Kerr-Schild Metric Research Articles

Kerr-Schild double copy for Kundt spacetimes of any dimension

We show that vacuum type N Kundt spacetimes in an arbitrary dimension admit a Kerr-Schild (KS) double copy. This is mostly done in a coordinate-independent way using the higher-dimensional Newman-Penrose formalism. We also discuss two kinds of non-uniqueness of an electromagnetic field corresponding to a given KS metric (i.e., its single copy) — these originate, respectively, from the rescaling freedom in the KS vector and from the non-uniqueness of the splitting of the KS metric in the flat part and the KS part. In connection to this, we show that the subset of KS pp-waves admits both null and non-null electromagnetic single copies. Since vacuum type N Kundt spacetimes are universal solutions of virtually any higher-order gravities and null fields in such backgrounds are immune to higher-order electromagnetic corrections, the KS-Kundt double copy demonstrated in the present paper also applies to large classes of modified theories.

On the discrete version of the Kerr geometry

In this paper, a Kerr-type solution in the Regge calculus is considered. It is assumed that the discrete general relativity, the Regge calculus, is quantized within the path integral approach. The only consequence of this approach used here is the existence of a length scale at which edge lengths are loosely fixed, as considered in our earlier paper. In addition, we previously considered the Regge action on a simplicial manifold on which the vertices are coordinatized and the corresponding piecewise constant metric is introduced, and found that for the simplest periodic simplicial structure and in the leading order over metric variations between four-simplices, this reduces to a finite-difference form of the Hilbert–Einstein action. The problem of solving the corresponding discrete Einstein equations (classical) with a length scale (having a quantum nature) arises as the problem of determining the optimal background metric for the perturbative expansion generated by the functional integral. Using a one-complex-function ansatz for the metric, which reduces to the Kerr–Schild metric in the continuum, we find a discrete metric that approximates the continuum one at large distances and is nonsingular on the (earlier) singularity ring. The effective curvature [Formula: see text], including where [Formula: see text] (gravity sources), is analyzed with a focus on the vicinity of the singularity ring.

A method for incorporating the Kerr–Schild metric in electromagnetic particle-in-cell code

An algorithm is presented that incorporates the Kerr–Schild metric in an electromagnetic particle-in-cell code. This paper describes the algorithm and implementation for the simulation of charged particles in the region of a spinning black hole. A description is given of the method used to calculate fields, currents, and particle motion using the tensor formalism. We test the overall model by using a ‘toy’ black hole and accretion disk system in a uniform magnetic field to produce bipolar jets.

Electromagnetic waves in the Kerr–Schild metric

A short electromagnetic wavelength approximation is used to obtain general solutions to the Maxwell equations for electromagnetic radiation in a Kerr–Schild plane gravitational wave metric. Their properties are then investigated for the specific case of a light beam in a weak, harmonic gravitational wave.

Kerr–Schild metric in topological massive (2+1) gravity

In the framework of the (2+1)-dimensional topologically massive theory of gravity put forward by Deser, Jackiw, and Templeton, we analyze the (2+1) Kerr–Schild metric, in order to search for exact solutions, and to obtain plane-fronted wave solutions, which in the standard (2+1) Einstein theory (whose dynamics is provided by a cosmological constant term) do not exist.

Energy distribution of a stationary beam of light

Aguirregabiria et al showed that Einstein, Landau and Lifshitz, Papapetrou, and Weinberg energy-momentum complexes coincide for all Kerr-Schild metric. Bringely used their general expression of the Kerr-Schild class and found energy and momentum densities for the Bonnor metric. We obtain these results without using Aguirregabiria et al results and verify that Bringley's results are correct. This also supports Aguirregabiria et al results as well as Cooperstock hypothesis. Further, we obtain the energy distribution of the space-time under consideration.

Integral Equation for the Spinor Amplitude of a Dirac Particle in a Curved Space–Time

We obtain a system of integral equations for the spinor amplitude of a wave packet describing a massive neutral Dirac particle in a curved space–time with an arbitrary geometry. This equation permits describing the spin dynamics of fermions in gravitational fields adequately to the quantum nature of spin. We consider a specific example of the Kerr–Schild metric. We also discuss the problem of massive neutrino oscillations in an external gravitational field.

