Quantum Fluctuations

Posted in Science | 38 Comments

We Suck (But We Can Be Better)

One day in grad school, a couple of friends and I were sitting at a table in a hallway in the astronomy building, working on a problem set. The professor who had assigned the problems walked by and noticed what we were doing — which was fine, working together was encouraged. But then he commented, “Hey, I’m confused — you’re all smart guys, so how come the girls have been scoring better than you on the problem sets?” Out loud we mumbled something noncommittal, but I remember thinking, “Maybe they are … also smart?”

This professor was a good-hearted guy, who would have been appalled and defensive at the suggestion that his wry remark perhaps reflected a degree of unconscious bias. Multiply this example by a million, and you get an idea of what it’s like to be a woman trying to succeed in science in a modern university. Not necessarily blatant abuse or discrimination, of the sort faced by Marie Curie or Emmy Noether, but a constant stream of reminders that many of your colleagues think you might not be good enough, that what counts as “confident” for someone else qualifies as “aggressive” or “bitchy” when it comes from you, that your successes are unexpected surprises rather than natural consequences of your talent.

But even today, as we’ve recently been reminded, the obstacles faced by women scientists can still be of the old-fashioned, blatant, every-sensible-person-agrees-it’s-terrible variety. A few months ago we learned that Geoff Marcy, the respected exoplanet researcher at Berkeley, had a long history of sexually harassing students. Yesterday a couple of other cases came to light. U.S. Representative Jackie Speier gave a speech before Congress highlighting the case of Timothy Slater, another astronomer (formerly at the University of Arizona, now at the University of Wyoming) with a track record of harassment. And my own institution, Caltech, has suspended Christian Ott, a professor of theoretical astrophysics, for at least a year, after an investigation concluded that he had harassed students. A full discussion can be found in this article by Azeen Ghorayshi at BuzzFeed, and there are also stories at Science, Nature, and Gizmodo. Caltech president Thomas Rosenbaum and provost Edward Stolper published a memo that (without mentioning names) talked about Caltech’s response to the findings. Enormous credit goes to the students involved, Io Kleiser and Sarah Gossan, who showed great courage and determination in coming forward. (I’m sure they would both much rather be doing science, as would we all.)

No doubt the specifics of these situations will be debated to death. There is a wider context, however. These incidents aren’t isolated; they’re just the ones that happened to come to light recently. And there are issues here that aren’t just about men and women; they’re about what kind of culture we have in academia generally, science in particular, and physics/astronomy especially. Not only did these things happen, but they happened over an extended period of time. They were allowed to happen. Part of that is simply because shit happens; but part is that we don’t place enough value, as working academic scientists, professors, and students, in caring about each other as human beings.

Academic science — and physics is arguably the worst, though perhaps parts of engineering and computer science are just as bad — engenders a macho, cutthroat, sink-or-swim culture. We valorize scoring well on tests, talking loudly, being cocky and fast, tearing others down, “technical” proficiency, overwork, speaking in jargon, focusing on research to the exclusion of all else. In that kind of environment, when someone who is supposed to be a mentor is actually terrorizing their students and postdocs, there is nowhere for the victims to turn, and heavy penalties when they do. “You think your advisor is asking inappropriate things of you? I guess you’re not cut out for this after all.”

In 1998, Jason Altom, a graduate student in chemistry at Harvard, took his own life. Renowned among his contemporaries as both an extraordinarily talented scientist and a meticulous personality, he left behind a pointed note:

“This event could have been avoided,” the note began. “Professors here have too much power over the lives of their grad students.” The letter recommended adoption of a three-member faculty committee to monitor each graduate student’s progress and “provide protection for graduate students from abusive research advisers. If I had such a committee now I know things would be different.” It was the first time, a columnist for The Crimson observed later, that a suicide note took the form of a policy memo.

Academia will always necessarily be, in some sense, competitive: there are more people who want to be researchers and professors than there will ever be jobs for everyone. Not every student will find an eventual research or teaching position. But none of that implies that it has to be a terrifying, tortuous slog — and indeed there are exceptions. My own memories of graduate school are that it was very hard, pulling a substantial number of all-nighters and struggling with difficult material, but that at the same time it was fun. Fulfilling childhood dreams, learning about the universe! That should be the primary feeling everyone has about their education as a scientist, but too often it’s not.

