Anisotropic spacetimes in $f(T,B)$ theory I: Bianchi I Universe

TL;DR

This paper explores the role of anisotropic spacetimes in the $f(T,B)$ teleparallel gravity theory, focusing on Bianchi I universes, analyzing their dynamics, solutions, and integrability properties.

Contribution

It investigates the global dynamics of $f(T,B)$-gravity in Bianchi I universes, especially for the case $f(T,B)=T+F(B)$, and examines the integrability of the field equations.

Findings

Identifies conditions under which $f(T,B)$-gravity addresses homogeneity.

Provides analytic solutions via Painlevé analysis.

Analyzes the integrability of the field equations.

Abstract

That is the first part of a series of studies on analyzing the higher-order teleparallel theory of gravity known as the $f(T, B)$-theory. This work attempts to understand how the anisotropic spacetimes are involved in the $f(T, B)$-theory. In this work, we review the previous analysis of $f(T, B)$-gravity, and we investigate the global dynamics in the case of a locally rotational Bianchi I background geometry. We focus in the case of $f\left( T,B\right) =T+F\left( B\right) $ theory and we determine the criteria where the $f\left( T,B\right) $-theory solves the homogeneity problem. Finally, the integrability properties for the field equations are investigated by applying the Painlev\'{e} analysis. The analytic solution is expressed by a right Painlev\'{e} expansion.