Extending the classical black hole inequality to the quantum domain

A recent study published in Physical Review Letters investigates quantum effects on the thermodynamics and geometry of black holes, focusing on extending two classical inequalities to the quantum realm.

Black holes have been thoroughly studied through classical approaches based on Einstein’s theory of general relativity. However, this approach does not take into account quantum effects such as Hawking radiation.

The goal of this study was for the researchers to refine classical theory by including quantum effects, thereby improving our understanding of black hole dynamics.

The research team included Dr. Antonia M. Frassino, Marie Curie Fellow at SISSA (Italy), Dr. Robie Hennigar, Assistant Professor and Wilmore Fellow at Durham University (UK), and Dr. Juan F. Pedraza, Assistant Professor at the Institute. I was there. Física Teórica UAM/CSIC (Spain) and King’s College London Research Fellow Dr. Andrew Svesko (UK).

Phys.org spoke to researchers about studying quantum inequalities to study the dynamics of black holes.

The motivation for their research was stated by Dr. Frassino, who said, “My interest in black hole thermodynamics dates back to my Ph.D.” “It helped establish boundaries.”

“I have been studying the influence of quantum effects on black holes for many years, but recently I have become interested in gravitational singularities and how quantum effects interact within them,” said Dr. Hennigar. said.

Dr. Pedraza said, “While my research for the past 15 years has focused on black holes, recent advances in holography allow us to study quantum effects on black hole physics in a more controlled and detailed manner. ”.

Dr. Svesko said, “For most of my career I have been interested in quantum effects on black holes as a window into quantum gravity, and I have finally found a team and approach to tackle this problem.” .

Forecasting space censorship

Inside a typical black hole, there exists a region of infinite density known as a singularity. At a singularity, quantum mechanics and gravity collapse, making it difficult to understand the laws of physics.

According to the cosmic censor’s predictions, the singularity is hidden behind the event horizon of a black hole. The event horizon marks the boundary where even light cannot escape the black hole’s strong gravity.

This conjecture helps maintain the predictability of the physics of the universe by ensuring that bare singularities are invisible and do not reveal breakdowns in physics.

In certain instances, classical physics cannot enforce cosmic censorship. For example, in a three-dimensional scenario (two spatial and one temporal dimension), a bare cone singularity can occur.

In such cases, scientists hypothesize that quantum effects cover the singularity by generating an event horizon. This leads to the Penrose inequality, which provides a framework for understanding the relationship between a black hole’s horizon and its spacetime mass.

Penrose and inverse isoperimetric inequality

“Roughly speaking, Penrose’s inequality prescribes a lower bound on the mass contained in spacetime with respect to the area of ​​the horizon of a black hole contained in spacetime,” the researchers explained.

In other words, the classic Penrose inequality provides a relationship between the mass of a black hole and the surface area of ​​the event horizon, placing a constraint or limit on the minimum mass a black hole can have.

The idea of ​​the quantum Penrose inequality could extend this concept and limit the space-time energy to the total entropy of the black hole and quantum matter. Extending this inequality to the quantum realm has been pursued in four dimensions and beyond, but remains computationally limited.

A related inequality, known as the inverse isoperimetric inequality, describes the relationship between the volume bounded by a black hole’s event horizon and its surface area. Similar to the Penrose inequality, researchers aim to extend this concept to the quantum regime.

Previous attempts have been difficult to apply to the three-dimensional case and have been successful only for small perturbations. Another significant limitation is the handling of strong quantum back-reactions.

Reverse reactions refer to the effects of matter and energy on the curvature of spacetime (the structure of the universe), as described by Einstein’s theory of general relativity. Simply put, it is a feedback loop between matter, energy, and the geometry of space-time.

Brainworld holography

The researchers used Braneworld holography, also known as double holography, as a framework for studying quantum black holes.

“Braneworld holography exploits holographic principles to obtain exact solutions to the semiclassical gravitational equations, including inverse reactions for all orders. This formalism reduces the problem to three dimensions, or “This is the only known method that can deal with all orders of magnitude in higher dimensions,” the researchers explained.

Based on the AdS/CFT correspondence, researchers studied quantum effects and corrections in AdS space. AdS (Anti-de Sitter space) is a spacetime with negative curvature (hyperbola), which is particularly useful when studying gravitational theories related to black holes. CFT (conformal field theory) is a type of quantum field theory that describes the behavior of fundamental particles without the influence of gravity.

The AdS/CFT correspondence suggests a duality between the study of gravity in AdS space and the behavior of fundamental particles in lower dimensions. Essentially, we can study gravity by examining quantum fields in low-dimensional space, and vice versa.

Furthermore, the AdS space allows a well-defined treatment of black holes and singularities at the boundaries.

They specifically focused on the BTZ (Banados-Teitelboim-Zanelli) black hole, a black hole in three-dimensional spacetime associated with AdS space. BTZ black holes are useful models to study quantum corrections and back-reaction effects due to their simplicity and well-understood behavior in a holographic framework.

Holographic approaches help explain quantum backreactions, which are the feedback effects of quantum matter on the curvature of spacetime.

address the gap

Researchers have successfully extended the classical Penrose and inverse isoperimetric inequalities to explain quantum effects. Their proposed version applies to all known black holes in three-dimensional AdS space, no matter what order of quantum back-reactions.

The quantum Penrose inequality suggests a type of quantum cosmic censorship.

“Our study provides two limits that apply not only to black hole entropy, but also to generalized entropy: a combination of black hole entropy and the entropy of the matter field outside it.

“This study suggests that a bare singularity is formed when the entropy of the black hole and matter exceeds the total energy of spacetime,” the researchers explained.

The researchers investigated the effect of dimensionality reduction on inequalities and suggested that Penrose-type inequalities can be derived for two-dimensional extended black holes. However, they pointed out that it is difficult to find an exact solution for higher-dimensional brainworld black holes.

In the case of the inverse isoperimetric inequality, researchers found that black holes that violate this inequality (known as superentropic black holes) are thermodynamically unstable. Even when quantum effects come into play, the stability of a black hole still strongly depends on its thermodynamic volume.

Regarding the impact of their work on the field of quantum information, the researchers said, “The consequences of both the quantum Penrose inequality and the quantum isoperimetric inequality can be understood as entropy limits.”

“Since entropy is an essentially information-theoretic quantity, it provides evidence of a fundamental limit of quantum information theory in the presence of gravity. These ideas can affect quantum information. It is quite possible.”

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