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.

arXiv comment15 pages

References in corpus (18)

• Black hole hair in generalized scalar-tensor gravity: An explicit example
• Relativistic conservation laws and integral constraints for large cosmological perturbations
• Second Order Scalar Invariants of the Riemann Tensor: Applications to Black Hole Spacetimes
• Conserved Charges in Einstein Gauss-Bonnet theory
• Effective actions for anomalous hydrodynamics
• Counterterms in Dimensionally Continued AdS Gravity
• Topological gravity and transgression holography
• Matching conditions for a brane of arbitrary codimension
• On the quasi-linearity of the Einstein- "Gauss-Bonnet" gravity field equations
• The Extended Cartan Homotopy Formula and a Subspace Separation Method for Chern--Simons Theory
• Eleven-Dimensional Gauge Theory for the M Algebra as an Abelian Semigroup Expansion of osp(32|1)
• Structure of Lanczos-Lovelock Lagrangians in Critical Dimensions
• A Novel Derivation of the Boundary Term for the Action in Lanczos-Lovelock Gravity
• Chern-Simons 5-form and Holographic Baryons
• Einstein-Gauss-Bonnet theory of gravity: The Gauss-Bonnet-Katz boundary term
• On eleven-dimensional Supergravity and Chern-Simons theory
• General aspects of Gauss-Bonnet models without potential in dimension four
• Even-dimensional topological gravity from Chern-Simons gravity

Cited by in corpus (5)

• Post-Newtonian dynamics and black hole thermodynamics in Einstein-scalar-Gauss-Bonnet gravity
• $d+1$ formalism in Einstein-scalar-Gauss-Bonnet gravity
• Einstein-Gauss-Bonnet theory of gravity: The Gauss-Bonnet-Katz boundary term
• On the topological character of metric-affine Lovelock Lagrangians in critical dimensions
• Dimensional aspects of Lovelock-Lanczos gravity