The Hidden Reality_ Parallel Universes and the Deep Laws of the Cosmos - Brian Greene [213]
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 10–5 or so times the Planck energy scale, where the latter is about 1019 times the energy equivalent of the mass 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 10–5 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 10–5 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 10–28 centimeters. The total mass-energy carried by such an inflaton field in a volume that is 10–26 centimeters on a side is thus: 10–5 grams/(10–28 centimeters)3 × (10–26 centimeters)3, which is about 10 grams. Readers of The Fabric of the Cosmos may recall that there I used a slightly different value. The difference came from the assumption that the energy scale of the inflaton was slightly higher.
4. Hans Moravec, Robot: Mere Machine to Transcendent Mind (New York: Oxford University Press, 2000). See also Ray Kurzweil, The Singularity Is Near: When Humans Transcend Biology (New York: Penguin, 2006).
5. See, for example, Robin Hanson, “How to Live in a Simulation,” Journal of Evolution and Technology 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.
7. Philosopher David Lewis developed a similar idea through what he called Modal Realism. See. On the Plurality of Worlds (Malden, Mass.: 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 Pi in the Sky (New York: Little, Brown, 1992).
9. As explained in 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.
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.
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.