The Fabric of the Cosmos: Space, Time, and the Texture of Reality

by Brian Greene

  3. For us to see anything, light has to travel to our eyes; similarly, for us to see light, the light itself would have to make the same journey. So, when I speak of Bart's seeing light that is speeding away, it is shorthand. I am imagining that Bart has a small army of helpers, all moving at Bart's speed, but situated at various distances along the path that he and the light beam follow. These helpers give Bart updates on how far ahead the light has sped and the time at which the light reached such distant locations. Then, on the basis of this information, Bart can calculate how fast the light is speeding away from him.

  4. There are many elementary mathematical derivations of Einstein's insights on space and time arising from special relativity. If you are interested, you can, for example, take a look at Chapter 2 of The Elegant Universe (together with mathematical details given in the endnotes to that chapter). A more technical but extremely lucid account is Edwin Taylor and John Archibald Wheeler, Spacetime Physics: Introduction to Special Relativity (New York, W. H. Freeman & Co., 1992).

  5. The stopping of time at light speed is an interesting notion, but it is important not to read too much into it. Special relativity shows that no material object can ever attain light speed: the faster a material object travels, the harder we'd have to push it to further increase its speed. Just shy of light speed, we'd have to give the object an essentially infinitely hard push for it to go any faster, and that's something we can't ever do. Thus, the "timeless" photon perspective is limited to massless objects (of which the photon is an example), and so "timelessness" is permanently beyond what all but a few types of particle species can ever attain. While it is an interesting and fruitful exercise to imagine how the universe would appear when moving at light speed, ultimately we need to focus on perspectives that material objects, such as ourselves, can reach, if we want to draw inferences about how special relativity affects our experiential conception of time.

  6. See Abraham Pais, Subtle Is the Lord, pp. 113-14.

  7. To be more precise, we define the water to be spinning if it takes on a concave shape, and not spinning if it doesn't. From a Machian perspective, in an empty universe there is no conception of spinning, so the water's surface would always be flat (or, to avoid issues of the lack of gravity pulling on the water, we can say that the tension on the rope tied between two rocks will always be slack). The statement here is that, by contrast, in special relativity there is a notion of spinning, even in an empty universe, so that the water's surface can be concave (and the tension on the rope tied between the rocks can be taut). In this sense, special relativity violates Mach's ideas.

  8. Albrecht Fölsing, Albert Einstein (New York: Viking Press, 1997), pp. 208-10.

  9. The mathematically inclined reader will note that if we choose units so that the speed of light takes the form of one space unit per one time unit (like one light-year per year or one light-second per second, where a light-year is about 6 trillion miles and a light-second is about 186,000 miles), then light moves through spacetime on 45-degree rays (because such diagonal lines are the ones which cover one space unit in one time unit, two space units in two time units, etc.). Since nothing can exceed the speed of light, any material object must cover less distance in space in a given interval of time than would a beam of light, and hence the path it follows through spacetime must make an angle with the centerline of the diagram (the line running through the center of the loaf from crust to crust) that is less than 45 degrees. Moreover, Einstein showed that the time slices for an observer moving with velocity v —all of space at one moment of such an observer's time—have an equation (assuming one space dimension for simplicity) given by t moving = (t stationary — (v/c 2 ) x stationary ), where = ( 1 — v 2 /c 2 ) -1/2 , and c is the velocity of light. In units where c = 1, we note that < 1 and hence a time slice for the moving observer—the locus where t moving takes on a fixed value—is of the form (t stationary — vx stationary ) = constant. Such time slices are angled with respect to the stationary time slices (the loci of the form t stationary = constant), and because v < 1, the angle between them is less than 45 degrees.

  10. For the mathematically inclined reader, the statement being made is that the geodesics of Minkowski's spacetime—the paths of extremal spacetime length between two given points—are geometrical entities that do not depend on any particular choice of coordinates or frame of reference. They are intrinsic, absolute, geometric spacetime features. Explicitly, using the standard Minkowski metric, the (timelike) geodesics are straight lines (whose angle with respect to the time axis is less than 45 degrees, since the speed involved is less than that of light).

  11. There is something else of importance that all observers, regardless of their motion, also agree upon. It's implicit in what we've described, but it's worth stating directly. If one event is the cause of another (I shoot a pebble, causing a window to break), all observers agree that the cause happened before the effect (all observers agree that I shot the pebble before the window broke). For the mathematically inclined reader, it is actually not difficult to see this using our schematic depiction of spacetime. If event A is the cause of event B, then a line drawn from A to B intersects each of the time slices (time slices of an observer at rest with respect to A) at an angle that is greater than 45 degrees (the angle between the space axes—axes that lie on any given time slice—and the line between A and B is greater than 45 degrees). For instance, if A and B take place at the same location in space (the rubber band wrapped around my finger [A] causes my finger to turn white [B]) then the line connecting A and B makes a 90-degree angle relative to the time slices. If A and B take place at different locations in space, whatever traveled from A to B to exert the influence (my pebble traveling from slingshot to window) did so at less than light speed, which means the angle differs from 90 degrees (the angle when no speed is involved) by less than 45 degrees—i.e. the angle with respect to the time slices (the space axes) is greater than 45 degrees. (Remember from endnote 9 of this chapter that light speed sets the limit and such motion traces out 45-degree lines.) Now, as in endnote 9, the different time slicings associated with an observer in motion are angled relative to those of an observer at rest, but the angle is always less than 45 degrees (since the relative motion between two material observers is always less than the speed of light). And since the angle associated with causally related events is always greater than 45 degrees, the time slices of an observer, who necessarily travels at less than light speed, cannot first encounter the effect and then later encounter the cause. To all observers, cause will precede effect.

