An exact vacuum solution of Einstein's field equations is presented, describing two isolated bodies balanced by their gravitational spin-spin interaction.
We present the Ernst potential and the full four-dimensional metric for a stationary axisymmetric solution of the vacuum Einstein equations, which we obtain by application of two rank-zero H.K.X.\ transformations (Hoenselaers, Kinnersley \& Xanthopoulos, J. Math. Phys. 20, 2530 (1979)) to the general static Weyl solution. A suitable Ehlers transformation ensures asymptotic flatness.
A method of solution of the static Einstein-Maxwell equations based on the separation of variables is presented.
A procedure is proposed for the construction of the gravitational multipoles. The full metric is given in the case of a mass possessing a quadrupole moment.
A solution of the Einstein-Maxwell magnetostatic equations describing the gravitational field of a mass endowed with a magnetic dipole moment is presented. It reduces to the Schwarzschild solution if the magnetic field vanishes.
A family of axisymmetric asymptotically flat solutions of the Einstein-Maxwell field equations is presented. In a particular case we obtain a magnetostatic solution which reduces to the well-known Schwarzschild metric in the absence of a magnetic field and describes the exterior field of a massive magnetic dipole moment.
A new series of solutions of the Einstein-Maxwell magnetostatic equations is presented. Two particular solutions are denoted which reduce to the Schwarzschild metric when the mangetic field vanishes.
New magnetostatic solutions reducing to the Schwarschild solution when the magnetic field vanishes are presented.
An electrovacuum solution of the general relativity equations reducing to the Schwarzschild solution in the case of vanishing electric field is obtained.
We present two new sttionary axisymmetric solutions of the Einstein-Maxwell equations which are asymptotically flat and have the Schwarzschild metric as a pure vacuum limit.
We present an asymptotically flat stationary vacuum solution of the Einstein equations reducing to the Schwarzschild metric in a static case.
An asymptotically flat solution of the Einstein field equations representing the exterior gravitational field of a stationary rotating mass and reducting to the Schwarzschild metric in the static limit is presented. We show that our solution has an event horizon.
It is show that a generalization of the procedure given in [1] for construction of the gravitational multipoles leads to the same Newtonian limit as the generalized Erez-Rosen solution [2].
A new asymptotically flat solution of the algebraic form for the Einstein-Maxwell equations referring to a massive magnetic dipole is presented. It reduces to the Schwarzschild metric when the magnetic field vanishes.
A new exact solution of the Einstein-Maxwell equations for the field of a massless magnetic dipole is presented.
An exact asymptotically flat solution of the vacuum Einstein equations is derived able to describe the exterior gravitational field of a rotating axisymmetric mass. This solution has the Schwarzschild solution as the static limit, and it contains the Kerr metric as a special case.
We present an asymptotically flat solution of the Einstein vacuum field equations which can describe correctly the exterior gravitational field of a rotating mass due to an infinite set of arbitrary multipole moments, and which reduces to the Schwarzschild metric in the static limit.
An exact asymptotically flat solution of the vacuum Einstein equations representing the exterior gravitational field of a stationary axisymmetric mass with an arbitrary mass-multipole structure is presented.
An exact asymptotically flat solution of the Einstein vacuum equations is presented. Similar to the well-known Kerr solution, it reduces to the Schwarzschild metric in the absence of rotation, and in addition it possesses an arbitrary multipole structure due to the seed Erez-Rosen metric.
It is shown that under "internal perturbations" of the Schwarzschild solution, the event horizon need not become singular everywhere.
A new exact asymptotically flat solution of Einstein's vacuum equations representing the exterior gravitational field of a static deformed mass with the entire set of multipole moments is obtained in the explicit form. It possesses the event horizon which is singular only on the equator (x=1,y=0).
We present a new representation of the asymptotically flat general static axisymmetric solution of Einstein's vacuum equations in which the gravitational multipoles are superimposed upon the Schwarzschild solution. When the parameter |a| > 1 the event horizon is regular and there is a naked point-like singularity outside the horizon.
New exact solutions of the algebraic form for the static Einstein-Maxwell equations representing the exterior gravitational field of a massive magnetic dipole are derived. They are then used for construction of the stationary electrovacuum solutions reducing to the Schwarzschild metric in a pure vacuum limit.
An exact asymptotically flat solution of the Einstein-Maxwell equations is derived in algebraic form. It describes the gravitational field of a stationary axisymmetric charged mass and has the Schwarzschild metric as its sttic vacuum limit, being distinct from the well-known Kerr-Newman solution.
