On General Solutions for Field Equations in Einstein and Higher Dimension Gravity

Abstract

We prove that the Einstein equations can be solved in a very general form for arbitrary spacetime dimensions and various types of vacuum and non-vacuum cases following a geometric method of anholonomic frame deformations for constructing exact solutions in gravity. The main idea of this method is to introduce on (pseudo) Riemannian manifolds an alternative (to the Levi-Civita connection) metric compatible linear connection which is also completely defined by the same metric structure. Such a canonically distinguished connection is with nontrivial torsion which is induced by some nonholonomy frame coefficients and generic off-diagonal terms of metrics. It is possible to define certain classes of adapted frames of reference when the Einstein equations for such an alternative connection transform into a system of partial differential equations which can be integrated in very general forms. Imposing nonholonomic constraints on generalized metrics and connections and adapted frames (selecting Levi-Civita configurations), we generate exact solutions in Einstein gravity and extra dimension generalizations.

latex 2e, 11pt, 40 pages; it is a generalizaton with modified title, including proofs and additional results for higher dimensional gravity of the letter v1, on 14 pages; v4, with new abstract, modified title and up-dated references is accepted by Int. J. Theor. Phys

References in corpus (9)

• Finsler and Lagrange Geometries in Einstein and String Gravity
• Fedosov Quantization of Lagrange-Finsler and Hamilton-Cartan Spaces and Einstein Gravity Lifts on (Co) Tangent Bundles
• Deformation Quantization of Nonholonomic Almost Kahler Models and Einstein Gravity
• Parametric Nonholonomic Frame Transforms and Exact Solutions in Gravity
• Nonholonomic Ricci Flows, Exact Solutions in Gravity, and Symmetric and Nonsymmetric Metrics
• Curve Flows in Lagrange-Finsler Geometry, Bi-Hamiltonian Structures and Solitons
• Ricci Flows and Solitonic pp--Waves
• Einstein Gravity in Almost Kahler Variables and Stability of Gravity with Nonholonomic Distributions and Nonsymmetric Metrics
• Nonholonomic Ricci Flows and Running Cosmological Constant: I. 4D Taub-NUT Metrics

Cited by in corpus (25)

• Principles of Einstein-Finsler Gravity and Perspectives in Modern Cosmology
• Modified Dispersion Relations in Horava-Lifshitz Gravity and Finsler Brane Models
• Fractional Dynamics from Einstein Gravity, General Solutions, and Black Holes
• Critical Remarks on Finsler Modifications of Gravity and Cosmology by Zhe Chang and Xin Li
• Off-Diagonal Deformations of Kerr Black Holes in Einstein and Modified Massive Gravity and Higher Dimensions
• Fractional Nonholonomic Ricci Flows
• Finsler Branes and Quantum Gravity Phenomenology with Lorentz Symmetry Violations
• On General Solutions of Einstein Equations
• Schwarzschild-like solutions in Finsler-Randers gravity
• Black Holes with MDRs and Bekenstein-Hawking and Perelman Entropies for Finsler-Lagrange-Hamilton Spaces
• Decoupling of Field Equations in Einstein and Modified Gravity
• New Classes of Off-Diagonal Cosmological Solutions in Einstein Gravity
• A geometric method of constructing exact solutions in modified f(R,T)-gravity with Yang-Mills and Higgs interactions
• Heterotic Supergravity with Internal Almost-Kähler Spaces, Instantons for $SO(32)$, or $E_8\times E_8$, Gauge Groups, and Deformed Black Holes with Soliton, Quasiperiodic and/or Pattern-forming Structures
• Gravitational Field on the Lorentz Tangent Bundle: Generalized Paths and Field Equations
• Off-Diagonal Deformations of Kerr Metrics and Black Ellipsoids in Heterotic Supergravity
• Hidden Symmetries for Ellipsoid-Solitonic Deformations of Kerr-Sen Black Holes and Quantum Anomalies
• The Algebraic Index Theorem and Fedosov Quantization of Lagrange-Finsler and Einstein Spaces
• Quasi-Stationary Solutions in Gravity Theories with Modified Dispersion Relations and Finsler-Lagrange-Hamilton Geometry
• Principles of Einstein-Finsler Gravity and Cosmology
• Metric Compatible or Noncompatible Finsler-Ricci Flows
• Super-Luminal Effects for Finsler Branes as a Way to Preserve the Paradigm of Relativity Theories
• Nonholonomic Jet Deformations and Exact Solutions for Modified Ricci Soliton and Einstein Equations
• General off-diagonal integrability of metric and nonmetric geometric flow and Finsler-Lagrange-Hamilton modified Einstein equations
• A 1+5-dimensional gravitational-wave solution: curvature singularity and spacetime singularity