1. The number I quoted, 10 The number I quoted, 1055 grams, accounts for the contents of the observable universe today, but at ever-earlier times, the temperature of these const.i.tuents would be larger and so they would contain higher energy. The number 10 grams, accounts for the contents of the observable universe today, but at ever-earlier times, the temperature of these const.i.tuents would be larger and so they would contain higher energy. The number 1065 grams is a better estimate of what you'd need to gather into a tiny speck to recapitulate the evolution of our universe from when it was roughly one second old. grams is a better estimate of what you'd need to gather into a tiny speck to recapitulate the evolution of our universe from when it was roughly one second old.

2. You might think that because your speed is constrained to be less than the speed of light, your kinetic energy will also be limited. But that's not the case. As your speed gets ever closer to that of light, your energy grows ever larger; according to special relativity, it has no bounds. Mathematically, the formula for your energy is: You might think that because your speed is constrained to be less than the speed of light, your kinetic energy will also be limited. But that's not the case. As your speed gets ever closer to that of light, your energy grows ever larger; according to special relativity, it has no bounds. Mathematically, the formula for your energy is:[image] , where , where c c is the speed of light and is the speed of light and v v is your speed. As you can see, as is your speed. As you can see, as v v approaches approaches c, E c, E grows arbitrarily large. Note too that the discussion is from the perspective of someone watching you fall, say someone stationary on the surface of the earth. From your perspective, while you are in free fall, you are stationary and all the surrounding matter is acquiring increasing speed. grows arbitrarily large. Note too that the discussion is from the perspective of someone watching you fall, say someone stationary on the surface of the earth. From your perspective, while you are in free fall, you are stationary and all the surrounding matter is acquiring increasing speed.

3. With our current level of understanding, there is significant flexibility in such estimates. The number "10 grams" comes from the following consideration: the energy scale at which inflation takes place is thought to be about 10 With our current level of understanding, there is significant flexibility in such estimates. The number "10 grams" comes from the following consideration: the energy scale at which inflation takes place is thought to be about 105 or so times the Planck energy scale, where the latter is about 10 or so times the Planck energy scale, where the latter is about 1019 times the energy equivalent of the ma.s.s of a proton. (If inflation happened at a higher energy scale, models suggest that evidence for gravitational waves produced in the early universe should already have been seen.) In more conventional units, the Planck scale is about 10 times the energy equivalent of the ma.s.s of a proton. (If inflation happened at a higher energy scale, models suggest that evidence for gravitational waves produced in the early universe should already have been seen.) In more conventional units, the Planck scale is about 105 grams (small by everyday standards, but enormous by the scales of elementary particle physics, where such energies would be carried by individual particles). The energy density of an inflaton field would therefore have been about 10 grams (small by everyday standards, but enormous by the scales of elementary particle physics, where such energies would be carried by individual particles). The energy density of an inflaton field would therefore have been about 105 grams packed in every cubic volume whose linear dimension is set by roughly 10 grams packed in every cubic volume whose linear dimension is set by roughly 105 times the Planck length (recall, from quantum uncertainty, that energies and lengths scale inversely proportional to each other), which is about 10 times the Planck length (recall, from quantum uncertainty, that energies and lengths scale inversely proportional to each other), which is about 1028 centimeters. The total ma.s.s-energy carried by such an inflaton field in a volume that is 10 centimeters. The total ma.s.s-energy carried by such an inflaton field in a volume that is 1026 centimeters on a side is thus: 10 centimeters on a side is thus: 105 grams/(10 grams/(1028 centimeters) centimeters)3 (10 (1026 centimeters) centimeters)3, which is about 10 grams. Readers of The Fabric of the Cosmos The Fabric of the Cosmos may recall that there I used a slightly different value. The difference came from the a.s.sumption that the energy scale of the inflaton was slightly higher. may recall that there I used a slightly different value. The difference came from the a.s.sumption that the energy scale of the inflaton was slightly higher.

