Boosting the Kerr metric

An exact solution of Einstein's equations describing the gravitational wave associated to a ring of massless particles is presented. It has been obtained by boosting the Kerr metric along the axis of symmetry, and considering the limit in which the speed of the boost tends to the velocity of light.     

Characterizations of the Kerr metric

For stationary, asymptotically flat solutions of Einstein's equations, covariant functionals of the metric variables are defined which characterize the Kerr metric uniquely. For instance, we obtain a generalization of the Bach tensor to stationary metrics, which vanishes if and only if the solution is Kerr. We also give a new interpretation of the Schwarzschild-to-Kerr-transformation. Our results might be applicable to simplify the proof of the uniqueness theorem for stationary black holes.     

A note on the derivation of the Kerr metric

An elementary derivation of the Kerr solution of Einstein's empty field equations is described.     

Causality violation in the Kerr metric

The conjecture that geodesics in the Kerr matric witha > m cannot violate causality in the sense of meeting a turning point before making up for lost time, is found to be untrue for null geodesics.     

De/re-constructing the Kerr metric

We derive the Kerr solution in a pedagogically transparent way, using physical symmetry and gauge arguments to reduce the candidate metric to just two unknowns. The resulting field equations are then easy to obtain, and solve. Separately, we transform the Kerr metric to Schwarzschild frame to exhibit its limits in that familiar setting.     

Kerr metric in the deSitter background

In addition to the Kerr metric with cosmological constant Λ several other metrics are presented giving a Kerr-like solution of Einstein’s equations in the background of deSitter universe. A new metric of what may be termed as rotating deSitter space-time—a space-time devoid of matter but containing null fluid with twisting null rays, has been presented. This metric reduces to the standard deSitter metric when the twist in the rays vanishes. Kerr metric in this background is the immediate generalization of Schwarzschild’s exterior metric with cosmological constant.     

Microgeon with a Kerr metric

The structure of electromagnetic and gravitational fields near the singular ring of the Kerr metric is analyzed. An approximate joint solution of the system of Maxwell-Einstein equations is given for a metric with the parameters of an electron; this solution describes a microgeon, a model of an electron. A new interpretation of the two-sheet nature of the Kerr metric is given.     

Kerr metric in eight dimensions

A Kerr-type metric is constructed in eight dimensions using the octonionic structure constants.     

A family of rotating disks as sources of the Kerr metric

A method of constructing a family of regular rotating disks as sources of the Kerr metric is discussed. The algebraic type of the energy-momentum tensor is analyzed, and it is found that none of the disks satisfies the dominant energy condition and, in some cases, even the weak energy condition is violated. Therefore, if the constructed family includes the totality of sources for the Kerr metric with a disk-like topology, then any physically satisfactory source will not have a disk structure.     

An “interior” of the Kerr metric

We show that the Kerr solution may be interpreted as arising form a thin rigidly rotating disk with a regular interior and a singular rim. The surface stress-energy tensor in the interior of the disk represents a fictitious material with positive isotropic stress and zero energy density.     

The Kerr metric in cosmological background

A metric satisfying Einstein’s equations is given which in the vicinity of the source reduces to the well-known Kerr metric and which at large distances reduces to the Robertson-Walker metric of a homogeneous cosmological model. The radius of the event horizon of the Kerr black hole in the cosmological background is found out.     

Ellipsoidal space-times,sources for the Kerr metric

The paper develops a systematic derivation of the Kerr metric and its possible sources in a clear geometric manner. It starts with a concise account of previous attempts at constructing an interior Kerr solution. Then a treatment of stationary-axisymmetric spacetimes, specially fitted to the needs of the following analysis, is presented. A new notion of an ellipsoidal space-time is introduced: it is a space-time in which local rest 3-spaces of some observers split naturally into congruences of concentric and coaxial ellipsoids. It is shown that these 3-spaces are natural spaces to consider the ellipsoidal figures of equilibrium. The investigation is carried out in detail for axially symmetric oblate confocal ellipsoids, but possible generalizations are indicated. The Kerr metric is found to be an ellipsoidal space-time of this special kind. Some remarks concerning an (unfound) explicit interior Kerr solution conclude the paper.     

A set of invariant quality factors measuring the deviation from the Kerr metric

A number of scalar invariant characterizations of the Kerr solution are presented. These characterizations come in the form of quality factors defined in stationary space-times. A quality factor is a scalar quantity varying in the interval $[0,1]$ with the value 1 being attained if and only if the space-time is locally isometric to the Kerr solution. No knowledge of the Kerr solution is required to compute these quality factors. A number of different possibilities arise depending on whether the space-time is Ricci-flat and asymptotically flat, just Ricci-flat, or Ricci non-flat. In each situation a number of quality factors are constructed and analysed. The relevance of these quality factors is clear in any situation where one seeks a rigorous formulation of the statement that a space-time is “close” to the Kerr solution, such as: its non-linear stability problem, the asymptotic settlement of a radiating isolated system undergoing gravitational collapse, or in the formulation of some uniqueness results.     

Numerical calculation of bound geodesics in the Kerr metric

Timelike geodesics, especially bound orbits in the equatorial plane (?=π/2) and spherical orbits (r=const), are calculated numerically. We plot the orbits using the Kerr-Schild coordinate system. The periastron advance and the dragging of nodes have the same values in any coordinate system and can be directly measured by an observer at infinity.     

The source of the Kerr geometry

It is shown that the volume Casimir effect for ultradense matter gives a negative contribution to the energy density and determines the specific structure of the energy-momentum tensor of the source of the Kerr metric. The source is assumed to have the form of a thin relativistic disk. The connection between the Kerr metric and models of strings and bags is discussed.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 80–86, May, 1988.In conclusion the author sincerely thanks P. I. Pronin and V. M. Mostepanenko, and also Prof. D. D. Ivanenko and the participants of his seminar for useful discussions supporting the work and for making preprints available.     

Stationary axisymmetric test fields on a Kerr metric

We determine explicitly the stationary axisymmetric Green's function of Teukolsky's equation.