The structural equations of Cartan and the wave solution Einstein's equation

TL;DR

This paper investigates the conditions under which wave space metrics in Riemann-Cartan geometry solve Einstein's vacuum equations, utilizing Cartan's structural equations and differential geometry tools.

Contribution

It derives explicit conditions and formulas for wave solutions of Einstein's equations within Riemann-Cartan spaces, extending geometric analysis with detailed calculations.

Findings

Derived formulas for connection coefficients in wave solutions

Calculated curvature tensors for Riemann-Cartan wave metrics

Established conditions for wave metrics to satisfy Einstein's vacuum equations

Abstract

Work consists of introduction, two chapters, conclusion and four applications. In this work is examined the condition, with which the wave space metrics of Riemann- Cartan is the solution of Einstein equation in the void. Geometric structures were for this purpose studied on the differentiated variety: connectedness, curvature and twisting connectedness. For this was used this analytical apparatus of differential geometry as the calculation of exterior forms. In the first chapter the following concepts are examined: - variety; - vectors and 1- forms on the varieties; - metric tensor, connectedness and the covariant derivative; - form with the values in the vector spaces; - the structural equations of Cartan. The second chapter is dedicated to the presence of condition, during which wave certificate is the solution of Einstein equation in the void. Using the first and second structural equations of Cartan for the variety without twisting of that allotted by wave certificate, they were calculated: the 1- forms of connectedness and the coefficients of connectedness, and also Riemann tensor, Ricci tensor and scalar of curvature. In appendix 1 the derivation of formula for the coefficients of connected-ness in the space with the twisting is given. In appendix 2 the calculation of the components of the 1- forms of connectedness is carried out. In appendix 3 the calculations of the components of the 2- forms of curvature are presented. In appendix 4 the components of Ricci tensor are calculated.