The 3+1 formalism in teleparallel and symmetric teleparallel gravity

Abstract

In this dissertation, both a tetrad and a metric 3+1 formulation for a general affine connection while also assuming metricity is developed. By employing a space and time split of the usual space time manifold, a spatial version of the fundamental variables is obtained. Finding the Gauss-like equations for any tensor through which gravity is expressed, a general foundation for the two formalisms is set up. Using this foundation the general form of the evolution equations of the 3-tetrad and 3- metric as they are dragged along the normal vector to the spatial foliations are derived. Finally through the choice of two different connections assuming metricity, and another case assuming the coincident gauge with non-metricity, the relevant 3+1 formulations for General Relativity, the Teleparallel Equivalent of General Relativity and the Symmetric Teleparallel Equivalent of General Relativity are respectively derived up to the latest state of the research. By obtaining the 3+1 formalisms with respect to each of these three different geometric interpretations of gravity we achieve what is called the 3+1 formalism in the geometric trinity of gravity. Building on the fully consistent system of equations obtained in the Symmetric Teleparallel Equivalent of General Relativity a more stable structure for this system is derived in the form of a BSSN-like formalism.