On the Gravitational Field of an Arbitrary Axisymmetric Mass Endowed with Magnetic Dipole Moment

Abstract

The well-known Birkhoff’s theorem¹, the proper understanding of which became possible in many respects thanks to the paper by Peter Bergmann et al²., establishes the uniqueness of the Schwarzschild spacetime as the only static spherically symmetric solution of Einstein’s equations in vacuum. Interestingly, until recently it has not been known any asymptotically flat magnetostatic generalization of this very important spacetime refering to a magnetic dipole. Only in a series of papers^3–6, 70 years, after the discovery by Schwarzschild, the first exact asymptotically flat solutions of the static Einstein-Maxwell equations representing the exterior field of a massive magnetic dipole and possessing the Schwarzschild limit have been obtained by application of the nonlinear superposition technique to the Bonnor magnetic dipole solution⁷ (the latter reduces to the Darmois metric⁸ in the absence of magnetism). From the point of view of astrophysics, mostly dealing with deformed objects, it would be more advantageous, however, to have an asymptotically flat metric which would describe the gravitational field of an arbitrary axisymmetric mass endowed with magnetic dipole moment. In the second Section of the present article we give a possible solution of this problem by generalizing the result of the paper³ in the case of an arbitrary set of mass-multipole moments. To obtain such generalization, we construct in the explicit form the full metric describing the nonlinear superposition of the solution³ with an arbitrary static vacuum Weyl field. In the third Section we derive a charged version of the magnetostatic solution obtained in the second Section; the new metric which also generalizes the stationary electrovacuum solution given by the formulae (6), (7) of the paper⁹ represents the field of an arbitrary axisymmetric rotating mass whose electric charge and magnetic dipole moment are defined by two independent parameters, in contradistinction to the Kerr-Newman metric¹⁰, the magnetic field of which, being caused by the rotation of a charged source, disappears in a static limit. At the same time it should be mentioned that our electrovacuum solution, like the solutions⁹, has no stationary pure vacuum limit: it becomes magnetostatic in the absence of electric charge, and electrostatic in the absence of magnetic dipole moment, describing in the latter case the field of a charged static arbitrary axisymmetric mass.