This paper investigates the gravitational equations on a 3-brane world in a 5-dimensional spacetime with Z2 symmetry, where all matter forces except gravity are confined to the brane. The effective gravitational equations on the brane are derived, which reduce to the conventional Einstein equations in the low energy limit. The authors conclude that the first Randall & Sundrum-type theory (RS1) predicts a brane with negative tension as an anti-gravity world and should be excluded. The second-type theory (RS2) with positive tension provides the correct gravity signature. In the case of an exactly anti-de Sitter bulk, 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 and the energy-momentum tensor. The effective Einstein equations on the brane are valid only when the cosmological constant on the brane is well-defined, such as in the case of a scalar field-dominated universe. The authors derive the effective Einstein equations on the 3-brane, assuming a 5-dimensional bulk spacetime. They use the Gauss and Codacci equations, and the 5-dimensional Einstein equations to obtain the 4-dimensional equations. The effective equations include terms from the 5-dimensional Weyl tensor and the energy-momentum tensor. The equations are shown to reduce to the conventional Einstein equations in the low energy limit. The presence of a well-defined cosmological constant is essential for the validity of the equations. The authors also show that the effective equations are not closed, as the Weyl tensor part, which carries information of the gravitational field outside the brane, is not freely specifiable but constrained by the matter on the brane. The equations are shown to reduce to the conventional Einstein equations in the low energy limit, with the effective gravitational constant depending on the vacuum energy. The authors conclude that the effective gravitational equations on the brane reduce to the conventional Einstein equations in the low energy limit, with the effective gravitational constant depending on the vacuum energy. The presence of a well-defined cosmological constant is essential for the validity of the equations.This paper investigates the gravitational equations on a 3-brane world in a 5-dimensional spacetime with Z2 symmetry, where all matter forces except gravity are confined to the brane. The effective gravitational equations on the brane are derived, which reduce to the conventional Einstein equations in the low energy limit. The authors conclude that the first Randall & Sundrum-type theory (RS1) predicts a brane with negative tension as an anti-gravity world and should be excluded. The second-type theory (RS2) with positive tension provides the correct gravity signature. In the case of an exactly anti-de Sitter bulk, 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 and the energy-momentum tensor. The effective Einstein equations on the brane are valid only when the cosmological constant on the brane is well-defined, such as in the case of a scalar field-dominated universe. The authors derive the effective Einstein equations on the 3-brane, assuming a 5-dimensional bulk spacetime. They use the Gauss and Codacci equations, and the 5-dimensional Einstein equations to obtain the 4-dimensional equations. The effective equations include terms from the 5-dimensional Weyl tensor and the energy-momentum tensor. The equations are shown to reduce to the conventional Einstein equations in the low energy limit. The presence of a well-defined cosmological constant is essential for the validity of the equations. The authors also show that the effective equations are not closed, as the Weyl tensor part, which carries information of the gravitational field outside the brane, is not freely specifiable but constrained by the matter on the brane. The equations are shown to reduce to the conventional Einstein equations in the low energy limit, with the effective gravitational constant depending on the vacuum energy. The authors conclude that the effective gravitational equations on the brane reduce to the conventional Einstein equations in the low energy limit, with the effective gravitational constant depending on the vacuum energy. The presence of a well-defined cosmological constant is essential for the validity of the equations.