Locating boosted Kerr and Schwarzschild apparent horizons

We describe a finite-difference method for locating apparent horizons and illustrate its capabilities on boosted Kerr and Schwarzschild black holes. Our model spacetime is given by the Kerr-Schild metric. We apply a Lorentz boost to this spacetime metric and then carry out a 3+1 decomposition. The result is a slicing of Kerr/Schwarzschild in which the black hole is propagated and Lorentz contracted. We show that our method can locate distorted apparent horizons efficiently and accurately.

Nonlinear spinor in a Kerr–Schild background

We study the nonlinear spinor field in a Kerr–Schild background by first looking for solitonic solutions in the absence of rotation, given that in the special relativistic limit there are solitonic solutions in this approximation. Since for the scalar field problem at least two independent radial functions are needed for the solitonic solutions to exist, we introduce a dilation field by deforming the Kerr–Schild metric by a Weyl factor. We find that for the parameter space studied there are no solitonic solutions in the spherically symmetric approximation. While in the present work we have studied the spinor field, we also had in mind a corresponding study of a scalar field in the same background where it may be of interest for describing a rotating stellar soliton. The proposed strategy in both cases begins by looking for solitonic solutions in the absence of rotation and then continues by perturbatively correcting for the rotation required by the angular dependence of the Kerr background. The strategy appears to be unsuccessful.

Massive neutral Dirac particle in a curved space-time with the Kerr-Schild metric

A method for constructing the solution of the general covariant Dirac equation is developed. The solution has the form of a wave packet in the case of a slightly (compared to the wave packet size) curved space-time with the Kerr-Schild metric and describes the evolution of the spin states of a massive neutral particle with a half-integer spin. The method allows reducing the Dirac equation to a system of ordinary differential equations for the spinor amplitudes (with the necessary accuracy). We propose an iterative procedure for solving the system based on expansion with respect to a small parameter equal to the ratio between the particle wave length and the characteristic spatial scale of the change of the metric. The characteristic dissipation time of a wave packet moving in a curved space-time is estimated.

Energy and angular momentum of charged rotating black holes

We show that the pseudotensors of Einstein, Tolman, Landau and Lifshitz, Papapetrou, and Weinberg (ETLLPW) give the same distributions of energy, linear momentum and angular momentum, for any Kerr-Schild metric. This result generalizes a previous work by G\"urses and G\"ursey that dealt only with the pseudotensors of Einstein and Landau and Lifshitz. We compute these distributions for the Kerr-Newman and Bonnor-Vaidya metrics and find reasonable results. All calculations are performed without any approximation in Kerr-Schild Cartesian coordinates. For the Reissner-Nordstr\"{o}m metric these definitions give the same result as the Penrose quasi-local mass. For the Kerr black hole the entire energy is confined to its interior whereas for the Kerr-Newman black hole, as expected, the energy is shared by its interior as well as exterior. The total energy and angular momentum of the Kerr-Newman black hole are $M$ and $ M a$, respectively ($M$ is the mass parameter and $a$ is the rotation parameter). The energy distribution for the Bonnor-Vaidya metric is the same as the Penrose quasi-local mass obtained by Tod.

On the Kerr–Newman metric in cosmological background

Using the method of null tetrad, the Kerr–Newman metric in the background of the Robertson–Walker universe is derived in terms of a metric that is conformal to a generalized Kerr–Schild metric. A new two–fluid interpretation of this metric is presented. The Kerr–Newman metric in the background of the Einstein–de Sitter universe is discussed in detail.

A radiating Kerr-Newman solution

A non-static exact solution of Einstein’s equations corresponding to a field of flowing null radiation plus an electromagnetic field is presented. The geometry of the solution is described by the Kerr-Schild metric. The solution admits a shear-free, geodetic null congruence. It has the symmetry of the Kerr-Newman solution and when a certain parameter is put equal to zero the solution becomes static and reduces to the Kerr-Newman solution.

Exact vacuum solutions of the Einstein equations

The vacuum Einstein equations for the Kerr-Schild metric are investigated. It is shown that they admit representation in the form of the double four-dimensional curl of the perturbation of the Euclidean metric, whereupon it is possible to note certain general directions in which to seek exact solutions. For spaces with a normal isotropic geodesic congruence the GR equations are rewritten with the application of a dyadic splitting of the metric; cases of two-dimensional subspaces of constant curvature are discussed. The investigation is illustrated by the exact nonstationary algebraic type N and anti-Schwarzschild solutions.