4. Hans Moravec, Hans Moravec, Robot: Mere Machine to Transcendent Mind Robot: Mere Machine to Transcendent Mind (New York: Oxford University Press, 2000). See also Ray Kurzweil, (New York: Oxford University Press, 2000). See also Ray Kurzweil, The Singularity Is Near: When Humans Transcend Biology The Singularity Is Near: When Humans Transcend Biology (New York: Penguin, 2006). (New York: Penguin, 2006).

5. See, for example, Robin Hanson, "How to Live in a Simulation," See, for example, Robin Hanson, "How to Live in a Simulation," Journal of Evolution and Technology Journal of Evolution and Technology 7, no. 1 (2001). 7, no. 1 (2001).

6. The Church-Turing thesis argues that any computer of the so-called universal Turing type can simulate the actions of another, and so it's perfectly reasonable for a computer that's within the simulation-and hence is itself simulated by the parent computer running the whole simulated world-to perform particular tasks equivalent to those undertaken by the parent computer. The Church-Turing thesis argues that any computer of the so-called universal Turing type can simulate the actions of another, and so it's perfectly reasonable for a computer that's within the simulation-and hence is itself simulated by the parent computer running the whole simulated world-to perform particular tasks equivalent to those undertaken by the parent computer.

7. Philosopher David Lewis developed a similar idea through what he called Modal Realism. See. Philosopher David Lewis developed a similar idea through what he called Modal Realism. See. On the Plurality of Worlds On the Plurality of Worlds (Malden, Ma.s.s.: Wiley-Blackwell, 2001). However, Lewis's motivation in introducing all possible universes differs from Nozick's. Lewis wanted a context where, for example, counterfactual statements (such as, "If Hitler had won the war, the world today would be very different") would be instantiated. (Malden, Ma.s.s.: Wiley-Blackwell, 2001). However, Lewis's motivation in introducing all possible universes differs from Nozick's. Lewis wanted a context where, for example, counterfactual statements (such as, "If Hitler had won the war, the world today would be very different") would be instantiated.

8. John Barrow has made a similar point in John Barrow has made a similar point in Pi in the Sky Pi in the Sky (New York: Little, Brown, 1992). (New York: Little, Brown, 1992).

9. As explained in As explained in endnote 10 endnote 10 of Chapter 7, the size of this infinity exceeds that of the infinite collection of whole numbers 1, 2, 3,...and so on. of Chapter 7, the size of this infinity exceeds that of the infinite collection of whole numbers 1, 2, 3,...and so on.

10. This is a variation on the famous Barber of Seville paradox, in which a barber shaves all those who don't shave themselves. The question then is: Who shaves the barber? The barber is usually stipulated to be male, to avoid the easy answer-the barber is a woman and so doesn't need to shave. This is a variation on the famous Barber of Seville paradox, in which a barber shaves all those who don't shave themselves. The question then is: Who shaves the barber? The barber is usually stipulated to be male, to avoid the easy answer-the barber is a woman and so doesn't need to shave.

11. Schmidhuber notes that an efficient strategy would be to have the computer evolve each simulated universe forward in time in a "dovetailed" manner: the first universe would be updated on every other time-step of the computer, the second universe would be updated on every other of the remaining time-steps, the third universe would be updated on every other time-step not already devoted to the first two universes, and so on. In due course, every computable universe would be evolved forward by an arbitrarily large number of time-steps. Schmidhuber notes that an efficient strategy would be to have the computer evolve each simulated universe forward in time in a "dovetailed" manner: the first universe would be updated on every other time-step of the computer, the second universe would be updated on every other of the remaining time-steps, the third universe would be updated on every other time-step not already devoted to the first two universes, and so on. In due course, every computable universe would be evolved forward by an arbitrarily large number of time-steps.

