I'm currently doing an internship on Kaluza-Klein theory. by reading some articles, I don't understand some things that seem easy to authors for example : In this article https://arxiv.org/abs/gr-qc/9805018 — very good article writted by Wesson and Overduin — They said
In page 7, << ... In particular, one should be able to recover a “ground state” solution consisting of four-dimensional Minkowski space plus a d-dimensional compact manifold. Although this is straightforward when d = 1 (Klein’s case) ... >>.
In page 15 << The fact that the action (3) leads to (9)... with source matter, constitutes the central miracle of Kaluza-Klein theory. >> I don't see in the equation matter term, is it the electromagnetic tensor F ? or is it the scalar field (dilaton)?