Yes, I know, I’m not very good at this hiatus thing. But there is important news that needs to be promulgated widely — the news of calculus. No more will innocent citizens cower in fear at the thought of derivatives and integrals, or flash back in horror to the days of terror and confusion in high-school math class. Because now there is a cure for these maladies — The Calculus Diaries: How Math Can Help You Lose Weight, Win in Vegas, and Survive a Zombie Apocalypse.

Yes, you read that subtitle correctly. Let’s be clear: this book is probably not for you. That’s because you, I have no doubt, already love calculus. You carry a table of integrals in your back pocket, and you practice substituting variables to while away the time in the DMV. This isn’t the book for people who already appreciate the austere beauty of a differential equation, or even for people who want to study up for their AP exam.

No, this is the book for people who hate math. It’s for people who look at you funny and turn away at parties when you mention that you enjoy science. It’s for your older relatives who think you’re crazy for appreciating all that technical stuff, or your nieces and nephews who haven’t yet been captivated by the beauty of mathematics. The Calculus Diaries is the book for people who need to be convinced that math isn’t an intimidating chore — that it can be fun.

Know anybody like that? Any gift-giving holidays coming up?

Now it’s true, I know the author. In fact, I appear as a character in the book (to a certain degree of comic effect). I’m the one who gets soaked when we ride Splash Mountain at Disneyland, but also the one who maximizes his winnings at craps by clever betting in Vegas. You get the idea: this isn’t a textbook, it’s a tour through the real world (and occasional fantasy worlds), pointing out that math is all around us, and that perceiving it is kind of cool.

When you understand math, how you think about the world changes. Every day, we all change position by accumulating velocity, or do informal optimization problems when making a decision. But most people don’t know about the wonderful insights that math can add to these processes. You know, because you are a mathphile. But you are outnumbered by the mathphobes. You have a secret that they don’t know, but now there’s a way to share it. What are you waiting for?

Caltech had its commencement ceremony last Friday, and I donned a cap and gown to march up on stage with the other faculty members. It’s always a great day, as years of work comes to fruition for several hundred students, ready to move on to the next stage of their careers.

Naturally, there was singing. The Glee Club sent spirits soaring with the Caltech alma mater, “Hail CIT.”

In southern California with grace and splendor bound,
Where the lofty mountain peaks look out to lands beyond,
Proudly stands our alma mater, glorious to see.
We raise our voices proudly, hailing, hailing thee.
Echos ringing while we’re singing, over land and sea.
The hall of fame resound thy name, noble CIT.

The one that got my attention, however, was the other song — Gaudeamus Igitur, apparently a “traditional college song.” How have I spent so many years in academia without coming across this one? It was sung in Latin, but a helpful translation into English was provided.

Therefore let us rejoice
While we are young
After pleasant youth,
After troublesome old age,
The earth will have us.

Where are they who before us
Were in the world?
You can cross the heavens,
You can go to hell,
If you wish to see them.

Our life is brief,
Shortly it will end.
Death comes quickly,
It snatches us cruelly,
No one is spared.

Long live the academy!
Long live the professors!
Long live each student!
Long live all students!
May they always flourish!

Cheerful, no? We’re all going to die, but at least the university will live on. Comforting.

And now Wikipedia informs me that a few verses were apparently left out of our version. To wit:

Long live all girls
Easy and beautiful!
Long live mature women also,
Tender and lovable
Good [and] productive,

Long live the state as well
And he who rules it!
Long live our city
[And] the charity of benefactors
Which protects us here!

Let sadness perish!
Let haters perish!
Let the devil perish!
Let whoever is anti-student
As well as the mockers!

So they left out the bits that were veering uncomfortably close to sexism, fascism, and serial killer-ism. I’m thinking they didn’t want the ceremony to drag on for too long.

Timothy Ferris, in The Science of Liberty:

In 1900 there was not a single liberal democracy in the world (since none yet had universal suffrage); by 1950 there were twenty-two.

Tyler Cowen at Marginal Revolution has an ongoing series of posts in which he highlights “good sentences.” At first the conceit bugged me a bit, as how good can a single sentence be? It’s not like you have space to develop a sensible argument or anything.