12. A more refined discussion of computable and noncomputable functions would also include A more refined discussion of computable and noncomputable functions would also include limit computable functions limit computable functions. These are functions for which there is a finite algorithm that evaluates them to ever greater precision. For instance, such is the case for producing the digits of : : a computer can produce each successive digit of a computer can produce each successive digit of , even though it will never reach the end of the computation. So, while is strictly speaking noncomputable, it is limit computable. Most real numbers, however, are not like is strictly speaking noncomputable, it is limit computable. Most real numbers, however, are not like . They are not just noncomputable, they are also not limit computable.

When we consider "successful" simulations, we should include those based on limit computable functions. In principle, a convincing reality could be generated by the partial output of a computer evaluating limit computable functions.

For the laws of physics to be computable, or even limit computable, the traditional reliance on real numbers would have to be abandoned. This would apply not just to s.p.a.ce and time, usually described using coordinates whose values can range over the real numbers, but also for all other mathematical ingredients the laws use. The strength of an electromagnetic field, for example, could not vary over real numbers, but only over a discrete set of values. Similarly for the probability that an electron is here or there. Schmidhuber has emphasized that all calculations that physicists have ever carried out have involved the manipulation of discrete symbols (written on paper, on a blackboard, or input to a computer). And so, even though this body of scientific work has always been viewed as invoking the real numbers, in practice it doesn't. Similarly for all quant.i.ties ever measured. No device has infinite accuracy and so our measurements always involve discrete numerical outputs. In that sense, all the successes of physics can be read as successes for a digital paradigm. Perhaps, then, the true laws themselves are, in fact, computable (or limit computable).

There are many different perspectives on the possibility of "digital physics." See, for example, Stephen Wolfram's A New Kind of Science A New Kind of Science (Champaign, Ill.: Wolfram Media, 2002) and Seth Lloyd's (Champaign, Ill.: Wolfram Media, 2002) and Seth Lloyd's Programming the Universe Programming the Universe (New York: Alfred A. Knopf, 2006). The mathematician Roger Penrose believes that the human mind is based on noncomputable processes and hence the universe we inhabit must involve noncomputable mathematical functions. From this perspective, our universe does not fall into the digital paradigm. See, for instance, (New York: Alfred A. Knopf, 2006). The mathematician Roger Penrose believes that the human mind is based on noncomputable processes and hence the universe we inhabit must involve noncomputable mathematical functions. From this perspective, our universe does not fall into the digital paradigm. See, for instance, The Emperor's New Mind The Emperor's New Mind (New York: Oxford University Press, 1989) and (New York: Oxford University Press, 1989) and Shadows of the Mind Shadows of the Mind (New York: Oxford University Press, 1994). (New York: Oxford University Press, 1994).

Chapter 11: The Limits of Inquiry.

1. Steven Weinberg, Steven Weinberg, The First Three Minutes The First Three Minutes (New York: Basic Books, 1973), p. 131. (New York: Basic Books, 1973), p. 131.

Suggestions for Further Reading The subject of parallel universes draws on a broad range of scientific material. There is a growing literature that focuses on various aspects of such material, mostly intended for the nonexpert, but often well-suited for those with more background. In addition to the references called out in the notes, here is a collection of books, from the many wonderful ones that have been written, through which the reader can continue exploring topics discussed in The Hidden Reality The Hidden Reality.

Albert, David. Quantum Mechanics and Experience Quantum Mechanics and Experience. Cambridge, Ma.s.s.: Harvard University Press, 1994.

Alexander, H. G. The Leibniz-Clarke Correspondence The Leibniz-Clarke Correspondence. Manchester: Manchester University Press, 1956.

Barrow, John. Pi in the Sky Pi in the Sky. Boston: Little, Brown, 1992.

_____. The World Within the World The World Within the World. Oxford: Clarendon Press, 1988.

Barrow, John, and Frank Tipler. The Anthropic Cosmological Principle The Anthropic Cosmological Principle. Oxford: Oxford University Press, 1986.