An exact asymptotically flat solution of the static Einstein-Maxwell equations describing the gravitational field of an arbitrary axisymmetric mass distribution endowed with a magnetic dipole moment is presented.
A new exact solution for the gravitational field of a massive magnetic dipole is presented.
An exact asymptotically flat solution of the Einstein-Maxwell equations describing the exterior gravitational field of a charged rotating axisymmetric mass possessing an arbitrary set of multipole moments is presented explicitly.
An exact 4-parameter solution of the Einstein-Maxwell equations is obtained which generalizes the well-known Kerr-Newman metric and takes into account the quadrupole deformation of the rotating charged mass.
An exact asymptotically flat solution of the Einstein-Maxwell equations referring to a charged rotating mass and differnet from the Kerr-Newman spacetime is presented in explicit form. The solution obtained has the Schwarzschild static pure vacuum limit.
New exact asymptotically flat solutions of Einstein's vacuum equations for the description of the exterior gravitational field of a static and stationary mass with an arbitrary mass-multipole structure are presented.
The full metric representing a charged generalization of the Gutsunaev-Manko stationary vacuum solution is given in the explicit form.
We present an asymptotically flat stationary generalization of the Schwarzschild spacetime describing the exterior gravitational field of a rotating source whose total angular momentum is equal to zero. The full metric is derived in an explicit form, and the weak field approximation for the obtained solution is considered.
The full metric representing a non-linear superposition of the Kerr solution with a massless magnetic dipole is obtained. This metric is asymptotically flat and describes correctly the exterior gravitational field of a magnetized spinning mass.
An exact asymptotically flat solution of the Einstein-Maxwell equations able to describe correctly the exterior gravitational field of a rotating mass endowed with an arbitrary electric charge and arbitrary magnetic dipole moment is obtained in an explicit form. Its static pure vacuum limit is the Schwarzschild spacetime.
We present in a concise analytical form two asymptotically flat metrics describing the superposition of the Kerr solution with an arbitrary static Weyl field which differ in their angular momentum distributions. They are then used for the construction of two asymptotically flat generalizations of the Kerr-Newman spacetime possessing the full set of mass-multipole moments able to describe the exterior gravitational field of a charged rotating arbitrary axisymmetric mass.
We present for the first time a physically realistic exact asymptotically flat solution of the Einstein-Maxwell equations representing the exterior gravitational field of a stationary rotating mass endowed with a magnetic dipole moment. Our solution reduces in its pure vacuum limit to the Schwarzschild metric.
The full metric representing the nonlinear superposition of the Kerr-Newman solution with a massless magnetic dipole is obtained. This metric is asymptotically flat and describes correctly the exterior gravitational field of a charged, magnetised, spinning mass.
A new exact asymptotically flat solution of the Einstein-Maxwell equations is presented. It contains four independent parameters associated with the mass, angular momentum, mass-quadrupole and magnetic moments of the source, and could be used for the description of the exterior field of an axisymmetric neutron star.
A new exact asymptotically flat solution of the Einstein equations able to describe the exterior gravitational field of a stationary rotating mass is presented. It contains two independent parameters associated with the total mass and total angular momentum of the source, and reduces in the static limit to the Schwarzschild space-time.
Four exact asymptotically flat solutions of the Einstein-Maxwell equations generalizing the well-known Curzon metric are presented in explicit form. A new generalization of the Dietz-Hoenselaers metric for two rotating isolated sources is obtained.
We present an exact stationary asymptotically flat solution of the Einstein-Maxwell equations that may describe the exterior gravitational field of an arbitrary-axisymmetric-mass distribution endowed with electric-charge and magnetic-dipole moments. The full metric for the solution obtained is constructed in explicit form.
An exact asymptotically flat four-parameter solution of the Einstein-Maxwell equations generalizing the well-known Kerr-Newman metric and appropriate for the description of the external field of a charged magnetized rotating mass is presented. A distinctive peculiar feature of this solution defined by remarkably simple expressions is its symmetry about the equatorial plane of the source.
An exact asymptotically flat solution of the Einstein-Maxwell equations generalizing the well known Kerr metric to the case of a rotating mass possessing an arbitrary magnetic dipole moment is presented. The new solution is symmetric with respect to the equatorial plane of the source and has a remarkably simple form.