A big problem is that, when problems like this arise, the natural reaction of people in positions of power is to get defensive. We deny that there is bias, or that it’s a problem, or that we haven’t been treating our students like human beings. We worry too much about the reputations of our institutions and our fields, and not enough about the lives of the people for whom we are responsible. I do it myself — nobody likes having their mistakes pointed out to them, and I’m certainly not an exception. It’s a constant struggle to balance legitimate justifications for your own views and actions against a knee-jerk tendency to defend everything you do (or don’t).

Maybe these recent events will be a wake-up call that provokes departments to take real steps to prevent harassment and improve the lives of students more generally. It’s unfortunate that we need to be shown a particularly egregious example of abuse before being stirred to action, but that’s often what it takes. In philosophy, the case of Colin McGinn has prompted a new dialogue about this kind of problem. In astronomy, President of the AAS Meg Urry has been very outspoken about the need to do better. Let’s see if physics will step up, recognize the problems we have, and take concrete steps to do better.

Posted in Academia, Women in Science | 40 Comments

That’s Just What They Would Say

The announcement we wait for every year has finally come in, and the American Dialect society has chosen their Word of the Year! That word is: “they”. It beat out other finalists such as “ammosexual.”

You might think that dubbing “they” as the Word of the Year is some sort of lifetime-achievement award, since the plucky pronoun has been part of English for quite a long time. But the prize has been given, not for the word itself, but for a particular usage that has been gaining ground for a while now: the singular “they.” We most commonly use the word to stand for the plural: “Jack and Jill went up the hill, but once there they realized they had forgotten their pail.” More and more, however, we’re seeing it used to denote one person at a time, when their sex is unknown to us: “The robber left no fingerprints, but they did leave a note to taunt the police.”

It would be somewhat more traditional, in some circumstances, to say “he or she did leave a note.” It’s a bit cumbersome, however, and to be honest, the real tradition is simply to act like women don’t exist, and say “he did leave a note.” The rise of “he or she” has reflected our gradual progress in remembering that human beings come in both male and female varieties, and our language should reflect that. (We can also try to make it reflect the full diversity of sex and gender roles, but while that’s an admirable goal, it might not be realistic in practice.)

Using “they” instead of “he or she” or just “he” is a very nice compromise. It sounds good, and it’s a word we’re already familiar with. Die-hard prescriptivists will complain that it’s simply a mistake, because when the God of English wrote the rules for our language, He (presumably) declared that “they” is only and always supposed to be plural. That view doesn’t accord with common sense, nor with the reality of the history of English. A long list of the best writers in the language, from Shakespeare and the authors of the King James Bible to Jane Austen and George Orwell, have deployed “they” as the correct pronoun to use when describing a single person whose sex is not known to us. Supporters of singular “they” are not revolutionaries twisting our language to the diabolical purposes of modern political correctness; we are just recalling a well-established and more correct way of speaking.

It’s long been argued that “he” served perfectly well as a generic singular pronoun, without any implication at all that the person being referred to is actually male. The problem with that view is that it is false. Studies have consistently shown that referring to unknown persons as “he” makes listeners envision a man much more often than a woman. To which one can scientifically reply, no duh. Pretending that “he” refers equally to men and women is just another strategy for pretending that sexism doesn’t exist — a tradition much more venerable than using “he” as a generic pronoun.

Minor fixes in our use of language aren’t going to make sexism go away. But they are steps in the right direction. I like to hope that, when the next young genius appears to revolutionize science, they will have had to deal with just a little bit less discrimination than their predecessors did.

Posted in Words | 45 Comments

Cosmic Maelstrom

I was doing some end-of-the-year housecleaning on my computer, and stumbled across this poem — an unrhymed sonnet on symmetry breaking in the early universe. (Always aiming at the least common denominator, what can I say?)

I have no misconceptions about my poetic abilities, which is no doubt why it sat privately on my hard drive for so long. But it’s the holidays, so here you go.

The cosmic maelstrom boiled bright and fierce,
A thousand fields did gambol nearly free.
Momentum was exchanged so high and hot
That couplings did asymptote to nil.
Amidst the glue and bosons ‘lectroweak
There stood our pensive scalar doublet, Phi
Surveying a potential all about
Like Buridan’s ass, secured by symmetry.
A longing pulled these spineless complex fields,
To rest where energy was minimized.
But held by finite temperature effects,
The quarks and leptons bound symmetric state.
Yet nothing perfect lasts through cosmic time,
The universe expands, illusion breaks.

