The $3+1$ Formalism in the Geometric Trinity of Gravity

Abstract: The geometric trinity of gravity offers a platform in which gravity can be formulated in three analogous approaches, namely curvature, torsion and nonmetricity. In this vein, general relativity can be expressed in three dynamically equivalent ways which may offer insights into the different properties of these decompositions such as their Hamiltonian structure, the efficiency of numerical analyses, as well as the classification of gravitational field degrees of freedom. In this work, we take a $3+1$ decomposition of the teleparallel equivalent of general relativity and the symmetric teleparallel equivalent of general relativity which are both dynamically equivalent to curvature based general relativity. By splitting the spacetime metric and corresponding tetrad into their spatial and temporal parts as well as through finding the Gauss-like equations, it is possible to set up a general foundation for the different formulations of gravity. Based on these results, general $3$-tetrad and $3$-metric evolution equations are derived. Finally through the choice of the two respective connections, the metric $3+1$ formulation for general relativity is recovered as well as the tetrad $3+1$ formulation of the teleparallel equivalent of general relativity and the metric $3+1$ formulation of symmetric teleparallel equivalent of general relativity. The approach is capable, in principle, of resolving common features of the various formulations of general relativity at a fundamental level and pointing out characteristics that extensions and alternatives to the various formulations can present.