But that’s the point, of course. A really good sentence packs a wallop because it fits an enormous amount into very few words. One technique for doing that is to exhibit an underlying assumption that is a remarkable claim in its own right. If I were to have tried to make the point that Ferris makes above, it would have been something like this:

Liberal democracies were established in fits and starts over a period of hundreds of years. The first major steps happened in countries like Britain, the United States, and France, where aristocratic systems were replaced (with different amounts of violence) by rule by popular vote. But I would argue that a true liberal democracy is one that features universal suffrage — every adult citizen has a right to participate. By that standard, there weren’t any liberal democracies in existence in the year 1900; but fifty years later, there were twenty-two.

Makes the point, but it’s a somewhat ponderous collection of mediocre sentences, rather than a single one of immense power. That’s the difference between someone who writes things, like me, and a true writer. I’m trying to learn.

Ferris’s book seems excellent, although I’ve just started reading it. It has a provocative thesis: the Enlightenment values of liberal democracy and scientific reasoning didn’t simply arise together. The emergence of science is rightfully understood as the cause of the democratic revolution. That’s the kind of thing I’d be happy to believe is true, so I’m especially skeptical, but I’m looking forward to the argument.

And we’ve reached the final installment of the From Eternity to Here book club. Chapter Fifteen is entitled “The Past Through Tomorrow,” in an oblique allusion to Robert Heinlein, my favorite author when I was younger. We’re going to throw in the Epilogue for good measure.

Excerpt:

What we’ve done is given the universe a way that it can increase its entropy without limit. In a de Sitter universe, space grows without bound, but the part of space that is visible to any one observer remains finite, and has a finite entropy—the area of the cosmological horizon. Within that space, the fields fluctuate at a fixed temperature that never changes. It’s an equilibrium configuration, with every process occurring equally as often as its time-reverse. Once baby universes are added to the game, the system is no longer in equilibrium, for the simple reason that there is no such thing as equilibrium. In the presence of a positive vacuum energy (according to this story), the entropy of the universe never reaches a maximum value and stays there, because there is no maximum value for the entropy of the universe—it can always increase, by creating new universes.

This is the chapter where we attempt to put it all together. The idea was that we had been so careful and thorough in the previous chapters that in this one we could be fairly terse, setting up ideas and knocking them down with our meticulously-prepared bludgeon of Science. I’m not sure if it actually worked that way; one could argue that it would have been more effective to linger lovingly over the implications of some of these scenarios. But there was already a lot of repetition throughout the book (intentionally, so that ideas remained clear), and I didn’t want to add to it.

Of course my own current favorite idea involves baby universes pinching off from a multiverse, and I’m certainly happy to explain my reasons in favor of it. But there are also good reasons to be skeptical, especially when it comes to our lack of knowledge concerning whether baby universes actually are formed in de Sitter space. What I hope comes across is the more generic scenario: a multiverse where entropy is increasing locally because it can always increase, and does so both toward the far past and the far future. While there’s obviously a lot of work to be done in filling in the details, I haven’t heard any other broad-stroke idea that sounds like a sensible dynamical origin for the arrow of time. (Which isn’t to say that one won’t come along tomorrow.)

Chapter 16 is the Epilogue, where I reflect on where we’ve been and what it all means. I talk a little about why thinking about the multiverse is a very respectable part of the scientific endeavor, and how we should think about the fact that we are a very tiny part of a very big cosmos. Finally, I wanted to quote the very last paragraph of text in the book, at the end of the Acknowledgments:

I’m the kind of person who grows restless working at home or in the office for too long, so I frequently gather up my physics books and papers and bring them to a restaurant or coffee shop for a change of venue. Almost inevitably, a stranger will ask me what it is I’m reading, and—rather than being repulsed by all the forbidding math and science—follow up with more questions about cosmology, quantum mechanics, the universe. At a pub in London, a bartender scribbled down the ISBN number of Scott Dodelson’s Modern Cosmology; at the Green Mill jazz club in Chicago, I got a free drink for explaining dark energy. I would like to thank every person who is not a scientist but maintains a sincere fascination with the inner workings of nature, and is willing to ask questions and mull over the answers. Thinking about the nature of time might not help us build better TV sets or lose weight without exercising, but we all share the same universe, and the urge to understand it is part of what makes us human.

Among those people who share a fascination with the inner workings of nature, I of course include people who regularly read this blog. So — thanks!

