Section 10: Fundamental Questions of Gravity
Figure 34: Some of Einstein's great insights began in thought experiments.
Source: © Marcelo Gleiser. More info
Quantum gravity is not a mystery that is, as yet, open to experiment. With a few exceptions such as inflationary cosmology and, quite possibly, the correct interpretation of dark energy (see Unit 11), thoughts about quantum gravity remain in the theoretical realm. This does not mean that theories of quantum gravity cannot be important and testable in some circumstances. In a given model, for instance, string theory models of particle physics make very definite statements about the origins of the mass hierarchy of fermions or the number of generations of Standard Model particles. But these problems may also have other solutions, insensitive to the structure of gravity at short distances; only in very few cases do we suspect that quantum gravity must be a part of the solution to a problem.
Here, we discuss some of these issues that are intrinsically gravitational. They have not yet confronted experiment directly. But we should remember that Einstein formulated special relativity by reconciling diﬀerent thought experiments, and that therefore even thought experiments about quantum gravity may eventually be useful.
Black holes and entropy
Black holes are objects so dense that the escape velocity from their surface exceeds the speed of light, c. Because of that, one would think that in a relativistic theory, outside observers performing classical experiments can never see their surfaces. As a rough-and-ready definition, we will call the surface defining the region where light itself can no longer escape from the gravitational attraction of a black hole, the event horizon. Nothing, in a classical theory, can be emitted from this horizon, though many things can fall through.
Figure 35: Artist's conception of a black hole accreting matter from a companion star.
Source: © Dana Berry (CfA, NASA). More info
Careful consideration of the theory of black holes in classical general relativity in the early 1970s led Jacob Bekenstein, Stephen Hawking, and others to a striking set of conclusions. They found that as a chargeless, non-rotating black hole accretes matter, its mass grows by an amount proportional to the strength of gravity at the black hole's surface and the change in its surface area. Also, the black hole's surface area (defined by its event horizon) cannot decrease under any circumstances, and usually increases in time.
At a heuristic level, Bekenstein and Hawking's laws for black holes seem reminiscent of the laws of thermodynamics and statistical mechanics: The change in energy is proportional to the change in entropy and the entropy (a measure of disorder) of a system can only increase. This is no coincidence. The results of general relativity imply what they seem to: A black hole does carry an entropy proportional to its surface area, and, of course, it has an energy that grows with its mass.
Figure 36: Black holes radiate by a quantum mechanical process.
Source: © Reprinted with permission from Nova Science Publishers, Inc. from: Sabine Hossenfelder, "What Black Holes Can Teach Us," in Focus on Black Hole Research, ed. Paul V. Kreitler (New York: Nova Publishers Inc., 2006), 121-58. More info
One mystery remains, however. In thermodynamics, the change in energy is proportional to the temperature times the change in entropy; and hot bodies radiate. Even though there is an analogous quantity—the surface gravity—in the black hole mechanics, no classical process can bring radiation through the horizon of a black hole. In a brilliant calculation in 1974, Stephen Hawking showed that, nevertheless, black holes radiate by a quantum process. This quantum effect occurs at just the right level to make the analogy between black hole thermodynamics and normal thermodynamics work perfectly.
Hawking's calculation reinforces our belief that a black hole's entropy should be proportional to its surface area. This is a bit confusing because most theories that govern the interactions of matter and force-carrying particles in the absence of gravity posit that entropy grows in proportion to the volume of the system. But in a gravity theory also containing these other degrees of freedom, if one tries to fill a region with enough particles so that their entropy exceeds the area bounding the region, one instead finds gravitational collapse into a black hole, whose entropy is proportional to its surface area. This means that at least in gravity theories, our naive idea that the entropy that can be contained in a space should scale with its volume must be incorrect.
Holography, multiple dimensions, and beyond
This concept that in every theory of quantum gravity, the full entropy is proportional only to the area of some suitably chosen boundary or "holographic screen" in the system, and not the full volume, carries a further implication: that we may be able to formulate a theory of gravity in D +1 spacetime dimensions in just D dimensions—but in terms of a purely non-gravitational quantum field theory. Dutch theorist Gerard 't Hooft and his American colleague Leonard Susskind put forward this loose idea, called holography, in the early 1990s. The idea, as stated, is a bit vague. It begs questions such as: On which "bounding surface" do we try to formulate the physics? Which quantum field theory is used to capture which quantum gravity theory in the "bulk" of the volume?
In the late 1990s, through the work of Argentine theorist Juan Maldacena and many others, this idea received its first very concrete realization. We will focus on the case of gravity in four dimensions; for diﬀerent reasons, much of the actual research has been focused on theories of gravity in five dimensions. This work gives, for the first time, a concrete non-perturbative formulation of quantum gravity in the 4D Anti de Sitter spacetime—the simplest solution of Einstein's theory with a negative cosmological constant.
