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18.8 Multipole moments 289of the potential equation ∆V = 0 in the flat 3-space ̂Σ 3 via the relations(18.34) and (18.22), i.e. in three-dimensional vector notation,e 2U =(V V ) −1 , curl A =i(V grad V − V grad V ), ∆V =0. (18.73)This class of solutions was discovered by Neugebauer (1969), Perjés(1971), and Israel and Wilson (1972). The fields are static if curl A =0.Inparticular, purely electric fields (Φ = Φ) are static. Papapetrou (1947) andMajumdar (1947) have given this special class of static Einstein–Maxwellfields without spatial symmetry.The asymptotic form of the electromagnetic potentials and the spacetimemetric shows that the source of the class under consideration satisfies|Q| = √ κ 0 /2M, µ = ± √ κ 0 /2J (M = mass, Q = charge, µ = magneticmoment, J = angular momentum). The conformastationary solutions arethe exterior fields of charged spinning sources in equilibrium under theirmutual electromagnetic and gravitational forces.The linearity of the differential equation for V allows a superpositionof solutions. An example of an exterior field of N isolated sources will begiven in (21.29).If the geometry is regular outside the sources, the condition∫(V grad V − V grad V )df = 0 (18.74)Sis satisfied for every exterior closed 2-surface S. From this regularity conditionone can derive restrictions on the source parameters which guaranteethat no stresses between the spinning sources occur (Israel and Spanos1973). For a discussion of the regularity conditions in stationary fields,and further references, see Ward (1976).Linet (1987) considered in the metricds 2 =e −2U ( dz 2 +dρ 2 + B 2 ρ 2 dϕ 2) − e 2U dt 2 , 0