Quantum Gravity as an Ordinary Gauge Theory

Today's paper, by Juan Maldacena, is prefaced by a very curious comment from guest editor, Paul Davies. Namely, that, "In 1975 Stephen Hawking stunned the physics community when he demonstrated mathematically that a black hole should not be perfectly black, but ought to glow with faint heat radiation. Curiously, the temperature of the black hole depends inversely on its mass, so as the hole radiates energy it gets hotter. The process is therefore unstable, and the hole will evaporate away at an accelerating rate, eventually exploding out of existence."

But where goeth the matter that was in the hole? Well, it seems that the jury is still out on that question. But Maldacena, viewing said matter as a kind of information, has a holographic answer. He says, "all information about the interior is stored on a surface of one less dimension (the boundary). This is similar to how a usual optical hologram works."

Indeed. Read on to find out more about just what that means!

—Editor

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In 1975 Stephen Hawking stunned the physics community when he demonstrated mathematically that a black hole should not be perfectly black, but ought to glow with faint heat radiation. Curiously, the temperature of the black hole depends inversely on its mass, so as the hole radiates energy it gets hotter. The process is therefore unstable, and the hole will evaporate away at an accelerating rate, eventually exploding out of existence.

Although the foregoing scenario is generally accepted, theorists remain sharply divided about the ultimate fate of the matter that fell into the hole in the first place. In the spirit of Wheeler's 'it from bit,' one would like to keep track of the information content of the hole. If the matter that imploded is forever beyond our ken, then it seems as if all information is irretrievably lost down the hole. That is Hawking's own favored interpretation. In which case, the entropy of the black hole can be associated with the lost information. Indeed, that was how Jacob Bekenstein, then a student of John Wheeler, first suggested a few years before Hawking's work that black holes have thermodynamic properties. However, Hawking's position has been challenged by Gerard 't Hooft and several others, who think that ultimately the evaporating black hole must give back to the universe all the information it swallowed. To be sure, it comes out again in a different form, as subtle correlations in the Hawking radiation. But according to this position information is never lost to the universe. By contrast Hawking claims it is.

Theorists have attacked this problem by pointing out that when matter falls into a black hole, from the point of view of a distant observer it gets frozen near the surface of the hole. (Measured in the reference frame of the distant observer it seems to take matter an infinite amount of time to reach the event horizon—the surface of the hole.) So from the informational point of view, the black hole is effectively two dimensional: everything just piles up on the surface. In principle one ought to be able to retrieve all the black hole's information content by examining all the degrees of freedom on the surface. This has led to the analogy with a hologram, in which a three-dimensional image is created by shining a laser on a two-dimensional plate.

Despite heroic attempts to recast the theory of quantum black holes in holographic language, the issue of the fate of the swallowed, or stalled, information is far from resolved. There is a general feeling, however, that this apparently esoteric technical matter conceals deeper principles that relate to string theory, unification and other aspects of quantum gravity. A paper by Juan Maldacena, a theoretical physicist from Princeton's Institute of Advanced Study, discusses a specific model that examines a black hole in a surrounding model universe. Whilst artificial, this so-called anti-de Sitter universe admits of certain transparent mathematical properties that help elucidate the status of the holographic analogy, and the nature of black hole information and entropy.

—Paul Davies

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Summary:

In order to get a consistent picture of nature we need to use quantum mechanics to describe the geometry of spacetime. In other words, we should construct a quantum theory of gravity. In such a theory the spacetime geometry becomes fuzzy, it is only defined in a probabilistic sense. This simple fact makes it very hard to define observables in a mathematically precise fashion.

So part of the problem is to understand what the good observables are. Important progress has been made understanding these observables and computing them in some cases. There are two cases where a natural set of observables can be defined. The first case is an asymptotically flat spacetime. In such a spacetime we can define scattering amplitudes (if the interactions die off fast enough). These scattering amplitudes are good observables.

A second case is when the spacetime is asymptotically anti-de-sitter. Anti-de-sitter is a spacetime with constant negative curvature. It is the Lorentzian analog of hyperbolic space. These spacetimes have a boundary at infinity. It is a boundary since a light ray can get there and back infinite time. In this case the obervables are measurements that one can do at the boundary of spacetime. In fact the observables are the same as those of a local quantum field theory on the boundary. So quantum gravity on an asymptotically anti-de-sitter spacetime is equivalent to a quantum field theory on the boundary.

The problem of quantizing gravity gets reduced to the problem of quantizing a quantum field theory, which we know how to do. Since the description only requires that the space is asymptotically anti-de-sitter we can describe black holes in the interior and we can have an explicit understanding of the microscopic degrees of freedom that give rise to their entropy. These degrees of freedom are just the microscopic degrees of freedom of the field theory living on the boundary. This description is holographic since all information about the interior is stored on a surface of one less dimension (the boundary). This is similar to how a usual optical hologram works.

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