Three Roads to Quantum Gravity (31 page)

This is not the only place where cosmological observation and fundamental theory are confronting each other. An even more exciting—and for some disturbing—case has to do with the
cosmological constant
. This refers to the possibility—first realized by Einstein—that empty space might have a non-zero energy density. This energy density would be observable in the effect it has on the expansion of the universe.

Once this possibility was accepted, it led to a major crisis in theoretical physics. The reason is that the most natural possibility allowed for the value of this empty space energy density is that it should be huge—more than a hundred powers of ten larger than is compatible with observation. The exact value—which is what the name
cosmological constant
refers to—cannot be predicted by current theory. In fact, we can adjust a parameter to get any value for the cosmological constant we want. The problem is that to avoid a huge cosmological constant, the parameter has to be adjusted to an accuracy of at least 120 decimal places. How such a precise adjustment is to be obtained is a mystery.

This is perhaps the most serious problem facing fundamental physics, and it recently got worse. Until a few years ago, it was almost universally believed that even if it required a very precise adjustment, in the end the cosmological constant would be exactly zero. We had no idea why the cosmological constant would be zero, but at least zero is a simple answer. However, recent observations have suggested that the cosmological constant is not zero; it has instead a very small, but positive, value. This value is tiny on the scales of fundamental physics; in Planck units it is around 10
^-120
(or .0000. . . .) with 120 zeros before one encounters a non-zero digit.

But even though tiny when measured in fundamental units, this value is large enough to have a profound effect on
the evolution of our universe. This cosmological constant would make the energy density of empty space equal to about twice the current value of the energy density of everything else that has been observed. This may seem surprising, but the point is that the energy density of all the kinds of matter that have been observed is currently very small. This is because the universe is very old. When measured in fundamental units, its present age is about 10
⁶⁰
Planck times. And it has been expanding all this time, thus diluting the density of matter.

The energy density due to the cosmological constant does not, as far as we know, dilute as the universe expands. This gives rise to a very troubling question: Why is it that we live at a time when the matter density has diluted to the point that it is of the same order of magnitude as the density due to the cosmological constant?

I do not know the answer to any of these questions. Neither, I think, does anyone else, although there are a few interesting ideas on the table.

However, the apparent fact that the cosmological constant is not zero has big implications for the quantum theory of gravity. One reason is that it seems to be incompatible with string theory. It turns out that a mathematical structure that is required for string theory to be consistent—which goes by the name
supersymmetry
—only permits the cosmological constant to exist if it has the opposite sign from the one that has apparently been observed. There are some interesting studies of string theory in the presence of a negative cosmological constant, but no one so far knows how to write down a consistent string theory when the cosmological constant is positive—as has apparently been observed.

I do not know if this obstacle will kill string theory—string theorists are very resourceful, and they have often expanded the definition of string theory to include cases once thought impossible. But string theorists are worried, for if string theory cannot be made compatible with a positive cosmological constant—and that continues to be what the astronomers observe—then the theory is dead.

But there is a second reason why a positive cosmological constant is troubling for quantum theories of gravity, including string theory. As the universe continues to expand, the energy density due to matter will continue to dilute. But the cosmological constant is believed to remain stable. This means that there will be a time in the future when the cosmological constant comprises most of the energy density in the universe. After this the expansion will accelerate—indeed the effect is very similar to the inflation proposed for the very early universe.

To be an observer in an inflating universe is to be in a very poor situation. As the universe inflates, we will see less and less of it. Light cannot keep up with the acceleration of the expansion, and light from distant galaxies will no longer be able to reach us. It would be as if large regions of the universe had fallen behind the horizon of a black hole. One by one distant galaxies will go over a horizon, to a zone from which their light will never again reach us. With the value apparently measured, it is only a matter of a few tens of billions of years before observers in a galaxy see nothing around them except their own galaxy surrounded by a void.

In such a universe, the considerations of Chapters 1-3 become crucial. A single observer can only see a small portion of the universe, and that small portion will only decrease over time. No matter how long we wait, we will never see more of the universe than we do now.

Tom Banks has expressed this principle beautifully. There is a finite limit to the amount of information that any observer in an inflating universe may ever see. The limit is that each observer can see no more than
bits of information, where G is Newton’s constant and L is the cosmological constant. Raphael Bousso called this the
N-bound
and argued that this principle may be derived by an argument that is closely related to Bekenstein’s bound, which is described in Chapters 8 and 12. The principle seems to be required by the second law of thermodynamics.

As the universe expands, we would expect that it contains more and more information. But, according to this principle,
any given observer can only see a fixed amount of information given by the N-bound.

In this circumstance, the traditional formulations of quantum theory fail because they assume that an observer can, given enough time, see anything that happens in the universe. It seems to me that there is then no alternative but to adopt the program I described in Chapter 3, which was proposed by Fotini Markopoulou—to reformulate physics in terms of only what observers inside the universe can actually see. As a result, Markopoulou’s proposal has been getting more attention from people on both sides of the string theory/ loop quantum gravity divide.