Posted in Words | 20 Comments

Reading List

Now that The Big Picture is complete, I have more time for fun things like blogging, but I have a bunch of research to catch up on before I can return as normal. So in the meantime, here’s another teaser from the book: my list of “Further Reading” keyed to the different sections. You should have enough time to read all of these between now and publication day, May 10.

Part One, Cosmos:

  • Adams, F., & Laughlin, G. (1999). The Five Ages of the Universe: Inside the Physics of Eternity. Free Press.
  • Albert, D.Z. (2003). Time and Chance. Harvard University Press.
  • Carroll, S. (2010). From Eternity to Here: The Quest for the Ultimate Theory of Time. Dutton.
  • Feynman, R.P. (1967). The Character of Physical Law. M.I.T. Press.
  • Greene, B. (2004). The Fabric of the Cosmos: Space, Time, and the Texture of Reality. A.A. Knopf.
  • Guth, A. (1997). The Inflationary Universe: The Quest for a New Theory of Cosmic Origins. Addison-Wesley Pub.
  • Hawking, S.W. and Mlodinow, L. (2010). The Grand Design. Bantam.
  • Pearl, J. (2009). Causality: Models, Reasoning, and Inference. Cambridge University Press.
  • Penrose, R. (2005). The Road to Reality: A Complete Guide to the Laws of the Universe. A.A. Knopf.
  • Weinberg, S. (2015). To Explain the World: The Discovery of Modern Science. HarperCollins.

Part Two, Understanding:

  • Ariely, D. (2008). Predictably Irrational: The Hidden Forces that Shape Our Decisions. HarperCollins.
  • Dennett, D.C. (2014) Intuition Pumps and Other Tools for Thinking. W.W. Norton.
  • Gillett, C. and Lower, B., eds. (2001). Physicalism and Its Discontents. Cambridge University Press.
  • Kaplan, E. (2014). Does Santa Exist? A Philosophical Investigation. Dutton.
  • Rosenberg, A. (2011). The Atheist’s Guide to Reality: Enjoying Life Without Illusions. W.W. Norton.
  • Sagan, C. (1995). The Demon-Haunted World: Science as a Candle in the Dark. Random House.
  • Silver, N. (2012). The Signal and the Noise: Why So Many Predictions Fail — But Some Don’t. Penguin Press.
  • Tavris, C. and Aronson, E. (2006). Mistakes Were Made (but not by me): Why We Justify Foolish Beliefs, Bad Decisions, and Hurtful Acts. Houghton Mifflin Harcourt.

Part Three, Essence:

  • Aaronson, S. (2013). Quantum Computing Since Democritus. Cambridge University Press.
  • Carroll, S. (2012). The Particle at the End of the Universe: How the Hunt for the Higgs Boson Leads Us to the Edge of a New World. Dutton.
  • Deutsch, D. (1997). The Fabric of Reality: The Science of Parallel Universes and Its Implications. Viking Adult.
  • Gefter, A. (2014). Trespassing on Einstein’s Lawn: A Father, a Daughter, the Meaning of Nothing, and the Beginning of Everything. Bantam.
  • Holt, J. (2012) Why Does the World Exist? An Existential Detective Story. Liveright Publishing.
  • Musser, G. (2015). Spooky Action at a Distance: The Phenomenon That Reimagines Space and Time–and What It Means for Black Holes, the Big Bang, and Theories of Everything. Scientific American / Farrar, Straus and Giroux.
  • Randall, L. (2011). Knocking on Heaven’s Door: How Physics and Scientific Thinking Illuminate the Universe and the Modern World. Ecco.
  • Wallace, D. (2014). The Emergent Multiverse: Quantum Theory According to the Everett Interpretation. Oxford University Press.
  • Wilczek, F. (2015). A Beautiful Question: Finding Nature’s Deep Design. Penguin Press.

Part Four, Complexity:

  • Bak, P. (1996). How Nature Works: The Science of Self-Organized Criticality. Copernicus.
  • Cohen, E. (2012). Cells to Civilizations: The Principles of Change that Shape Life. Princeton University Press.
  • Coyne, J. (2009). Why Evolution is True. Viking.
  • Dawkins, R. (1986). The Blind Watchmaker: Why the Evidence of Evolution Reveals a Universe without Design. W.W. Norton.
  • Dennett, D.C. (1995). Darwin’s Dangerous Idea: Evolution and the Meanings of Life. Simon & Schuster.
  • Hidalgo, C. (2015). Why Information Grows: The Evolution of Order, from Atoms to Economies. Basic Books.
  • Hoffman, P. (2012). Life’s Ratchet: How Molecular Machines Extract Order from Chaos. Basic Books.
  • Krugman, P. (1996). The Self-Organizing Economy. Wiley-Blackwell.
  • Lane, N. (2015). The Vital Question: Energy, Evolution, and the Origins of Complex Life. W.W. Norton.
  • Mitchell, M. (2009). Complexity: A Guided Tour. Oxford University Press.
  • Pross, A. (2012). What Is Life? How Chemistry Becomes Biology. Oxford University Press.
  • Rutherford, A. (2013). Creation: How Science is Reinventing Life Itself. Current.
  • Shubin, N. (2008). Your Inner Fish: A Journey into the 3.5-Billion-Year History of the Human Body. Pantheon.