Figure 37: The AdS/CFT duality relates a theory on the boundary of a region to a theory with gravity in the interior.
It turns out that the symmetries of the 4D space—that is, in 3+1 dimensions—match those of a quantum field theory of a very particular sort, called a "conformal field theory," in 2+1 spacetime dimensions. However, we also expect that quantum gravity in 3+1 dimensions should have the same behavior of its thermodynamic quantities (and in particular, its entropy) as a theory in 2+1 dimensions, without gravity. In fact, these two observations coincide in a beautiful story called the Anti de Sitter/Conformal Field Theory (AdS/CFT) correspondence. Certain classes of quantum field theories in 2+1 dimensions are exactly equivalent to quantum gravity theories in 3+1 dimensional Anti de Sitter space. Physicists say that these two theories are dual to one another.
The precise examples of this duality that we recognize come from string theory. Compactifications of the theory to four dimensions on diﬀerent compact spaces give rise to diﬀerent examples of AdS4 gravity and to diﬀerent dual field theories. While we do not yet know the gravity dual of every 2+1 dimensional conformal field theory or the field theory dual of every gravity theory in AdS4, we do have an infinite set of explicit examples derived from string theory.
The value of duality
This duality has a particularly interesting, useful, and, on reflection, necessary aspect. The 2+1 dimensional field theories analogous to electromagnetism have coupling constants g analogous to the electron charge e. The gravity theory also has a natural coupling constant, given by the ratio of the curvature radius of space to the Planck length, which we will call L. In the known examples of the duality between AdS space and quantum field theories, large values of L, for which the gravity theory is weakly curved, and hence involves only weak gravity, correspond to very large values of g for the quantum field theory. Conversely, when the quantum field theory is weakly coupled (at small values of g), the gravity theory has very small L; it is strongly coupled in the sense that quantum gravity corrections (which are very hard to compute, even in string theory) are important.
This kind of duality, between a strongly coupled theory on the one hand and a weakly coupled theory on the other, is actually a common (though remarkable and beautiful) feature in physical theories. The extra shock here is that one of the theories involves quantum gravity in a diﬀerent dimension.
This duality has had two kinds of uses to date. One obvious use is that it provides a definition of quantum gravity in terms of a normal field theory for certain kinds of gravitational backgrounds. Another use, however, that has so far proved more fruitful, is that it gives a practical way to compute in classes of strongly coupled quantum field theories: You can use their weakly curved gravitational dual and compute the dual quantities in the gravity theory there.
Figure 38: The aftermath of a heavy ion collision at RHIC, the Relativistic Heavy Ion Collider.
Source: © Courtesy of Brookhaven National Laboratory. More info
Physicists have high hopes that such explorations of very strongly coupled quantum field theories based on gravity duals may provide insight into many of the central problems in strongly coupled quantum field theory; these include a proper understanding of quark confinement in QCD, the ability to compute transport in the strongly coupled QCD plasma created at present-day accelerators like Brookhaven National Laboratory's Relativistic Heavy Ion Collider (RHIC) that smash together heavy ions, the ability to solve for quantities such as conductivity in strongly correlated electron systems in condensed matter physics, and an understanding of numerous zero-temperature quantum phase transitions in such systems. However, we must note that, while this new approach is orthogonal to old ones and promises to shed new light in various toy models of those systems, it has not yet helped to solve any of the central problems in those subjects.
The initial singularity
Even more intriguing is the question of the initial cosmological singularity. In general relativity, one can prove powerful theorems showing that any expanding cosmology (of the sort we inhabit) must have arisen in the distant past from a point of singularity in which the energy density and curvature are very large and the classical theory of relativity is expected to break down. Intuitively, one should just run the cosmological expansion backwards; then the matter we currently see ﬂying apart would grow into an ever-denser and more highly curved state.
Figure 39: If we run time backwards, the entire universe collapses into a single, infinitely dense point.
Source: © NASA, WMAP Science Team. More info
Our current picture of the Big Bang, including the successes of Big Bang nucleosynthesis and the CMB, gives us confidence that we can safely extrapolate our current cosmology back to temperatures of order MeV. In most inflationary theories, the Big Bang engendered temperatures far above the TeV scale. But at some still higher energy scale, probing the start of inflation and beyond, we do not know what happened; we do not even have any ideas that provide hints of testable predictions.
What is the origin of our observable universe? Is it one of many, coming from quantum tunneling events out of regions of space with diﬀerent macroscopic laws of physics? Or is it a unique state, arising from some unknown initial condition of quantum gravity that we have yet to unravel? And, how does this physics eventually deal with the singularity theorems of general relativity, which assure us that extrapolation backwards into the past will lead to a state of high density and curvature, where little can be reliably calculated? These mysteries remain at the forefront of modern research in quantum gravity.