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The problem of comparison of the stationary axisymmetric vacuum solutionsobtained within the framework of exact and approximate approaches for thedescription of the same general relativistic systems is considered. We suggesttwo ways of carrying out such comparison: (i) through the calculation of theErnst complex potential associated with the approximate solution whose form onthe symmetry axis is subsequently used for the identification of the exactsolution possessing the same multipole structure, and (ii) the generation ofapproximate solutions from exact ones by expanding the latter in series ofpowers of a small parameter. The central result of our paper is the derivationof the correct approximate analogues of the double-Kerr solution possessing thephysically meaningful equilibrium configurations. We also show that theinterpretation of an approximate solution originally attributed to it on thebasis of some general physical suppositions may not coincide with its truenature established with the aid of a more accurate technique.Comment: 32 pages, 5 figure

How can exact and approximate solutions of Einstein's field equations be compared?