So far there is no proposal for how to reformulate string theory in such terms. One possible step toward such a formulation is Andrew Strominger’s new proposal, which applies the holographic principle to spacetimes, with a positive cosmological constant.

At the same time, loop quantum gravity is clearly compatible with such a reformulation of quantum theory—it is already background-independent and expressed in a language in which the causal structure exists all the way down to the Planck scale.

In fact, Bank’s N-bound is easy to derive in loop quantum gravity, using the same methods that led to the description of the quantum states on black hole horizons. Moreover, in loop quantum gravity there is a complete description of a quantum universe filled with nothing but a positive cosmological constant. This is given by a certain mathematical expression, discovered by the Japanese physicist Hideo Kodama. Using Kodama’s result, we are able to answer previously unsolvable questions, such as exactly how the solutions of Einstein’s general relativity theory emerge from the quantum theory. Thus, at least in our present stage of knowledge, while string theory has trouble incorporating the apparently observed positive value of the cosmological constant, loop quantum gravity seems to prefer that case.

Beyond this, there has continued to be steady progress in loop quantum gravity. The work of two young physicists,
Chopin Soo and Martin Bojowald, has led to a greatly improved understanding of how classical cosmology emerges from loop quantum gravity. New calculational methods for spin foams have given us very satisfactory results. Large classes of calculations, for example, turn out to give finite, well-defined answers, where conventional quantum theories gave infinities. These results present more evidence that loop quantum gravity provides a consistent framework for a quantum theory of gravity.

 

Before closing I want to emphasize again that this book describes science
in the making
. There are some people who think that popular science should be restricted to reporting discoveries that have been completely confirmed experimentally, leaving no room for controversy among experts. But restricting popular science in this way blurs the line between science and dogma, and dictates how we believe the public should think. To communicate how science really works, we must open the door and let the public watch as we go about searching for the truth. Our task is to present all the evidence and invite the readers to think for themselves.

But this is the paradox of science: It is an organized, even ritualized, community designed to support the process of a large number of people thinking for themselves and discussing and arguing the conclusions they come to.

Exposing the debates in a field like quantum gravity to the public is also bound to raise controversy among experts. In this book, I tried to treat the different approaches to quantum gravity as evenhandedly as possible. Still, some experts have told me I do not praise string theory enough, whereas others have told me I did not emphasize its shortcomings nearly enough. Some colleagues complained that I did not champion my own field of loop quantum gravity strongly enough, given that string theorists generally fail to even mention loop quantum gravity—or anything other than string theory—in their own books and public talks. Indeed, one string theorist who reviewed the book called me a “maverick” for even
mentioning that many of the leading people who made key discoveries in quantum gravity did not work on string theory. I take the fact that this kind of criticism came from both sides as evidence that I did not completely fail to present an evenhanded view of the successes and failures of loop quantum gravity, string theory, and the other approaches to quantum gravity.

At the same time, I cannot help but notice that as time goes on, it appears that the close-mindedness that characterizes the thinking of some (of course, not all) string theorists does appear to have inhibited progress. Many string theorists seem disinterested in thinking about questions that cannot be sensibly posed within the existing framework for string theory. This is perhaps because they are convinced that supersymmetry is more fundamental than the lesson from general relativity that spacetime is a dynamical, relational entity. Nevertheless, I suspect this is the main reason for the slow progress on key questions such as making string theory background independent, or understanding the role of the dynamics of causal structure, problems that cannot be addressed without going beyond current string theory. Of course, other people can and do work on this problem, and we are making progress on it, even if we are not considered by the orthodox to be “real string theorists.”

My own view remains optimistic. I believe that we have on the table all the ingredients we need to make the quantum theory of gravity and that it is mostly a matter of putting the pieces together. So far, nothing has changed my understanding that loop quantum gravity is a consistent framework for a complete quantum theory of spacetime, and string theory does not yet provide more than a background-dependent approximation to such a theory. I believe that some aspects of string theory might nevertheless play a role, as an approximation to the real theory, but given a choice between the two, loop quantum gravity is certainly the deeper and more comprehensive theory. Furthermore, if the atomic structure of spacetime predicted by loop quantum gravity requires modifications of special relativity such as a variation in the
speed of light with energy, this is a challenge for string theory, which in its current form assumes the theory makes sense without such effects. So if—as conjectured in Chapter 14—a form of string theory can be derived from loop quantum gravity, it may be in a modified form.

But what is important above all is that it doesn’t matter what I or any other theorist thinks. Experiment will decide. And quite possibly in the next few years.

 

Lee Smolin
March 3, 2002
Waterloo, Canada

Terms in italics have their own glossary entries.