When the (3 + 1) Einstein equations with matter are regarded as a subs
et of the (4 + 1) Kaluza-Klein equations in apparent vacuum, the recov
ery of appropriate properties of matter requires in general that the (
4 + 1) metric depend on all the coordinates including the extra one. W
e display some consequences of (4 + 1) covariance through the use of e
xact solutions and coordinate transformations. We conclude that 4-D ph
ysics can be regarded as taking place on a hypersurface in 5-D, a math
ematical condition which may be set by a physical one on the rest mass
es of particles.