Gravitational Effects of Varying Alpha Strings
  We study spatial variations of the fine-structure constant in the presence of static straight cosmic strings in the weak-field approximation in Einstein gravity. We work in the context of a generic Bekenstein-type model and consider a gauge kinetic function linear in the scalar field. We determine an analytical form for the scalar field and the string metric at large distances from the core. We show that the gravitational effects of $\alpha$-varying strings can be seen as a combination of the gravitational effects of global and local strings. We also verify that at large distances to the core the space-time metric is similar to that of a global string. We study the motion of test particles approaching from infinity and show that photons are scattered to infinity while massive particles are trapped in bounded trajectories. We also calculate an overall limit on the magnitude of the variation of $\alpha$ for a GUT string, by considering suitable cosmological constraints coming from the Equivalence Principle. 
J. Menezes, P. P. Avelino, C. Santos