The series solution to the metric of stationary vacuum with axisymmetry

The multipole moment method not only conduces to the understanding of the deformation of the space–time, but also serves as an effective tool to approximately solve the Einstein field equation with. However, the usual multipole moments are recursively determined by a sequence of symmetric and trace-free tensors, which is inconvenient for practical resolution. In this paper, we develop a simplified procedure to generate the series solutions to the metric of the stationary vacuum with axisymmetry, and show its validity. In order to understand the free parameters in the solution, we propose to take the Schwarzschild metric as a standard ruler, and some well-known examples are analysed and compared with the series solutions in detail