The paper investigates the gravitational equations in a 3-brane world, where matter forces are confined to the brane in a 5-dimensional spacetime with $Z_2$ symmetry. The authors derive the effective gravitational equations on the brane, which reduce to the conventional Einstein equations in the low energy limit. They conclude that the first Randall-Sundrum-type theory (RS1) predicts an anti-gravity world with negative brane tension, which is physically unallowable. The second Randall-Sundrum-type theory (RS2) with positive brane tension provides the correct signature of gravity. In the case of an exact anti-de Sitter bulk spacetime, the matter on the brane must be spatially homogeneous due to Bianchi identities. Deviations from anti-de Sitter in the bulk relax this condition, and the Bianchi identities relate the Weyl tensor to the energy-momentum tensor. The effective Einstein equations cease to be valid when the cosmological constant on the brane is not well-defined, such as in scenarios with scalar field potential energy. The paper also discusses the role of the $\pi_{\mu \nu}$ term, which is quadratic in the energy-momentum tensor, and the $E_{\mu \nu}$ term, which carries information about the gravitational field outside the brane. The $E_{\mu \nu}$ term is constrained by the motion of matter on the brane and is not freely specifiable. The paper estimates the effects of each term on the right-hand side of the effective gravitational equation, showing that the Einstein equations reduce to the conventional 4-dimensional Einstein gravity in the low energy limit. The presence of a well-defined cosmological constant is essential for this reduction. Finally, the paper discusses the implications of the constraint on the matter on the brane when the bulk spacetime is pure anti-de Sitter, leading to the conclusion that only spatially homogeneous universes are allowed.The paper investigates the gravitational equations in a 3-brane world, where matter forces are confined to the brane in a 5-dimensional spacetime with $Z_2$ symmetry. The authors derive the effective gravitational equations on the brane, which reduce to the conventional Einstein equations in the low energy limit. They conclude that the first Randall-Sundrum-type theory (RS1) predicts an anti-gravity world with negative brane tension, which is physically unallowable. The second Randall-Sundrum-type theory (RS2) with positive brane tension provides the correct signature of gravity. In the case of an exact anti-de Sitter bulk spacetime, the matter on the brane must be spatially homogeneous due to Bianchi identities. Deviations from anti-de Sitter in the bulk relax this condition, and the Bianchi identities relate the Weyl tensor to the energy-momentum tensor. The effective Einstein equations cease to be valid when the cosmological constant on the brane is not well-defined, such as in scenarios with scalar field potential energy. The paper also discusses the role of the $\pi_{\mu \nu}$ term, which is quadratic in the energy-momentum tensor, and the $E_{\mu \nu}$ term, which carries information about the gravitational field outside the brane. The $E_{\mu \nu}$ term is constrained by the motion of matter on the brane and is not freely specifiable. The paper estimates the effects of each term on the right-hand side of the effective gravitational equation, showing that the Einstein equations reduce to the conventional 4-dimensional Einstein gravity in the low energy limit. The presence of a well-defined cosmological constant is essential for this reduction. Finally, the paper discusses the implications of the constraint on the matter on the brane when the bulk spacetime is pure anti-de Sitter, leading to the conclusion that only spatially homogeneous universes are allowed.