Modification of the Coulomb Potential from a Kaluza-Klein Model with a Gauss-Bonnet Term in the Action

Abstract In four dimensions a Gauss-Bonnet term in the action corresponds to a total derivative, and therefore it does not contribute to the classical equations of motion. For higher-dimensional geometries this term has the interesting property (which it shares with other dimensionally continued Euler densities) that when the action is varied with respect to the metric, it gives rise to a symmetric, covariantly conserved tenser of rank two which is a function of the metric and its first- and second-order derivatives. Here we review the unification of general relativity and electromagnetism in the classical five-dimensional, restricted (with g55 = 1) Kaluza-Klein model. Then we discuss the modifications of the Einstein-Maxwell theory that results from adding the Gauss-Bonnet term in the action. The resulting four-dimensional theory describes a non-linear U(1) gauge theory non-minimally coupled to gravity. For a point charge at rest we find a perturbative solution for large distances which gives a mass-dependent correction to the Coulomb potential. Near the source we find a power-law solution which seems to cure the short-distance divergency of the Coulomb potential. Possible ways to obtain an experimental upper limit to the coupling of the hypothetical Gauss-Bonnet term are also considered.