Primary constraints in general teleparallel quadratic gravity

The primary constraints for general teleparallel quadratic gravity are presented. They provide a basic classification of teleparallel theories from the perspective of the full nonlinear theory and represent the first step towards a full-fledged Hamiltonian analysis. The results are consistent with the limit of metric and symmetric teleparallel quadratic gravity. In the latter case we also present novel results, since symmetric teleparallel theories have only been partially studied so far. Apart from the general results, we also present the special cases of teleparallel theories classically equivalent to general relativity, which differ by a boundary term from the formulation of Einstein and Hilbert. This affects the constraint algebra as the primary constraints involve a mix of torsion and nonmetricity, implying that the symmetries of general relativity are realized in a more intricate way compared to the teleparallel case. In this context, a more detailed understanding will provide insights for energy and entropy in gravity, quantum gravity, and numerical relativity of this alternative formulation of general relativity. The primary constraints are presented both in the standard formulation and in irreducible parts of torsion and nonmetricity. The special role of axial torsion and its connection to the one-parameter of viable new general relativity is confirmed. Furthermore, we find that one of the irreducible parts of nonmetricity affects the primary constraint for shift but not lapse.

Figure 1

Visualization the constraints that G1–G19 theories satisfy, respectively. The lower-left region (NC) is the most generic theory with no constraint. The lower-right corner (NG) stands for “no gravity” and is constrained such that the spin-2 field is nonpropagating. The five corners, i.e., $\alpha ,\beta ,\mathrm{V},\mathrm{A},\mathrm{S}$, represent the parameter spaces that have primary constraints for lapse, shift, vector, antisymmetric, and symmetric, respectively. The combinations for coefficients that impose primary constraints are collected to the right.