Kerr-NUT Solution Research Articles

Inverse scattering and the Geroch group

We study the integrability of gravity-matter systems in D=2 spatial dimensions with matter related to a symmetric space G/K using the well-known linear systems of Belinski-Zakharov (BZ) and Breitenlohner-Maison (BM). The linear system of BM makes the group structure of the Geroch group manifest and we analyse the relation of this group structure to the inverse scattering method of the BZ approach in general. Concrete solution generating methods are exhibited in the BM approach in the so-called soliton transformation sector where the analysis becomes purely algebraic. As a novel example we construct the Kerr-NUT solution by solving the appropriate purely algebraic Riemann-Hilbert problem in the BM approach.

Gravitational duality and rotating solutions

We study how gravitational duality acts on rotating solutions, using the Kerr-NUT black hole as an example. After properly reconsidering how to take into account both electric (i.e. mass-like) and magnetic (i.e. NUT-like) sources in the equations of general relativity, we propose a set of definitions for the dual Lorentz charges. We then show that the Kerr-NUT solution has non-trivial such charges. Further, we clarify in which respect Kerr's source can be seen as a mass M with a dipole of NUT charges.

Kerr–Schild structure and harmonic 2-forms on (A)dS–Kerr–NUT metrics

We demonstrate that the general (A)dS–Kerr–NUT solutions in D dimensions with ([D/2],[(D+1)/2]) signature admit [D/2] linearly-independent, mutually-orthogonal and affinely-parameterised null geodesic congruences. This enables us to write the metrics in a multi-Kerr–Schild form, where the mass and all of the NUT parameters enter the metrics linearly. In the case of D=2n, we also obtain n harmonic 2-forms, which can be viewed as charged (A)dS–Kerr–NUT solution at the linear level of small-charge expansion, for the higher-dimensional Einstein–Maxwell theories. In the BPS limit, these 2-forms reduce to n−1 linearly-independent ones, whilst the resulting Calabi–Yau metric acquires a Kähler 2-form, leaving the total number the same.

On the invariant symmetries of the D-metrics

We analyze the symmetries and other invariant qualities of the D-metrics (type D aligned Einstein-Maxwell solutions with cosmological constant whose Debever null principal directions determine shear-free geodesic null congruences). We recover some properties and deduce new ones about their isometry group and about their quadratic first integrals of the geodesic equation, and we analyze when these invariant symmetries characterize the family of metrics. We show that the subfamily of the Kerr-NUT solutions are those admitting a Papapetrou field aligned with the Weyl tensor.

The two NUT-like solutions of Ernst equation

By applying Ehlers transformation to Schwarzschild and Kerr solutions of Ernst equation and choosing the suitable coordinate transformations, the two NUT-like solutions, i.e., the so called NUT–Taub-like and the Kerr–NUT-like solutions are obtained which not only can, respectively, reduce to Schwarzschild and Kerr solutions when the parameter l′=0, but also can also reduce to the NUT–Taub metric and Kerr–NUT metric, respectively, when l′ satisfies the some approximation. Meanwhile it is shown that in the NUT–Taub and Kerr–NUT solutions the range of value for the parameter l interpreted as the gravomagnetic monopole cannot be arbitrary and should be confined by mass of the source to ∣l∣⪡m. Furthermore, the differences in the geometrical structure between the NUT–Taub-like and NUT–Taub solutions are discussed and the physical properties of the horizons on the Kerr–NUT-like space–time are analyzed.

The two generalized solutions for the stationary axisymmetric vacuum garvitational field

In this paper by applying Ehlers transformation to Schwarzschild and Kerr soluti ons of Ernst equation and introducing the proper coordinate transformations, the two solutions of the Ernst equation, i.e., the so called generalized NUT-Taub ( GNT) solution and generalized Kerr-NUT (GKN) solution are obtained, which not on ly can reduce to the well-known Schwarzschild and Kerr solutions when the parame ter L=0, but also can also reduce to the NUT-Taub metric and Kerr-NUT metric res pectively when the parameter LM and if taking 1-LM2≈1. It is showe d that in the NUT-Taub and Kerr-NUT solutions the range of value for the paramet er l (interpreted as the gravomagnetic monopole) can't be arbitrary and should be limited by mass of the source to |l|M.

Letter: Parametrization of the Kerr-NUT Solution

The dragging of the Kerr-NUT solution does not tend to zero at infinity. To modify this solution in order to produce a good asymptotic behaviour we transform it by introducing two further parameters with the aid of a SU(1,1) transformation followed by a unitary transformation. By imposing a certain relation between these parameters we obtain a new solution with a good asymptotic behaviour for any value of l, the NUT parameter. The new solution corresponds to a parametrized Kerr solution and we show that l is linked to the form of its ergosphere.

The most general axially symmetric electrovac spacetime admitting separable equations of motion

We obtain the most general solution of the Einstein electro-vacuum equation for the stationary axially symmetric spacetime in which the Hamilton–Jacobi and Klein–Gordon equations are separable. The most remarkable feature of the solution is its invariance under the duality transformation involving mass and NUT parameter, and the radial and angle coordinates. It is the general solution for a rotating (gravitational dyon) particle which is endowed with both gravitoelectric and gravitomagnetic charges, and a duality transformation exists from one to the other. It also happens to be a transform of the Kerr–NUT solution. Like the Kerr family, it is also possible to make this solution radiating which asymptotically conforms to the Vaidya null radiation.

A STATIONARY VACUUM SOLUTION DUAL TO THE KERR SOLUTION

We present a stationary axially symmetric two-parameter vacuum solution which could be considered as "dual" to the Kerr solution. It is obtained by removing the mass parameter from the function of the radial coordinate and introducing a dimensionless parameter in the function of the angle coordinate in the metric functions. It turns out that it is in fact the massless limit of the Kerr–NUT solution.

