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