Gravitational field equations in a braneworld with Euler–Poincaré term

Abstract

We present the effective gravitational field equations in a 3-braneworld with Euler–Poincaré term and a cosmological constant in the bulk spacetime. The similar equations on a 3-brane with symmetry embedded in a five-dimensional bulk spacetime were obtained earlier by Maeda and Torii using the Gauss–Codazzi 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 manner 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 parts of the bulk Riemann 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.