Kerr-Schild Metrics Research Articles

Regular sources of the Kerr-Schild class for rotating and nonrotating black hole solutions

A unified approach to regular interiors of black holes with smooth matter distributions in the core region is given. The approach is based on a class of Kerr-Schild metrics representing minimal deformations of the Kerr-Newman solution, and allows us to give a common treatment for (charged and uncharged) rotating and nonrotating black holes. It is shown that the requirement of smoothness of the source constrains the structure of the core region in many respects: in particular, for Schwarzschild holes a de Sitter core can be selected, which is surrounded by a smooth shell giving a leading contribution to the total mass of the source. In the rotating, noncharged case the source has a similar structure, taking the form of a (anisotropic and rotating) de Sitter--like core surrounded by a rotating elliptic shell. The Kerr singular ring is regularized by anisotropic matter rotating in the equatorial plane, so that the negative sheet of the Kerr geometry is absent. In the charged case the sources take the form of ``bags,'' which can have de Sitter or anti--de Sitter interiors and a smooth domain wall boundary, with a tangential stress providing charge confinement. The Arnowitt-Deser-Misner and Tolman relations are used to calculate the total mass of the sources.

Generalized Kerr-Schild metrics and the gravitational field of a massless particle on the horizon

We investigate the structure of the gravitational field generated by a massless particle moving on the horizon of an arbitrary (stationary) black hole. This is done by employing the generalized Kerr-Schild class where we take the null generators of the horizon as the geodetic null vector field and a scalar function which is concentrated on the horizon.

Letter: A Note on Kerr-Schild Metrics with Geodesic Null Vector

We calculate the Einstein and Landau-Lifshitzpseudo-tensors for Kerr-Schild metrics with geodesicnull vector and show that the first of these vanishesidentically but that the second does not. These results correct a statement by Gurses and Gursey whoclaimed that both were zero. We also show that thesub-class of metrics for which both pseudotensors vanishincludes the Kinnersley photon rocket metric for anarbitrarily accelerating point mass.

Curvature-free asymmetric metric connections and Lanczos potentials in Kerr–Schild spacetimes

The set of spinor equations linking the curvature spinors of a Riemann–Cartan space with the curvature spinors of the corresponding Riemann space is derived, and these are used to establish a simple relationship between Ψ-flat connections of the Riemann–Cartan space and Lanczos potentials of the Riemann space. This not only yields, very easily, a recent result of Bergqvist for the Kerr metric, but also enables Bergqvist’s result to be generalized; specifically we show that a curvature-free connection, associated with a class of Kerr–Schild metrics, can be identified as a Lanczos potential for the Weyl conformal curvature spinor of these spaces.

Kerr–Schild metrics revisited. II. The complete vacuum solution

The complete solution of Einstein’s gravitational equations with a vacuum–vacuum Kerr–Schild pencil of metrics gab+Vlalb is obtained. Our result generalizes the solution of the Kerr–Schild problem with a flat metric gab (represented by the Kerr theorem) to the case when gab is the metric of a curved space-time.

Kerr–Schild metrics revisited. I. The ground state

The Kerr–Schild pencil of metrics gab+Λlalb is investigated in the generic case when it maps an arbitrary vacuum space-time with metric gab to a vacuum space-time. The theorem is proved that this generic case, with the field l shearing, does not contain the shear-free subclass as a smooth limit. It is shown that one of the Kóta–Perjés metrics is a solution in the shearing class.

Single kerr-schild metrics: A double view

Real-vacuum single Kerr-Schild (ISKS) metrics are discussed and new results proved. It is shown that if the Weyl tensor of such a metric has a twist-free expanding principal null direction, then it belongs to the Schwarzschild family of metrics — there are no Petrov type-II Robinson-Trautman metrics of Kerr-Schild type. If such a metric has twist then it belongs either to the Kerr family or else its Weyl tensor is of Petrov type II. The main part of the paper is concerned with complexified versions of Kerr-Schild metrics. The general real ISKS metric is written in double Kerr-Schild (IDKS) form. TheH andl potentials which generate IDKS metrics are determined for the general vacuum ISKS metric and given explicitly for the Schwarzschild and Kerr families of metrics.

Complex relativity and real solutions. V. The flat space background

One of the problems in dealing with double (and single) Kerr-Schild metrics is that there is a good deal of combined coordinate and tetrad freedom in writing any particular metric of this type in a standard form, and this freedom is not yet properly understood. This paper investigates part of this problem by examining the freedom in choosing the background flat metric for the standard form of a vacuum IDKS (integrable double Kerr-Schild) metric and by looking at the freedom in choosing the potentialH, which determines IDKS flat metrics in this standard form. Examples are given of different forms ofH which generate flat space. A slight generalization of the vacuum IDKS metric is also given in which thesurface equations andcurvature scalar are zero (i.e.,Φ00=Φ01=Φ11=Λ=0 in Newman-Penrose language) but the othercentral equations and theresidual equations are nonzero.

Complex relativity and real solutions II: Classification of complex bivectors and metric classes

This paper continues the examination of real metrics and their properties from the viewpoint of complex relativity as initiated by McIntosh and Hickman [1]. Tetrads of real metrics can be formally complexified by complex coordinate transformations and tetrad rotations and their properties investigated from the viewpoint of complex relativity. First, complex bivectors are examined and classified, partly by using the fundamental quadric surface of a metric in projective complex 3-space Pℂ3-an elegant but not well-known method of investigating the null structure of a metric. A generalization of the Mariot-Robinson theorem from real relativity is then given and related to various canonical forms of complex bivectors. The second part of the paper discusses four classes of complex metrics. Real metrics of the first class are ones with a null congruence whose wave surfaces have equal curvature. The second class, a subcase of the first one, is the main one; it contains integrable double Kerr-Schild metrics. Different, but equivalent, definitions of such metrics are given from various viewpoints. Two other subcasses are also discussed. The nonexpanding typs-D vacuum metric is considered and it is shown how complex transformations may be made to write it (and subcases) in double (or single) Kerr-Schild form.

Self-dual Kerr–Schild metrics and null Maxwell fields

A construction for self-dual solutions of Einstein’s equations with metric in Kerr–Schild form from null electromagnetic fields in Minkowski space is described. Two examples are given.

Spin strings in gravitation

Vibrating annular singular filaments in the Kerr-Schild metrics which are analogues of the strings of the dual models are considered. The spin excitation of the filament is related to the propagation of an electromagnetic wave along it. The model is described by a point source undergoing periodic motion in the complex domain.

Complex potential equations II. An application to general relativity

AbstractThe results of a previous paper are applied to a study of the class of Kerr–Schild metrics in general relativity. These metrics have the formwhere η is the flat (Minkowski) space-time metric, m is an arbitrary real number, and 1 is a null covector. It is already known that for a certain restricted subclass of these metrics, the vacuum Einstein field equations, viz.can be written in the formwhere γ is a complex potential. Using the methods developed in a previous paper, such space-times are characterized by means of a special family of complex surfaces in three-dimensional Euclidean space, and the exact solutions for the metric g are consequently recovered. It is also shown that the field equations for a much wider class of Kerr–Schild metrics can be expressed in terms of a potential formalism, not only in the vacuum case, but also for many electrovacuum solutions.