by Brian Greene
Chapter 9: Black Holes and Holograms
1. Einstein undertook calculations within general relativity to prove mathematically that Schwarzschild’s extreme configurations—what we would now call a black hole—could not exist. The mathematics underlying his calculations was invariably correct. But he made additional assumptions that, given the intense folding of space and time that would be caused by a black hole, turn out to be too restrictive; in essence, the assumption left out the possibility of matter imploding. The assumptions meant that Einstein’s mathematical formulation did not have the latitude to reveal black holes as possibly real. But this was an artifact of Einstein’s approach, not an indication of whether black holes might actually form. The modern understanding makes clear that general relativity allows for black hole solutions.
2. Once a system reaches a maximal entropy configuration (such as steam, at a fixed temperature, that is uniformly spread throughout a vat), it will have exhausted its capacity for yet further entropic increase. So, the more precise statement is that entropy tends to increase, until it reaches the largest value the system can support.
3. In 1972, James Bardeen, Brandon Carter, and Stephen Hawking worked out the mathematical laws underlying the evolution of black holes, and found that the equations looked just like those of thermodynamics. To translate between the two sets of laws, all one needed to do was substitute “area of black hole’s horizon” for “entropy” (and vice versa), and “gravity at the surface of the black hole” for “temperature.” So, for Bekenstein’s idea to hold—for this similarity to not just be a coincidence, but to reflect the fact that black holes have entropy—black holes would also need to have a nonzero temperature.
4. The reason for the apparent change in energy is far from obvious; it relies on an intimate connection between energy and time. You can think of a particle’s energy as the vibrational speed of its quantum field. Noting that the very meaning of speed invokes the concept of time, a relationship between energy and time becomes apparent. Now, black holes have a profound effect on time. From a distant vantage point, time appears to slow for an object approaching the horizon of a black hole, and comes to a stop at the horizon itself. Upon crossing the horizon, time and space interchange roles—inside the black hole, the radial direction becomes the time direction. This implies that within the black hole, the notion of positive energy coincides with motion in the radial direction toward the black hole’s singularity. When the negative energy member of a particle pair crosses the horizon, it does indeed fall toward the black hole’s center. Thus the negative energy it had from the perspective of someone watching from afar becomes positive energy from the perspective of someone situated within the black hole itself. This makes the interior of the black hole a place where such particles can exist.
5. When a black hole shrinks, the surface area of its event horizon shrinks too, conflicting with Hawking’s pronouncement that total surface area increases. Remember, however, that Hawking’s area theorem is based on classical general relativity. We are now taking account of quantum processes and coming to a more refined conclusion.
6. To be a little more precise, it’s the minimum number of yes-no questions whose answers uniquely specify the microscopic details of the system.
7. Hawking found that the entropy is the area of the event horizon in Planck units, divided by four.
8. For all the insights that will be described as this chapter unfolds, the issue of a black hole’s microscopic makeup has yet to be fully resolved. As I mentioned in Chapter 4, in 1996, Andrew Strominger and Cumrun Vafa discovered that if one (mathematically) gradually turns down the strength of gravity, then certain black holes morph into particular collections of strings and branes. By counting the possible rearrangements of these ingredients, Strominger and Vafa recovered, in the most explicit manner ever achieved, Hawking’s famous black hole entropy formula. Even so, they were not able to describe these ingredients at stronger gravitational strength, i.e., when the black hole actually forms. Other authors, such as Samir Mathur and various of his collaborators, have put forward other ideas, such as the possibility that black holes are what they call “fuzz balls,” accumulations of vibrating strings strewn throughout the black hole’s interior. These ideas remain tentative. The results we discuss later in this chapter (in the section “String Theory and Holography”) provide some of the sharpest insight into this question.
9. More precisely, gravity can be canceled in a region of space by going into a freely falling state of motion. The size of the region depends on the scales over which the gravitational field varies. If the gravitational field varies only over large scales (that is, if the gravitational field is uniform, or nearly so), your free-fall motion will cancel gravity over a large region of space. But if the gravitational field varies over short-distance scales—the scales of your body, say—then you might cancel gravity at your feet and yet still feel it at your head. This becomes particularly relevant later in your fall because the gravitational field gets ever stronger ever closer to the black hole’s singularity; its strength rises sharply as your distance from the singularity decreases. The rapid variation means there is no way to cancel the effects of the singularity, which will ultimately stretch your body to its breaking point since the gravitational pull on your feet, if you jump in feetfirst, will be ever stronger than the pull on your head.
10. This discussion exemplifies the discovery, made in 1976 by William Unruh, that links one’s motion and the particles one encounters. Unruh found that if you accelerate through otherwise empty space, you will encounter a bath of particles at a temperature determined by your motion. General relativity instructs us to determine one’s rate of acceleration by comparing with the benchmark set by free-fall observers (see Fabric of the Cosmos, Chapter 3). A distant, non-free-fall observer thereby sees radiation emerging from a black hole; a free-fall observer does not.
