Physical identification of static and axially-symmetric vacuum metric tensors

We present a method of a possible physical identification of the static and axially symmetric Weyl-type vacuum γ and nγ metrics. This method, in which no interior solutions of any kind are involved, is based on the comparison of the far-field forms of the γ and nγ metrics, and of the far-field form of the metric tensor due to a bounded gravitating perfect-fluid source given correctly to post-Newtonian accuracy. The parameters of the vacuum solutions are expressed in terms of physical parameters of a prolate fluid source, namely total mass-energy, semiaxes and eccentricity, defined consistently to post-Newtonian accuracy. The results, based on the otherwise arbitrary fluid source, appear physically general. Possible astrophysical candidates for the far-field γ andnγ metrics are proposed, based on the conditions imposed on them by the identification method. The advantages and deficiencies of the identification method are briefly discussed.