Five-Dimensional Gravitational Toy Model with Bounded Non-Compact Extra Dimensions and Orthogonal Decoupling Dimensional Reduction

To avoid defects such as poor experimental consistency and moduli field issues arising from extra-dimensional compactification in traditional Kaluza-Klein theory, this paper constructs a five-dimensional gravitational toy model with bounded non-compact extra dimensions, focusing on theoretical derivations of orthogonal decoupling dimensional reduction with "five-dimensional to four-dimensional reduction" as the core premise. The model takes the Cartesian product of a four-dimensional Minkowski spacetime and a one-dimensional bounded open interval as the topological basis of the five-dimensional manifold. With dimensional reduction as the physical premise, it introduces the orthogonal decoupling assumption to constrain the cross terms of the five-dimensional metric to be identically zero, realizing dynamic decoupling between the four-dimensional submanifold and the extra dimension; combining tools from Riemannian geometry, it derives the five-dimensional Christoffel symbols and geodesic equations, and strictly verifies the mathematical consistency of the four-dimensional spacetime as a geodesic submanifold isometrically embedded in the five-dimensional spacetime; furthermore, through the five-dimensional Einstein-Hilbert action and Einstein field equations, a quantitative relationship between the four-dimensional and five-dimensional gravitational constants is established, ultimately achieving clean dimensional reduction of higher-dimensional gravity to the four-dimensional physical spacetime. The model in this paper serves the dimensional reduction process with orthogonal decoupling constraints throughout, with no inter-dimensional dynamic coupling, ensuring that four-dimensional particles have no tendency to move in the extra-dimensional direction, which is consistent with experimental observations. It provides a concise toy model reference for the low-energy effective theory of higher-dimensional gravity and has good extensibility—it can be further extended to incorporate the Yang-Mills action to construct a prototype of higher-dimensional unification of gravity and gauge fields.