Part Five, Thinking:

  • Alter, T. and Howell, R.J. (2009). A Dialogue on Consciousness. Oxford University Press.
  • Chalmers, D.J. (1996). The Conscious Mind: In Search of a Fundamental Theory. Oxford University Press.
  • Churchland, P.S. (2013). Touching a Nerve: The Self as Brain. W.W. Norton.
  • Damasio, A. (2010). Self Comes to Mind: Constructing the Conscious Brain. Pantheon.
  • Dennett, D.C. (1991). Consciousness Explained. Little Brown & Co.
  • Eagleman, D. (2011). Incognito: The Secret Lives of the Brain. Pantheon.
  • Flanagan, O. (2003). The Problem of the Soul: Two Visions of Mind and How to Reconcile Them. Basic Books.
  • Gazzaniga, M.S. (2011). Who’s In Charge? Free Will and the Science of the Brain. Ecco.
  • Hankins, P. (2015). The Shadow of Consciousness.
  • Kahneman, D. (2011). Thinking, Fast and Slow. Farrah, Straus and Giroux.
  • Tononi, G. (2012). Phi: A Voyage from the Brain to the Soul. Pantheon.

Part Six, Caring:

  • de Waal, F. (2013). The Bonobo and the Atheist: In Search of Humanism Among the Primates. W.W. Norton.
  • Epstein, G.M. (2009). Good Without God: What a Billion Nonreligious People Do Believe. William Morrow.
  • Flanagan, O. (2007). The Really Hard Problem: Meaning in a Material World. The MIT Press.
  • Gottschall, J. (2012). The Storytelling Animal: How Stories Make Us Human. Houghton Mifflin Harcourt.
  • Greene, J. (2013). Moral Tribes: Emotion, Reason, and the Gap Between Us and Them. Penguin Press.
  • Johnson, C. (2014). A Better Life: 100 Atheists Speak Out on Joy & Meaning in a World Without God. Cosmic Teapot.
  • Kitcher, P. (2011). The Ethical Project. Harvard University Press.
  • Lehman, J. and Shemmer, Y. (2012). Constructivism in Practical Philosophy. Oxford University Press.
  • May, T. (2015). A Significant Life: Human Meaning in a Silent Universe. University of Chicago Press.
  • Ruti, M. (2014). The Call of Character: Living a Life Worth Living. Columbia University Press.
  • Wilson, E.O. (2014). The Meaning of Human Existence. Liveright.


Posted in Big Picture, Words | 33 Comments

The Big Picture: Table of Contents

Greetings, surface-dwellers! I have finally emerged from the secret underground laboratory where I have been polishing the manuscript for The Big Picture: On the Origins of Life, Meaning, and the Universe Itself. We pushed up the publication date to May 10, so you’ll get it in plenty of time for your summer beach reading. Evidence that it exists, all 145,000 glorious words:


As will happen in the writing process, the organization of the book has changed since I first mentioned it. Here is the final table of contents. As you might gather, I went with an organization of many short chapters. Hopefully that will help give the book the feeling of a light and enjoyable read.


    0. Prologue

* Part One: Cosmos

  • 1. The Fundamental Nature of Reality
  • 2. Poetic Naturalism
  • 3. The World Moves By Itself
  • 4. What Determines What Will Happen Next?
  • 5. Reasons Why
  • 6. Our Universe
  • 7. Time’s Arrow
  • 8. Memories and Causes

* Part Two: Understanding

  • 9. Learning About the World
  • 10. Updating Our Knowledge
  • 11. Is It Okay to Doubt Everything?
  • 12. Reality Emerges
  • 13. What Exists, and What Is Illusion?
  • 14. Planets of Belief
  • 15. Accepting Uncertainty
  • 16. What Can We Know About the Universe Without Looking at It?
  • 17. Who Am I?
  • 18. Abducting God