  12. The notion that causes precede their effects (see the preceding note) would, among other things, be challenged if influences could travel faster than the speed of light.

  13. Isaac Newton, Sir Isaac Newton's Mathematical Principles of Natural Philosophy and His System of the World, trans. A. Motte and Florian Cajori (Berkeley: University of California Press, 1962), vol. 1, p. 634.

  14. Because the gravitational pull of the earth differs from one location to another, a spatially extended, freely falling observer can still detect a residual gravitational influence. Namely, if the observer, while falling, releases two baseballs—one from his outstretched right arm and the other from his left—each will fall along a path toward the earth's center. So, from the observer's perspective, he will be falling straight down toward the earth's center, while the ball released from his right hand will travel downward and slightly toward the left, while the ball released from his left hand will travel downward and slightly toward the right. Through careful measurement, the observer will therefore see that the distance between the two baseballs slowly decreases; they move toward one another. Crucial to this effect, though, is that the baseballs were released in slightly different locations in space, so that their freely falling paths toward earth's center were slightly different as well. Thus, a more precise statement of Einstein's realization is that the smaller the spatial extent of an object, the more fully it can eliminate gravity by goin
g into free fall. While an important point of principle, this complication can be safely ignored throughout the discussion.

  15. For a more detailed, yet general-level, explanation of the warping of space and time according to general relativity, see, for example, Chapter 3 of The Elegant Universe.

  16. For the mathematically trained reader, Einstein's equations are G = (8 G/c 4 ) T , where the left-hand side describes the curvature of spacetime using the Einstein tensor and the right-hand side describes the distribution of matter and energy in the universe using the energy-momentum tensor.

  17. Charles Misner, Kip Thorne, and John Archibald Wheeler, Gravitation (San Francisco: W. H. Freeman and Co., 1973), pp. 544-45.

  18. In 1954, Einstein wrote to a colleague: "As a matter of fact, one should no longer speak of Mach's principle at all" (as quoted in Abraham Pais, Subtle Is the Lord, p. 288).

  19. As mentioned earlier, successive generations have attributed the following ideas to Mach even though his own writings do not phrase things explicitly in this manner.

  20. One qualification here is that objects which are so distant that there hasn't been enough time since the beginning of the universe for their light—or gravitational influence—to yet reach us have no impact on the gravity we feel.

  21. The expert reader will recognize that this statement is, technically speaking, too strong, as there are nontrivial (that is, non-Minkowski space) empty space solutions to general relativity. Here I am simply using the fact that special relativity can be thought of as a special case of general relativity in which gravity is ignored.

  22. For balance, let me note that there are physicists and philosophers who do not agree with this conclusion. Even though Einstein gave up on Mach's principle, during the last thirty years it has taken on a life of its own. Various versions and interpretations of Mach's idea have been put forward, and, for example, some physicists have suggested that general relativity does fundamentally embrace Mach's ideas; it's just that some particular shapes that spacetime can have—such as the infinite flat spacetime of an empty universe—don't. Perhaps, they suggest, any spacetime that is remotely realistic—populated by stars and galaxies, and so forth—does satisfy Mach's principle. Others have offered reformulations of Mach's principle in which the issue is no longer how objects, such as rocks tied by a string or buckets filled with water, behave in an otherwise empty universe, but rather how the various time slicings—the various three-dimensional spatial geometries—relate to one another through time. An enlightening reference on modern thinking about these ideas is Mach's Principle: From Newton's Bucket to Quantum Gravity, Julian Barbour and Herbert Pfister, eds. (Berlin: Birkhäuser, 1995), which is a collection of essays on the subject. As an interesting aside, this reference contains a poll of roughly forty physicists and philosophers regarding their view on Mach's principle. Most (more than 90 percent) agreed that general relativity does not fully conform to Mach's ideas. Another excellent and extremely interesting discussion of these ideas, from a distinctly pro-Machian perspective and at a level suited to general readers, is Julian Barbour's book The End of Time: The Next Revolution in Physics (Oxford: Oxford University Press, 1999).

