Block-Structure Method for the Solution of the Matrix System of Equations g{ij}g{jk}=delta{i}{k} in the N-dimensional Case

TL;DR

This paper introduces a novel block-structure method to transform the matrix system g_{ij}g_{jk}=delta_{ik} into a linear algebraic system in N dimensions, facilitating solutions in gravity theory and higher-dimensional models.

Contribution

The paper presents a new analytical block-structure method for solving the matrix system in N-dimensional gravity theory, revealing its structure for symmetrical matrices and potential applications in Kaluza-Klein theories.

Findings

Structured linear system in N-dimensional case derived

Method applicable to symmetrical matrices in gravity theory

Potential extension to non-symmetrical matrices and graviton modes

Abstract

In this paper a new block-structure method is presented for the solution of the well-known from gravity theory matrix system of equations g{ij}g{jk}=delta{i}{k} (with respect to the unknown covariant components g{ij} and by known contravariant ones g{jk}) by transforming this matrix system into a linear algebraic system of equations in the general N-dimensional case. Although powerful computer methods exist for the solution of this problem for a given (fixed) dimension of the matrices g{ij} and especially for numerical elements of g{ij}, the structure of the obtained linear algebraic system in the general N-dimensional case and for arbitrary elements of g{ij} (functions) has not been known. The proposed new analytical block-structure method for the case of symmetrical matrices g{ij} and g{jk} (the standard case in gravity theory) is based on the construction of a block-structure matrix, whose "elements" are again matrices. The method allows to obtain the structure of this linear system in the general N-dimensional case, after multiplication (to the left) with the transponed matrix. Some arguments are given why the proposed method may be applied, after some refinement and generalization for the case of non-symmetrical matrices g{ij} and g{jk}, for finding the graviton modes in the Kaluza-Klein expansion in theories with extra dimensions.