Welcome to this week’s installment of the From Eternity to Here book club. We’re on to Chapter Fourteen, “Inflation and the Multiverse.” Only one more episode to go! It’s like the upcoming finale of Lost, with a slightly lower level of message-board frenzy.

Excerpt:

There is a lot to say about eternal inflation, but let’s just focus on one consequence: While the universe we see looks very smooth on large scales, on even larger (unobservable) scales the universe would be very far from smooth. The large-scale uniformity of our observed universe sometimes tempts cosmologists into assuming that it must keep going like that infinitely far in every direction. But that was always an assumption that made our lives easier, not a conclusion from any rigorous chain of reasoning. The scenario of eternal inflation predicts that the universe does not continue on smoothly as far as it goes; far beyond our observable horizon, things eventually begin to look very different. Indeed, somewhere out there, inflation is still going on. This scenario is obviously very speculative at this point, but it’s important to keep in mind that the universe on ultra-large scales is, if anything, likely to be very different than the tiny patch of universe to which we have immediate access.

This is a fairly straightforward chapter, trying to explain how inflation works. Given that by this point the reader already is familiar with dark energy making the universe accelerate, and with the fine-tuning problem represented by the low entropy of the early universe, the basic case isn’t that hard to put together. Of course we have an additional non-traditional goal as well: to illuminate the tension between the usual story we tell about inflation and the “information-conserving evolution of our comoving patch” story we told in the last chapter. Here’s where I argue that inflation is not the panacea it’s sometimes presented as, primarily because it’s not that easy to take all the degrees of freedom within the universe we observe and pack them delicately into a tiny patch dominated by false vacuum energy. Put that way, it doesn’t seem all that surprising, but too many people don’t want to get the message.

This is also the chapter where we first introduce the idea of the multiverse. (The multiverse occupies less than 15 pages or so in the entire book, but to read some reactions you would think it was the dominant theme. The publicists and I must share some of the blame for that perspective, as it is an irresistible thing to mention when talking about the book.) Mostly I wanted to demystify the idea of the multiverse, presenting it as a perfectly natural outgrowth of the idea of inflation. What we’re supposed to make of it is of course a different story.

Looking back, I think the chapter is a mixed success. I like the gripping narrative of the opening pages. But the actual explanation of inflation is kind of workmanlike and uninspiring. I really put a lot of effort into coming up with novel explanations of entropy and quantum mechanics, which didn’t simply rehash the expositions found in other books; but for inflation I didn’t try as hard. Partly simply because of looming deadlines, partly because I was eager to get to the rest of the book. Hopefully the basic points are more or less clear.

Welcome to this week’s installment of the From Eternity to Here book club. Today we have a look at Chapter Thirteen, “The Life of the Universe.”

Excerpt:

If our comoving patch defines an approximately closed system, the next step is to think about its space of states. General relativity tells us that space itself, the stage on which particles and matter move and interact, evolves over time. Because of this, the definition of the space of states becomes more subtle than it would have been in if spacetime were absolute. Most physicists would agree that information is conserved as the universe evolves, but the way that works is quite unclear in a cosmological context. The essential problem is that more and more things can fit into the universe as it expands, so—naively, anyway—it looks as if the space of states is getting bigger. That would be in flagrant contradiction to the usual rules of reversible, information-conserving physics, where the space of states is fixed once and for all.

Of course we’ve already looked a bit at the life of the universe, way back in Chapter Three. The difference is that we’re now focusing on how entropy evolves, given our hard-acquired understanding of what entropy is and how it works for black holes. This is where we review Roger Penrose’s well-known-yet-still-widely-ignored argument that the low entropy of the early universe is something that needs to be explained.

In a sense, this is pretty straightforward stuff, following directly from what we’ve already done in the book. But it’s also somewhat controversial among professional cosmologists. The reason why can be found in the slightly technical digression that begins on page 292, “Conservation of information in an expanding universe.”

The point is that physicists often think of “the space of states in a region of spacetime” as being equal to “the space of states we can describe by quantum field theory.” They know that’s not right, because gravity doesn’t fit into that description, but these are the states they know how to deal with. This collection of states isn’t fixed; it grows with time as the universe expands. You will therefore sometimes hear cosmologists talk about the high entropy of the early universe, under the misguided assumption that there were fewer states that could “fit” into the universe at that time. (Equivalently, that gravity can be ignored.) This approach has, in my opinion anyway, done great damage to how cosmologists think about fine-tuning problems. One of the major motivations for writing the book was to explain these issues, not only to the general reader but also to my scientist friends.

