First-order formalism of gravity: Symmetries and exact solutions

Cristóbal Corral (ICN-UNAM)

RESUMEN: The first-order formalism of gravity, where there are two independent gravitational fields, namely, the vielbein and Lorentz connection, is a framework where gravity can be considered as a gauge theory. Its geometrical structure considers curvature and torsion as two independent entities that characterize the spacetime manifold and it allows one to consider the spin density of matter as a gravitational source. The most basic realization, the Einstein-Cartan theory, has been regarded as the simplest classical extension of GR, passing all the experimental tests up to date. Moreover, this formalism allows one to include more general internal symmetries than local Lorentz transformations. In this talk, a relation among infinitesimal diffeomorphisms, internal symmetries, and a third symmetry dubbed local translations, is presented. It is found that the structure of local translations is sensitive to the internal symmetry group, however, the algebra of local translations and the internal group always closes. The relation among symmetries implies that when one of them is broken, at least another must be affected. As an example, a theory known as unimodular gravity, which is only invariant under restricted diffeomorphisms, is presented. In addition, the aforementioned results can be extended to theories with additional degrees of freedom such as in scalar-tensor theories. In a particular example of these theories, i.e. Chern--Simons modified gravity, the transformation law of (pseudo)scalar fields under local translations is found. Finally, exact four-dimensional vacuum solutions of Chern--Simons modified gravity with nontrivial torsion are presented which represent the black string extension of the Banados-Teitelboim-Zanelli black hole. These configurations may be useful to see whether torsion can cure the Gregory-Laflame instability, rooted in higher dimensional black strings.