Constraints on a brane-world from the vanishing of the cosmological constant

We derive the analogue of the vanishing of the cosmological constant in /3+1 dimensions, T₀⁰=0, in terms of an integral over components of the energy-momentum tensor of a /4+1 dimensional universe with parallel three-branes, and an additional constraint local to the branes. The basic ingredients are the existence of a static solution of the Einstein equations, and the compactness of the 5th dimension. The corresponding constraints are applied to a general action of scalar fields with arbitrary potentials in the bulk and on the branes. The equations of motion are solved in a linearized approximation in the 5th dimension, whereupon they require the search for extrema of an ``effective potential'', which depends nonlinearly on the action in the bulk and on the branes. The previous constraints then turn into the vanishing of this ``effective potential'' at the extremum.