2 research outputs found
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
neighborhood 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 generalize the SMS effective field equations on and
off a 3-brane 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 the terms determined by both the nonvanishing acceleration of
normals in the nongeodesic slicing of spacetime and the presence of other
fields in the bulk.Comment: 22 pages, REVTEX