Global existence of solutions to the coupled Einstein and Maxwell-Higgs system in the spherical symmetry

We prove the global unique existence of classical solutions to the Einstein equations coupled with Maxwell-Higgs system for small initial data under the spherical symmetry. We also obtain the decay estimates of the solutions, and find that the corresponding space-time is time-like and null geodesically complete toward the future. For the proof we reduce the system to a single first order integro-differential equation, and use the contraction mapping theorem in the appropriate function spaces. We also obtain the completeness of space-time along the future directed time-like lines exterior to a region which resembles the even horizon of the Reissner-Nordstrbm black hole.