25 years of Maldacena’s bridge

Twenty-five years go, in 1997, an Argentine physicist named Juan Martin Maldacena published what would become the most highly cited physics paper in history (more than 20,000 to date). In the paper, Maldacena described a ‘bridge’ between two theories that describe how our world works, but separately, without meeting each other. These are the field theories that describe the behaviour of energy fields (like the electromagnetic fields) and subatomic particles, and the theory of general relativity, which deals with gravity and the universe at the largest scales.

Field theories have many types and properties. One of them is a conformal field theory: a field theory that doesn’t change when it undergoes a conformal transformation – i.e. one which preserves angles but not lengths pertaining to the field. As such, conformal field theories are said to be “mathematically well-behaved”.

In relativity, space and time are unified into the spacetime continuum. This continuum can broadly exist in one of three possible spaces (roughly, universes of certain ‘shapes’): de Sitter space, Minkowski space and anti-de Sitter space. de Sitter space has positive curvature everywhere – like a sphere (but is empty of any matter). Minkowski space has zero curvature everywhere – i.e. a flat surface. Anti-de Sitter space has negative curvature everywhere – like a hyperbola.

A sphere, a hyperbolic surface and a flat surface. Credit: NASA

Because these shapes are related to the way our universe looks and works, cosmologists have their own way to understand these spaces. If the spacetime continuum exists in de Sitter space, the universe is said to have a positive cosmological constant. Similarly, Minkowski space implies a zero cosmological constant and anti-de Sitter space a negative cosmological constant. Studies by various space telescopes have found that our universe has a positive cosmological constant, meaning ‘our’ spacetime continuum occupies a de Sitter space (sort of, since our universe does have matter).

In 1997, Maldacena found that a description of quantum gravity in anti-de Sitter space in N dimensions is the same as a conformal field theory in N – 1 dimensions. This – called the AdS/CFT correspondence – was an unexpected but monumental discovery that connected two kinds of theories that had thus far refused to cooperate. (The Wire Science had a chance to interview Maldacena about his past and current work in 2018, in which he provided more insights on AdS/CFT as well.)

In his paper, Maldacena demonstrated his finding by using the example of string theory as a theory of quantum gravity in anti-de Sitter space – so the finding was also hailed as a major victory for string theory. String theory is a leading contender for a theory that can unify quantum mechanics and general relativity. However, we have found no experimental evidence of its many claims. This is why the AdS/CFT correspondence is also called the AdS/CFT conjecture.

Nonetheless, thanks to the correspondence, (mathematical) physicists have found that some problems that are hard on the ‘AdS’ side are much easier to crack on the ‘CFT’ side, and vice versa – all they had to do was cross Maldacena’s ‘bridge’! This was another sign that the AdS/CFT correspondence wasn’t just a mathematical trick but could be a legitimate description of reality.

So how could it be real?

The holographic principle

In 1997, Maldacena proved that a string theory in five dimensions was the same as a conformal field theory in four dimensions. However, gravity in our universe exists in four dimensions – not five. So the correspondence came close to providing a unified description of gravity and quantum mechanics, but not close enough. Nonetheless, it gave rise to the possibility that an entity that existed in some number of dimensions could be described by another entity that existed in one fewer dimensions.

Actually, in fact, the AdS/CFT correspondence didn’t give rise to this possibility but proved it, at least mathematically; the awareness of the possibility had existed for many years until then, as the holographic principle. The Dutch physicist Gerardus ‘t Hooft first proposed it and the American physicist Leonard Susskind in the 1990s brought it firmly into the realm of string theory. One way to state the holographic principle, in the words of physicist Matthew Headrick, is thus:

“The universe around us, which we are used to thinking of as being three dimensional, is actually at a more fundamental level two-dimensional and that everything we see that’s going on around us in three dimensions is actually happening in a two-dimensional space.”

This “two-dimensional space” is the ‘surface’ of the universe, located at an infinite distance from us, where information is encoded that describes everything happening within the universe. It’s a mind-boggling idea. ‘Information’ here refers to physical information, such as, to use one of Headrick’s examples, “the positions and velocities of physical objects”. In beholding this information from the infinitely faraway surface, we apparently behold a three-dimensional reality.

It bears repeating that this is a mind-boggling idea. We have no proof so far that the holographic principle is a real description of our universe – we only know that it could describe our reality, thanks to the AdS/CFT correspondence. This said, physicists have used the holographic principle to study and understand black holes as well.

In 1915, Albert Einstein’s general theory of relativity provided a set of complicated equations to understand how mass, the spacetime continuum and the gravitational force are related. Within a few months, physicists Karl Swarzschild and Johannes Droste, followed in subsequent years by Georges Lemaitre, Subrahmanyan Chandrasekhar, Robert Oppenheimer and David Finkelstein, among others, began to realise that one of the equations’ exact solutions (i.e. non-approximate) indicated the existence of a point mass around which space was wrapped completely, preventing even light from escaping from inside this space to outside. This was the black hole.

Because black holes were exact solutions, physicists assumed that they didn’t have any entropy – i.e. that its insides didn’t have any disorder. If there had been such disorder, it should have appeared in Einstein’s equations. It didn’t, so QED. But in the early 1970s, the Israeli-American physicist Jacob Bekenstein noticed a problem: if a system with entropy, like a container of hot gas, was thrown into the black hole, and the black hole doesn’t have entropy, where does the entropy go? It had to go somewhere; otherwise, the black hole would violate the second law of thermodynamics – that the entropy of an isolated system, like our universe, can’t decrease.

Bekenstein postulated that black holes must also have entropy, and that the amount of entropy is proportional to the black hole’s surface area, i.e. the area of the event horizon. Bekenstein also worked out that there is a limit to the amount of entropy a given volume of space can contain, as well as that all black holes could be described by just three observable attributes: their mass, electric charge and angular momentum. So if a black hole’s entropy increases because it has swallowed some hot gas, this change ought to manifest as a change in one, some or all of these three attributes.

Taken together: when some hot gas is tossed into a black hole, the gas would fall into the event horizon but the information about its entropy might appear to be encoded on the black hole’s surface, from the point of view of an observer located outside and away from the event horizon. Note here that the black hole, a sphere, is a three-dimensional object whereas its surface is a flat, curved sheet and therefore two-dimensional. That is, all the information required to describe a 3D black hole could in fact be encoded on its 2D surface – which evokes the AdS/CFT correspondence!

However, that the event horizon of a black hole preserves information about objects falling into the black hole gives rise to another problem. Quantum mechanics requires all physical information (like “the positions and velocities of physical objects”, in Headrick’s example) to be conserved. That is, such information can’t ever be destroyed. And there’s no reason to expect it will be destroyed if black holes lived forever – but they don’t.

Stephen Hawking found in the 1970s that black holes should slowly evaporate by emitting radiation, called Hawking radiation, and there is nothing in the theories of quantum mechanics to suggest that this radiation will be encoded with the information preserved on the event horizon. This, fundamentally, is the black hole information loss problem: either the black hole must shed the information in some way or quantum mechanics must be wrong about the preservation of physical information. Which one is it? This is a major unsolved problem in physics, and it’s just one part of the wider context that the AdS/CFT correspondence inhabits.

For more insights into this discovery, do read The Wire Science‘s interview of Maldacena.

I’m grateful to Nirmalya Kajuri for his feedback on this article.