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

by Brian Greene

If spacetime were really fundamental, most physicists expect that everyone, regardless of perspective—regardless of the language or theory used—would agree on its geometrical properties. But the fact that, at least within string theory, this need not be the case, suggests that spacetime may be a secondary phenomenon.

Wherefore the Entropy of Black Holes?

The three distinguishing features of a black hole are its mass (which determines how big it is—the distance from its center to its event horizon, the enshrouding surface of no return), its electric charge, and how fast it’s spinning.

When it comes to black holes, even though we can’t say what their constituents actually are— since we don’t know what happens when matter is crushed at the black hole’s center—we can say confidently that rearranging these constituents will no more affect a black hole’s mass, charge, or spin than rearranging the pages in War and Peace will affect the weight of the book. And since mass, charge, and spin fully determine the face that a black hole shows the external world, all such manipulations go unnoticed and we can say a black hole has maximal entropy.

Thus, the amount of entropy contained within a black hole not only tells us a fundamental feature of the black hole, it also tells us something fundamental about space itself: the maximum entropy that can be crammed into a region of space—any region of space, anywhere, anytime—is equal to the entropy contained within a black hole whose size equals that of the region in question.

Were you to double the radius of a black hole, its volume would increase by a factor of 8 (23) while its surface area would increase by only a factor of 4 (22); were you to increase its radius by a factor of a hundred, its volume would increase by a factor of a million (1003), while its surface area would increase only by a factor of 10,000 (1002). Big black holes have much more volume than they do surface area.2 Thus, even though black holes contain the greatest entropy among all things of a given size, Bekenstein and Hawking showed that the amount of entropy they contain is less than what we’d naïvely guess.

We’ve seen that black holes set a limit to the amount of entropy that, even in principle, can be crammed into a region of space: take a black hole whose size precisely equals that of the region in question, figure out how much entropy the black hole has, and that is the absolute limit on the amount of entropy the region of space can contain.

All right, but why should we care? There are two reasons.

First, the entropy bound gives yet another clue that ultramicroscopic space has an atomized structure.

Bekenstein and Hawking found that if you imagine drawing a checkerboard pattern on the event horizon of a black hole, with each square being one Planck length by one Planck length (so each such “Planck square” has an area of about 10−66 square centimeters), then the black hole’s entropy equals the number of such squares that can fit on its surface.4 It’s hard to miss the conclusion to which this result strongly hints: each Planck square is a minimal, fundamental unit of space, and each carries a minimal, single unit of entropy. This suggests that there is nothing, even in principle, that can take place within a Planck square...

Second, for a physicist, the upper limit to the entropy that can exist in a region of space is a critical, almost sacred quantity.

the more disorder the region can contain—even in principle—the more things the theory must be capable of keeping track of. Thus, the maximum entropy a region can contain provides a simple but incisive litmus test: physicists expect that a truly fundamental theory is one that is perfectly matched to the maximum entropy in any given spatial region. The theory should be so tightly in tune with nature that its maximum capacity to keep track of disorder exactly equals the maximum disorder a region can possibly contain, not more and not less.

a theory that includes gravity is, in some sense, simpler than a theory that doesn’t. There are fewer “degrees of freedom”—fewer things that can change and hence contribute to disorder—that the theory must describe.

If the maximum entropy in any given region of space is proportional to the region’s surface area and not its volume, then perhaps the true, fundamental degrees of freedom—the attributes that have the potential to give rise to that disorder—actually reside on the region’s surface and not within its volume. Maybe, that is, the universe’s real physical processes take place on a thin, distant surface that surrounds us, and all we see and experience is merely a projection of those processes. Maybe, that is, the universe is rather like a hologram.

Is the Universe a Hologram?

In the early 1990s, the Dutch Nobel laureate Gerard ’t Hooft and Leonard Susskind, the same physicist who coinvented string theory, suggested that the universe itself might operate in a manner analogous to a hologram. They put forward the startling idea that the comings and goings we observe in the three dimensions of day-to-day life might themselves be holographic projections of physical processes taking place on a distant, two-dimensional surface. In their new and peculiar-sounding vision, we and everything we do or see would be akin to holographic images.