  23. The mathematically inclined reader might find it enlightening to learn that Einstein believed that spacetime had no existence independent of its metric (the mathematical device that gives distance relations in spacetime), so that if one were to remove everything—including the metric—spacetime would not be a something. By "spacetime" I always mean a manifold together with a metric that solves the Einstein equations, and so the conclusion we've reached, in mathematical language, is that metrical spacetime is a something.

  24. Max Jammer, Concepts of Space, p. xvii.

  Chapter 4

  1. More accurately, this appears to be a medieval conception with historical roots that go back to Aristotle.

  2. As we will discuss later in the book, there are realms (such as the big bang and black holes) that still present many mysteries, at least in part owing to extremes of small size and huge densities that cause even Einstein's more refined theory to break down. So, the statement here applies to all but the extreme contexts in which the known laws themselves become suspect.

  3. An early reader of this text, and one who, surprisingly, has a particular expertise in voodoo, has informed me that something is imagined to go from place to place to carry out the voodoo practitioner's intentions—namely, a spirit. So my example of a fanciful nonlocal process may, depending on your take on voodoo, be flawed. Nevertheless, the idea is clear.

  4. To avoid any confusion, let me reemphasize at the outset that when I say, "The universe is not local," or "Something we do over here can be entwined with something over there," I am not referring to the ability to exert an instantaneous intentioned control over something distant. Instead, as will become clear, the effect I am referring to manifests itself as correlations between events taking place—usually, in the form of correlations between results of measurements—at distant locations (locations for which there would not be sufficient time for even light to travel from one to the other). Thus, I am referring to what physicists call nonlocal correlations. At first blush, such correlations may not strike you as particularly surprising. If someone sends you a box containing one member of a pair of gloves, and sends the other member of the pair to your friend thousands of miles away, there will be a correlation between the handedness of the glove each of you sees upon opening your respective box: if you see left, your friend will see right; if you see right, your friend will see left. And, clearly, nothing in these correlations is at all mysterious. But, as we will gradually describe, the correlations apparent in the quantum world seem to be of a very different character. It's as if you have a pair of "quantum gloves" in which each member can be either left-handed or right-handed, and commits to a definite handedness only when appropriately observed or interacted with. The weirdness arises because, although each glove seems to choose its handedness randomly when observed, the gloves work in tandem, even if widely separated: if one chooses left, the other chooses right, and vice versa.

  5. Quantum mechanics makes predictions about the microworld that agree fantastically well with experimental observations. On this, there is universal agreement. Nevertheless, because the detailed features of quantum mechanics, as discussed in this chapter, differ significantly from those of common experience, and, relatedly, as there are different mathematical formulations of the theory (and different formulations of how the theory spans the gap between the microworld of phenomena and the macroworld of measured results), there isn't consensus on how to interpret various features of the theory (and various puzzling data which the theory, nevertheless, is able to explain mathematically), including issues of nonlocality. In this chapter, I have taken a particular point of view, the one I find most convincing based on current theoretical understanding and experimental results. But, I stress here that not everyone agrees with this view, and in a later endnote, after explaining this perspective more fully, I will briefly note some of the other perspectives and indicate where you can read more about them. Let me also stress, as we will discuss later, that the experiments contradict Einstein's belief that the data could be explained solely on the basis of particles always possessing definite, albeit hidden, properties without any use or mention of nonlocal entanglement. However, the failure of this perspective only rules out a local universe. It does not rule out the possibility that particles have such definite hidden features.

  6. For the mathematically inclined reader, let me note one potentially misleading aspect of this description. For multiparticle systems, the probability wave (the wavefunction, in standard terminology) has essentially the same interpretation as just described, but is defined as a function on the configuration space of the particles (for a single particle, the configuration space is isomorphic to real space, but for an N-particle system it has 3N dimensions). This is important to bear in mind when thinking about the question of whether the wavefu
nction is a real physical entity or merely a mathematical device, since if one takes the former position, one would need to embrace the reality of configuration space as well—an interesting variation on the themes of Chapters 2 and 3. In relativistic quantum field theory, the fields can be defined in the usual four spacetime dimensions of common experience, but there are also somewhat less widely used formulations that invoke generalized wavefunctions—so-called wavefunctionals defined on an even more abstract space, field space.

  7. The experiments I am referring to here are those on the photoelectric effect, in which light shining on various metals causes electrons to be ejected from the metal's surface. Experimenters found that the greater the intensity of the light, the greater the number of electrons emitted. Moreover, the experiments revealed that the energy of each ejected electron was determined by the color—the frequency—of the light. This, as Einstein argued, is easy to understand if the light beam is composed of particles, since greater light intensity translates into more light particles (more photons) in the beam—and the more photons there are, the more electrons they will hit and hence eject from the metallic surface. Furthermore, the frequency of the light would determine the energy of each photon, and hence the energy of each electron ejected, precisely in keeping with the data. The particlelike properties of photons were finally confirmed by Arthur Compton in 1923 through experiments involving the elastic scattering of electrons and photons.