* Part Three: Essence

  • 19. How Much We Know
  • 20. The Quantum Realm
  • 21. Interpreting Quantum Mechanics
  • 22. The Core Theory
  • 23. The Stuff of Which We Are Made
  • 24. The Effective Theory of the Everyday World
  • 25. Why Does the Universe Exist?
  • 26. Body and Soul
  • 27. Death Is the End

* Part Four: Complexity

  • 28. The Universe in a Cup of Coffee
  • 29. Light and Life
  • 30. Funneling Energy
  • 31. Spontaneous Organization
  • 32. The Origin and Purpose of Life
  • 33. Evolution’s Bootstraps
  • 34. Searching Through the Landscape
  • 35. Emergent Purpose
  • 36. Are We the Point?

* Part Five: Thinking

  • 37. Crawling Into Consciousness
  • 38. The Babbling Brain
  • 39. What Thinks?
  • 40. The Hard Problem
  • 41. Zombies and Stories
  • 42. Are Photons Conscious?
  • 43. What Acts on What?
  • 44. Freedom to Choose

* Part Six: Caring

  • 45. Three Billion Heartbeats
  • 46. What Is and What Ought to Be
  • 47. Rules and Consequences
  • 48. Constructing Goodness
  • 49. Listening to the World
  • 50. Existential Therapy
  • Appendix: The Equation Underlying You and Me
  • Acknowledgments
  • Further Reading
  • References
  • Index

A lot of ground gets covered. In Part One we set the stage, seeing how discoveries in science have revealed a universe that runs under unbreakable, impersonal laws of nature. In Part Two we think about how to conceptualize such a universe: how to learn about it (Bayesian inference, abduction) and how to talk about it (emergence and overlapping theoretical vocabularies). In Part Three we get down and dirty with quantum mechanics, the Core Theory, and effective field theories. In Part Four we start down the road of connecting to our macroscopic world, seeing how complexity and life can arise due to the arrow of time. In Part Five we think about the leading challenge to a physicalist worldview: the existence of consciousness. And in Part Six we recognize that the universe isn’t going to tell us how to behave, and acknowledge that the creation of meaning and purpose is ultimately our job.

Now back to being a scientist with me. I have drafts of four different papers on my computer that need to be kicked out and onto the arxiv!

Posted in Big Picture, Philosophy, Science, Words | 40 Comments


This year we give thanks for an area of mathematics that has become completely indispensable to modern theoretical physics: Riemannian Geometry. (We’ve previously given thanks for the Standard Model Lagrangian, Hubble’s Law, the Spin-Statistics Theorem, conservation of momentum, effective field theory, the error bar, gauge symmetry, Landauer’s Principle, and the Fourier Transform. Ten years of giving thanks!)

Now, the thing everyone has been giving thanks for over the last few days is Albert Einstein’s general theory of relativity, which by some measures was introduced to the world exactly one hundred years ago yesterday. But we don’t want to be everybody, and besides we’re a day late. So it makes sense to honor the epochal advance in mathematics that directly enabled Einstein’s epochal advance in our understanding of spacetime.

Highly popularized accounts of the history of non-Euclidean geometry often give short shrift to Riemann, for reasons I don’t quite understand. You know the basic story: Euclid showed that geometry could be axiomatized on the basis of a few simple postulates, but one of them (the infamous Fifth Postulate) seemed just a bit less natural than the others. That’s the parallel postulate, which has been employed by generations of high-school geometry teachers to torture their students by challenging them to “prove” it. (Mine did, anyway.)

It can’t be proved, and indeed it’s not even necessarily true. In the ordinary flat geometry of a tabletop, initially parallel lines remain parallel forever, and Euclidean geometry is the name of the game. But we can imagine surfaces on which initially parallel lines diverge, such as a saddle, or ones on which they begin to come together, such as a sphere. In those contexts it is appropriate to replace the parallel postulate with something else, and we end up with non-Euclidean geometry.


Historically, this was first carried out by Hungarian mathematician János Bolyai and the Russian mathematician Nikolai Lobachevsky, both of whom developed the hyperbolic (saddle-shaped) form of the alternative theory. Actually, while Bolyai and Lobachevsky were the first to publish, much of the theory had previously been worked out by the great Carl Friedrich Gauss, who was an incredibly influential mathematician but not very good about getting his results into print.

