to consistently describe nature.

While the 1974 paper was predated by Bekenstein’s finding that black holes resemble thermodynamical systems, Hawking’s derivation was the starting point for countless later revelations. Thanks to it, physicists understand today that black holes are a melting pot for many different fields of physics – besides general relativity and quantum field theory, there is thermodynamics and statistical mechanics, and quantum information and quantum gravity. Let’s not forget astrophysics, and also mix in a good dose of philosophy. In 2017, “black hole physics” could be a subdiscipline in its own right – and maybe it should be. We owe much of this to Stephen Hawking.

In the 1980s, Hawking worked with Jim Hartle on the no-boundary proposal according to which our universe started in a time-less state. It’s an appealing idea whose time hasn’t yet come, but I believe this might change within the next decade or so.

After this, Hawking tries several times to solve the riddle of black hole information loss that he posed himself, most recently in early 2016. While his more recent work has been met with interest in the community, it hasn’t been hugely impactful – it attracts significantly more attention by journalists than by physicists.

As a physicist myself, I frequently get questions about Stephen Hawking: “What’s he doing these days?” – I don’t know. “Have you ever met him?” – He slept right through it. “Do you also work on the stuff that he works on?” – I try to avoid it. “Will he win a Nobel Prize?” – Ah. Good question.

Hawking’s shot at the Nobel Prize is the Hawking radiation. The astrophysical black holes which we can presently observe have a temperature way too small to be measured in the foreseeable future. But since the temperature increases for smaller mass, lighter black holes are hotter, and could allow us to measure Hawking radiation.

Black holes of sufficiently small masses could have formed from density fluctuations in the early universe and are therefore referred to as “primordial black holes.” However, none of them have been seen, and we have tight observational constraints on their existence from a variety of data. It isn’t yet entirely excluded that they are around, but I consider it extremely unlikely that we’ll observe one of these within my lifetime.

For what the Nobel is concerned, this leaves the Hawking radiation in gravitational analogues. In this case, one uses a fluid to mimic a curved space-time background. The mathematical formulation of this system is (in certain approximations) identical to that of an actual black hole, and consequently the gravitational analogues should also emit Hawking radiation. Indeed, Jeff Steinhauer claims that he has measured this radiation.

At the time of writing, it’s still somewhat controversial whether Steinhauer has measured what he thinks he has. But I have little doubt that sooner or later this will be settled – the math is clear: The radiation should be there. It might take some more experimental tinkering, but I’m confident sooner or later it’ll be measured.

Sometimes I hear people complain: “But it’s only an analogy.” I don’t understand this objection. Mathematically it’s the same. That in the one case the background is an actually curved space-time and in the other case it’s an effectively curved space-time created by a flowing fluid doesn’t matter for the calculation. In either situation, measuring the radiation would demonstrate the effect is real.

However, I don’t think that measuring Hawking radiation in an analogue gravity system would be sufficient to convince the Nobel committee Hawking deserves the prize. For that, the finding would have to have important implications beyond confirming a 40-years-old computation.

One way this could happen, for example, would be if the properties of such condensed matter systems could be exploited as quantum computers. This isn’t as crazy as it sounds. Thanks to work built on Hawking’s 1974 paper we know that black holes are both extremely good at storing information and extremely efficient at distributing it. If that could be exploited in quantum computing based on gravitational analogues, then I think Hawking would be in line for a Nobel. But that’s a big “if.” So don’t bet on it.

Besides his scientific work, Hawking has been and still is a master of science communication. In 1988,
“A Brief History of Time” was a daring book about abstract ideas in a fringe area of theoretical physics. Hawking, to everybody’s surprise, proved that the public has an interest in esoteric problems like what happens if you fall into a black hole, what happed at the Big Bang, or whether god had any choice when he created the laws of nature.

Since 1988, the popular science landscape has changed dramatically. There are more books about theoretical physics than ever before and they are more widely read than ever before. I believe that Stephen Hawking played a big role in encouraging other scientists to write about their own research for the public. It certainly was an inspiration for me.

So, Happy Birthday, Stephen, and thank you.

