The paper by Lisa Randall and Raman Sundrum challenges the conventional wisdom that Newton's force law implies only four non-compact dimensions. They demonstrate that this is not necessarily true in the presence of a non-factorizable background geometry. Specifically, they study a single 3-brane embedded in five dimensions and show that four-dimensional Newtonian and general relativistic gravity can be reproduced with high precision, even without a gap in the Kaluza-Klein spectrum. The authors argue that the properties of gravity in higher dimensions rely on a factorizable geometry, where the metric of the four familiar dimensions is independent of the extra dimensions. However, when this assumption is dropped, the story can change significantly. They show that a curved background can support a "bound state" of the higher-dimensional graviton, localized in the extra dimensions. This bound state mode reproduces conventional four-dimensional gravity, while other Kaluza-Klein modes provide only small corrections. The setup involves a single 3-brane with positive tension embedded in a five-dimensional bulk spacetime. The authors carefully quantize the system, treating non-normalizable modes that appear in the Kaluza-Klein reduction. They derive the effective four-dimensional Planck scale, \( M_{Pl} \), and show that it is determined by the higher-dimensional curvature rather than the size of the extra dimension. This curvature is consistent with four-dimensional Poincaré invariance. The paper also discusses the implications of this scenario for the moduli problem and the possibility of a dual description in terms of a cutoff conformal field theory in four dimensions. The authors conclude that their framework provides a well-defined alternative to geometric compactification and opens up new perspectives for solving unresolved issues in quantum gravity and cosmology.The paper by Lisa Randall and Raman Sundrum challenges the conventional wisdom that Newton's force law implies only four non-compact dimensions. They demonstrate that this is not necessarily true in the presence of a non-factorizable background geometry. Specifically, they study a single 3-brane embedded in five dimensions and show that four-dimensional Newtonian and general relativistic gravity can be reproduced with high precision, even without a gap in the Kaluza-Klein spectrum. The authors argue that the properties of gravity in higher dimensions rely on a factorizable geometry, where the metric of the four familiar dimensions is independent of the extra dimensions. However, when this assumption is dropped, the story can change significantly. They show that a curved background can support a "bound state" of the higher-dimensional graviton, localized in the extra dimensions. This bound state mode reproduces conventional four-dimensional gravity, while other Kaluza-Klein modes provide only small corrections. The setup involves a single 3-brane with positive tension embedded in a five-dimensional bulk spacetime. The authors carefully quantize the system, treating non-normalizable modes that appear in the Kaluza-Klein reduction. They derive the effective four-dimensional Planck scale, \( M_{Pl} \), and show that it is determined by the higher-dimensional curvature rather than the size of the extra dimension. This curvature is consistent with four-dimensional Poincaré invariance. The paper also discusses the implications of this scenario for the moduli problem and the possibility of a dual description in terms of a cutoff conformal field theory in four dimensions. The authors conclude that their framework provides a well-defined alternative to geometric compactification and opens up new perspectives for solving unresolved issues in quantum gravity and cosmology.