arXiv:1312.7828v2 [gr-qc] 3 Apr 2014 First order gravity: Actions, topological terms and boundaries Alejandro Corichi, 1, 2, ∗ Irais Rubalcava-García, 3,1, † and Tatjana Vukašinac 4, ‡ 1 Centro de Ciencias Matemáticas, Universidad Nacional Autónoma de México, UNAM-Campus Morelia, A. Postal 61-3, Morelia, Michoacán 58090, Mexico 2 Center for Fundamental Theory, Institute for Gravitation and the Cosmos, Pennsylvania State University, University Park PA 16802, USA 3 Instituto de Física y Matemáticas, Universidad Michoacana de San Nicolás de Hidalgo, Morelia, Michoacán, Mexico 4 Facultad de Ingeniería Civil, Universidad Michoacana de San Nicolás de Hidalgo, Morelia, Michoacán 58000, Mexico We consider first order gravity in four dimensions. In particular, we consider formulations where the fundamental variables are a tetrad e and a SO(3,1) connection ω. We study the most general action principle compatible with diffeomorphism invariance. This implies, in particular, considering besides the standard Einstein- Hilbert-Palatini term, other terms that either do not change the equations of motion, or are topological in nature. Having a well defined action principle also implies adding additional boundary terms, whose detailed form may depend on the particular boundary conditions at hand. We consider spacetimes that include a boundary at infinity, satisfying asymptotically flat boundary conditions and/or an internal boundary satisfying isolated horizons boundary conditions. For our analysis we employ the covariant Hamiltonian formalism where the phase space Γ is given by solutions to the equation of motion. For each of the possible terms contributing to the action we study the well posedness of the action, its finiteness, the contribution to the symplectic structure, and the Hamiltonian and Noether charges. We show that for the chosen boundary conditions, standard boundary terms warrant a well posed theory. Furthermore, we show that the boundary and topological terms do not contribute to the symplectic structure, nor the Hamiltonian conserved charges. The Noether conserved charges, on the other hand, do depend on such additional terms. The aim of the paper is to present a comprehensive and self-contained treatment of the subject, so the style is somewhat pedagogical. Furthermore, we point out and clarify some issues that have not been clearly understood in the literature. PACS numbers: 04.20.Fy, 04.20.Ha, 04.70.Bw * Electronic address: corichi@matmor.unam.mx † Electronic address: irais@matmor.unam.mx ‡ Electronic address: tatjana@umich.mx