## 44 comments:

I had the privilege to hear him give his talk "The Universe in a nutshell" in Seattle many years ago. It is something I tell my grandchildren about. I continue to be inspired by the man.

A Brief History of Time got me into physics in the first place, it's one of those reads that has really shaped many of my decisions. :)

"our universe started in a time-less state. It’s an appealing idea whose time hasn’t yet come" Hmmmm!

As a math graduate student at Princeton, I attended a Hawking lecture at the IAS. He was still speaking through an interpreter in those days. After the lecture, a young man asked a long and very technical question about something Hawking had said in his previous day's seminar. Hawking's answer: "I certainly didn't say anything about that today." The young man was Ed Witten. It was a memorable day!

The problem with the black holes analogues is that we don't know if the mathematics is the same. We conjecture that the maths decribing the black holes is the same as our model, but unlike the model we can't experiment on a black hole to prove it.

https://www.youtube.com/watch?v=L-28jAoTPFw

...Happy birthday, Stephen!

"

experimental tinkering" Physics describes aspirin in deep. punctilious, universal ways that chemistry cannot. A physicist with a headache still needs a gram of chemistry. Art is elegant, craft builds the world.Net matter versus antimatter is a good place for physics to begin rather than begin to patch a

lomcovák.You forgot to mention his triumph over the muscle degenerative disease he has fought over all those years since he was first diagnosed and doctors gave him a two year prognosis of the terminal illness.

Excellent article, manages to be both clear and accurate. I would add only that the answer to the question "will hawking win a nobel" is that the question is irrelevant. Like any award, the Nobel in physics has certain criteria that are harder for some phyicists to satisfy than others. His accomplishments are secure and need no validation By yet more awards.

John,

No, you misunderstood this. We know the math is the same.

Happy birthday, Stephen

Besides his scientific work, Hawking has been and still is a master of science communication. In 1988, “A Brief History of Time” was a daring book about abstract ideas in a fringe area of theoretical physics. Hawking, to everybody’s surprise, proved that the public has an interest in esoteric problems like what happens if you fall into a black hole, what happed at the Big Bang, or whether god had any choice when he created the laws of nature.While the book (which I have and have read) was a bestseller, I think that there are many popular-science books covering the same territory which do it better. Also, not everyone who bought Hawking's book has read it. (To be sure, that might be true of Hawking and Ellis as well.)

Like John, I misunderstood this. Even when the analogous systems will be finally known to emit Hawking radiation: how will you "know" then that the emission process of Hawking radiation in BHs is described by the same mathematics? Do you have holes in your hands?

(Haha, just a joke, that's how we used to say as kids.)

True, you believed for good reasons that it is even before the work on the analogous systems started. But how can experimental results on analogous systems strengthen this belief in any fundamental manner?

Maurice,

Please note that you have changed the question. You are asking how do I know that the (real) black hole radiation is described by the mathematics that Hawking used. I don't know. Nobody knows. That isn't the point. The point is that the mathematics that Hawking used to describe black holes is the same as that which applies to the fluid analogues. It's the same calculation (in certain approximations).

Having said that, there's no known reason why Hawking's calculation should not apply for real black holes. Indeed, it's exceedingly hard to get rid of it. Keep in mind that this calculation is done in the near horizon region, but that region can have vanishingly small space-time curvature. It can be as small as right here. We have tested both general relativity and quantum field theory at such small curvature to extremely high accuracy. It's hard to change anything about that without screwing up the success of our present theories. Best,

B.

Bee,

"

No, you misunderstood this. We know the math is the same."Perhaps it would help to try to define our terms more explicitly.

