Stationary electromagnetic fields in the exterior of a slowly rotating relativistic star: a description beyond the low-frequency approximation

We investigate the electromagnetic fields in the vacuum exterior to a rotating relativistic star endowed with a magnetic dipole moment, and with the stellar surface behaving as a perfect conductor. While the stellar rotation is treated in the slow approximation of general relativity, we do not restrict our attention to slowly rotating electromagnetic fields, and take our analysis beyond the low-frequency approximation considered so far. When the dipole moment is misaligned with the rotational axis, our approach does not yield analytic solutions, but determines the properties of the electromagnetic fields approximately and semi-analytically by computing the coefficients of simple expressions for the fields through the numerical solution of two partial differential equations. Because our approach provides a solution that is in principle valid throughout space, we can evaluate the accuracy and or invalidity of previously known analytic expressions at different distances from the stellar surface. Overall, the solutions found in this way represent an efficient way of bridging in a single semi-analytic formalism the strongly relativistic and the asymptotic regimes for which analytic solutions have been found.