Teleparallel equivalent of Gauss-Bonnet gravity and its modifications
Abstract
Inspired by the teleparallel formulation of General Relativity, whose Lagrangian is the torsion invariant T, we have constructed the teleparallel equivalent of Gauss-Bonnet gravity in arbitrary dimensions. Without imposing the Weitzenbock connection, we have extracted the torsion invariant TG, equivalent (up to boundary terms) to the Gauss-Bonnet term G. TG is constructed by the vielbein and the connection, it contains quartic powers of the torsion tensor, it is diffeomorphism and Lorentz invariant, and in four dimensions it reduces to a topological invariant as expected. Imposing the Weitzenbock connection, TG depends only on the vielbein, and this allows us to consider a novel class of modified gravity theories based on F(T,TG), which is not spanned by the class of F(T) theories, nor by the F(R,G) class of curvature modified gravity. Finally, varying the action we extract the equations of motion for F(T,TG) gravity.