We carefully investigate the gravitational equations of the brane world, in
which all the matter forces except gravity are confined on the 3-brane in
a 5-dimensional spacetime with Z(2) symmetry. We derive the effective gravi
tational equations on the brane, which reduce to the conventional Einstein
equations in the low energy limit. From our general argument we conclude th
at the first Randall-Sundrum-type theory predicts that the brane with a neg
ative tension is an antigravity world and hence should be excluded from the
physical point of view. Their second-type theory where the brane has a pos
itive tension provides the correct signature of gravity. In this latter cas
e, if the bulk spacetime is exactly anti-de Sitter spacetime, generically t
he matter on the brane is required to be spatially homogeneous because of t
he Bianchi identities. By allowing deviations from anti-de Sitter spacetime
in the bulk, the situation will be relaxed and the Bianchi identities give
just the relation between the Weyl tensor and the energy momentum tensor.
In the present brane world scenario, the effective Einstein equations cease
to be valid during an era when the cosmological constant on the brane is n
ot well defined, such as in the case of the matter dominated by the potenti
al energy of the scalar field.