Modified teleparallel theories of gravity in symmetric spacetimes

Abstract

Teleparallel gravity theories employ a tetrad and a Lorentz spin connection as independent variables in their covariant formulation. In order to solve their field equations, it is helpful to search for solutions which exhibit certain amounts of symmetry, such as spherical or cosmological symmetry. In this article we present how to apply the notion of spacetime symmetries known from Cartan geometry to teleparallel geometries. We explicitly derive the most general tetrads and spin connections which are compatible with axial, spherical, cosmological and maximal symmetry. For homogeneous and isotropic spacetime symmetry we find that the tetrads and spin connection found by the symmetry constraints are universal solutions to the anti-symmetric part of the field equations of any teleparallel theory of gravity. In other words, for cosmological symmetry we find what has become known as "good tetrads" in the context of f (T )

Connection independent formulation of general relativity

“…The gravity action (12) has been presented by using the connection-independence in gravity. The action contains the scalar W in addition to the Ricci scalar R. The scalar W can be written in terms of the torsion tensor and the non-metricity tensor, as shown in (26).…”

“…We investigate the variation of the action (12) with respect to the connection and the metric independently. It is convenient to rewrite (12), instead of varying (12) itself. Using (11), we can write the R + W as…”

“…3. The formulation of this paper: The action is defined by (12). The connection and the metric are regarded as independent variables.…”

“…This means that the connection is uniquely determined as the Levi-Civita connection. Therefore, the action (12) reduces to S GR , which is given by the first term in (14).…”

“…Teleparallel gravity [4] is a special case of (32). Its theoretical structure, extensions, and physical applications have been studied in the literature [5][6][7][8][9][10][11][12][13]. It uses the Weitzenböck connection [14][15][16], which leads to nonzero torsion but zero curvature [15].…”