Phenomenology for an extra dimension from gravitational waves propagation on a Kaluza-Klein space-time

In the present work we analyze the behavior of 5-dimensional (5-d) gravitational waves propagating on a Kaluza-Klein background. We face separately the two cases in which the waves are generated before and after the process of dimensional compactification respectively. We show that if the waves are originated on a 5-d space-time which fulfills the principle of general relativity, then the process of compactification cannot reduce the dynamics to the pure 4-d scalar, vector and tensor degrees of freedom. In particular, while the electromagnetic waves evolve independently, the scalar and tensor fields couple to each other; this feature appears because when the gauge conditions are split, the presence of the scalar ripple prevents the 4-d gravitational waves from becoming traceless. The phenomenological issue of this scheme consists of an anomalous relative amplitude of the two independent polarizations which characterize the 4-d gravitational waves. Such profile of polarization amplitudes, if detected, would outline the extra dimension in a very reliable way, because a wave with non-zero trace cannot arise from ordinary matter sources. We discuss the above mentioned phenomenon either in the case of a unit constant value of the background scalar component (when the geodesic deviation is treated with precise outputs), or assuming such background field as a dynamical degree. Only qualitative conclusion are provided here, because the details of the polarization amplitudes depend on the choice of specific metric forms. Finally, we perturb a real Kaluza-Klein theory showing that in this context, while the electromagnetic waves propagate independently, the 4-d gravitational waves preserve their ordinary structure, while the scalar plays for the role of source.