Do you mean that the appearance of Hawking radiation in the flowing-fluid analogy would constitute evidence that BH radiation occurs

in the theory of GR? (I guess few people would contest that.)But despite the extensive experimental evidence supporting GR in other situations

notinvolving quantum phenomena, shouldn't we distinguish the above evidence from evidence for Hawking radiation fromreal black holes in nature? The radiation is a quantum phenomenon, and you stated "we need a theory of quantum gravity in order to consistently describe nature.~~~

"

Black holes of sufficiently small masses could have formed from density fluctuations in the early universe and are therefore referred to as “primordial black holes.”Does your qualification "sufficiently small masses" arise only within (some version of) the theory of inflation? (If so, which version?) As you may have noticed, Paul Frampton is enthusiastic about PBHs as dark matter, and stated in arXiv:1608.0509 that "The formation of PBHs with masses as large as Eq. (1) [10⁵ solar masses] and much larger is known to be mathematically possible during the radiation era." He likes PBHs as a possible explanation for the super-massive black holes at the centre of some galaxies [see Eq. (5)], which I seem to recall are not easily explained in the Concordance Model.

Thanks for the delightful blog article.

Andy

Sorry Bee, I read carelessly; "

sufficiently small masses" obviously means only sufficiently small to evaporate rapidly enough to detect.Andy

Sabine,

Maurice has nicely summarised my point. Observing radiation from the analogue systems will be a great result and will certainly strongly suggest Hawking radiation exists. But then I'd guess most of us (including me) already believe Hawking radiation exists for the reasons you mention above.

I just don't think observing radiation in an analogue system would be the cue to hand out a Nobel prize.

"While the 1974 paper was predated by Bekenstein’s finding that black holes resemble thermodynamical systems, Hawking’s derivation was the starting point for countless later revelations. Thanks to it, physicists understand today that black holes are a melting pot for many different fields of physics – besides general relativity and quantum field theory, there is thermodynamics and statistical mechanics, and quantum information and quantum gravity."Quantum gravity or quantum mechanics in curved spacetime? :-)

John,

Yes, you are both putting words into my mouth that I didn't use. Why not first try to understand what I said before complaining about it? I didn't say that measuring Hawking radiation in black hole analogues would show that real black holes emit radiation. Why would I say such a thing? What I said is that it would prove that the effect which Hawking calculated exists. As I've now repeated sufficiently often, it's the same calculation.

Neither did I say that observing the radiation in analogue systems would be sufficient for a Nobel Prize. Instead I said explicitly that it would not be sufficient. Maybe read it again before wasting more of my time. Best,

B.

Phillip,

Not sure what you mean. It has arguably lead to a lot of effort to try and find a theory of quantum gravity that would solve the problem. A lot of papers on quantum gravity are actually about black hole evaporation.

Sabine,

My apologies. I've obviously offended you and I didn't mean to. I'm a long time reader of your blog and enjoyed reading this post. My comment wasn't meant as a contradiction or to imply that anything in your post was wrong. It was just, well, a comment.

John,

I'm not offended. I'm tired of constantly having to repeat myself. Anyways, apology accepted :o)

Sabine,

Perhaps something interesting might be salvaged from this misunderstanding and wasted time.

Now we understand that you propose that the physical fluid analogue can prove that Hawking's

mathematicaltheorem calculated in the context of GR (and QFT?) is correct. Perhaps one reason why some of us misunderstood you is that it seems highly unusual to resort to a physical analogue to prove a mathematical theorem.− Why not use logic as mathematicians would? Where does this approach fail? I.e., ...

− What do you think the analogue experiment is really testing?

− Could the fluid analogue falsify the Hawking theory? You seemed to imply that if radiation is not seen in the fluid, the experiment might be wrong. Could one ever demonstrate that the experiment is correct, and a negative finding means the theorem is false? If not, why would one believe a positive finding? Has any confirmed expert in fluid dynamics approved the claims being made for the fluid model?

Bee, parenthetically, I believe from experience that one can save time by being a bit more explicit than one thinks is necessary. When some of us see "effect exists" in a physics context, we assume that a physical effect is meant, and to some of us, a physical effect is always in nature. So yes, we wondered "Why would you say such a thing?", and so we asked and wasted your time, which indeed is a pity. I hope I'm not in danger of wasting more.

Andy

@Sabine

I did not change the question I directly

responded to your statement "It is more than an analogy."

As to your bashing of John: If it really were

more than an analogy (it is not in our opinion) and if it were firmly established

I (and I guess u also John?) would think Hawking would

get his Noble prize.

