Gravitational field equations in a braneworld with Euler-Poincare term

TL;DR

This paper derives effective gravitational field equations for a 3-brane in a five-dimensional bulk with Euler-Poincare term, using differential forms and ADM formalism, revealing how bulk effects influence brane gravity.

Contribution

It generalizes previous brane-world equations by incorporating ADM slicing and Euler-Poincare term, providing a more flexible framework for bulk-brane gravitational interactions.

Findings

Derived effective gravitational equations on the brane with ADM formalism.

Expressed bulk effects via projected Weyl curvature forms.

Presented evolution equations for bulk Weyl fields with acceleration effects.

Abstract

We present the effective gravitational field equations in a 3-brane world with Euler-Poincare term and a cosmological constant in the bulk spacetime. The similar equations on a 3-brane with $\mathbb{Z}_2$ symmetry embedded in a five dimensional bulk spacetime were obtained earlier by Maeda and Torii using the Gauss-Coddazzi projective approach in the framework of the Gaussian normal coordinates. We recover these equations on the brane in terms of differential forms and using a more general coordinate setting in the spirit of Arnowitt, Deser and Misner (ADM). The latter allows for acceleration of the normals to the brane surface through the lapse function and the shift vector. We show that the gravitational effects of the bulk space are transmitted to the brane through the projected ``electric'' 1-form field constructed from the conformal Weyl curvature 2-form of the bulk space. We also derive the evolution equations into the bulk space for the electric 1-form field, as well as for the ``magnetic'' 2-form field part of the bulk Weyl curvature 2-form. As expected, unlike on-brane equations, the evolution equations involve terms determined by the nonvanishing acceleration of the normals in the ADM-type slicing of spacetime.