Gravitational field equations on and off a 3-brane world

The effective gravitational field equations on and off a 3-brane world possessing a Z(2) mirror symmetry and embedded in a five-dimensional bulk spacetime with cosmological constant were derived by Shiromizu, Maeda and Sasaki (SMS) in the framework of the Gauss-Codazzi projective approach with the subsequent specialization to the Gaussian normal coordinates in the neighbourhood of the brane. However, the Gaussian normal coordinates imply a very special slicing of spacetime and clearly, the consistent analysis of the brane dynamics would benefit from complete freedom in the slicing of spacetime, pushing the layer surfaces in the fifth dimension at any rates of evolution and in arbitrary positions. We rederive the SMS effective gravitational field equations on a 3-brane and generalize the off-brane equations to the case where there is an arbitrary energy-momentum tensor in the bulk. We use a more general setting to allow for acceleration of the normals to the brane surface through the lapse function and the shift vector in the spirit of Arnowitt, Deser and Misner. We show that the gravitational influence of the bulk spacetime on the brane may be described by a traceless second-rank tensor W-ij, constructed from the 'electric' part of the bulk Riemann tensor. We also present the evolution equations for the tensor W-ij, as well as for the corresponding 'magnetic' part of the bulk curvature. These equations involve terms determined by both the nonvanishing acceleration of normals in the nongeodesic slicing of spacetime and the presence of other fields in the bulk.