Black diholes with unbalanced magnetic charges

We present a technique that can be used to generate a static, axisymmetric solution of the Einstein-Maxwell-dilaton equations from a stationary, axisymmetric solution of the vacuum Einstein equations. Starting from the Kerr solution, Davidson and Gedalin have previously made use of this technique to obtain a pair of oppositely charged, extremal dilatonic black holes, known as a black dihole. In this paper, we shall instead start from the Kerr-NUT (Newman-Unti-Tamburino) solution. It will be shown that the new solution can also be interpreted as a dihole, but with the black holes carrying unbalanced magnetic charges. The effect of the NUT parameter is to introduce a net magnetic charge into the system. Finally, we uplift our solution to ten dimensions to describe a system consisting of $D6$ and anti--$D6$-branes with unbalanced charges. The limit in which they coincide agrees with a solution recently derived by Brax et al.

Exact Disk Solution of the Einstein Equation

The 4-soliton solution with complex pole trajectories to the vacuum Einstein equation is studied. The solution describes a static and axisymmetric gravitational field with asymptotic flatness, which is produced by a thin disk. We obtain the energy-momentum tensor of the to reveal its properties. The soliton technique 1 ) to solve the gravitational field equation gave us a new viewpoint on the exact solutions of the vacuum Einstein equation. There is an infinite series of solutions which are characterized by the soliton number n. In the stationary case with axial symmetry the series includes the Kerr-NUT solution as a 2-soliton solution and the Tomimatsu-Sato solution as a certain limit of 4-soliton solution. In the static case 2 ),3) there is an infinite series of the Weyl solutions, which is interpreted as a nonlinear superposition of multi-Weyl solutions. Two of the present authors 3 ),4) showed that the soliton technique gives not only these series of solutions but a series of ring-type solutions by adopting complex pole trajectories in the static case. The first non-trivial example of this series is the Weyl solution expressed in the oblate spheroidal coordinates,5) which has a ring-type curvature singularity. In our classification this solution is interpreted as the 2-soliton solution with non-complex-conjugate pole trajectories. The next example is the 4-soliton solution in which there are two types. One is just a nonlinear superposition of the above 2-soliton solutions, which has two ring-type singularities with different radii. The other, which is more important, is the 4-soliton solution constructed from two pairs of complex conjugate pole trajectories. This was referred to as the two­ ring solution, because the curvature invariants have finite limits at two rings although the metric coefficients diverge there. This fact contrasts with the fact that the curvature invariants diverge at the ring in the 2-soliton case. The purpose of this paper is to study the detailed structure of the two-ring solution. It will be shown that the solution should be called a solution rather than a ring solution. The properties of the will be also clarified by calculating the energy-momentum tensor. In the next section we give the solution in an explicit form and investigate its mathematical structure. From this we find that the plane containing the rings is separated into a regular region inside the inner ring and a region with the discontinuity of curvature outside the inner ring. We shall call this outer region a disk because of its non-zero energy density as will be.discussed in § 3. The is further divided into two parts by the outer ring. In § 3 we obtain the

SL(3, [formula omitted]) representation for invariance transformations in five-dimensional gravity

The invariance transformations of the axisymmetric five-dimensional vacuum Einstein equations in a represenation of the group SL(3, R ) in the potential space (analogous to the Ernst potential space). Using this formulation, an exact class of stationary axisymmetric solutions is generated, which contains among others Belinsky-Ruffini, Kramer and the Kerr-NUT solutions, as particular cases.

Two Kerr-NUT constituents in equilibrium

The equilibrium conditions for the superposition of two Kerr-NUT solutions are revisited. The new derivation of these conditions leads to formulas which include also the hyperextreme case. In the symmetric model of two hyperextreme constituents [3–5,8] the surfacef=0 of infinite red shift is investigated. It turns out that, for large enough distance parameter, the surfacef=0 consists of disconnected parts surrounding each source separately.

Conditions for Regularity on the Symmetry Axis in a Superposition of Two Kerr-NUT Solutions

A property of the symmetry axis of the metric describing a superposition of two Kerr·NUT solutions is examined. We give some conditions to the parameters included in the metric, in order to assure that the two constituent masses are dynamically balanced by the rotational repulsion against the gravitational attraction, and the total mass becomes positive. It is shown that a suitable choice of the two NUT terms of the constituent solutions is especially important. The space-time structure of single black hole has been extensively studied. However, we have little knowledge about the metric which describes many interacting black holes. Usually, such a metric is dependent on time, and the radiation of gravitational waves cannot be neglected. There exists another possibility that several black holes align along their common rotational axis and take a stationary configuration due to the dynamical balance between the gravi­ tational attraction and the rotational repulsion. Recent investigationsi),2) of stationary and axi-symmetric solutions of the vacuum Einstein equation have presented an interesting class of metrics which may represent such an equilibrium configuration. It is obvious that these metrics can be interpreted as a nonlinear superposition of several Kerr-NUT solutions. However, on the symmetry (rotational) axis between the horizons of the Kerr-NUT constituents there may exist any weak singularities of violation of local Euclidean behavior. In order to see whether this superposition is physically acceptable or not, it is necessary to examine the space-time properties on the symmetry axis. The line element of stationary and axi-symmetric vacuum space-time can be written in the form

The superposition of two Kerr solutions

We present a new exact solution of Einstein's field equations: the nonlinear superposition of two Kerr-NUT solutions. The Tomimatsu-Sato δ = 2 solution is contained as a limiting case.