Palatini gravity with nonmetricity, torsion, and boundaries in metric and connection variables

Abstract

We prove the equivalence in the covariant phase space of the metric and connection formulations for Palatini gravity, with nonmetricity and torsion, on a spacetime manifold with boundary. To this end, we will rely on the cohomological approach provided by the relative bicomplex framework. Finally, we discuss some of the physical implications derived from this equivalence in the context of singularity identification through curvature invariants.

On-shell equivalence of general relativity and Holst theories with nonmetricity, torsion, and boundaries

“…Plugging the solutions (3.14) into (3.11) leads to This result proves that the metric sector of the metric-HMS theory is equivalent to the metric-EH theory as explained in [37,38]. We have the following on shell identities, Thus, the metric-HMS presymplectic form d HMS ðmÞ defined over the space of solutions Sol ðmÞ HMS is the same as the one of metric-Palatini d HMS ðmÞ which, in turn, has the same functional form as the metric-EH presymplectic form d EH ðmÞ (see [37,38]). In fact, if we define the projection π ðmÞ ðg; QÞ ¼ g, the presymplectic forms canonically associated with the three actions can be related as vol αβμν Riem αβμν vol;…”

“…where we recall that q μ ¼ vol μαβν Q αβν . The first term of the boundary Lagrangian is the generalized Hawking-Gibbons-York term introduced by Obukhov in [40] (see also [37])…”

“…As we will see, the presence of the parity breaking terms does not change the solution spaces with respect to the ones corresponding to the original Palatini models. In [37], the metric-Palatini action was proved to give GR on the metric sector, hence the metric sector of the HMS theory is also necessarily GR. In [38], the metric and tetrad formulations were proved to be equivalent, so the same result holds for the tetrad formulation.…”

Proof of the equivalence of the symplectic forms derived from the canonical and the covariant phase space formalisms

“…We devote this section to summarizing how to define a presymplectic form canonically associated with a local action. For a detailed discussion and some applications see [23][24][25][26].…”

Dualized gravity beyond linear approximation

“…We would like to point out that to use the spin connection as a dynamical gauge connection [25,26], we had to use Palatini formalism and view the spin connection and vierbein as independent variables [58,59]. Then we can view this theory as a non-abelian guage theory on curved spacetime, with spin connection as the gauge potential (for which a dual gauge potential will be constructed).…”