Vacuum Einstein-Maxwell Equations Research Articles

ROTATING RELATIVISTIC THIN DISKS AS SOURCES OF CHARGED AND MAGNETIZED KERR–NUT SPACE–TIMES

A family of models of counterrotating and rotating relativistic thin disks of infinite extension based on a charged and magnetized Kerr–NUT metric are constructed using the well-known "displace, cut and reflect" method extended to solutions to vacuum Einstein–Maxwell equations. The metric considered has as limiting cases a charged and magnetized Taub–NUT solution and the well-known Kerr–Newman solutions. We show that for Kerr–Newman fields the eigenvalues of the energy–momentum tensor of the disk are for all the values of the parameters' real quantities so that these disks do not present heat flow in any case, whereas for charged and magnetized Kerr–NUT and Taub–NUT fields we always find regions with heat flow. We also find a general constraint on the counterrotating tangential velocities needed to cast the surface energy–momentum tensor of the disk as the superposition of two counterrotating charged dust fluids. We show that, in general, it is not possible to take the two counterrotating fluids as circulating along electrogeodesics or take the two counterrotating tangential velocities as equal and opposite.

Finite axisymmetric charged dust disks in conformastatic spacetimes

An infinite family of axisymmetric charged dust disks of finite extension is presented. The disks are obtained by solving the vacuum Einstein-Maxwell equations for conformastatic spacetimes, which are characterized by only one metric function. In order to obtain the solutions, a functional relationship between the metric function and the electric potential is assumed. It is also assumed that the metric function is functionally dependent on another auxiliary function, which is taken as a solution of the Laplace equation. The solutions for the auxiliary function are then taken as given by the infinite family of generalized Kalnajs disks recently obtained by Gonzalez and Reina [G. A. Gonzalez and J. I. Reina, Mon. Not. R. Astron. Soc. 371, 1873 (2006).], expressed in terms of the oblate spheroidal coordinates and corresponding to a family of well-behaved Newtonian axisymmetric thin disks of finite radius. The obtained relativistic thin disks have a charge density that is equal, except maybe by a sign, to their mass density, in such a way that the electric and gravitational forces are in exact balance. The energy density of the disks is everywhere positive and well behaved, vanishing at the edge. Accordingly, as the disks are made of dust, theirmore » energy-momentum tensor agrees with all the energy conditions.« less

Rotating and counterrotating relativistic thin disks as sources of stationary electrovacuum spacetimes

A detailed study is presented of the counterrotating model (CRM) for electrovacuum stationary axially symmetric relativistic thin disks of infinite extension without radial stress, in the case when the eigenvalues of the energy-momentum tensor of the disk are real quantities, so that there is not heat flow. We find a general constraint over the counterrotating tangential velocities needed to cast the surface energy-momentum tensor of the disk as the superposition of two counterrotating charged dust fluids. We then show that, in some cases, this constraint can be satisfied if we take the two counterrotating tangential velocities as equal and opposite or by taking the two counterrotating streams as circulating along electro-geodesics. However, we show that, in general, it is not possible to take the two counterrotating fluids as circulating along electro-geodesics nor take the two counterrotating tangential velocities as equal and opposite. A simple family of models of counterrotating charged disks based on the Kerr-Newman solution are considered where we obtain some disks with a CRM well behaved. We also show that the disks constructed from the Kerr-Newman solution can be interpreted, for all the values of parameters, as a matter distribution with currents and purely azimuthal pressure without heat flow. The models are constructed using the well-known "displace, cut and reflect" method extended to solutions of vacuum Einstein-Maxwell equations. We obtain, in all the cases, counterrotating Kerr-Newman disks that are in agreement with all the energy conditions.

Electrovacuum static counterrotating relativistic dust disks

A detailed study is presented of the counterrotating model (CRM) for generic electrovacuum static axially symmetric relativistic thin disks without radial pressure. We find a general constraint over the counterrotating tangential velocities needed to cast the surface energy-momentum tensor of the disk as the superposition of two counterrotating charged dust fluids. We also find explicit expressions for the energy densities, charge densities and velocities of the counterrotating fluids. We then show that this constraint can be satisfied if we take the two counterrotating streams as circulating along electrogeodesics. However, we show that, in general, it is not possible to take the two counterrotating fluids as circulating along electrogeodesics nor take the two counterrotating tangential velocities as equal and opposite. Four simple families of models of counterrotating charged disks based on Chazy-Curzon-type, Zipoy-Voorhees-type, Bonnor-Sackfield-type, and Kerr-type electrovacuum solutions are considered where we obtain some disks with a CRM well behaved. The models are constructed using the well-known ``displace, cut and reflect'' method extended to solutions of vacuum Einstein-Maxwell equations.

Shear-free, twisting Einstein-Maxwell metrics in the Newman-Penrose formalism

The problem of finding algebraically special solutions of the vacuum Einstein-Maxwell equations is investigated using the spin coefficient formalism of Newman and Penrose. The general case, in which the degenerate null vectors are not hypersurface orthogonal, is reduced to a problem of solving five coupled differential equations that are no longer dependent on the affine parameter along the degenerate null directions. It is shown that the most general regular, shearfree, nonradiating solution of these equations is the Kerr-Newman metric.

Global analysis of the Kerr-Taub-NUT metric

The Kerr-Taub-NUT metric is a local analytic solution of the vacuum Einstein-Maxwell equations. When the metric is expressed in Schwarzschild-like coordinates, two types of coordinate singularity are present. One occurs at certain values of the ``radial'' coordinate where grr becomes infinite and corresponds to bifurcate Killing horizons; the other occurs at θ=0,π, where the determinant of the components of the metric vanishes. It is shown that for nonzero NUT parameter the fixed points of the bifurcate Killing horizons and the degeneracies at θ=0,π cannot all be covered in one manifold. A maximal analytic manifold is constructed which covers the degeneracies at θ=0,π. It is non-Hausdorff but contains maximal Hausdorff subspaces, topologically S3×R, which reduce to Taub-NUT space for vanishing Kerr parameter. Kerr-Taub space can be interpreted as a closed, inhomogeneous electromagnetic-gravitational wave undergoing gravitational collapse. Another maximal analytic manifold is constructed which covers the fixed points of the bifurcate Killing horizons and the degeneracy at θ=0. It is suggested that this manifold represents the superposition of the Kerr geometry and a massless source of angular momentum at θ=π characterized by the NUT parameter.