Time-dependent generalizations of the standard static, spherically symmetric solutions of the empty Kaluza–Klein or (4+1) Einstein equations are considered herein. Such time-dependent solutions exist because Birkhoff’s theorem does not hold in (4+1) gravity. The solutions depend on the time and radius but not the fifth or internal coordinate, and fall into four classes with distinct mathematical properties. Physically, some of our solutions represent a static 4D space–time with an oscillating or contracting extra dimension, which suggests their relevance to particle physics. Others represent an expanding 3D space with a contracting extra dimension, which suggests their relevance to cosmology.