# Studying quantum gravity with the holographic principle

**29 Jul. 2021**

Mathematical representation of an hyperbolic disk, corresponding to a two-dimensionnal anti-de Sitter space.

Charles Marteau won the honorable mention of the Daniel Guinier Prize of the French Physical Society (SFP). He completed his doctorate at the Centre for Theoretical Physics (CPHT*) under the supervision of Marios Petropoulos. His work, carried out with another CPHT doctoral student, Luca Ciambelli, is part of the general framework quantum gravitation, a field where a complete theory is yet to be found.

On the one hand, 'classical' gravitation is now described by the theory of general relativity developed by Albert Einstein over a century ago. In this well-established theory, space-time is distorted by matter and energy, which explains many phenomena such as the movements of stars or the existence of black holes. On the other hand, quantum theory governs the microscopic laws of the particles that make up our universe (photons, electrons, quarks, etc.) and builds the standard model of particle physics. However, these two aspects of physical reality remain to be put together into a coherent theory. A few approaches to quantum gravitation exist, such as loop quantum gravity or string theory, but none of them is satisfactory - let alone experimentally confirmed. In fact, it is difficult to study these theories in depth. "For example, trying to describe a black hole using string theory is an extremely difficult task," explains Charles Marteau.

Holography

In his thesis, he was interested in the holographic principle, another angle of attack established by the physicist Juan Maldacena and developed extensively in recent years. To understand it, it is necessary to recall the notion of curvature of a space-time. The abstract spacetimes studied in physics have high dimensions (more than four) and can have positive, zero, or negative curvature. In particular, spacetimes with negative curvature have a kind of edge and are called "anti-de Sitter" spaces. This boundary is itself a three-dimensional spacetime. The holographic principle states that any theory of quantum gravitation in an anti de Sitter space is mathematically equivalent to another theory defined only on the boundary. This is called conformal field theory (CFT). "In other words, to probe quantum gravitation, a highly unknown object, we can study a slightly simpler object instead," summarises Charles Marteau.

There are two reasons why this principle is fascinating. Firstly, this somewhat simpler conformal theory is located only at the boundary of space-time, without losing any information. This is what gives the holographic principle its name by analogy with optical holography, in which a suitably illuminated two-dimensional surface projects the image of a three-dimensional object. "The second reason is that we know this CFT well. It is quite similar to the Standard Model, in other words its elementary constituents are particles of the same nature as photons or electrons, but it does not include gravity as a fundamental ingredient. To know that hidden in this theory is gravitation and black holes is almost unbelievable.”

Beware, anti-de Sitter spaces do not correspond to our Universe, whose curvature is slightly positive and very close to zero according to observations. Holography is not reality: it offers a tool to physicists, a kind of virtual laboratory in which they can carry out thought experiments by manipulating, for example, quantum black holes, in order to try to reveal their profound nature. While 'classical' black holes already exist in the theory of general relativity, many scientists believe that they are not the end of the story. A quantum description of black holes would, among other things, make it possible to solve the information paradox.

Carrollian fluids

Using the holographic principle, scientists are looking for the equivalent of a quantum black hole in the conformal field theory. "It is a simpler object, a kind of hot body composed of particles at a certain temperature, called the Hawking temperature," explains Charles Marteau. However, the problem remains complex. This is why we are studying limiting cases that allow us to simplify the problem even further. At low energy, we can analyse how a black hole reacts to a perturbation, when energy is added to it, for example. "In the conformal theory, we find that the dynamics of this perturbation is the same as that of a fluid. Charles Marteau's thesis, which was awarded a prize by the SFP, is based in depth on this fluid/gravity correspondence.

Charles Marteau's work also relates this to another limiting case, reached when the curvature of the anti-de Sitter space tends to zero, in order to approximate to the Universe as observed. They show that, in the case where spacetime is nearly flat, one can nevertheless define a boundary on which the conformal theory and the tools of holography are still operational. In doing so, this work also points out that at this limit, the speed of light in the conformal theory becomes zero. This is called the "Carrollian" limit in reference to Lewis Carroll, because in such a world it would be difficult to distinguish cause from effect, as in some passages of the novel Alice in Wonderland. Charles Marteau is currently pursuing his post-doctoral research at the University of British Columbia in Vancouver.

>More information on SFP's website

*CPHT: a joint research unit CNRS, École Polytechnique - Institut Polytechnique de Paris