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

TL;DR

This paper compares exact and approximate solutions of Einstein's field equations for stationary axisymmetric systems, introducing methods to relate and generate solutions, and clarifies the physical interpretation of approximate solutions.

Contribution

It proposes two methods for comparing and generating solutions, and derives correct approximate analogues of the double-Kerr solution with meaningful equilibrium configurations.

Findings

Two comparison methods for exact and approximate solutions are introduced.

Correct approximate analogues of the double-Kerr solution are derived.

Interpretations of approximate solutions can differ from their true physical nature.

Abstract

The problem of comparison of the stationary axisymmetric vacuum solutions obtained within the framework of exact and approximate approaches for the description of the same general relativistic systems is considered. We suggest two ways of carrying out such comparison: (i) through the calculation of the Ernst complex potential associated with the approximate solution whose form on the symmetry axis is subsequently used for the identification of the exact solution possessing the same multipole structure, and (ii) the generation of approximate solutions from exact ones by expanding the latter in series of powers of a small parameter. The central result of our paper is the derivation of the correct approximate analogues of the double-Kerr solution possessing the physically meaningful equilibrium configurations. We also show that the interpretation of an approximate solution originally attributed to it on the basis of some general physical suppositions may not coincide with its true nature established with the aid of a more accurate technique.