Spacetime Curvature Backreaction Corrects BTZ Black Hole Properties, Revealing Mass-Dependent Geodesics

The interplay between quantum mechanics and gravity remains one of the most challenging problems in modern physics, and recent research explores how quantum effects might subtly alter the fabric of spacetime itself. Partha Nandi from Stellenbosch University, Mainak Roy from Ramakrishna Mission Vivekananda Educational and Research Institute, and Langa Horoto, along with Frederik G. Scholtz and Biswajit Chakraborty, investigate this connection by examining how the curvature of momentum space, a fundamental concept in quantum theory, influences classical gravity. Their work demonstrates that these quantum kinematic effects can indeed backreact on spacetime, leading to modifications of black hole geometry and potentially observable signatures in the behaviour of particles. This research provides a concrete framework for understanding how Planck-scale quantum phenomena leave measurable imprints on the classical world, offering a crucial step towards a more complete theory of quantum gravity.

Curved Momentum Space and Quantum Gravity

This body of work explores the intriguing idea that momentum space, traditionally considered flat and infinite, may possess its own geometric structure. Scientists investigate how this “curved momentum space” could fundamentally alter our understanding of quantum gravity, non-commutative geometry, and even special relativity. The central argument proposes that the standard picture of momentum space is not necessarily fundamental, and its geometry could arise from underlying quantum gravity effects or the principles of non-commutative geometry. This curvature of momentum space can lead to modifications of special relativity, often manifesting as alterations to the relationship between energy and momentum.

These changes can result in phenomena like doubly special relativity, where a minimum length scale exists, or modified dispersion relations, which affect how quickly different wavelengths of light travel. Non-commutative geometry provides a powerful mathematical framework for describing this curved momentum space, allowing scientists to represent its geometry using algebraic tools where the order of operations matters. This framework has significant implications for understanding extreme environments like black holes, and for phenomena like Hawking radiation and the Unruh effect. The research connects several key concepts, starting with the idea of curved momentum space, where momentum is treated as a coordinate on a curved manifold.

Non-commutative geometry provides the mathematical language to describe this curvature, while doubly special relativity proposes modifications to special relativity based on a minimum length scale. Modified dispersion relations, arising from curved momentum space or doubly special relativity, change the relationship between energy and momentum, potentially leading to observable effects. The spectral action principle and asymptotically safe gravity offer potential avenues for realizing these ideas. This research draws upon foundational work by pioneers like Golfand, Tamm, and Connes, as well as contemporary contributions from researchers such as Amelino-Camelia and Kowalski-Glikman. Overall, this work suggests that curved momentum space is a promising avenue for exploring the fundamental nature of spacetime and gravity, offering a highly theoretical but potentially insightful approach to understanding the universe.

Momentum Curvature and Classical Backreaction Effects

Scientists have developed a novel approach to investigate how the curvature of momentum space influences classical physics. Working within a two-dimensional framework with a negative cosmological constant, they engineered a first-order action, which allowed them to derive an effective configuration-space action. This revealed that geodesics are mass-dependent, and that the equivalence principle experiences a subtle violation. This method provides a detailed understanding of how spacetime features “backreact” on classical systems, linking the geometry of momentum space to observable physical effects.

The team developed a Lagrangian for a massive point particle moving in classical spacetime, incorporating a constraint linking momentum and mass. This formulation naturally yields a symplectic structure, defining the relationships between position and momentum, and allowing the researchers to identify canonical coordinates on momentum space. These coordinates were then used to calculate Lorentz generators, confirming the preservation of Poincaré symmetry, even with the introduced deformations. Scientists then defined a deformed dispersion relation, linking particle mass to the geometry of momentum space, by solving a modified d’Alembertian operator. This technique reveals how the curvature of momentum space modifies the relationship between energy and momentum, leading to corrections in a black hole’s mass, temperature, and entropy. The team computed the return time of a massless particle along null geodesics, illustrating observable signatures of the underlying spacetime structures and providing a framework for understanding Planck-scale kinematic modifications.

Momentum Space Curvature and Black Hole Corrections

Scientists demonstrate how features of momentum space curvature can influence classical gravity within a semiclassical framework, working with a two-dimensional system possessing a negative cosmological constant. Starting with a specific noncommutative spacetime algebra, the team reveals a duality between noncommutative spacetime and curved momentum space, establishing an anti-de Sitter geometry on momentum space itself. This work builds upon Max Born’s principle of phase space reciprocity, suggesting symmetry between position and momentum as a fundamental link between quantum theory and gravity. The research yields a corrected black hole solution when coupling the modified matter source to Einstein gravity, demonstrating explicit corrections to the ADM mass, Hawking temperature, and entropy, all dictated by the geometry of momentum space.

These corrections represent a significant departure from standard black hole thermodynamics. Furthermore, calculations of the return time of a massless particle traveling along null geodesics reveal tangible semiclassical effects stemming from the underlying quantum-spacetime features. Experiments show that Planck-scale kinematic modifications can indeed leave imprints on classical geometry, providing a framework to connect quantum gravity ideas with observable consequences. The findings confirm that the mass of fundamental particles is bounded, potentially by the Planck mass, if the curved momentum space originates from a Lie-algebraic type of noncommutative spacetime. This research establishes a novel connection between quantum dynamics, geometric structures, and the emergence of Lorentz covariance, offering a promising avenue for exploring the foundations of quantum gravity.

Momentum Space Curvature Corrects Black Hole Properties

This research investigates how the curvature of momentum space influences classical physics within a simplified, two-dimensional framework. The study demonstrates that features of momentum space can affect classical behaviour, leading to mass-dependent geodesics and a subtle deviation from the equivalence principle. By incorporating these modifications into calculations involving black holes, the team derived a corrected model of a rotating black hole, revealing that its mass, temperature, and entropy all experience corrections linked to the geometry of momentum space. The results provide a concrete, semiclassical framework illustrating how modifications at the Planck scale can leave observable imprints on classical geometry. Specifically, the team calculated the return time of a massless particle travelling along specific paths, revealing potential observable signatures of the underlying momentum-space structure.