Black Hole Horizons Exhibit Subtle Geometry at the Planck Scale

Philipp Dorau and colleagues at the Institut für Theoretische Physik, Universität Leipzig, have created a deformed algebraic field theory describing black hole horizons, incorporating principles of noncommutative geometry. The theory constructs this deformation using dilations and rotations, mirroring established techniques, and calculates the relative entropy between quantum states within this framework. The calculations reveal a key correction to the relative entropy dependent on the deformation parameter, suggesting that Planck-scale effects become sharply more important for black holes with smaller horizon areas, offering new insights into quantum gravity at these boundaries.

Quantifying information loss near black hole event horizons via second-order relative entropy

The relative entropy, a fundamental concept in quantum information theory quantifying the distinguishability between two quantum states, now exhibits a second-order correction of up to 2 in the deformation parameter, a substantial increase from previous calculations. Traditionally, calculations of relative entropy in the context of black hole horizons have relied on semiclassical approximations, neglecting the inherent quantum fluctuations expected at the Planck scale. These existing theoretical frameworks were unable to model these subtle effects. This threshold allows, for the first time, quantification of quantum information behaviour at the event horizon of black holes. This new level of detail reveals how Planck-scale affects modify black hole thermodynamics and provides a pathway to explore the interaction between gravity and quantum mechanics in extreme environments. The significance lies in the potential to resolve long-standing paradoxes, such as the information loss paradox, by providing a more complete description of the horizon’s quantum properties.

An algebraic framework for describing quantum fields on bifurcate Killing horizons has been established, modelling the horizon’s geometry as noncommutative. A bifurcate Killing horizon is a specific type of horizon found in stationary, axisymmetric spacetimes, such as those describing rotating black holes. Modelling the horizon as noncommutative implies that the coordinates describing the horizon do not commute, that is, xy ≠ yx, fundamentally altering the geometric structure at the Planck scale. A second-order correction of up to 2 in the deformation parameter was calculated when determining relative entropy, a parameter measuring information loss in theoretical models. This correction is particularly noteworthy as it indicates that the effects of the deformation become increasingly significant as the deformation parameter increases, suggesting a strong coupling between the noncommutative geometry and the quantum information content of the horizon. Building upon previous work unable to capture these subtle effects, this finding allows quantification of quantum information behaviour at the event horizon of black holes, a region previously inaccessible to detailed analysis. The team employed techniques from algebraic field theory, a rigorous mathematical framework for describing quantum fields, to ensure the consistency and validity of their results.

The team constructed a deformed algebraic field theory on bifurcate Killing horizons, modelling the horizon’s geometry as noncommutative through affine dilations and rotations; these symmetries form the basis of a new mathematical framework. Affine dilations represent transformations that scale the coordinates of the horizon, while rotations describe changes in orientation. These symmetries are crucial for preserving the underlying physics of the spacetime. Furthermore, the computed relative entropy remained strictly positive, indicating a stable quantum state even with these modifications to established theory. A strictly positive relative entropy is essential for ensuring the physical consistency of the model, as a negative value would imply the existence of non-normalizable states. While this work provides upward corrections to the Page curve, reflecting Planck-scale changes to black hole thermodynamics, it does not yet bridge the gap between these theoretical calculations and observable phenomena or practical applications of quantum gravity. The Page curve describes the evolution of entanglement entropy in a black hole, and modifications to this curve can provide insights into the fate of information falling into the black hole.

Algebraic deformation quantization unlocks novel black hole horizon modelling

This research offers a strong new tool for modelling quantum fields on black hole horizons, but it relies on an algebraic approach to deformation quantization, a technique with several established, yet competing, formulations. Fedosov, Kontsevich and Waldmann have each proposed distinct methods for achieving this ‘quantization’, altering the precise mathematical structures employed and potentially leading to different physical interpretations, as noted by Naren Manjunath from the Perimeter Institute and his colleagues. The choice of deformation quantization method can influence the specific form of the noncommutative geometry and the resulting physical predictions. Fedosov’s approach, for example, relies on a star product defined through differential forms, while Kontsevich’s method utilizes graph cohomology. Despite differing approaches to deformation quantization, a mathematical technique used to blend classical and quantum descriptions, this research delivers a key advance in understanding black hole horizons. The researchers’ specific implementation of deformation quantization, based on affine dilations and rotations, provides a unique perspective on the horizon’s quantum structure.

A novel algebraic framework successfully models horizons, effectively introducing a ‘noncommutative geometry’ where standard spatial relationships break down. This noncommutative geometry arises from the deformation of the classical algebra of functions on the horizon, replacing it with a deformed algebra where the product of functions is modified. Calculations of ‘relative entropy’, a measure of distinguishability between quantum states, reveal significant effects at smaller black hole sizes, potentially linking gravity to quantum phenomena. The fact that these effects are more pronounced for smaller black holes suggests that the Planck scale becomes increasingly relevant as the horizon area decreases. By deforming established quantum field theory using symmetries related to dilation, stretching or shrinking, and rotation, relative entropy was calculated to measure information loss and distinguish between quantum states. In particular, the resulting calculations reveal a second-order correction to this relative entropy, demonstrating the influence of quantum effects at the horizon and allowing for quantification previously impossible with existing theoretical tools. This work establishes a new algebraic framework for describing quantum fields on black hole event horizons, modelling the horizon’s geometry as ‘noncommutative’, meaning the usual rules of spatial relationships no longer apply. The implications of this noncommutative geometry extend beyond black hole physics, potentially offering insights into the fundamental nature of spacetime itself.