Kerr-Schild geometry, spinors, and instantons

The spin connection is studied for the Kerr-Schild metric. Our choice of tetrad leads to particularly simple results. The associated complex SU(2) gauge fields for many important particular cases, including the axially symmetric stationary Kerr metric, are thus presented in a unified fashion. Static spherically symmetric cases are studied in detail, including a cosmological term. A Lorentz gauge transformation is introduced for this class such that after a further (inverse Finkelstein) coordinate transformation a very simple form is obtained in the diagonal static metric. Using this, we study the passage to the Euclidean section, leading to real (anti) self-dual SU(2) gauge fields for the uncharged case. The role of the cosmological constant concerning the Pontryagin indices is elucidated. Finally a class of solutions of the zero-mass Dirac equation is studied in an appendix. The relation of such solutions possessing stringlike singularities, to similar ones in flat space, in the presence of non-Abelian monopoles and instantons is pointed out.

Some special Kerr-Schild metrics

Spinor representation of the Schwarzschild metric is given. It is shown how the Trautman-Newman complex translation leads to the Kerr metric in this representation. Some new solutions are found when λμ appearing in the Kerr-Schild metric is a constant null vector. In this case, it is observed that the Kerr-Schild metric is the solution of the Einstein-Maxwell field equations with a null fluid. It is also found that, for some special cases, this metric is a solution for the electrovacuum gravitational-field equations with a cosmological constant.

A class of solutions of the Dirac equation in the Kerr–Newman space

In a region of spacetime that may be described by the Kerr–Schild metric, the gravitational field equations define a field of O(3) matrices. By examining the spin representations of these rotations it is first shown how the gravitational field equations define a spinor field, and it is then shown how this spinor field is related to special solutions of the massless Dirac equation in the Kerr–Newman space. These special solutions have arbitrary angular momentum about the axis of rotation and in the classical limit correspond to orbits that coincide with the principal null congruences.

High frequency gravitational radiation in Kerr-Schild space-times

Vaidya has obtained general solutions of the Einstein equationsR ab=σξ a ξ b by means of the Kerr-Schild metricsg ab=η ab +Hξ a ξ b . The vector field ξ a generates a shear free null geodetic congruence both in Minkowski space and in the Kerr-Schild space-time. If in addition it is hypersurface orthogonal, the Kerr-Schild metric may be interpreted as the “background metric” in a space-time perturbed by a high frequency gravitational wave. It is shown that Vaidya's solutions satisfying this additional condition are of only two types: (1) Kinnersley's accelerating point mass solution and (2) a similar solution where a space-like curve plays the role of the time-like curve describing the world line of the accelerating mass. The solution named by Vaidya as the radiating Kerr metric does not satisfy the hypersurface orthogonal condition.

The Kerr solution: A geometrical construction

We derive an expression for the metric tensorη ij of Minkowski space-time in terms of a co-ordinate system adapted to a timelike world-tube and a shear-free, twisting congruence of null geodesicsk i emanating into the future from every event on the world-tube. By requiring the tube to represent the history of a rigid, unaccelerated, nonrotating, axially symmetric ellipsoid (in a technical sense) the null congruence is found to coincide with the null congruence of the Kerr solution which is tangent to the multiple Debever-Penrose direction of that solution. If one takes the Minkowski metric tensor as a «background» for a Kerr-Schild metric,η ij +2H k i k j , the Kerr solution is readily obtained by solving Einstein's vacuum field equations for the only remaining unknown—the scalar «potential»H. We prove that if the world-tube is static (unaccelerated and rigid) and has spatially compact normal sections, which are not necessarily axially symmetric, then the background geometry is flatif and only if the normal sections are axially symmetric ellipsoids. In so far as our work emphasizes the role of the properties of the basic world-tube in our construction, it may be considered as complementing the recent work of Newman and Winicour and of Vaidya on the same topic.

The field of a rotating charged body emitting null fluid

An exact solution of Einstein's equations for an Electromagnetic field plus pure radiation is given in terms of the Kerr-Schild metric. The radiating Kerr solution and the Debney-Kerr-Schild solution are derived as particular cases.