Universal Gibbons-Hawking-York term for theories with curvature, torsion and non-metricity

TL;DR

This paper introduces a new method for calculating Gibbons-Hawking-York boundary terms in gravitational theories with curvature, torsion, and non-metricity, simplifying the process and extending results to new theories.

Contribution

A systematic and efficient method for deriving GHY terms in complex gravitational theories with non-metricity and torsion, applicable to various models including Lovelock-Chern-Simons and metric-affine gravity.

Findings

Confirmed existing GHY results for Einstein-Hilbert and Chern-Simons gravity.

Derived new GHY terms for Lovelock-Chern-Simons and metric-affine gravity.

Facilitated a systematic approach for holographic renormalization in extended gravity theories.

Abstract

Motivated by establishing holographic renormalization for gravitational theories with non-metricity and torsion, we present a new and efficient general method for calculating Gibbons-Hawking-York (GHY) terms. Our method consists of linearizing any nonlinearity in curvature, torsion or non-metricity by introducing suitable Lagrange multipliers. Moreover, we use a split formalism for differential forms, writing them in $(n-1)+1$ dimensions. The boundary terms of the action are manifest in this formalism by means of Stokes' theorem, such that the compensating GHY term for the Dirichlet problem may be read off directly. We observe that only those terms in the Lagrangian that contain curvature contribute to the GHY term. Terms polynomial solely in torsion and non-metricity do not require any GHY term compensation for the variational problem to be well-defined. We test our method by confirming existing results for Einstein-Hilbert and four-dimensional Chern-Simons modified gravity. Moreover, we obtain new results for Lovelock-Chern-Simons and metric-affine gravity. For all four examples, our new method and results contribute to a new approach towards a systematic hydrodynamic expansion for spin and hypermomentum currents within AdS/CFT.