11. A black hole forms if the mass M within a sphere of radius R exceeds c2R/2G, where c is the speed of light and G is Newton’s constant.
12. In actuality, as the matter collapsed under its own weight and a black hole formed, the event horizon would generally be located within the boundary of the region we’ve been discussing. This means that we would not have so far maxed out the entropy that the region itself could contain. This is easily remedied. Throw more material into the black hole, causing the event horizon to swell out to the region’s original boundary. Since entropy would again increase throughout this somewhat more elaborate process, the entropy of the material we put within the region would be less than that of the black hole that fills the region, i.e., the surface area of the region in Planck units.
13. G. ’t Hooft, “Dimensional Reduction in Quantum Gravity.” In Salam Festschrift, edited by A. Ali, J. Ellis, and S. Randjbar-Daemi (River Edge, N.J.: World Scientific, 1993), pp. 284–96 (QCD161:C512:1993).
14. We’ve discussed that “tired” or “exhausted” light is light whose wavelength is stretched (redshifted) and vibrational frequency reduced by virtue of its having expended energy climbing away from a black hole (or climbing away from any source of gravity). Like more familiar cyclical processes (the earth’s orbit around the sun; the earth’s rotation on its axis, etc.), the vibrations of light can be used to define elapsed time. In fact, the vibrations of light emitted by excited Cesium-133 atoms are now used by scientists to define the second. The tired light’s slower vibrational frequency thus implies that the passage of time near the black hole—as viewed by the faraway observer—is slower too.
15. With most important discoveries in science, the pinnacle result relies on a collection of earlier works. Such is the case here. In addition to ’t Hooft, Susskind, and Maldacena, the researchers who helped blaze the trail to this result and develop its consequences include Steve Gubser, Joe Polchinski, Alexander Polyakov, Ashoke Sen, Andy Strominger, Cumrun Vafa, Edward Witten, and many others.
For the mathematically inclined reader, the more precise statement of Mald
acena’s result is the following. Let N be the number of three-branes in the brane stack, and let g be the value of the coupling constant in the Type IIB string theory. When gN is a small number, much less than one, the physics is well described by low-energy strings moving on the brane stack. In turn, such strings are well described by a particular four-dimensional supersymmetric conformally invariant quantum field theory. But when gN is a large number, this field theory is strongly coupled, making its analytical treatment difficult. However, in this regime, Maldacena’s result is that we can use the description of strings moving on the near horizon geometry of the brane stack, which is AdS5 × S5 (anti-de Sitter five-space times the five sphere). The radius of these spaces is controlled by gN (specifically, the radius is proportional to (gN)¼), and thus for large gN, the curvature of AdS5 × S5 is small, ensuring that string theory calculations are tractable (in particular, they are well approximated by calculations in a particular modification of Einsteinian gravity). Therefore, as the value of gN varies from small to large values, the physics morphs from being described by four-dimensional supersymmetric conformally invariant quantum field theory to being described by ten-dimensional string theory on AdS5 × S5. This is the so-called AdS/CFT (anti-de Sitter space/conformal field theory) correspondence.
16. Although a full proof of Maldacena’s argument remains beyond reach, in recent years the link between the bulk and boundary descriptions has become increasingly well understood. For example, a class of calculations has been identified whose results are accurate for any value of the coupling constant. The results can therefore be explicitly tracked from small to large values. This provides a window onto the “morphing” process by which a description of physics from the bulk perspective transforms into a description in the boundary perspective, and vice versa. Such calculations have shown, for instance, how chains of interacting particles from the boundary perspective can transform into strings in the bulk perspective—a particularly convincing interpolation between the two descriptions.
17. More precisely, this is a variation on Maldacena’s result, modified so that the quantum field theory on the boundary is not the one that originally arose in his investigations, but instead closely approximates quantum chromodynamics. This variation also entails parallel modifications to the bulk theory. Specifically, following the work of Witten, the high temperature of the boundary theory translates into a black hole in the interior description. In turn, the dictionary between the two descriptions shows that the difficult viscosity calculations of the quark-gluon plasma translate into the response of the black hole’s event horizon to particular deformations—a technical but tractable calculation.
18. Another approach to providing a full definition of string theory emerged from earlier work in an area called Matrix theory (another possible meaning of the “M” in M-theory), developed by Tom Banks, Willy Fischler, Steve Shenker, and Leonard Susskind.
Chapter 10: Universes, Computers, and Mathematical Reality
1. The number I quoted, 1055 grams, accounts for the contents of the observable universe today, but at ever-earlier times, the temperature of these constituents 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.
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: , where c is the speed of light and v is your speed. As you can see, as v approaches 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.
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.
12. A more refined discussion of computable and noncomputable functions would also include 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 ψ: a computer can produce each successive digit of ψ, even though it will never reach the end of the computation. So, while ψ is strictly speaking noncomputable, it is limit computable. Most real numbers, however, are not like ψ. 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 space 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 quantities 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).