That you think even this would not suffice doesn't invalidate

John's point.

@John

John, u certainly did not insult Sabine

even if you had written something incorrect.

But Sabine offended you.

Or what else but an offence is: "stop wasting more of my time"?

Maurice,

I'll not fight with you over the use of the word 'analogy'. I have explained sufficiently often now what I mean.

Andy,

You don't prove a mathematical theorem by making an observation. Hawking did a calculation that demonstrates particle production happens in certain background fields and the radiation has certain properties, is thermal and has correlations etc. You measure these particles, you know that the effect exists and the particles are like what Hawking predicted, that's what I'm saying. Whether that background field is a fluid or a black hole geometry doesn't make a difference for the math, hence showing that particles are produced with the specific properties that Hawking said they have demonstrates the effect indeed exists. This isn't mysterious: It's a prediction, you go and test it. I am merely saying that Hawking's prediction holds for analogue gravity systems.

Just because the long distance, classical, description is the same (analogous) does not imply that the leading quantum correction is the same. The leading correction to fluid dyanmics near your bath tub drain is not quantized water waves, it is quantum kinetics of atoms and molecules. The situation is only marginally better in BEC's. There are many classical and quantum corrections to classical fluid dynamics near a sonic horizon that are more important than analog Hawking radiation.

Hawking may not have created a completely new field of science but rather worked in one. But most of the progress made in science and society in general was not made by the original innovations, but by their usage and improvement. And Hawking surely improved general relativity.

So, happy birthday, Stephen Hawking.

Thomas,

In water the pertubations one uses aren't quantized, so there isn't strictly speaking a Hawking effect. Excuse me for being somewhat sketchy on that here, there are different types of analogue gravity systems, some quantized, some not, some using fluids, others solids. But yes, as I said, even with BECs the situation isn't clear and it'll take more time to sort out what exactly one is observing there.

"In the 1980s, Hawking worked with Jim Hartle on the no-boundary proposal according to which our universe started in a time-less state. "

Hi Sabine,

Do you happen to have a link handy to a decent reference/paper on this subject? Sounds fascinating, would love to learn more.

Thanks!

Thanks Bee for reminding us that great visionaries still exist among us!

ok, I'm not a theoretical or mathematical physicist, but when I read something like:

"... with Roger Penrose, Hawking proved that singularities form under quite general conditions in General Relativity,...

Before Hawking and Penrose’s work, physicists had hoped that the singularities which appeared in certain solutions to General Relativity were mathematical curiosities of little relevance for physical reality. But the two showed that this was not so ... "

It seems a singularity is still just an exercise in mathematical physics, a mathematical curiosity with no chance of any "physical reality" ever being observed, directly or indirectly.

Seems all we can ever observe are effects and events _outside_ the event horizon(s). What am I missing?

-- TomH

Hi Bee,

Thanks for the great post. I was following the discussion above with Andy and Maurice and John and this is what continues to confuse me at least (maybe they had similar issues)

Let me illustrate my confusion with an analogy (um..)

Suppose I model the motion of a pendulum using Simple Harmonic Motion (that's like using QFT in curved space to model behaviour near the event horizon).

I use my model to derive a formula for the time period of the pendulum (that's prediction of Hawking radiation)

Then someone points out that SHM also describes the motion of a loaded spring (fluid analogy).

So I set up a loaded spring and measure its oscillation time period (that's what the fluid guys are doing)

Now suppose it turns out that the oscillation time for the spring matches my formula.

Does it prove:

- That SHM describes the motion of the spring ? Probably. But I wanted to describe the pendulum, not the spring !

- That my time period derivation is correct ? Very likely. But wouldn't I already know that if I had done the math right ?

Suppose the experiment contradicts my prediction. Does it prove:

- My math was wrong ? Could be, but maybe best to check the math itself ?

- That SHM doesn't describe the spring ? Probably. But is that a problem ? After all, I am interested in the pendulum, not the spring.

So, what exactly is gained/lost from the spring experiment ?

