Magnetospheres of neutron stars are anchored in the rigid crust and can be twisted by sudden crustal motions ( “ starquakes ” ) . The twisted magnetosphere does not remain static and gradually untwists , dissipating magnetic energy and producing radiation . The equation describing this evolution is derived , and its solutions are presented . Two distinct regions coexist in untwisting magnetospheres : a potential region where \nabla \times { \mathbf { B } } = 0 ( “ cavity ” ) and a current-carrying bundle of field lines with \nabla \times { \mathbf { B } } \neq 0 ( “ j-bundle ” ) . The cavity has a sharp boundary , which expands with time and eventually erases all of the twist . In this process , the electric current of the j-bundle is sucked into the star . Observational appearance of the untwisting process is discussed . A hot spot forms at the footprints of the j-bundle . The spot shrinks with time toward the magnetic dipole axis , and its luminosity and temperature gradually decrease . As the j-bundle shrinks , the amplitude of its twist \psi can grow to the maximum possible value \psi _ { max } \sim 1 . The strong twist near the dipole axis increases the spindown rate of the star and can generate a broad beam of radio emission . The model explains the puzzling behavior of magnetar XTE J1810-197 — a canonical example of magnetospheric evolution following a starquake . We also discuss implications for other magnetars . The untwisting theory suggests that the nonthermal radiation of magnetars is preferentially generated on a bundle of extended closed field lines near the dipole axis .