Covariant phase space for gravity with boundaries: metric vs tetrad formulations J. Fernando Barbero G.c,d Juan Margalef-Bentabola,c Valle Varob,c Eduardo J.S. Villaseñorb,c aInstitute for Gravitation and the Cosmos and Physics Department. Penn State University. PA 16802, USA. bDepartamento de Matemáticas, Universidad Carlos III de Madrid. Avda. de la Universidad 30, 28911 Leganés, Spain. cGrupo de Teorías de Campos y Física Estadística. Instituto Gregorio Millán (UC3M). Unidad Asociada al Instituto de Estructura de la Materia, CSIC, Madrid, Spain. dInstituto de Estructura de la Materia, CSIC, Serrano 123, 28006, Madrid, Spain E-mail:
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[email protected] Abstract: We use covariant phase space methods to study the metric and tetrad formula- tions of General Relativity in a manifold with boundary and compare the results obtained in both approaches. Proving their equivalence has been a long-lasting problem that we solve here by using the cohomological approach provided by the relative bicomplex framework. This setting provides a clean and ambiguity-free way to describe the solution spaces and associated symplectic structures. We also compute several relevant charges in both schemes and show that they are equivalent, as expected. arXiv:2103.06362v2 [gr-qc] 14 Jun 2021 Contents I Introduction1 II The geometric arena2 II.1 M as a space-time2 II.2 From M-vector fields to F-vector fields3 II.3 CPS-Algorithm4 IIIGeneral Relativity in terms of metrics5 IV General relativity in terms of tetrads8 V Metric vs Tetrad formulation 13 V.1 Space of solutions 13 V.2 Presymplectic structures 14 VI Conclusions 15 A Ancillary material 16 A.1 Mathematical background 16 A.2 Some computations in the metric case 19 A.3 Some tetrad computations 21 Bibliography 22 I Introduction Space-time boundaries play a prominent role in classical and quantum General Relativity (GR).