Newtonian gravitational multipoles as group-invariant solutions

Abstract

A family of vector fields that are the infinitesimal generators of determined one-parameter groups of transformations are constructed. It is shown that these vector fields represent symmetries of the system of differential equations interrelated by the axially symmetric Laplace equation and a certain supplementary equation. Group-invariant solutions of this system of equations are obtained by means of two alternative methods, and it is proved that these solutions turn out to be the family of axisymmetric potentials related to specific gravitational multipoles. The existence of these symmetries provides us with a generalization of the fact that the Newtonian Monopole is defined by the solution of the Laplace equation with spherical symmetry, and it allows us to extract from all solutions of this equation those with the prescribed Newtonian Multipole Moments.

On the calculation and interpretation of MSA coordinates

“…Could we write this function u analytically equal to the Newtonian gravitational series but in terms of Relativistic Multipole Moments (RMM) ? We are concerned with these questions for several reasons, in particular because such a description of the relativistic gravitational solution would recover the benefits of the classical interpretation of the gravitational potential (see [10] for details). Moreover, the Weyl family of solutions depends on arbitrary constants, a n , in principle without any physical criteria to choose one or another solution from them, whereas the function u would allow us to deal, in a very simple form, with the Relativistic Multipole Solutions.…”

“…The possibility of extrapolating the symmetries obtained in NG [10] to GR, as well as characterizing the solutions with a finite number of RMM by means of group-invariant solutions, are the relevant features of these coordinate systems and the reason for their proposed name.…”

“…In a work published recently [10], the existence of some kinds of symmetries in NG has been proved, which makes it possible to extract from all solutions of the axially symmetric Laplace equation those with the prescribed Newtonian Multipole Moments. A family of vector fields that are the infinitesimal generators of certain one-parameter groups of transformations can be constructed.…”

“…By introducing these coordinates, the function u, which describes the static and axially symmetric vacuum solution with a finite number of RMM, should satisfy the same system of differential equations as the classical potential in NG, and the symmetries of these equations [10] thus allow us to describe and determine the Multipole Solutions in GR analogously to the Newtonian case.…”

“…In other words, can the symmetry groups obtained for the Laplace equation and the supplementary equation [10] be extrapolated to the Ernst equation in that system of coordinates? And if so, could we establish theorems relating the existence of the symmetry of a system of differential equations to that system of coordinates?…”