The new geometry developed by Bolyai and Lobachevsky described what we would now call “spaces of constant negative curvature.” Such a space is curved, but in precisely the same way at every point; there is no difference between what’s happening at one point in the space and what’s happening anywhere else, just as had been the case for Euclid’s tabletop geometry.

Real geometries, as takes only a moment to visualize, can be a lot more complicated than that. Surfaces or solids can twist and turn in all sorts of ways. Gauss thought about how to deal with this problem, and came up with some techniques that could characterize a two-dimensional curved surface embedded in a three-dimensional Euclidean space. Which is pretty great, but falls far short of the full generality that mathematicians are known to crave.

Georg_Friedrich_Bernhard_Riemann.jpeg Fortunately Gauss had a brilliant and accomplished apprentice: his student Bernard Riemann. (Riemann was supposed to be studying theology, but he became entranced by one of Gauss’s lectures, and never looked back.) In 1853, Riemann was coming up for Habilitation, a German degree that is even higher than the Ph.D. He suggested a number of possible dissertation topics to his advisor Gauss, who (so the story goes) chose the one that Riemann thought was the most boring: the foundations of geometry. The next year, he presented his paper, “On the hypotheses which underlie geometry,” which laid out what we now call Riemannian geometry.

With this one paper on a subject he professed not to be all that interested in, Riemann (who also made incredible contributions to analysis and number theory) provided everything you need to understand the geometry of a space of arbitrary numbers of dimensions, with an arbitrary amount of curvature at any point in the space. It was as if Bolyai and Lobachevsky had invented the abacus, Gauss came up with the pocket calculator, and Riemann had turned around a built a powerful supercomputer.

Like many great works of mathematics, a lot of new superstructure had to be built up along the way. A subtle but brilliant part of Riemann’s work is that he didn’t start with a larger space (like the three-dimensional almost-Euclidean world around us) and imagine smaller spaces embedded with it. Rather, he considered the intrinsic geometry of a space, or how it would look “from the inside,” whether or not there was any larger space at all.

Next, Riemann needed a tool to handle a simple but frustrating fact of life: “curvature” is not a single number, but a way of characterizing many questions one could possibly ask about the geometry of a space. What you need, really, are tensors, which gather a set of numbers together in one elegant mathematical package. Tensor analysis as such didn’t really exist at the time, not being fully developed until 1890, but Riemann was able to use some bits and pieces of the theory that had been developed by Gauss.

Finally and most importantly, Riemann grasped that all the facts about the geometry of a space could be encoded in a simple quantity: the distance along any curve we might want to draw through the space. He showed how that distance could be written in terms of a special tensor, called the metric. You give me segment along a curve inside the space you’re interested in, the metric lets me calculate how long it is. This simple object, Riemann showed, could ultimately be used to answer any query you might have about the shape of a space — the length of curves, of course, but also the area of surfaces and volume of regions, the shortest-distance path between two fixed points, where you go if you keep marching “forward” in the space, the sum of the angles inside a triangle, and so on.

Unfortunately, the geometric information implied by the metric is only revealed when you follow how the metric changes along a curve or on some surface. What Riemann wanted was a single tensor that would tell you everything you needed to know about the curvature at each point in its own right, without having to consider curves or surfaces. So he showed how that could be done, by taking appropriate derivatives of the metric, giving us what we now call the Riemann curvature tensor. Here is the formula for it:


This isn’t the place to explain the whole thing, but I can recommend some spiffy lecture notes, including a very short version, or the longer and sexier textbook. From this he deduced several interesting features about curvature. For example, the intrinsic curvature of a one-dimensional space (a line or curve) is alway precisely zero. Its extrinsic curvature — how it is embedded in some larger space — can be complicated, but to a tiny one-dimensional being, all spaces have the same geometry. For two-dimensional spaces there is a single function that characterizes the curvature at each point; in three dimensions you need six numbers, in four you need twenty, and it goes up from there.

There were more developments in store for Riemannian geometry, of course, associated with names that are attached to various tensors and related symbols: Christoffel, Ricci, Levi-Civita, Cartan. But to a remarkable degree, when Albert Einstein needed the right mathematics to describe his new idea of dynamical spacetime, Riemann had bequeathed it to him in a plug-and-play form. Add the word “time” everywhere we’ve said “space,” introduce some annoying minus signs because time and space really aren’t precisely equivalent, and otherwise the geometry that Riemann invented is the same we use today to describe how the universe works.