One case I can imagine is that:

- Maybe my time period derivation wasn't rigorous. I made some approximations, appeal to intuition and special cases, etc in the difficult bits.

- I am really very confident that my SHM model applies to the spring.

In that case, a positive result would show that my shaky derivation is indeed correct and I "got the essentials right". After that we bring in the mathematicians :)

Could that be some of the motivation here ? From the little I know QFT in curved space is not very rigorously founded.

Thanks in advance,

Anindya

senanindya,

If your model correctly describes the observation, then it describes the observation. If it doesn't, it doesn't. I don't understand your problem. Sure, maybe you originally wanted to describe something else. So what?

The problem is with your sentence:

"But wouldn't I already know that if I had done the math right ?"

If that would make sense, then why are we doing any experiments for which we can calculate predictions? Answer: Because physics isn't math. Best,

B.

Unknown,

Singularities were believed to appear in the math, but not in solutions than can occur under realistic circumstances. The singularity theorems proved that isn't so. Indeed, singularities can be formed under quite general, physically realistic, situations. To avoid them it's hence not enough to just believe they don't occur, one has to really modify the theory. Hence the relevance of Penrose and Hawking's work. Best,

B.

senanindya,

Nice discussion! I do not see "the case you can imagine". Indeed the derivation of Hawking radiation is not completely rigorous.

See e.g. this paper from highly respected colleagues https://arxiv.org/abs/gr-qc/0408009

who conclude after a discussion of some deep conceptual issues involving ultra-high energies:

"However, we also demonstrate counter-examples, which do not appear to be unphysical or artificial, displaying strong deviations from Hawking's result. Therefore, whether real black holes emit Hawking radiation remains an open question and could give non-trivial information about Planckian physics."

But needless to say, seeing Hawking radiation in some analogous system would not clear up the above deep issues...

"It seems a singularity is still just an exercise in mathematical physics, a mathematical curiosity with no chance of any "physical reality" ever being observed, directly or indirectly.

Seems all we can ever observe are effects and events _outside_ the event horizon(s). What am I missing?"

Following up Sabine's comment: You are confusing several things. First, there are singularities which are mathematical in the sense that they are an artifact of the coordinate system. In GR the best known one involves the Schwarzschild solution, but think of the north (or south) pole of the Earth: even though the distance of a degree of longitude goes to zero there, nothing strange happens physically. In fact, if you are near the pole, you don't want a map in polar coordinates, but probably cartesian coordinates with the pole at the centre. Second, Hawking and Penrose showed that GR leads to real singularities, not just coordinate singularities. Whether they actually exist physically (i.e. whether GR is still valid here) is a separate question. Third, even though we can't directly observe things beyond the horizon, that doesn't mean that we can't say anything about them.

Is a translation of the singularity theorems into plain language available somewhere?

Ambi Valent,

I hope Sabine doesn't mind the blatant publicising, but if you have questions about physics then the Physics Stack Exchange is a place where you can have real physicists answer your questions about physics. I won't add the link because I'm not sure about the rules on putting links in comments, but a quick Google will find the site.

The answer is that it isn't really possible to translate the singularity theorems into plain English, though we can give a feel for how they work.

Bee:

1) "Sure maybe you wanted to describe something else, so what ?"

So everything. As far as I understand, mathematical modeling in physics only makes sense with respect to some system being modelled.

So "Is SHM a correct model of a pendulum ?" is a meaningful question.

But if I aimed to describe a pendulum and ended up describing a spring - I have achieved...what exactly ?

Shown that something in the world is described by SHM ? Okaay, but NOW so what ?

Have I shown that "SHM is correct" ? Its just a differential equation with a solution - how can it be wrong or correct?

2) "Why are we doing any experiments for which we can calculate predictions ?"

In order to test if a given mathematical model *correctly describes a particular physical system*.

That's why I can't see what follows from doing experiments for a *different physical system*

Because:

- If you just wanted to check the consistency or predictions of the model itself, then that's a purely mathematical question.

- If you wanted to check if the model describes System A, then checking it for System B doesn't answer your question.

