Kerr-Schild 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.

Compactifying the Kerr-Schild double copy

We show that the classical double copy relationship for Kerr-Schild spacetimes can be dimensionally reduced to give a natural notion of the double copy for Kaluza-Klein theory with gravity coupled to a gauge field and a dilaton. Under dimensional reduction the Kerr-Schild (KS) ansatz becomes the stringy Kerr-Schild (sKS) ansatz introduced by Wu. This ansatz captures many Kaluza-Klein black hole solutions, including single-charge black holes arising in both gauged and ungauged supergravity theories. We identify the single copy gauge field and scalar field of an arbitrary sKS solution. We show that the boost-reduction procedure for generating charged black hole solutions can be generalized to any stationary KS seed, and used to combine the metric with the zeroth and single copies of that seed into a single sKS solution. Furthermore, we comment on multi-charge solutions that can be written in a multi-sKS form, proposing a double copy interpretation involving multiple single copy sectors.

Heterotic Kerr-Schild double field theory and classical double copy

We discuss the generalization of the Kerr-Schild (KS) formalism for general relativity and double field theory (DFT) to the heterotic DFT and supergravity. We first introduce a heterotic KS ansatz by introducing a pair of null O (d, d+G) generalized tangent vectors. The pair of null vectors are represented by a pair of d-dimensional vector fields, and one of the vector fields is not a null vector. This implies that the null property of the usual KS formalism, which plays a crucial role in linearizing the field equations, can be partially relaxed in a consistent way. We show that the equations of motion under the heterotic KS ansatz in a flat background can be reduced to linear equations. Using the heterotic KS equations, we establish the single and zeroth copy for heterotic supergravity and derive the Maxwell and Maxwell-scalar equations. This agrees with the KLT relation for heterotic string theory.

GRay2: A General Purpose Geodesic Integrator for Kerr Spacetimes

Abstract Fast and accurate integration of geodesics in Kerr spacetimes is an important tool in modeling the orbits of stars and the transport of radiation in the vicinities of black holes. Most existing integration algorithms employ Boyer–Lindquist (BL) coordinates, which have coordinate singularities at the event horizon and along the poles. Handling the singularities requires special numerical treatment in these regions, often slows down the calculations, and may lead to inaccurate geodesics. We present here a new general-purpose geodesic integrator, GRay2, that overcomes these issues by employing the Cartesian form of Kerr–Schild (KS) coordinates. By performing particular mathematical manipulations of the geodesic equations and several optimizations, we develop an implementation of the Cartesian KS coordinates that outperforms calculations that use the seemingly simpler equations in BL coordinates. We also employ the OpenCL framework, which allows GRay2 to run on multicore CPUs as well as on a wide range of graphics processing units hardware accelerators, making the algorithm more versatile. We report numerous convergence tests and benchmark results for GRay2 for both time-like (particle) and null (photon) geodesics.

Kerr-Schild double field theory and classical double copy

The Kerr-Schild (KS) formalism is a powerful tool for constructing exact solutions in general relativity. In this paper, we present a generalization of the conventional KS formalism to double field theory (DFT) and supergravities. We introduce a generalized KS ansatz for the generalized metric in terms of a pair of null vectors. Applying this ansatz to the equations of motion of DFT, we construct the generalized KS field equation. While the generalized KS equations are quadratic in the fields, we show that it is possible to find solutions by considering linear equations only. Furthermore, we construct a Killing spinor equation under the generalized KS ansatz. Based on this formalism, we show that the classical double copy structure, which represents solutions of the Einstein equation in terms of solutions of the Maxwell equation, can be extended to the entire massless string NS-NS sector. We propose a supersymmetric classical double copy which shows that solutions of the Killing spinor equation can be realized in terms of solutions of the BPS equation of the supersymmetric Maxwell theory.

Approach of background metric expansion to a new metric ansatz for gauged and ungauged Kaluza-Klein supergravity black holes

In a previous paper [S.Q. Wu, Phys. Rev. D 83, 121502(R) (2011)], a new kind of metric ansatz was found to fairly describe all already known black hole solutions in the ungauged Kaluza-Klein (KK) supergravity theories. That metric ansatz somewhat resembles to the famous Kerr-Schild (KS) form, but it is different from the KS one in two distinct aspects. That is, apart from a global conformal factor, the metric ansatz can be written as a vacuum background spacetime plus a "perturbation" modification term, the latter of which is associated with a timelike geodesic vector field rather than a null geodesic congruence in the usual KS ansatz. In this paper, we shall study this novel metric ansatz in detail, aiming at achieving some inspiration as to the construction of rotating charged AdS black holes with multiple charges in other gauged supergravity theories. In order to investigate the metric properties of the general KK-AdS solutions, in this paper we devise a new effective method, dubbed the background metric expansion method, which can be thought of as a generalization of the perturbation expansion method, to deal with the Lagrangian and all equations of motion. In addition to two previously known conditions, namely the timelike and geodesic properties of the vector, we get the three additional constraints via contracting the Maxwell and Einstein equations once or twice with this timelike geodesic vector. In particular, we find that these are a simple set of sufficient conditions to determine the vector and the dilaton scalar around the background metric, which is helpful in obtaining new exact solutions. With these five simplified equations in hand, we rederive the general rotating charged KK-(A)dS black hole solutions with spherical horizon topology and obtain new solutions with planar topology in all dimensions.

Extended Kerr—Schild spacetimes of any dimension

We study geometric and algebraic properties of extended Kerr-Schild space times(xKS), i.e. an extension of the Kerr-Schild(KS) ansatz where, in addition to the null KS vector, a spacelike vector field appears in the metric. In contrast to the KS case, it turns out that xKS space times with a geodetic KS vector are not necessarily algebraically special and we obtain, in general, only a necessary condition under which the KS vector is geodetic. However, it is shown that this condition becomes sufficient if we appropriately restrict the geometry of the null and spacelike vector fields. Examples of xKS space times belonging to the Kundt class and also expanding xKS space times, namely the CCLP black hole, are provided and briefly discussed.

