Source integrals of multipole moments for static space-times
J.L. Hernandez-Pastora, J. Martin, E. Ruiz
TL;DR
This paper introduces a method to compute relativistic multipole moments of static space-times using volume and surface integrals, connecting classical and relativistic definitions.
Contribution
It generalizes the classical Gauss theorem to relativistic contexts, enabling the calculation of asymptotic multipole moments in static space-times.
Findings
Derivation of volume integrals for Relativistic Multipole Moments
Reformulation of multipole moments as surface integrals at infinity
Connection between relativistic and classical multipole definitions
Abstract
The definition of Komar for the mass of a relativistic source is used as a starting point to introduce volume integrals for Relativistic Multipole Moments (RMM). A certain generalization of the classical Gauss theorem is used to rewrite these multipole moments as integrals over a surface at the infinity. Therefore it is shown that the above generalization leads to Asymptotic Relativistic Multipole Moments (ARMM), recovering the multipoles of Geroch or Thorne, when the integrals are evaluated in asympotically cartesian harmonic coordinates. Relationships regarding the Thorne definition and the classical theory of moments are shown.
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