[Paper Review] A Method of Constructing Off-Diagonal Solutions in Metric-Affine and String Gravity

Sergiu I. Vacaru, Evghenii Gaburov+1 more|ArXiv.org|Oct 14, 2003

Advanced Differential Geometry Research7 references10 citations

TL;DR

This paper develops a generalized anholonomic frame method to construct exact off-diagonal solutions in metric-affine gravity (MAG) and string gravity, using nonlinear connection (N-connection) structures to model generic off-diagonal metrics and non-Riemannian geometries. The key contribution is proving that field equations reduce to effective Einstein-Proca systems with torsion, nonmetricity, and generalized Finsler-affine geometry, enabling exact solutions with 3–5 variables and locally anisotropic configurations.

ABSTRACT

The anholonomic frame method is generalized for non--Riemannian gravity models defined by string corrections to the general relativity and metric-affine gravity (MAG) theories. Such spacetime configurations are modeled as metric-affine spaces provided with generic off-diagonal metrics (which can not be diagonalized by coordinate transforms) and anholonomic frames with associated nonlinear connection (N-connection) structure. We investigate the field equations of MAG and string gravity with mixed holonomic and anholonomic variables. There are proved the main theorems on irreducible reduction to effective Einstein-Proca equations with respect to anholonomic frames adapted to N-connections. String corrections induced by the antisymmetric H-fields are considered. There are also proved the theorems and criteria stating a new method of constructing exact solutions with generic off-diagonal metric ansatz depending on 3-5 variables and describing various type of locally anisotropic gravitational configurations with torsion, nonmetricity and/or generalized Finsler-affine effective geometry. We analyze solutions, generated in string gravity, when generalized Finsler-affine metrics, torsion and nonmetricity interact with three dimensional solitons.

Motivation & Objective

• To extend the anholonomic frame method to non-Riemannian gravity models in metric-affine and string gravity with generic off-diagonal metrics.
• To formulate a variational formalism for field equations in metric-affine spaces equipped with N-connection structures.
• To prove that field equations irreducibly reduce to effective Einstein-Proca systems adapted to N-connections.
• To analyze string gravity corrections via H-fields and their role in generating generalized Finsler-affine geometries.
• To establish criteria and theorems for constructing exact solutions with 3–5 independent variables and nontrivial torsion, nonmetricity, and anisotropy.

Proposed method

• Generalizes the anholonomic frame method to metric-affine gravity (MAG) and string gravity by introducing nonlinear connection (N-connection) structures on generic off-diagonal metrics.
• Uses d-connection formalism with adapted frames to decompose geometric objects (torsion, curvature, nonmetricity) into N-connection-distinguished components.
• Derives field equations from Lagrangians in generalized Finsler-affine and metric-affine geometries, incorporating string corrections via H-fields.
• Applies irreducible decomposition theorems to reduce the full MAG and string gravity field equations to effective Einstein-Proca systems with N-connection adapted variables.
• Introduces generalized Lagrange- and Finsler-affine spaces (GLA, FA) with d-metrics and d-connections distorted by arbitrary torsion and nonmetricity fields.
• Employs d-tensor calculus and anholonomic structure to ensure all geometric quantities (curvature, torsion, nonmetricity) are N-adapted, preserving consistency in field equations.

Experimental results

Research questions

• RQ1 How can exact solutions be systematically constructed in metric-affine gravity with generic off-diagonal metrics and nontrivial torsion and nonmetricity?
• RQ2 In what way does the inclusion of N-connection structures allow for the reduction of MAG and string gravity field equations to effective Einstein-Proca systems?
• RQ3 What are the conditions under which string gravity corrections via H-fields generate generalized Finsler-affine geometries with nontrivial anholonomic structures?
• RQ4 How do the anholonomic frame method and N-connection formalism enable the construction of exact solutions depending on 3–5 variables in non-Riemannian gravity?
• RQ5 What are the criteria for distinguishing Einstein-Cartan and Einstein gravity limits within the generalized Finsler-affine framework?

Key findings

• The field equations of metric-affine gravity and string gravity with N-connection structures reduce irreducibly to effective Einstein-Proca systems, preserving anholonomic and anisotropic features.
• Exact solutions are constructed using generic off-diagonal metrics that cannot be diagonalized by coordinate transformations, depending on 3–5 independent variables.
• String gravity corrections via H-fields induce generalized Finsler-affine geometries, with interacting torsion, nonmetricity, and 3D solitonic configurations.
• The method proves that generalized Finsler-affine metrics, torsion, and nonmetricity can be consistently generated in string gravity through anholonomic configurations.
• The canonical d-connection and N-connection curvature are explicitly computed, showing that all geometric quantities (torsion, curvature, nonmetricity) are N-adapted, ensuring consistency in the field equations.
• The framework allows for the construction of exact solutions in MAG with effective variable cosmological constants and in string gravity with 3D solitons coupled to anisotropic gravitational fields.