What we experience in the “volume” of the universe—in the bulk, as physicists often call it—would be determined by what takes place on the bounding surface, much as what we see in a holographic projection is determined by information encoded on a bounding piece of plastic. The laws of physics would act as the universe’s laser, illuminating the real processes of the cosmos—processes taking place on a thin, distant surface—and generating the holographic illusions of daily life.

So, where would the supposed “bounding holographic surface” be located?

Maldacena’s result is amazing. He found a concrete, albeit hypothetical, realization of holography within string theory. He showed that a particular quantum theory without gravity is a translation of—is indistinguishable from—another quantum theory that includes gravity but is formulated with one more space dimension.

And, as with the case of geometric translations described earlier, this provides yet another hint that spacetime is not fundamental. Not only can the size and shape of spacetime change in translation from one formulation of a theory to another, equivalent form, but the number of space dimensions can change, too.

The Constituents of Spacetime

We picture strings as vibrating in space and through time, but without the spacetime fabric that the strings are themselves imagined to yield through their orderly union, there is no space or time. In this proposal, the concepts of space and time fail to have meaning until innumerable strings weave together to produce them.

Thus, to make sense of this proposal, we would need a framework for describing strings that does not assume from the get-go that they are vibrating in a preexisting spacetime. We would need a fully spaceless and timeless formulation of string theory, in which spacetime emerges from the collective behavior of strings.

If the three-dimensional space we experience is a three-brane, is the brane itself indecomposable or is it made from combining the theory’s other ingredients? Are branes, for example, made from strings, or are branes and strings both elementary? Or should we consider yet another possibility, that branes and strings might be made from some yet finer ingredients?

Tom Banks of Rutgers University and Willy Fischler of the University of Texas at Austin, together with Leonard Susskind and Stephen Shenker, both now at Stanford, have formulated a version of string/M-theory in which zero-branes are the fundamental ingredients that can be combined to generate strings and the other, higher dimensional branes. This proposal, known as Matrix theory—still another possible meaning for the “M” in “M-theory”—has generated an avalanche of follow-up research, but the difficult mathematics involved has so far prevented scientists from bringing the approach to completion.

Three Roads to Quantum Gravity),

However, as just noted, the main failing of current formulations of string theory is that they presuppose a background spacetime within which strings move and vibrate. By contrast, the main achievement of loop quantum gravity—an impressive one—is that it does not assume a background spacetime. Loop quantum gravity is a background-independent framework.

Inner and Outer Space

After centuries of thought, we still can only portray space and time as the most familiar of strangers. They unabashedly wend their way through our lives, but adroitly conceal their fundamental makeup from the very perceptions they so fully inform and influence.

Until our theories make contact with observable, testable phenomena, they remain in limbo—they remain promising collections of ideas that may or may not have relevance for the real world.

Endnotes

because there are examples, involving relatively esoteric particles (such as K-mesons and B-mesons), which show that the so-called weak nuclear force does not treat past and future fully symmetrically.

Thus, while it is technically an overstatement, I will assume throughout that the error made in asserting that the laws treat past and future on equal footing is minimal—at least as far as explaining the puzzle of time’s arrow is concerned.

Edwin Taylor and John Archibald Wheeler, Spacetime Physics: Introduction to Special Relativity (New York, W. H. Freeman & Co., 1992).

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.

Suggestions for Further Reading

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

Barrow, John. The Book of Nothing. New York: Pantheon, 2000.

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

Davies, Paul. About Time. New York: Simon & Schuster, 1995.

How to Build a Time Machine. New York: Allen Lane, 2001.

Space and Time in the Modern Universe. Cambridge, Eng.: Cambridge Un...

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

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

Feynman, Richard. QED. Princeton: Princeton Univer...

Gott, J. Richard. Time Travel in Einstein’s Universe. Boston: Houghton Mifflin, 2001. Guth, Alan. The Inflationary Universe. Reading, Mass.: Perseus, 1997.

Greene, Brian. The Elegant Universe. New York: Vintage, 2000.

Gribbin, John. Schrödinger’s Kittens and the Search for Reality. Boston...

Hawking, Stephen. The Universe in a Nutshell. New York: Bantam, 2001.

and Roger Penrose. The Nature of Space and Time. Princeton: Princeton University Press, 1996.