Simplest static and stationary vacuum quadrupolar metrics

Abstract

In the present paper we argue that a special case of the Bach-Weyl metric describing a static configuration of two Schwarzschild black holes gives rise, after extending its parameter space to complex values, to a very simple 2-parameter model for the gravitational field of a static deformed mass. We compare this model, which has no restrictions on the quadrupole parameter, with the well-known Zipoy-Voorhees δ-metric and show in particular that the mass quadrupole moment in the latter solution cannot take arbitrary negative values. We subsequently add an arbitrary angular momentum to our static model and study some properties of the resulting 3-parameter stationary solitonic spacetime, which permits us to introduce the notion of the Fodor-Hoenselaers-Perjés relativistic multipole moments.

Metric of a rotating charged magnetized sphere

“…As has been observed in the paper [7], it is the GH multipole moments P n that should be considered approximations to the FHP quantities m n , and not the contrary. This is because the knowledge of the axis value of the Ernst potential, which is uniquely determined by the FHP multipoles m n , is sufficient for its holomorphic continuation to the whole space and for calculating the corresponding metric functions, and the GH moments P n are completely dropped from such a modern solution generating procedure.…”

“…In the limit q = 0, the axis data (3) defines the solution describing the exterior field of a rotating sphere because in this case all the corresponding FHP mass multipole moments, except for the monopole one, become equal to zero. We find it likely, in view of the potential interest for physical and astrophysical applications, to generalize the results of the paper [7] to the electrovacuum case by introducing the additional parameters of electric charge and magnetic dipole moment. Then we must consider the axis data of the form…”

“…In our recent work [7] we have introduced the notion of the Fodor-Hoenselaers-Perjés (FHP) multipole moments [8] for vacuum spacetimes as an alternative to the Geroch-Hansen (GH) multipoles [9,10]. The objective of the present paper is to extend our previous results to the electrovacuum case and in particular give arguments in favor of introducing the HP multipole moments instead of the Simon's ones.…”

“…For all practical applications, the knowledge of Simon's moments higher than n = 3 is not actually needed, and in this respect it would be worthy to note that even the expressions of P n and Q n for n = 4 and 5 were given in [4] with errors detected only fourteen years later [5]. In our recent work [7] we have introduced the notion of the Fodor-Hoenselaers-Perjés (FHP) multipole moments [8] for vacuum spacetimes as an alternative to the Geroch-Hansen (GH) multipoles [9,10]. The objective of the present paper is to extend our previous results to the electrovacuum case and in particular give arguments in favor of introducing the HP multipole moments instead of the Simon's ones.…”

On the properties of a deformed extension of the NUT space-time

“…This has led many authors in the past to consider 'rotating' generalization of static axially symmetric solutions (see for example [21][22][23][24][25][26]). In particular, stationary generalizations of the ZV metric have been studied in [27][28][29][30]. However, as it turns out, not all such generalizations describe rotating objects.…”