Geometry: The Implications of Homogeneity and Isotropy

Eugene Shubert

I’m wondering if it’s mathematically permissible, if space is homogeneous and isotropic, for a moving rod to experience a uniform expansion or contraction during the time it’s not in its stationary frame of reference. What’s preventing a moving rod from returning to its point of origin smaller or larger? The expanding or shrinking effect could go like f(v)exp(kt) for as long as the rod maintains a constant velocity v. Conceivably, this might be made to work in 3 spatial dimensions. Every observer could say that every other frame is shrinking uniformly in time and, akin to the twin paradox, all returning twins could end up YOUNGER and smaller.

Can you prove that homogeneity and isotropy alone disallows this possibility?