The Einstein equations on a brane world

We present effective equations to describe the four-dimensional gravity of a brane world, assuming that a five-dimensional bulk spacetime satisfies the Einstein equations and gravity is confined on the Z₂ symmetric brane. Applying this formalism to the Randall-Sundrum brane world model, we find two new additional terms in the Einstein equations: One is the quadratic term of the energy-momentum tensor of matter field and the other is the 5-dimensional Weyl curvature term. Although the four-dimensional effective equations are not closed and further information from the bulk is required, those terms may provide us windows to search for extra dimensions. We discuss some effects of those terms on cosmology and compact objects. Since those additional terms appear naturally from the present reduction, we apply it to other brane models. We study three models; the model with a bulk dilaton field motivated by Hořava-Witten model, the induced gravity brane model of Dvali, Gabadadze and Porrati, and the R² inflationary model proposed by Starobinsky.