Riemann died of tuberculosis before he reached the age of forty. He didn’t do bad for such a young guy; you know you’ve made it when you not only have a Wikipedia page for yourself, but a separate (long) Wikipedia page for the list of things named after you. We can all be thankful that Riemann’s genius allowed him to grasp the tricky geometry of curved spaces several decades before Einstein would put it to use in the most beautiful physical theory ever invented.

Posted in Math, Science | 15 Comments

Long-Term Forecast

This xkcd cartoon is undeniably awesome as-is, but the cosmologist in me couldn’t resist adding one more row at the bottom.


Looks like the forecast calls for Boltzmann Brains! I guess Hilbert space is finite-dimensional after all.

Posted in Humor | 12 Comments

The Universe Never Expands Faster Than the Speed of Light

Breaking my radio silence here to get a little nitpick off my chest: the claim that during inflation, the universe “expanded faster than the speed of light.” It’s extraordinarily common, if utterly and hopelessly incorrect. (I just noticed it in this otherwise generally excellent post by Fraser Cain.) A Google search for “inflation superluminal expansion” reveals over 100,000 hits, although happily a few of the first ones are brave attempts to squelch the misconception. I can recommend this nice article by Tamara Davis and Charlie Lineweaver, which tries to address this and several other cosmological misconceptions.

This isn’t, by the way, one of those misconceptions that rattles around the popular-explanation sphere, while experts sit back silently and roll their eyes. Experts get this one wrong all the time. “Inflation was a period of superluminal expansion” is repeated, for example, in these texts by by Tai-Peng Cheng, by Joel Primack, and by Lawrence Krauss, all of whom should certainly know better.

The great thing about the superluminal-expansion misconception is that it’s actually a mangle of several different problems, which sadly don’t cancel out to give you the right answer.

1.The expansion of the universe doesn’t have a “speed.” Really the discussion should begin and end right there. Comparing the expansion rate of the universe to the speed of light is like comparing the height of a building to your weight. You’re not doing good scientific explanation; you’ve had too much to drink and should just go home.The expansion of the universe is quantified by the Hubble constant, which is typically quoted in crazy units of kilometers per second per megaparsec. That’s (distance divided by time) divided by distance, or simply 1/time. Speed, meanwhile, is measured in distance/time. Not the same units! Comparing the two concepts is crazy.

Admittedly, you can construct a quantity with units of velocity from the Hubble constant, using Hubble’s law, v = Hd (the apparent velocity of a galaxy is given by the Hubble constant times its distance). Individual galaxies are indeed associated with recession velocities. But different galaxies, manifestly, have different velocities. The idea of even talking about “the expansion velocity of the universe” is bizarre and never should have been entertained in the first place.

2. There is no well-defined notion of “the velocity of distant objects” in general relativity. There is a rule, valid both in special relativity and general relativity, that says two objects cannot pass by each other with relative velocities faster than the speed of light. In special relativity, where spacetime is a fixed, flat, Minkowskian geometry, we can pick a global reference frame and extend that rule to distant objects. In general relativity, we just can’t. There is simply no such thing as the “velocity” between two objects that aren’t located in the same place. If you tried to measure such a velocity, you would have to parallel transport the motion of one object to the location of the other one, and your answer would completely depend on the path that you took to do that. So there can’t be any rule that says that velocity can’t be greater than the speed of light. Period, full stop, end of story.

Except it’s not quite the end of the story, since under certain special circumstances it’s possible to define quantities that are kind-of sort-of like a velocity between distant objects. Cosmology, where we model the universe as having a preferred reference frame defined by the matter filling space, is one such circumstance. When galaxies are not too far away, we can measure their cosmological redshifts, pretend that it’s a Doppler shift, and work backwards to define an “apparent velocity.” Good for you, cosmologists! But that number you’ve defined shouldn’t be confused with the actual relative velocity between two objects passing by each other. In particular, there’s no reason whatsoever that this apparent velocity can’t be greater than the speed of light.

Sometimes this idea is mangled into something like “the rule against superluminal velocities doesn’t refer to the expansion of space.” A good try, certainly well-intentioned, but the problem is deeper than that. The rule against superluminal velocities only refers to relative velocities between two objects passing right by each other.

3. There is nothing special about the expansion rate during inflation. If you want to stubbornly insist on treating the cosmological apparent velocity as a real velocity, just so you can then go and confuse people by saying that sometimes that velocity can be greater than the speed of light, I can’t stop you. But it can be — and is! — greater than the speed of light at any time in the history of the universe, not just during inflation. There are galaxies sufficiently distant that their apparent recession velocities today are greater than the speed of light. To give people the impression that what’s special about inflation is that the universe is expanding faster than light is a crime against comprehension and good taste.

