Extending classical black hole inequalities into the quantum realm

A recent study in Physical Review Letters explores quantum effects on black hole thermodynamics and geometry, focusing on extending two classical inequalities into the quantum regime.

Black holes have been thoroughly studied through a classical approach based on Einstein’s general theory of relativity. However, this approach does not account for quantum effects like Hawking radiation.

The goal of the study was for the researchers to refine classical theories by including quantum effects, thereby offering an improved 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 Willmore Fellow at Durham University (UK), Dr. Juan F. Pedraza, Assistant Professor at the Instituto de Física Teórica UAM/CSIC (Spain), and Dr. Andrew Svesko, Research Associate at King’s College London (UK).

Phys.org spoke to the researchers about their work with quantum inequalities for studying black hole dynamics.

The motivation for their study was expressed by Dr. Frassino, who said, “My fascination with black hole thermodynamics goes back to my Ph.D. This project helped us establish universal bounds to guide studies of quantum effects in curved spacetime.”

Dr. Hennigar said, “I’ve long researched the influence of quantum effects on black holes, and lately, I’ve taken an interest in gravitational singularities and how quantum effects may play a role in them.”

Dr. Pedraza remarked, “My research over the past 15 years has focused on black holes, and recent advancements in holography have allowed us to study quantum effects on black hole physics in a more controlled and detailed way.”

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

The cosmic censorship conjecture

Inside a typical black hole exists a region of infinite density known as a singularity. At singularities, the breakdown of quantum mechanics and gravity challenges our comprehension of the laws of physics.

According to the cosmic censorship conjecture, singularities are hidden behind black hole event horizons. An event horizon marks the boundary beyond which not even light can escape the strong gravitational pull of the black hole.

The conjecture helps maintain the predictability of physics in the universe by ensuring that naked singularities aren’t visible and don’t expose the breakdown of physics.

In specific instances, classical physics fails to enforce cosmic censorship. For example, in a three-dimensional scenario (two spatial dimensions and one temporal dimension), naked conical singularities can occur.

In such cases, scientists hypothesize that quantum effects would cover the singularities by creating event horizons. This leads us to the Penrose inequality, which provides a framework for understanding the relationship between black hole horizons and spacetime mass.

The Penrose and reverse isoperimetric inequalities

“Roughly speaking, the Penrose inequality provides a lower bound on the mass contained in spacetime in terms of the area of black hole horizons contained within said spacetime,” explained the researchers.

In other words, the classical Penrose inequality provides a relationship between the mass of a black hole and the surface area of the event horizon, placing a constraint or bound on the minimum mass a black hole can possess.

The idea of a quantum Penrose inequality extends this concept, potentially bounding spacetime energy with the total black hole and quantum matter entropy. Extending this inequality to the quantum regime has been pursued in dimensions 4 and higher but remains computationally limited.

A related inequality, known as the reverse isoperimetric inequality, provides a relationship between the volume enclosed by the event horizon of a black hole and its surface area. Like the Penrose inequality, researchers aim to extend this concept to the quantum regime.

Previous attempts have struggled when applied to three-dimensional cases and only succeeded for small perturbations. Another significant limitation has been handling strong quantum backreactions.

Backreaction refers to the effect that matter and energy have on the curvature of spacetime (the fabric of the universe), as described by Einstein’s theory of general relativity. In simple terms, it’s the feedback loop between matter, energy, and spacetime geometry.

Braneworld holography

The researchers used a framework to study quantum black holes, as it were, using braneworld holography, also known as double holography.

“Braneworld holography leverages the holographic principle to obtain an exact solution to semi-classical gravitational equations, including backreaction to all orders. This formalism is the only known way this problem can be addressed to all orders in three, or, in principle, higher dimensions,” explained the researchers.

The researchers used the AdS/CFT correspondence as a foundation to study quantum effects or corrections in AdS space. AdS (Anti-de Sitter space) is a spacetime having a negative curvature (hyperbolic) and is particularly helpful while studying gravitational theories associated with black holes. CFT (Conformal field theory) is a type of quantum field theory that describes the behavior of fundamental particles but without the influence of gravity.

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

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

They specifically focused on BTZ (Banados-Teitelboim-Zanelli) black holes, which are black holes in three-dimensional spacetime associated with AdS space. BTZ black holes are a useful model for studying quantum corrections and backreaction effects because of their simplicity and well-understood behavior in the holographic framework.

The holographic approach helps them account for quantum backreactions, which are the feedback effects of quantum matter on the curvature of spacetime.

Addressing the gaps

The researchers successfully extended the classical Penrose and reverse isoperimetric inequalities to account for quantum effects. Their proposed version holds for all known black holes in three-dimensional AdS space, even with any order of quantum backreaction.

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

“Our work provides two bounds that hold not just for black hole entropy but for generalized entropy—the combination of black hole entropy and the entropy of matter fields to the outside of it.

“The research suggests that if the entropy of black holes plus matter were to exceed the total energy of the spacetime, then a naked singularity would form,” explained the researchers.

The researchers explored the effects of dimensional reduction on the inequalities, suggesting that Penrose-type inequalities can be derived for two-dimensional dilatonic black holes. However, they noted difficulty in finding exact solutions to braneworld black holes in higher dimensions.

For the reverse isoperimetric inequality, the 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 black holes still hinges significantly on the thermodynamic volume.

Speaking of the effect of their work on the field of quantum information, the researchers said, “Both our results—the quantum Penrose inequality and the quantum isoperimetric inequality—can be understood as entropy bounds.

“Entropy is inherently an information-theoretic quantity, and we therefore provide evidence for fundamental bounds in quantum information theory when gravity is present. It’s entirely plausible that these ideas could have a bearing on quantum information.”

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