Dilaton gravity on the brane

We derive the four-dimensional field equations for the induced metric and scalar field on the world-volume of a 3-brane in a five-dimensional bulk with Einstein gravity plus a self-interacting scalar field. We calculate the effective four-dimensional gravitational constant and cosmological constant for arbitrary forms of the brane tension and self-interaction potential for the scalar field in the bulk. In addition to the canonical energy-momentum tensor for the scalar held and ordinary matter on the brane, the effective four-dimensional Einstein equations include terms due to the scalar field and gravitational waves in the bulk. We present solutions corresponding to static Minkowski brane worlds and also dynamical Friedmann-Robertson-Walker brane world cosmologies. We discuss the induced coupling of the scalar field to ordinary matter on the brane.