[Paper Review] Noncommutative Symmetries and Stability of Black Ellipsoids in Metric--Affine and String Gravity

Sergiu I. Vacaru, Evghenii Gaburov|ArXiv.org|Oct 14, 2003

Advanced Differential Geometry Research10 references4 citations

TL;DR

This paper constructs exact solutions for static and stationary black ellipsoid configurations in metric–affine and string gravity using off–diagonal metrics with noncommutative symmetries via anholonomic frames. It demonstrates that such anisotropic black holes are stable under small eccentricity perturbations, governed by one-dimensional Schrödinger-type equations, and are compatible with cosmic censorship and black hole thermodynamics despite lacking Killing symmetries.

ABSTRACT

We construct new classes of exact solutions in metric--affine gravity (MAG) with string corrections by the antisymmetric $H$--field. The solutions are parametrized by generic off--diagonal metrics possessing noncommutative symmetry associated to anholonomy framerelations and related nonlinear connection (N--connection) structure. We analyze the horizon and geodesic properties of a class of off--diagonal metrics with deformed spherical symmetries. The maximal analytic extension of ellipsoid type metrics are constructed and the Penrose diagrams are analyzed with respect to adapted frames. We prove that for small deformations (small eccentricities) there are such metrics that the geodesic behaviour is similar to the Schwarzcshild one. The new class of spacetimes do not possess Killing symmetries even in the limits to the general relativity and, in consequence, they are not prohibited by black hole uniqueness theorems. Such static ellipsoid (rotoid) configurations are compatible with the cosmic cenzorship criteria. We study the perturbations of two classes of static black ellipsoid solutions of four dimensional gravitational field equations. We conclude that such anisotropic black hole objects may be stable with respect to the perturbations parametrized by the Schrodinger equations in the framework of the one--dimensional inverse scattering theory.

Motivation & Objective

• To construct exact solutions for black ellipsoids in metric–affine and string gravity with generic off–diagonal metrics.
• To analyze the stability of these black ellipsoid configurations under small eccentricity deformations.
• To investigate whether such solutions, lacking Killing symmetries, can satisfy cosmic censorship and black hole thermodynamics.
• To extend the inverse scattering method for perturbations to off–diagonal, anholonomic spacetimes with noncommutative symmetries.
• To explore the geometric and thermodynamic properties of anisotropic horizons in vacuum gravitational backgrounds with off–diagonal structure.

Proposed method

• Utilizes the anholonomic frame method to generate exact solutions for off–diagonal metrics depending on 2–3 variables in gravity theories.
• Introduces nonlinear connection (N–connection) structures and anholonomic relations to model noncommutative symmetries in spacetime geometry.
• Applies the formalism to metric–affine gravity (MAG) with string corrections, including the antisymmetric H–field and cosmological constant terms.
• Performs maximal analytic extensions of static ellipsoid-type metrics and constructs Penrose diagrams for horizon and geodesic analysis.
• Reduces perturbation equations to one-dimensional Schrödinger-type equations using adapted frames and inverse scattering theory.
• Models perturbations via axial and polar metric deformations, preserving the structure of the wave equations under small eccentricity approximations.

Experimental results

Research questions

• RQ1 Can static black ellipsoid solutions exist in metric–affine and string gravity without Killing symmetries?
• RQ2 Do such black ellipsoids remain stable under small eccentricity deformations, and can their perturbations be described by Schrödinger-type equations?
• RQ3 How do geodesic behaviors and horizon structures of off–diagonal ellipsoid metrics compare to those of Schwarzschild and Reissner–Nordström spacetimes?
• RQ4 Can a consistent thermodynamics be formulated for black ellipsoids with anisotropic horizons and off–diagonal metrics?
• RQ5 Is the inverse scattering method applicable to stability analysis of off–diagonal, anholonomic spacetimes with noncommutative symmetries?

Key findings

• New exact solutions for black ellipsoids are constructed in metric–affine and string gravity using off–diagonal metrics with noncommutative symmetries and nonlinear connection structures.
• For small eccentricities, geodesic behavior of black ellipsoids closely resembles that of the Schwarzschild solution, supporting their physical viability.
• The maximal analytic extension of ellipsoid-type metrics is achieved, and Penrose diagrams confirm causal structure compatibility with known black hole solutions.
• Stability under axial and polar perturbations is proven via one-dimensional Schrödinger equations derived from the inverse scattering framework.
• Black ellipsoids are compatible with cosmic censorship and can be assigned a horizon area and entropy, enabling a thermodynamic description despite anisotropy.
• Perturbations of the vacuum gravitational configurations remain governed by similar equations as in diagonal holonomic cases, with modifications due to anholonomic frames and anisotropy functions.