ONCE upon a time, there lived a man who was fascinated by the phenomenon of gravity. In his mind he imagined experiments in rocket ships and elevators, eventually concluding that gravity isn’t a conventional “force” at all — it’s a manifestation of the curvature of spacetime. He threw himself into the study of differential geometry, the abstruse mathematics of arbitrarily curved manifolds. At the end of his investigations he had a new way of thinking about space and time, culminating in a marvelous equation that quantified how gravity responds to matter and energy in the universe.
Not being one to rest on his laurels, this man worked out a number of consequences of his new theory. One was that changes in gravity didn’t spread instantly throughout the universe; they traveled at the speed of light, in the form of gravitational waves. In later years he would change his mind about this prediction, only to later change it back. Eventually more and more scientists became convinced that this prediction was valid, and worth testing. They launched a spectacularly ambitious program to build a technological marvel of an observatory that would be sensitive to the faint traces left by a passing gravitational wave. Eventually, a century after the prediction was made — a press conference was called.
Chances are that everyone reading this blog post has heard that LIGO, the Laser Interferometric Gravitational-Wave Observatory, officially announced the first direct detection of gravitational waves. Two black holes, caught in a close orbit, gradually lost energy and spiraled toward each other as they emitted gravitational waves, which zipped through space at the speed of light before eventually being detected by our observatories here on Earth. Plenty of other places will give you details on this specific discovery, or tutorials on the nature of gravitational waves, including in user-friendly comic/video form.
What I want to do here is to make sure, in case there was any danger, that nobody loses sight of the extraordinary magnitude of what has been accomplished here. We’ve become a bit blasé about such things: physics makes a prediction, it comes true, yay. But we shouldn’t take it for granted; successes like this reveal something profound about the core nature of reality.
Some guy scribbles down some symbols in an esoteric mixture of Latin, Greek, and mathematical notation. Scribbles originating in his tiny, squishy human brain. (Here are what some of those those scribbles look like, in my own incredibly sloppy handwriting.) Other people (notably Rainer Weiss, Ronald Drever, and Kip Thorne), on the basis of taking those scribbles extremely seriously, launch a plan to spend hundreds of millions of dollars over the course of decades. They concoct an audacious scheme to shoot laser beams at mirrors to look for modulated displacements of less than a millionth of a billionth of a centimeter — smaller than the diameter of an atomic nucleus. Meanwhile other people looked at the sky and tried to figure out what kind of signals they might be able to see, for example from the death spiral of black holes a billion light-years away. You know, black holes: universal regions of death where, again according to elaborate theoretical calculations, the curvature of spacetime has become so pronounced that anything entering can never possibly escape. And still other people built the lasers and the mirrors and the kilometers-long evacuated tubes and the interferometers and the electronics and the hydraulic actuators and so much more, all because they believed in those equations. And then they ran LIGO (and other related observatories) for several years, then took it apart and upgraded to Advanced LIGO, finally reaching a sensitivity where you would expect to see real gravitational waves if all that fancy theorizing was on the right track. Continue reading →
“Understanding quantum gravity” is on every physicist’s short list of Big Issues we would all like to know more about. If there’s been any lesson from last half-century of serious work on this problem, it’s that the answer is likely to be something more subtle than just “take classical general relativity and quantize it.” Quantum gravity doesn’t seem to be an ordinary quantum field theory.
In that context, it makes sense to take many different approaches and see what shakes out. Alongside old stand-bys such as string theory and loop quantum gravity, there are less head-on approaches that try to understand how quantum gravity can really be so weird, without proposing a specific and complete model of what it might be.
Grant Remmen, a graduate student here at Caltech, has been working with me recently on one such approach, dubbed entropic gravity. We just submitted a paper entitled “What Is the Entropy in Entropic Gravity?” Grant was kind enough to write up this guest blog post to explain what we’re talking about.
Meanwhile, if you’re near Pasadena, Grant and his brother Cole have written a musical, Boldly Go!, which will be performed at Caltech in a few weeks. You won’t want to miss it!
One of the most exciting developments in theoretical physics in the past few years is the growing understanding of the connections between gravity, thermodynamics, and quantum entanglement. Famously, a complete quantum mechanical theory of gravitation is difficult to construct. However, one of the aspects that we are now coming to understand about quantum gravity is that in the final theory, gravitation and even spacetime itself will be closely related to, and maybe even emergent from, the mysterious quantum mechanical property known as entanglement.
This all started several decades ago, when Hawking and others realized that black holes behave with many of the same aspects as garden-variety thermodynamic systems, including temperature, entropy, etc. Most importantly, the black hole’s entropy is equal to its area [divided by (4 times Newton’s constant)]. Attempts to understand the origin of black hole entropy, along with key developments in string theory, led to the formulation of the holographic principle – see, for example, the celebrated AdS/CFT correspondence – in which quantum gravitational physics in some spacetime is found to be completely described by some special non-gravitational physics on the boundary of the spacetime. In a nutshell, one gets a gravitational universe as a “hologram” of a non-gravitational universe.
