Induced matter and particle motion in non-compact Kaluza-Klein gravity (Q5959874)

The generalizations of the five-dimensional canonical metric by including a dependence of the extra coordinate \(x^{4}=\ell \) in the four-dimensional metric are examined. The 4D induced energy-momentum tensor from \(g_{\alpha \beta }\), decomposing it into a false vacuum component, matter component and scalar field component (if present) are derived. An appropriate interpretation of the four-dimensional energy-momentum tensor induced from the five-dimensional space-time is given, which can lead to quite different physical situations depending on the interpretation chosen. It is shown that the assumption of five-dimensional null trajectories in Kaluza-Klein gravity can correspond to either four-dimensional massive or null trajectories when the path parameterization is chosen properly. Retaining the extra-coordinate dependence in the metric, the possibility of a cosmological variation in the rest masses of particles and a consequent departure from four-dimensional geodesic motion by a geometric force is demonstrated. In the examples given, it is shown that at late times it is possible for particles traveling along 5D null geodesics to be in a frame consistent with the induced matter scenario.