What’s special about inflation is that the universe is accelerating. During inflation (as well as today, since dark energy has taken over), the scale factor, which characterizes the relative distance between comoving points in space, is increasing faster and faster, rather than increasing but at a gradually diminishing rate. As a result, if you looked at one particular galaxy over time, its apparent recession velocity would be increasing. That’s a big deal, with all sorts of interesting and important cosmological ramifications. And it’s not that hard to explain.

But it’s not superluminal expansion. If you’re sitting at a stoplight in your Tesla, kick it into insane mode, and accelerate to 60 mph in 3.5 seconds, you won’t get a ticket for speeding, as long as the speed limit itself is 60 mph or greater. You can still get a ticket — there’s such a thing as reckless driving, after all — but if you’re hauled before the traffic judge on a count of speeding, you should be able to get off scot-free.

Many “misconceptions” in physics stem from an honest attempt to explain technical concepts in natural language, and I try to be very forgiving about those. This one, I believe, isn’t like that; it’s just wrongity-wrong wrong. The only good quality of the phrase “inflation is a period of superluminal expansion” is that it’s short. It conveys the illusion of understanding, but that can be just as bad as straightforward misunderstanding. Every time it is repeated, people’s appreciation of how the universe works gets a little bit worse. We should be able to do better.

Posted in Science | 140 Comments

Core Theory T-Shirts

Way back when, for purposes of giving a talk, I made a figure that displayed the world of everyday experience in one equation. The label reflects the fact that the laws of physics underlying everyday life are completely understood.

So now there are T-shirts. (See below to purchase your own.)

Core Theory T-shirt

It’s a good equation, representing the Feynman path-integral formulation of an amplitude for going from one field configuration to another one, in the effective field theory consisting of Einstein’s general theory of relativity plus the Standard Model of particle physics. It even made it onto an extremely cool guitar.

I’m not quite up to doing a comprehensive post explaining every term in detail, but here’s the general idea. Our everyday world is well-described by an effective field theory. So the fundamental stuff of the world is a set of quantum fields that interact with each other. Feynman figured out that you could calculate the transition between two configurations of such fields by integrating over every possible trajectory between them — that’s what this equation represents. The thing being integrated is the exponential of the action for this theory — as mentioned, general relativity plus the Standard Model. The GR part integrates over the metric, which characterizes the geometry of spacetime; the matter fields are a bunch of fermions, the quarks and leptons; the non-gravitational forces are gauge fields (photon, gluons, W and Z bosons); and of course the Higgs field breaks symmetry and gives mass to those fermions that deserve it. If none of that makes sense — maybe I’ll do it more carefully some other time.

Gravity is usually thought to be the odd force out when it comes to quantum mechanics, but that’s only if you really want a description of gravity that is valid everywhere, even at (for example) the Big Bang. But if you only want a theory that makes sense when gravity is weak, like here on Earth, there’s no problem at all. The little notation k < Λ at the bottom of the integral indicates that we only integrate over low-frequency (long-wavelength, low-energy) vibrations in the relevant fields. (That's what gives away that this is an "effective" theory.) In that case there's no trouble including gravity. The fact that gravity is readily included in the EFT of everyday life has long been emphasized by Frank Wilczek. As discussed in his latest book, A Beautiful Question, he therefore advocates lumping GR together with the Standard Model and calling it The Core Theory.

I couldn’t agree more, so I adopted the same nomenclature for my own upcoming book, The Big Picture. There’s a whole chapter (more, really) in there about the Core Theory. After finishing those chapters, I rewarded myself by doing something I’ve been meaning to do for a long time — put the equation on a T-shirt, which you see above.

I’ve had T-shirts made before, with pretty grim results as far as quality is concerned. I knew this one would be especially tricky, what with all those tiny symbols. But I tried out Design-A-Shirt, and the result seems pretty impressively good.

So I’m happy to let anyone who might be interested go ahead and purchase shirts for themselves and their loved ones. Here are the links for light/dark and men’s/women’s versions. I don’t actually make any money off of this — you’re just buying a T-shirt from Design-A-Shirt. They’re a little pricey, but that’s what you get for the quality. I believe you can even edit colors and all that — feel free to give it a whirl and report back with your experiences.

Posted in Big Picture, Miscellany | 29 Comments