Chern-Weil theorem, Lovelock Lagrangians in critical dimensions, and boundary terms in gravity actions

Abstract

In this paper we show how to translate into tensorial language the Chern-Weil theorem for the Lorentz symmetry, which equates the difference of the Euler densities of two manifolds to the exterior derivative of a transgression form. For doing so we need to introduce an auxiliary, hybrid, manifold whose geometry we construct explicitely. This allows us to find the vector density, constructed out of spacetime quantities only, whose divergence is the exterior derivative of the transgression form. As a consequence we can show how the Einstein-Hilbert, Gauss-Bonnet and, in general, the Euler scalar densities can be written as the divergences of genuine vector densities in the critical dimensions D = 2, 4, etc. As Lovelock gravity is a dimensional continuation of Euler densities, these results are of relevance for Gauss-Bonnet and, in general, Lovelock gravity. Indeed, these vectors which can be called generalized Katz vectors ensure, in particular, a well-posed Dirichlet variational principle.

Post-Newtonian dynamics and black hole thermodynamics in Einstein-scalar-Gauss-Bonnet gravity

“…where R is the Ricci scalar, g = det g µν denotes the metric determinant, and the integral of the Gauss-Bonnet scalar over spacetime d D x √ −gR 2 GB is a boundary term in dimension D 4 (see e.g. [29,30]). The coupling constant α (which is chosen to be positive without loss of generality) has dimensions of length squared, and f (ϕ) is a dimensionless function defining the theory.…”

Einstein-Gauss-Bonnet theory of gravity: The Gauss-Bonnet-Katz boundary term

“…As shown in Ref. [25], this ensures that the curvature comes from a well-defined Lorentz gauge connection and satisfies the Bianchi identities. We also notice thatΩ AB is antisymmetric by construction and that the antisymmetrization bracket has been omitted in the derivative term of Eq.…”

“…4 The geometric properties and the proof thatω AB does transform as a spin connection under local Lorentz transformations can be found in Ref. [25]. 5 Consistency between the fully covariant versions of the Katz action (2.7) and (3.10)-(3.12) and its expression in Gaussian coordinates (2.9) is obtained by taking into account that, with our conventions, the Gauss' theorems for a vector A µ and a three-form Q are given, respectively, by…”

“…• In the language of forms To rewrite the previous expressions in the vielbein language we start again with the Chern-Weil theorem, this time in four dimensions (see Refs. [7,25]): 8) and where the expression inside the exterior derivative on the rhs of Eq. (4.7) is the transgression form.…”

Null boundary terms for Lanczos–Lovelock gravity

“…The non-null boundary term for the Gauss-Bonnet Lagrangian, the first non-trivial correction to the Einstein-Hilbert Lagrangian that is quadratic in the curvature tensor, was derived by Bunch [51]. For general Lanczos-Lovelock theories, the non-null boundary terms were derived by Myers [52] (see also [53][54][55][56][57]). However, the structure of the full boundary variation on a non-null boundary, including the total surface derivative term and the Dirichlet variation term that tells us the degrees of freedom to be fixed on the boundary and the corresponding conjugate momenta, was not known for Lanczos-Lovelock theories.…”