At the end of the chapter I deviate from Penrose’s argument a bit. He believes that a high-entropy state of the universe would be one that was highly inhomogeneous, full of black holes and white holes and what have you. I think that’s right if you are thinking about a very dense configuration of matter. But matter doesn’t have to be dense — the expansion of the universe can dilute it away. So I argue that the truly highest-entropy configuration is one where space is essentially empty, with nothing but vacuum energy. This is also very far from being widely accepted, and certainly relies on a bit of hand-waving. But again, I think the failure to appreciate this point has distorted how cosmologists think about the problems presented by the early universe. So hopefully they read this far in the book!

Welcome to this week’s installment of the From Eternity to Here book club. Part Four opens with Chapter Twelve, “Black Holes: The Ends of Time.”

Excerpt:

Unlike boxes full of atoms, we can’t make black holes with the same size but different masses. The size of a black hole is characterized by the “Schwarzschild radius,” which is precisely proportional to its mass. If you know the mass, you know the size; contrariwise, if you have a box of fixed size, there is a maximum mass black hole you can possibly fit into it. But if the entropy of the black hole is proportional to the area of its event horizon, that means there is a maximum amount of entropy you can possibly fit into a region of some fixed size, which is achieved by a black hole of that size.

That’s a remarkable fact. It represents a dramatic difference in the behavior of entropy once gravity becomes important. In a hypothetical world in which there was no such thing as gravity, we could squeeze as much entropy as we wanted into any given region; but gravity stops us from doing that.

It’s not surprising to find a chapter about black holes in a book that talks about relativity and cosmology and all that. But the point here is obviously a slightly different one than usual: we care about the entropy of the black hole, not the gruesome story of what happens if you fall into the singularity.

Black holes are important to our story for a couple of reasons. One is that gravity is certainly important to our story, because we care about the entropy of the universe and gravity plays a crucial role in how the universe evolves. But that raises a problem that people love to bring up: because we don’t understand quantum gravity (and in particular we don’t have a complete understanding of the space of microstates), we’re not really able to calculate the entropy of a system when gravity is important. The one shining counterexample to this is when the system is a black hole; Bekenstein and Hawking gave us a formula that allows us to calculate the entropy with confidence. It’s a slightly weird situation — we know how to calculate the entropy of a system when gravity is completely irrelevant, and we also know how to calculate the entropy when gravity is completely dominant and you have a black hole. It’s only the messy in-between situations that give us trouble.

The other reason black holes are important, of course, is that the answer that Bekenstein and Hawking derive is somewhat surprising, and ultimately game-changing. The entropy is not proportional to the volume inside the black hole (whatever that might have meant, anyway) — it’s proportional to the area of the event horizon. That’s the origin of the holographic principle, which is perhaps the most intriguing result yet to come out of the thought-experiment-driven world of quantum gravity.

The holographic principle is undoubtedly going to have important consequences for our ultimate understanding of spacetime and entropy, but how it will all play out is somewhat unclear right now. I felt it was important to cover this stuff in the book, although it doesn’t really lead to any neat resolutions of the problems we are tackling. Still, hopefully it was somewhat comprehensible.

There used to be a Twitter account called Best of Wikipedia — it was a wonderful source for quirky things you might not have chanced upon in your normal browsing. Alas, it’s been quiet since November, so we’re left to our own devices. For some reason or another I was reading about Scholasticism, the dominant approach to teaching and learning in medieval Europe. Its early days came to pass during the Carolingian Renaissance in the late 700′s under Charlemagne.

Besides uniting Central Europe, Charlemagne was also a patron of learning, and used his influence to bring scholars from across the continent to his court. Most importantly, he recognized that the decline of literacy and the splintering of Latin into mutually incomprehensible regional dialects caused difficulties for the administration of an empire, so he ordered that every abbey in his domain should start a school. The idea of widespread schooling was a novel one at the time, and the long-term impact of this decision is probably incalculable. Sure, most of the scholarship may have been devoted to the interpretation of classic texts rather than the production of new knowledge, but you have to think that all that learning helped lay the groundwork for the eventual climb out of the Dark Ages. Start people thinking, and you never know where they will go.