If gravity can emerge from, or be equivalent to, a set of physical laws without gravity, then something special about that non-gravitational physics has to make it happen. Physicists have now found that that special something is quantum entanglement: the special correlations among quantum mechanical particles that defies classical description. As a result, physicists are very interested in how to get the dynamics describing how spacetime is shaped and moves – Einstein’s equation of general relativity – from various properties of entanglement. In particular, it’s been suggested that the equations of gravity can be shown to come from some notion of entropy. As our universe is quantum mechanical, we should think about the entanglement entropy, a measure of the degree of correlation of quantum subsystems, which for thermal states matches the familiar thermodynamic notion of entropy.
The general idea is as follows: Inspired by black hole thermodynamics, suppose that there’s some more general notion, in which you choose some region of spacetime, compute its area, and find that when its area changes this is associated with a change in entropy. (I’ve been vague here as to what is meant by a “change” in the area and what system we’re computing the area of – this will be clarified soon!) Next, you somehow relate the entropy to an energy (e.g., using thermodynamic relations). Finally, you write the change in area in terms of a change in the spacetime curvature, using differential geometry. Putting all the pieces together, you get a relation between an energy and the curvature of spacetime, which if everything goes well, gives you nothing more or less than Einstein’s equation! This program can be broadly described as entropic gravity and the idea has appeared in numerous forms. With the plethora of entropic gravity theories out there, we realized that there was a need to investigate what categories they fall into and whether their assumptions are justified – this is what we’ve done in our recent work.
In particular, there are two types of theories in which gravity is related to (entanglement) entropy, which we’ve called holographic gravity and thermodynamic gravity in our paper. The difference between the two is in what system you’re considering, how you define the area, and what you mean by a change in that area.
In holographic gravity, you consider a region and define the area as that of its boundary, then consider various alternate configurations and histories of the matter in that region to see how the area would be different. Recent work in AdS/CFT, in which Einstein’s equation at linear order is equivalent to something called the “entanglement first law”, falls into the holographic gravity category. This idea has been extended to apply outside of AdS/CFT by Jacobson (2015). Crucially, Jacobson’s idea is to apply holographic mathematical technology to arbitrary quantum field theories in the bulk of spacetime (rather than specializing to conformal field theories – special physical models – on the boundary as in AdS/CFT) and thereby derive Einstein’s equation. However, in this work, Jacobson needed to make various assumptions about the entanglement structure of quantum field theories. In our paper, we showed how to justify many of those assumptions, applying recent results derived in quantum field theory (for experts, the form of the modular Hamiltonian and vacuum-subtracted entanglement entropy on null surfaces for general quantum field theories). Thus, we are able to show that the holographic gravity approach actually seems to work!
On the other hand, thermodynamic gravity is of a different character. Though it appears in various forms in the literature, we focus on the famous work of Jacobson (1995). In thermodynamic gravity, you don’t consider changing the entire spacetime configuration. Instead, you imagine a bundle of light rays – a lightsheet – in a particular dynamical spacetime background. As the light rays travel along – as you move down the lightsheet – the rays can be focused by curvature of the spacetime. Now, if the bundle of light rays started with a particular cross-sectional area, you’ll find a different area later on. In thermodynamic gravity, this is the change in area that goes into the derivation of Einstein’s equation. Next, one assumes that this change in area is equivalent to an entropy – in the usual black hole way with a factor of 1/(4 times Newton’s constant) – and that this entropy can be interpreted thermodynamically in terms of an energy flow through the lightsheet. The entropy vanishes from the derivation and the Einstein equation almost immediately appears as a thermodynamic equation of state. What we realized, however, is that what the entropy is actually the entropy of was ambiguous in thermodynamic gravity. Surprisingly, we found that there doesn’t seem to be a consistent definition of the entropy in thermodynamic gravity – applying quantum field theory results for the energy and entanglement entropy, we found that thermodynamic gravity could not simultaneously reproduce the correct constant in the Einstein equation and in the entropy/area relation for black holes.
So when all is said and done, we’ve found that holographic gravity, but not thermodynamic gravity, is on the right track. To answer our own question in the title of the paper, we found – in admittedly somewhat technical language – that the vacuum-subtracted von Neumann entropy evaluated on the null boundary of small causal diamonds gives a consistent formulation of holographic gravity. The future looks exciting for finding the connections between gravity and entanglement!
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.
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.
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.
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.
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.
THE BIG PICTURE: ON THE ORIGINS OF LIFE, MEANING, AND THE UNIVERSE ITSELF
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!
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.
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.
