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General relativity (GR) as described in terms of curvature by theEinstein-Hilbert action is dynamically equivalent to theories of gravityformulated in terms of spacetime torsion or non-metricity. This forms what iscalled the geometrical trinity of gravity. The theories corresponding to thistrinity are, apart from GR, the teleparallel (TEGR) and symmetric teleparallel(STEGR) equivalent theories of general relativity, respectively, and theiractions are equivalent to GR up to boundary terms $B$. We investigate how theGibbons-Hawking-York (GHY) boundary term of GR changes under the transition toTEGR and STEGR within the framework of metric-affine gravity. We show that $B$is the difference between the GHY term of GR and that of metric-affine gravity.Moreover, we show that the GHY term for both TEGR and STEGR must vanish forconsistency of the variational problem. Furthermore, our approach allows toextend the equivalence between GR, TEGR and STEGR beyond the Einstein-Hilbertaction to any action built out of the curvature two-form, thus establishing thegeneralized geometrical trinity of gravity. We argue that these results will beparticularly useful in view of studying gauge/gravity duality for theories withtorsion and non-metricity.

Gibbons-Hawking-York boundary terms and the generalized geometrical trinity of gravity