Stringlike structures in the real and complex Kerr-Schild geometry

Four-dimensional Kerr-Schild (KS) geometry displays remarkable relationships with quantum world and theory of superstrings. In particular, the Kerr-Newman (KN) solution has gyromagnetic ratio g = 2, as that of the Dirac electron and represents a consistent background for gravitational and electromagnetic field of the electron. As a consequence of very big spin/mass ratio, black hole horizons disappear, exposing the naked Kerr singular ring. We consider four-decade history of development of this structure which took finally the form of a point-string-membrane-bubble complex which is reminiscent of the enhancon model of string/M-theory. A complex string obtained in the complex structure of the Kerr geometry gives an extra dimension to the world-sheet of the real Kerr string, forming a membrane by analogue with the string/M-theory unification. By analysis of the orientifold parity of the complex Kerr string, we obtain that the determined by the Kerr theorem principal null congruence of the Kerr geometry is described by a quartic equation in projective twistor space CP3, and therefore, it creates the known Calabi- Yau twofold (K3 surface) in twistor space of the 4d KS geometry. We connect it with N=2 superstring which has (complex) critical dimension two and may be embedded into complex KS geometry.

Extended Kerr–Schild spacetimes: general properties and some explicit examples

We study extended Kerr–Schild (xKS) spacetimes, i.e. extensions of the Kerr–Schild (KS) ansatz where, in addition to the null KS vector, a spacelike vector field appears in the metric. In contrast to the KS case, we obtain only a necessary condition under which the KS vector is geodetic. However, it is shown that this condition becomes sufficient if we appropriately restrict the geometry of the null KS and spacelike vector. It turns out that xKS spacetimes with a geodetic KS vector are of Weyl type I and the KS vector has the same optical properties in the full and background spacetimes. In the case of Kundt xKS spacetimes, the compatible Weyl types are further restricted and a few examples of such metrics are provided. The relation of pp-waves to the class of Kundt xKS spacetimes is briefly discussed. We also show that the CCLP black hole, as an example of an expanding xKS spacetime, is of Weyl type Ii, which specializes to type D in the uncharged or non-rotating limit, and that its optical matrix satisfies the optical constraint.

Is the flip-flop behaviour of accretion shock cones on to black holes an effect of coordinates?

We study numerically the relativistic Bondi-Hoyle accretion of an ideal gas onto a Kerr fixed background space-time on the equatorial plane with s-lab symmetry. We use both Kerr-Schild (KS) and Boyer-Lindquist (BL) coordinates. We particularly focus on the study of the flip-flop motion of the shock cone formed when the gas is injected at supersonic speed. The development of the flip-flop instability of the shock cone in the relativistic regime was reported recently for the first time. We reproduce the flip-flop behaviour found in the past when BL coordinates are used, and perform similar numerical experiments using horizon penetrating KS coordinates. We find that when using KS coordinates the shock cone oscillates, however such oscillations are not of the flip-flop type and their amplitude decrease with resolution.

Higher Dimensional Kerr-Schild Spacetimes with (A)dS Background

Geometric properties of higher dimensional Kerr-Schild (KS) spacetimes with (A)dS background are summarised. For Einstein spaces algebraic types of the Weyl tensor compatible with the KS ansatz are established, optical properties of the KS vector are given. Important examples of exact solutions belonging to this class, such as Kerr-de Sitter metrics in arbitrary dimension are also briefly discussed.

Twistor-beam excitations of black holes and prequantum Kerr-Schild geometry

Exact Kerr-Schild (KS) solutions for electromagnetic excitations of black holes are singular beams supported on twistor lines of the KS geometry. These beams have a very strong back-reaction on the metric and horizon and create a fluctuating KS geometry occupying an intermediate position between the classical and quantum gravities. We consider the Kerr theorem, which determines the twistor structure of the KS geometry and the corresponding holographic prequantum space-time adapted to a subsequent quantum treatment.

Fluctuating twistor-beam solutions and Pre-Quantum Kerr-Schild geometry

Kerr-Schild (KS) geometry is based on a congruence of twistors which is determined by the Kerr theorem. We describe time-dependent KS solutions for electromagnetic excitations of black-holes taking into account the consistent back-reaction to metric. The exact solutions have the form of singular beam-like pulses supported on twistor null lines of the Kerr congruence. The twistor-beams have very strong back reaction to metric and BH horizon and produce a fluctuating KS geometry which takes an intermediate position between the Classical and Quantum gravity.

Higher dimensional Kerr–Schild spacetimes

We investigate general properties of Kerr–Schild (KS) metrics in n > 4 spacetime dimensions. First, we show that the Weyl tensor is of type II or more special if the null KS vector k is geodetic (or, equivalently, if Tabkakb = 0). We subsequently specialize to vacuum KS solutions, which naturally split into two families of non-expanding and expanding metrics. After demonstrating that non-expanding solutions are equivalent to the known class of vacuum Kundt solutions of Weyl type N, we analyze expanding solutions in detail. We show that they can only be of the type II or D, and we characterize optical properties of the multiple Weyl aligned null direction (WAND) k. In general, k has caustics corresponding to curvature singularities. In addition, it is generically shearing. Nevertheless, we arrive at a possible ‘weak’ n > 4 extension of the Goldberg–Sachs theorem, limited to the KS class, which matches previous conclusions for general type III/N solutions. In passing, properties of Myers–Perry black holes and black rings related to our results are also briefly discussed.