Multipole Moments in General Relativity

Abstract

Multipole moments in general relativity are defined as coefficients of a multipole expan­ sion of appropriate potentials, as they are so in Newton's theory of gravitation. The essential point is the introduction of Fock's harmonic coordinate system in which the potentials are expanded in inverse powers of the distance from the source. First several moments are obtained for the Kerr, Tomimatsu-Sato and a class of the Weyl solutions of the Einstein equation, and then are inferred all moments for the Kerr and Weyl solutions. § l. Introduction The problem of obtaining multipole moments of a solution of the Einstein equation is the problem of interpreting the solution in terms of its Newtonian limit. In particular the knowledge of multipole moments serves to infer a possible source distribution which produces the gravitational field in question. Since the interior solutions which may be considered to describe the interior metric of the source of the Weyl,n Kerr') or Tomimatsu-Sato (T-S) 3l gravitational field have not been discovered at this stage, it is desirable to have a systematic method of obtain­ ing multipole moments of these fields. One of the methods was developed by Geroch using conformal Killing vec­ tors,<)· 5) and by means of this method Hansen was able to obtain multipole moments of the Kerr solution.6l Although it is not so easy to find the necessary conformal factor, once it is found, Geroch's method allows us to calculate all moments in principle. There are also other methods of finding multipole moments, although they are in many cases not adequate to find higher moments. For example, Voor­ hees obtained the quadrupole moment of a W eyl solution ;n Hernandez obtained all moments of the Kerr solutions ;8l Tomimatsu and Sa to obtained the quadrupole moments of their solutions. 3l The purpose of this paper is to present a new method of calculating multipole moments which, we hope, is complementary to Geroch's method and containing all the results obtained by the above authors. The idea is to introduce Fock's harmonic coordinate system 9l in which appropriate potentials are expanded in inverse powers of the distance from the source. This expansion will be hoped to be of the form of a multipole expansion in the Newtonian limit. Of cource, there is, a priori, no assurance that such a expansion will be just of the form of a multipole expansion. Then the main purpose of this paper is to show that this is the case