Bartusiak, Marcia. The Day We Found the Universe The Day We Found the Universe. New York: Vintage, 2010.

Bell, John. Speakable and Unspeakable in Quantum Mechanics Speakable and Unspeakable in Quantum Mechanics. Cambridge, Eng.: Cambridge University Press, 1993.

Bronowski, Jacob. The Ascent of Man The Ascent of Man. Boston: Little, Brown, 1973.

Byrne, Peter. The Many Worlds of Hugh Everett III The Many Worlds of Hugh Everett III. New York: Oxford University Press, 2010.

Callender, Craig, and Nick Huggett. Physics Meets Philosophy at the Planck Scale Physics Meets Philosophy at the Planck Scale. Cambridge, Eng.: Cambridge University Press, 2001.

Carroll, Sean. From Eternity to Here From Eternity to Here. New York: Dutton, 2010.

Clark, Ronald. Einstein: The Life and Times Einstein: The Life and Times. New York: Avon, 1984.

Cole, K. C. The Hole in the Universe The Hole in the Universe. New York: Harcourt, 2001.

Crease, Robert P., and Charles C. Mann. The Second Creation The Second Creation. New Brunswick, N.J.: Rutgers University Press, 1996.

Davies, Paul. Cosmic Jackpot Cosmic Jackpot. Boston: Houghton Mifflin, 2007.

Deutsch, David. The Fabric of Reality The Fabric of Reality. New York: Allen Lane, 1997.

DeWitt, Bryce, and Neill Graham, eds. The Many-Worlds Interpretation of Quantum Mechanics The Many-Worlds Interpretation of Quantum Mechanics. Princeton: Princeton University Press, 1973.

Einstein, Albert. The Meaning of Relativity The Meaning of Relativity. Princeton: Princeton University Press, 1988.

_____. Relativity Relativity. New York: Crown, 1961.

Ferris, Timothy. Coming of Age in the Milky Way Coming of Age in the Milky Way. New York: Anchor, 1989.

_____. The Whole Shebang The Whole Shebang. New York: Simon & Schuster, 1997.

Feynman, Richard. The Character of Physical Law The Character of Physical Law. Cambridge, Ma.s.s.: MIT Press, 1995.

_____. QED QED. Princeton: Princeton University Press, 1986.

Gamow, George. Mr. Tompkins in Paperback Mr. Tompkins in Paperback. Cambridge, Eng.: Cambridge University Press, 1993.

Gleick, James. Isaac Newton Isaac Newton. New York: Pantheon, 2003.

Gribbin, John. In Search of the Multiverse In Search of the Multiverse. Hoboken, N.J.: Wiley, 2010.

_____. Schrodinger's Kittens and the Search for Reality Schrodinger's Kittens and the Search for Reality. Boston: Little, Brown, 1995.

Guth, Alan H. The Inflationary Universe The Inflationary Universe. Reading, Ma.s.s.: Addison-Wesley, 1997.

Hawking, Stephen. A Brief History of Time A Brief History of Time. New York: Bantam Books, 1988.

_____. The Universe in a Nutsh.e.l.l The Universe in a Nutsh.e.l.l. New York: Bantam Books, 2001.

Isaacson, Walter. Einstein Einstein. New York: Simon & Schuster, 2007.

Kaku, Michio. Parallel Worlds Parallel Worlds. New York: Anchor, 2006.

Kirschner, Robert. The Extravagant Universe The Extravagant Universe. Princeton: Princeton University Press, 2002.

Krauss, Lawrence. Quintessence Quintessence. New York: Perseus, 2000.

Kurzweil, Ray. The Age of Spiritual Machines The Age of Spiritual Machines. New York: Viking, 1999.

_____. The Singularity Is Near The Singularity Is Near. New York: Viking, 2005.

Lederman, Leon, and Christopher Hill. Symmetry and the Beautiful Universe Symmetry and the Beautiful Universe. Amherst, N.Y.: Prometheus Books, 2004.

