Electromagnetic generalized quasitopological gravities in ( 2+1 ) dimensions

Abstract

The construction of quasitopological gravities in three-dimensions requires coupling a scalar field to the metric. As shown in [arXiv:2104.10172], the resulting ``electromagnetic'' quasitopological (EQT) theories admit charged black hole solutions characterized by a single-function for the metric, $\ensuremath{-}{g}_{tt}={g}_{rr}^{\ensuremath{-}1}\ensuremath{\equiv}f(r)$, and a simple azimuthal form for the scalar. Such black holes, whose metric can be determined fully analytically, generalize the Ba\~nados-Teiteilboim-Zanelli black hole (BTZ) solution in various ways, including singularity-free black holes without any fine-tuning of couplings or parameters. In this paper we extend the family of EQT theories to general curvature orders. We show that, beyond linear order, $f(r)$ satisfies a second-order differential equation rather than an algebraic one, making the corresponding theories belong to the electromagnetic generalized quasitopological (EGQT) class. We prove that at each curvature order, the most general EGQT density is given by a single term which contributes nontrivially to the equation of $f(r)$ plus densities which do not contribute at all to such equation. The proof relies on the counting of the exact number of independent order-$n$ densities of the form $\mathcal{L}({R}_{ab},{\ensuremath{\partial}}_{a}\ensuremath{\phi})$, which we carry out. We study some general aspects of the new families of EGQT black-hole solutions, including their thermodynamic properties and the fulfillment of the first law, and explicitly construct a few of them numerically.