An exact asymptotically flat 5-paramter solution of the Einstein-Maxwell equations generalizing the well-known Kerr-Newman metric is obtained in an explicit form. Besides the independent parameters of mass, angular momentum and electric charge, it also contains two other arbitrary parameters associated with the magnetic dipole and mass-quadrupole moments of the source. An important peculiar feature of this solution that differs (sic) it from other solutions for a magntetized rotating mass recently discussed in the literature is its symmetry with respect to the equatorial plane of the source in the general case.
A compact analytical form of the solution of the Einstein-Maxwell equations able to describe the exterior gravitational field of a charged rotating massive source possessing arbitrary magnetic dipole and mass-quadrupole moments is presented.
The simplest three-parameter member of the Kramer-Neugebauer family of solutions representing the exterior gravitational field of two identical Kerr black holes is identified. We show that the condition for the existence of two separated horizons is that binary system is less stringent than in the case of a single black hole.
An exact asymptotically flat axisymmetric solution of the Einstein-Maxwell equations representing the exterior field of two arbitrary Kerr-Newman masses located on the symmetry axis is constructed in explicit form. In a particular case, when the solution describes two identical Kerr-Newman sources, simple analytic formulas are obtained which allow a straightforward analysis of the equilibrium of two charged rotating masses. Some arguments are given in favor of the possibility of the balance of stationary black holes due to their gravitational spin-spin interaction.
An extended N-soliton solution of the Einstein-Maxwell equations which involve 6N arbitrary real parameters corresponding to 6N arbitrary relativistic multipole moments is presented.
The complex Ernst potentials for a family of two-soliton solutions of the Einstein-Maxwell equations in which 6 arbitrary complex constants correspond to 12 arbitrary relativistic multipole moments are constructed. Two new asymptotically flat members of this family are pointed out representing the exterior fields of binary systems of identical Kerr-Newman masses and of Kerr magnetized masses.
Sibgatullin's integral method is applied to construct the extended N-soliton solution of the Einstein-Maxwell field equations whose Ernst complex potentials and corresponding metric functions are obtained explicitly in a simple determinant form. The reported solution involves 6N arbitrary real parameters determining 6N arbitrary relativistic multipole moments.
We consider the general asymptotically flat two-soliton solution of the Einstein-Maxwell equations symmetric about the equatorial plane. The complex potentials of the solution and the corresponding metric functions are obtained in the compact form. The balance of two symmetric charged spinning objects is analysed.
New exact solution of the Einstein-Maxwell equations for the field of a charged rotating mass different from the Kerr-Newman metric is presented.
An exact asymptotically flat four-parameter solution of the Einstein-Maxwell equations representing the exterior field of two identical Kerr-Newman sources with angular momenta oppositely orientated is constructed in a simple analytic form. We show that there are no equilibrium states in such compact systems except for the case when the masses of the sources are equal to their charges. The condition for the existence of two horizons in the "antisymmetric" binary system turns out to be more stringent than in the case of a single Kerr-Newman black hole.
An exact and simple enough solution of the Einstein-Maxwell field equations is presented. This electrovacuum static axisymmetric solution possesses a clear physical interpretation: it is an external field of a nonrotating uncharged mass immersed in a homogeneous external electromagnetic and gravitational field---the Bertotti-Robinson universe. Unlike to the well known Ernst solution for a black hole in the Melvin universe, in our solution the black hole is immersed in a space-time with completely spatially homogeneous magnetic (or electric) field and with different (R2 × S2) topology. The influence of this specific background space-time topology, the structure of curvature singularities, the relation between the laws of motion and the condition of the absence of any unphysical noncurvature singularities as well as some other questions are considered.
A brief sketch of the solution contruction method used here and of various its applications, the comparison with other methods as well as a general discussion concerning the construction of solutions for interacting fields are presented in the Appendix.
We point out that the results recently presented by M. P. Lyadenko are, firstly, not original and, secondly, contain some erroneous physical statements.
Extension of the parameters in the well-known Tomimatsu-Sato family of solutions is presented which makes this family simultaneously applicable for the description of the exterior fields of sub- and super-extreme sources.
The full metric describing a Kerr black hole in an arbitrary static and axisymmetric gravitational field is presented in a concise analytical form which allows a straightforward verification of the mass formula for black holes. A sufficient condition of the regularity of the metric in the region exterior to the black hole horizon is formulated.
An exact asymptotically flat 3-parameter solution of the Einstein-Maxwell equations is presented that reduces to the Bonnor magnetic dipole solution in the magnetostatic limit, and to the Tomimatsu-Sato d = 2 solution in the stationary pure vacuum limit. This solution is the simplest possible one admitting the polynomial representation in the spheroidal coordinates (x,y) and able to describe the exterior field of a magnetized spinning mass. A multipole criterion on the choice of the parameters in the Einstein-Maxwell spacetimes is also formulated.
