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, T_0^0 = 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.