So I was especially fascinated to read about Alcuin of York, one of Charlemagne’s greatest scholars. He was a respected teacher in Northumbria before being brought to court, where he had an enormous effect on the scholarship — establishing the liberal arts (the trivium and quadrivium) as the basis for the curriculum, and convincing Charlemagne not to put pagans to death if they refused to convert. He also produced a textbook of math problems with solutions, from which we learn that medieval word problems were more colorful than those we have today — these include the problem of the three jealous husbands and the problem of the wolf, goat and cabbage.

But it’s clear to me what Alcuin’s greatest achievement really was: he’s the guy who invented lower case letters. Can you imagine a world in which everything was written in ALL CAPS? Every time we read a crazy person ranting on the internet, we should give thanks to Alcuin that not everybody sounds like that.

Welcome to this week’s installment of the From Eternity to Here book club. Part Three of the book concludes with Chapter Eleven, “Quantum Time.”

Excerpt:

This distinction between “incomplete knowledge” and “intrinsic quantum indeterminacy” is worth dwelling on. If the wave function tells us there is a 75 percent chance of observing the cat under the table and a 25 percent chance of observing her on the sofa, that does not mean there is a 75 percent chance that the cat is under the table and a 25 percent chance that she is on the sofa. There is no such thing as “where the cat is.” Her quantum state is described by a superposition of the two distinct possibilities we would have in classical mechanics. It’s not even that “they are both true at once”; it’s that there is no “true” place where the cat is. The wave function is the best description we have of the reality of the cat.

It’s clear why this is hard to accept at first blush. To put it bluntly, the world doesn’t look anything like that. We see cats and planets and even electrons in particular positions when we look at them, not in superpositions of different possibilities described by wave functions. But that’s the true magic of quantum mechanics: What we see is not what there is. The wave function really exists, but we don’t see it when we look; we see things as if they were in particular ordinary classical configurations.

Title notwithstanding, the point of the chapter is not that there’s some “quantum” version of time that we have to understand. Some people labor under the impression that the transition from classical mechanics to quantum mechanics ends up “quantizing” everything, and turning continuous parameters into discrete ones, perhaps even including time. It doesn’t work that way; the conventional formalism of quantum mechanics (such as the Schrödinger equation) implies that time should be a continuous parameter. Things could conceivably change when we eventually understand quantum gravity, but they just as conceivably might not. In fact, I’d argue that the smart money is on time remaining continuous once all is said and done. (As a small piece of evidence, the context in which we understand quantum gravity the best is probably the AdS/CFT correspondence, where the Schrödinger equation is completely conventional and time is perfectly continuous.)

However, we still need to talk about quantum mechanics for the purposes of this book, for one very good reason: we’ve been making a big deal about how the fundamental laws of physics are reversible, but wave function collapse (under the textbook Copenhagen interpretation) is an apparent counterexample. Whether it’s a real counterexample, or simply an artifact of an inadequate interpretation of quantum mechanics, is a matter of much debate. I personally come down on the side that believes that there’s no fundamental irreversibility, only apparent irreversibility, in quantum mechanics. That’s basically the many-worlds interpretation, so I felt the book needed a chapter on what that was all about.

Along the way, I get to give my own perspective on what quantum mechanics really means. Unlike certain parts of the book, I’m pretty happy with how this one came out — feel free to correct me if you don’t completely agree. Quantum mechanics can certainly be tricky to understand, for the basic reason that what we see isn’t the same as what there is. I’m firmly convinced that most expositions of the subject make it seem even more difficult than it should be, by speaking as if “what we see” really does reflect “what there is,” even if we should know better.

So I present a number of colorful examples of two-state systems involving cats and dogs. Experts will recognize very standard treatments of the two-slit experiment and the EPR experiment, but in very different words. Things that seem very forbidding when phrased in terms of interference fringes and electron spins hopefully become a bit more accessible when we’re asking whether the cat is on the sofa or under the table. I did have to treat complicated macroscopic objects with many moving parts as if they could be described as very simple systems, but I judged that to be a worthwhile compromise in the interests of pedagogy. And no animals were harmed in the writing of this chapter! Let me know how you think the strategy worked.

A sad but true fact:

Receiving a balloon animal is only Act 1 of an inevitable three act tragedy.