It is pointed out that the recent pape by Chaudhuri and Das does not present a generalization of known results.
An exact axisymmetric solution of the Einstein-Maxwell equations possessing two infinite sets of arbitrary real parameters and able to describe a deformed Kerr-Newman black hole in an external gravitational field is presented in a concise analytic form. The validity of Smarr's mass formula is demonstrated for a Kerr-Newman black hole surrounded by an external static gravitational field.
Extended soliton solutions of the Einstein field equations obtained within the framework of Sibgatullin's integral method are further analysed. We write the metric defining such solutions in a concise form suitable for concrete applications. The general formulae relating the parameters of the solution to the axis data and corresponding relativistic multipole moments are derived.
An exact, axially symmetric, asymptotically flat four-parameter solution of the Einstein-Maxwell equations representing a charged, magnetized generalization of the Tomimatsu-Sato d = 2 spinning mass solution is presented.
The problem of the equilibrium of charged aligned masses is discussed within the framework of the electrostatic multi-soliton solution of the Einstein-Maxwell equations. The equilibrium states in a charged two-body system are considered in detail. We demonstrate the validity of Smarr's mass formula for a charged static black hole when the latter is in electrostatic equilibrium with a superextreme object.
The correspondence of arbitrary parameters in exact axisymmetric solutions of the Einstein-Maxwell equations constructed with the aid of different generating methods to the analytically extended parameter sets is discussed and examples of the extended solutions are given.
The full metric describing a charged, magnetized generalization of the Tomimatsu-Sato (TS) d = 2 solution is presented in a concise explicit form. We use it to investigate some physical properties of the solution: in particular, we point out the existence of naked ring singularities in the hyperextreme TS metrics, the fact previously overlooked by the researchers, and we also demonstrate that the ring singularities can be eliminated by sufficiently strong magnetic fields in the subextreme case, while in the hyperextreme case the magnetic field can move singularities to the equatorial plane.
The equilibrium states in the axisymmetric systems of charged, magnetized, aligned, spinning masses are investigated. We show tht in the stationary binary systems consisting of a black hole and a hyperextreme object, or of two hyperextreme objects, the equilibrium can exist for the constituents possessing positive Komar masses; at the same time, Smarr's mass formula does not generally hold for the former systems due to a specific behaviour of the magnetic field.
The hyperextreme Tomimatsu-Sato solutions are shown to have ring singularities.
We present a solution to the Ernst equation which represents an infinitesimally thin dust disk consisting of two streams of particles circulating with constant angular velocities in opposite directions. These streams have the same density distribution but their relative density may vary continuously. In the limit of only one component of dust, we get the solution for the rigidly rotating dust disk of Neugebauer and Meinel; in the limit of identical densities, the static disk of Morgan and Morgan is obtained. We discuss the Newtonian and the ultrarelativistic limit, the occurrence of ergospheres, and the regularity of the solution.
A four-parameter class of exact asymptotically flat solutions of the Einstein-Maxwell equations involving only rational functions is presented. It is able to describe the exterior field of a slowly or rapidly rotating neutron star with a poloidal magnetic field.
In this note we point out that a large class of stationary, axisymmetric, vacuum solutions of the Einstein equations which are representable as rational functions of some coordinates has a factor structure similar to that of Tomimatsu-Sato metrics.
An analytically extended nine-parameter family of the electrovac rational function solutions of the Einstein-Maxwell equations generalizing the Chen-Guo-Ernst class of hyperextreme spacetimes is presented. The general four-soliton asymptotically flat solution possessing the equatorial symmetry and involving five independent real parameters is derived in a concise analytical form and its relevance to the equilibrium problem of two extreme particles is discussed.
The results of our previous paper are applied to solving analytically the balance problem in the double-Kerr solution for all three possible types of binary systems, i.e., when a binary system is composed of two non-extreme black holes, of a non-extreme black hole and a hyperextreme object and of two hyperextreme objects. We also construct a new stationary electrovacuum metric representing binary systems of charged, magnetized, rotating, aligned masses involving one extreme object and on the basis of the numerical study oof balance equations we conjecture that the equilibrium states in such systems are impossible.
We show that equilibrium of two Kerr black holes can be achieved by placing between them a relativistic disk or a third Kerr black hole, the latter case demonstrating the existence of equilibrium configurations in the purely black hole systems with a number of constituents more than two.
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