by Lee Smolin
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 dynami
cs 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
GLOSSARY
Terms in italics have their own glossary entries.
absolute space and time
Newton’s view of space and time according to which they exist eternally, independent of whether anything is in the universe or not and of what happens inside the universe.
angular momentum
A measure of rotational motion, analogous to momentum. The total angular momentum of an isolated system is conserved.
background
A scientific model or theory often describes only part of the universe. Some features of the rest of the universe may be included as necessary to define the properties of that part of the universe that is studied. These features are called the background. For example, in Newtonian physics space and time are part of the background because they are taken to be absolute.
background dependent
A theory, such as Newtonian physics, that makes use of a background.
background independent
A theory that does not make use of a division of the universe into a part that is modelled and the rest, which is taken to be part of the background. General relativity is said to be background independent because the geometry of space and time is not fixed, but evolves in time just as any other field, such as the electromagnetic field.
Bekenstein bound
The relationship between the area of a surface and the maximum amount of information about the universe on one side of it that can pass through it to an observer on the other side. The relationship states that the number of bits of information the observer can gain cannot be greater than one-quarter the area of the surface in Planck units.
black hole
A region of space and time that cannot send signals to the outside world because all light emitted comes back. Among the ways a black hole may be formed is by the collapse of a very massive star when it runs out of its nuclear fuel.
black hole horizon
The surface surrounding a black hole, within which is the region from which light signals cannot escape.
boson
A particle whose angular momentum comes in integer multiples of Planck’s constant. Bosons do not obey the Pauli exclusion principle.
brane
A possible feature of geometry, as described in string theory, which consists of a surface of some dimensions embedded in space, which evolves in time. For example, strings are one-dimensional branes.
causality
The principle that events are influenced by those in their past. In relativity theory one event can have a causal influence on another only if energy or information sent from the first reaches the second.
causal structure
Because there is a maximum speed at which energy and information can be transmitted, the events in the history of the universe can be organized in terms of their possible causal relations. To do this one indicates, for every pair of events, whether the first is in the causal future of the second, or vice versa, or whether there is no possible causal relation between them because no signal could have travelled between them. Such a complete description defines the causal structure of the universe.
classical theory
Any physical theory that shares certain features with Newtonian physics, including the assumption that the future is completely determined by the present and that the act of observation has no effect on the system studied. The term is used mainly to label any theory that is not part of quantum theory. Einstein’s general theory of relativity is considered to be a classical theory.
classical physics
The collection of classical theories.
consistent histories
An approach to the interpretation of quantum theory which asserts that the theory makes predictions about the probabilities for sets of alternative histories, when these can be done consistently.
continuous
Describing a smooth and unbroken space which has the property of the number line, which is that it can be quantified in terms of coordinates expressed in real numbers. Any region of continuous space having a finite volume contains an infinitely uncountable number of points.
continuum
Any space that is continuous.
curvature tensor
The basic mathematical object in Einstein’s general theory of relativity. It determines how the tipping of light cones changes from time to time and place to place in the history of the universe.
degree of freedom
Any variable in a physical theory that may be specified independently of the other variables, which once specifies evolves in time according to a dynamical law. Examples are the positions of particles and the values of the electric and magnetic fields.
diffeomorphism
An operation that moves the points of space around, preserving only those relationships between them that are used to define which points are near to one another.
discrete
Describing a space that is made of a finite number of points.
duality
The principle of duality applies when two descriptions are different ways of looking at the same thing. In particle physics it usually refers to a description in terms of strings and a description in terms of the flux of the electric field or some generalization of it.
Einstein equations
The basic equations of the general theory of relativity. They determine how light cones tip and how they are related to the distribution of matter in the universe.
electromagnetism
The theory of electricity and magnetism, including light, developed by Michael Faraday and James Clerk Maxwell in the nineteenth century.
entropy
A measure of the disorder of a physical system. It is defined as the amount of information about the microscopic motion of the atoms making up the system which is not determined by a description of the macroscopic state of that system.
equilibrium
A system is defined to be in equilibrium, or thermodynamic equilibrium, when it has the maximum possible amount of entropy.
event
In relativity theory, something that happens at a particular point of space and moment of time.
exclusion principle
see Pauli exclusion principle.
fermion
A particle whose angular momentum comes in integer multiples of one-half of Planck’s constant. Fermions satisfy the Pauli exclusion principle.
Feynman diagram
A depiction of a possible process in the interaction
of several elementary particles. Quantum theory assigns to each diagram the probability amplitude for that process to occur. The total probability is proportional to the square of the sum of the amplitudes of the possible processes, each of which is depicted by a Feynman diagram.
field
A physical entity that is described by specifying the value of some quantity at every point of space and time; examples are the electric and magnetic fields.
future
The future, or causal future, of an event consists of all those events that it can influence by sending energy or information to it.
future light cone
For a specific event, all other events that can be reached from it by a signal travelling at the speed of light. Since the speed of light is the maximum speed at which energy or information can travel, the future light cone of an event marks the limits of the causal future of that event. See also light cone.
general theory of relativity
Einstein’s theory of gravity, according to which gravity is related to the influence the distribution of matter has on the causal structure of spacetime.
graph
A diagram consisting of a set of points, called vertices, connected by lines, called edges. See also lattice.
Hawking radiation
The thermal radiation black holes are predicted to give off, having a temperature which is inversely proportional to the black hole’s mass. Hawking radiation is caused by quantum effects.
