Semiclassical Gravitational Effects in the Spacetime of a Magnetic Flux Cosmic String

It is well-known that some physical effects may arise in the spacetime of a straigth cosmic string due to its global conic properties. Among these effects, the vacuum polarization effect has been extensively studied in the litterature. In papers of reference [4] a more general situation has been considered in which the cosmic string carries a magnetic flux $\Phi$ and interacts with a charged scalar field. In this case, the vacuum polarization arises both via non-trivial gravitational interaction (i.e, the conical structure) and via Aharonov-Bohm interaction. In papers [4] the non vanishing VEV of the energy-momentum tensor of the scalar field were computed. However, this energy-momentum tensor should, in principle, be taken into account to determine the spacetime associated with the magnetic flux cosmic string. Using the semiclassical approach to the Einstein eqs. we find the first-order (in $\hbar$) metric associated to the cosmic string and we show that the gravitational force resulting from the backreaction of the $ $ is attractive or repulsive depending on whether the magnetic flux is absent or present, respectively.