A.

senanindya,

"As far as I understand, mathematical modeling in physics only makes sense with respect to some system being modelled."

As I said, the fluid analogue is such a system.

You write

"If you wanted to check if the model describes System A, then checking it for System B doesn't answer your question."

That's right, as I have now said several times: You do not check the prediction on system B to see if it describes system A. You check the prediction on system B to see if it describes system B. I don't know why you think checking a prediction is interesting and should be done for system A but not for system B.

Look, just forget that Hawking ever said he does this calculation for black holes. Imagine he'd just have said you take a quantum field theory and put it on a background described so-and-so, then I predict there should be a radiation with properties so-and-so. That's what is being tested here. Best,

B.

"Look, just forget that Hawking ever said he does this calculation for black holes. Imagine he'd just have said you take a quantum field theory and put it on a background described so-and-so, then I predict there should be a radiation with properties so-and-so. That's what is being tested here"

Which brings me to a question I have also asked several times and you have ignored/dismissed:

Why does anything need to be *experimentally tested* here ??

To elaborate:

"Imagine he'd just have said you take a quantum field theory and put it on a background described so-and-so, then I predict there should be a radiation with properties so-and-so."

So I imagine that basically involves setting up a bunch of PDE's on some kind of manifold.

The "radiation with XYZ property" would then be a solution of those equations under some kind of boundary condition.

And this is my point - Everything so far is just mathematics, there is *absolutely nothing here which needs to be experimentally tested*.

If I have a system with two variables x, t satisfying dx/dt = x, do I need to then *empirically test* that x is exponential in t ?

Now suppose I have some physical system which I *believe* is described by the equations and yaay, the effect shows up.

That might be a little fun - like actually measuring angles of a triangle and adding them up to get 180 degrees.

But it tells me *absolutely nothing* about whether the radiation solution to my QFT model was correctly derived or not.

At best it confirms my model describes a certain system. At worst, my model doesn't really describe the system AND my radiation derivation was also wrong - we just got a happy cancellation ( I admit this is unlikely).

So, to summarize, my experiment really didn't add anything to my understanding of the model, UNLESS (and this is a question I asked in my first post):

My derivation of the radiation solution was mathematically suspect.

Maybe I could only derive the solution in a "close to linear case" or for small values of a parameter or for an infinitely distant observer and then I used heuristics, intuitive arguments etc to say the solution is always true.

In THIS case, I can see the value of doing the experiment - If I see the effect, it gives me confidence that my dodgy derivation was nevertheless correct and my intuition was justified.

So is THAT why people are excited about the fluid models and whether they really show a Hawking effect analogue?

Another possibility:

If I do repeated experiments with a fluid and they always match predictions from the QFT model, then I can do NEW experiments which don't correspond to anything which has been already derived mathematically. That might help me explore the model more.

And THEN if I am also independently confident that the QFT model applies to black holes, I could start thinking about the black hole equivalent of my new fluid experiment results.

Maybe that's another motivation ?

Anyway, that's all I had to say/ask on this.

A.

senanindya,

I have already answered these questions, but it seems you didn't read my reply. I'll do it once again but I am already getting tired of this and have no interest in continuing this fruitless exchange.

You are asking "why does this have to be experimentally tested". Once again: It's a prediction for a physical system. You test it because that's how science works. You have a theory for a system, you calculate observables, you go and test the theory. No mathematical proof can demonstrate that your system is actually described by the theory.

You seem to have a difficulty understanding what's the difference between math and physics.

Having said that, of course the experiments always match predictions from the QFT model. What were you thinking, that it only works Wednesdays? Really I think you should first try to understand what you are talking about before insisting something I said is wrong. Best,

B.

Pendulum equations can omit the bob (swing, but not in free fall), have mass (torsion, moment of inertia), have mass and weight (spring). Pendulum equations are empirically validated.

Exact theory (math) is not applied theory (science). All quantum gravitations are empirically sterile. SUSY is sterile or fudged (e.g., proton decay and Super-Kamiokande). Something is fundamentally wrong (physics) though the math is rigorous, elegant, and profuse.

Post a Comment