Livio, Mario. The Accelerating Universe The Accelerating Universe. New York: Wiley, 2000.

Lloyd, Seth. Programming the Universe Programming the Universe. New York: Knopf, 2006.

Moravec, Hans. Robot Robot. New York: Oxford University Press, 1998.

Pais, Abraham. Subtle Is the Lord Subtle Is the Lord. Oxford: Oxford University Press, 1982.

Penrose, Roger. The Emperor's New Mind The Emperor's New Mind. New York: Oxford University Press, 1989.

_____. Shadows of the Mind Shadows of the Mind. New York: Oxford University Press, 1994.

Randall, Lisa. Warped Pa.s.sages Warped Pa.s.sages. New York: Ecco, 2005.

Rees, Martin. Before the Beginning Before the Beginning. Reading, Ma.s.s.: Addison-Wesley, 1997.

_____. Just Six Numbers Just Six Numbers. New York: Basic Books, 2001.

Schrodinger, Erwin. What Is Life? What Is Life? Cambridge, Eng.: Canto, 2000. Cambridge, Eng.: Canto, 2000.

Siegfried, Tom. The Bit and the Pendulum The Bit and the Pendulum. New York: John Wiley & Sons, 2000.

Singh, Simon. Big Bang Big Bang. New York: Fourth Estate, 2004.

Susskind, Leonard. The Black Hole War The Black Hole War. New York: Little, Brown, 2008.

_____. The Cosmic Landscape The Cosmic Landscape. New York: Little, Brown, 2005.

Thorne, Kip. Black Holes and Time Warps Black Holes and Time Warps. New York: W. W. Norton, 1994.

Tyson, Neil deGra.s.se. Death by Black Hole Death by Black Hole. New York: W. W. Norton, 2007.

Vilenkin, Alexander. Many Worlds in One Many Worlds in One. New York: Hill and w.a.n.g, 2006.

von Weizsacker, Carl Friedrich. The Unity of Nature The Unity of Nature. New York: Farrar, Straus and Giroux, 1980.

Weinberg, Steven. Dreams of a Final Theory Dreams of a Final Theory. New York: Pantheon, 1992.

_____. The First Three Minutes The First Three Minutes. New York: Basic Books, 1993.

Wheeler, John. A Journey into Gravity and s.p.a.cetime A Journey into Gravity and s.p.a.cetime. New York: Scientific American Library, 1990.

Wilzcek, Frank. The Lightness of Being The Lightness of Being. New York: Basic Books, 2008.

Wilzczek, Frank, and Betsy Devine. Longing for the Harmonies Longing for the Harmonies. New York: W. W. Norton, 1988.

Yau, Shing-Tung, and Steve Nadis. The Shape of Inner s.p.a.ce The Shape of Inner s.p.a.ce. New York: Basic Books, 2010.

ABOUT THE AUTHOR.

Brian Greene received his undergraduate degree from Harvard University and his doctorate from Oxford University, where he was a Rhodes Scholar. He joined the physics faculty at Cornell University in 1990, was appointed to a full professorship in 1995, and in 1996 joined Columbia University, where he is professor of physics and mathematics. He has lectured at both a general and a technical level in more than twenty-five countries and is widely regarded for a number of groundbreaking discoveries in superstring theory. His first book, The Elegant Universe The Elegant Universe, has sold more than 1 million copies worldwide and was a finalist for the Pulitzer Prize. His second book, The Fabric of the Cosmos The Fabric of the Cosmos, spent twenty-five weeks on The New York Times The New York Times best-seller list. He lives in Andes, New York, and New York City. best-seller list. He lives in Andes, New York, and New York City.

ALSO BY BRIAN GREENE.

Icarus at the Edge of Time

The Fabric of the Cosmos

The Elegant Universe