The 3+1 formalism in teleparallel and symmetric teleparallel gravity

Abstract

Teleparallel and symmetric teleparallel gravity offer platforms in which gravity can be formulated in interesting geometric approaches, respectively given by 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 $$3+1$$ 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 $$3+1$$ 3 + 1 formulation for general relativity is recovered as well as the tetrad $$3+1$$ 3 + 1 formulation of the teleparallel equivalent of general relativity and the metric $$3+1$$ 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.

Comparing equivalent gravities: common features and differences

“…Another fundamental ingredient of TG theory is represented by the superpotential, whose expression is [93…”

Comparing Equivalent Gravities: common features and differences

“…Another fundamental ingredient of TG theory is represented by the superpotential, whose expression is [96…”

“…We have now all the elements to write the STEGR field equations (obtained varying the STEGR action with respect to the metric tensor), which reads as [79,96]…”

“…where G µν is the STEGR Einstein tensor. The variation of STEGR action with respect to the connection produces the connection field equations [79,96] ∇…”

“…In this perspective, performing the calculations to settle the solutions in the three gravity scenarios is useful also in view of extensions of the theories. Recently, it has been proposed a 3+1 splitting formalism in the geometric trinity of gravity [96] entailing the following advantages: (1) simplicity in carrying out numerical analyses;…”