Kerr Theorem Research Articles

Twistor space origins of the Newman-Penrose map

Recently, we introduced the “Newman-Penrose map”, a novel correspondence between a certain class of solutions of Einstein’s equations and self-dual solutions of the vacuum Maxwell equations, which we showed was closely related to the classical double copy. Here, we give an alternative definition of this correspondence in terms of quantities that are defined naturally on twistor space, and a shear-free null geodesic congruence on Minkowski space whose twistorial character is articulated by the Kerr theorem. The advantage of this reformulation is that it is purely geometrical in nature, being manifestly invariant under both spacetime diffeomorphisms and projective transformations on twistor space. While the original formulation of the map may be more convenient for most explicit calculations, the twistorial formulation we present here may be of greater theoretical utility.

The Geometry of Marked Contact Engel Structures

A contact twisted cubic structure({mathcal M},mathcal {C},{varvec{upgamma }}) is a 5-dimensional manifold {mathcal M} together with a contact distribution mathcal {C} and a bundle of twisted cubics {varvec{upgamma }}subset mathbb {P}(mathcal {C}) compatible with the conformal symplectic form on mathcal {C}. The simplest contact twisted cubic structure is referred to as the contact Engel structure; its symmetry group is the exceptional group mathrm {G}_2. In the present paper we equip the contact Engel structure with a smooth section sigma : {mathcal M}rightarrow {varvec{upgamma }}, which “marks” a point in each fibre {varvec{upgamma }}_x. We study the local geometry of the resulting structures ({mathcal M},mathcal {C},{varvec{upgamma }}, sigma ), which we call marked contact Engel structures. Equivalently, our study can be viewed as a study of foliations of {mathcal M} by curves whose tangent directions are everywhere contained in {varvec{upgamma }}. We provide a complete set of local invariants of marked contact Engel structures, we classify all homogeneous models with symmetry groups of dimension ge 6 up to local equivalence, and we prove an analogue of the classical Kerr theorem from Relativity.

Maxwell, Yang–Mills, Weyl and eikonal fields defined by any null shear-free congruence

We show that (specifically scaled) equations of shear-free null geodesic congruences on the Minkowski space-time possess intrinsic self-dual, restricted gauge and algebraic structures. The complex eikonal, Weyl 2-spinor, [Formula: see text] Yang–Mills and complex Maxwell fields, the latter produced by integer-valued electric charges (“elementary” for the Kerr-like congruences), can all be explicitly associated with any shear-free null geodesic congruence. Using twistor variables, we derive the general solution of the equations of the shear-free null geodesic congruence (as a modification of the Kerr theorem) and analyze the corresponding “particle-like” field distributions, with bounded singularities of the associated physical fields. These can be obtained in a straightforward algebraic way and exhibit nontrivial collective dynamics simulating physical interactions.

Role of symmetries in the Kerr-Schild derivation of the Kerr black hole

In this work we explore the consequences of considering from the very beginning the stationary and axisymmetric properties of the Kerr black hole as one attempts to derive this solution through the Kerr-Schild ansatz. The first consequence is kinematical and is based on a new stationary and axisymmetric version of the Kerr theorem that yields to the precise shear-free and geodesic null congruence of flat spacetime characterizing the Kerr solution. A straightforward advantage of this strategy is that now the parameter $a$ appears naturally as associated to the conserved angular momentum of the geodesics due to axisymmetry. The second consequence is dynamical and takes into account the circularity theorem. In fact, a stationary-axisymmetric Kerr-Schild ansatz is in general incompatible with the circularity property warranted by vacuum Einstein equations unless the remaining angular dependence in the Kerr-Schild profile appears fixed in a precise way. Thanks to these two ingredients, the integration of the Einstein equations reduces to a simple ordinary differential equation on the radial dependence, whose integration constant is precisely the mass $m$. This derivation of the Kerr solution is simple but rigorous, and it may be suitable for any textbook.

Emergence of the Dirac equation in the solitonic source of the Kerr spinning particle

Gravitational and electromagnetic (EM) field of the Dirac electron is described by the Kerr-Newman (KN) solution. We elaborate a regular source of the KN solution which satisfies the requirement of flat space-time inside the source and realization of the exact KN solution outside the source. This requirement removes conflict between gravity and quantum theory and determines many details of the source structure. In particular, we obtain that the KN source should forms a gravitating bag model, similar to the known MIT and SLAC bag models. As opposite to the known bag models, the self-interacting Higgs field should be confined inside the bag, while outside the bag the gauge symmetry should be unbroken to provide the external KN gravity. We show that twistorial structure of the Kerr geometry (the Kerr theorem) determines structure of the Dirac equation, resulting in a variable mass term, which is generated inside the bag through interaction with the confined Higgs field. Similar to the other bag models, ellipsoidal deformations of the KN bag creates a string-like structure of the dressed electron -- circular string positioned along perimeter of the KN bag.

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.

On the six-dimensional Kerr theorem and twistor equation

The Kerr theorem is revisited as part of the twistor program in six dimensions. The relationship between pure spinors and integrable 3-planes is investigated. The real condition for Lorentzian spacetimes is seen to induce a projective property in the space of solutions, reminiscent of the quaternionic structure of the six-dimensional Lorentz group. The twistor equation (or Killing spinor equations generically) also has an interpretation as integrable null planes, and a family of Einstein spacetimes with this property are presented in the Kerr–Schild fashion.

Stringlike structures in Kerr-Schild geometry: The N=2 string, twistors, and the Calabi-Yau twofold

The four-dimensional Kerr-Schild geometry contains two stringy structures. The first is the closed string formed by the Kerr singular ring, and the second is an open complex string obtained in the complex structure of the Kerr-Schild geometry. The real and complex Kerr strings together form a membrane source of the over-rotating Kerr-Newman solution without a horizon, a = J/m ≫ m. It was also recently found that the principal null congruence of the Kerr geometry is determined by the Kerr theorem as a quartic in the projective twistor space, which corresponds to an embedding of the Calabi-Yau twofold into the bulk of the Kerr geometry. We describe this embedding in detail and show that the four sheets of the twistorial K3 surface represent an analytic extension of the Kerr congruence created by antipodal involution.

Complex Structure of the Four-Dimensional Kerr Geometry: Stringy System, Kerr Theorem, and Calabi-Yau Twofold

The 4D Kerr geometry displays many wonderful relations with quantum world and, in particular, with superstring theory. The lightlike structure of fields near the Kerr singular ring is similar to the structure of Sen solution for a closed heterotic string. Another string, open and complex, appears in the complex representation of the Kerr geometry initiated by Newman. Combination of these strings forms a membrane source of the Kerr geometry which is parallel to the structure of M-theory. In this paper we give one more evidence of this relationship, emergence of the Calabi-Yau twofold (K3 surface) in twistorial structure of the Kerr geometry as a consequence of the Kerr theorem. Finally, we indicate that the Kerr stringy system may correspond to a complex embedding of the criticalN=2superstring.

Electromagnetic fields in Kerr-Schild space-times

Making use of twistor structures and the Kerr theorem for shear-free null geodesic congruences, an infinite family of electromagnetic fields satisfying the homogeneous Maxwell equations in flat Minkowski and the associated curved Kerr-Schild backgrounds is obtained for any such congruence in a purely algebraic way. Simple examples of invariant axisymmetric Maxwell fields are presented.

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.

A nonstationary generalization of the Kerr congruence

Making use of the Kerr theorem for shear-free null congruences and of Newman's representation for a virtual charge ``moving'' in complex space-time, we obtain an axisymmetric time-dependent generalization of the Kerr congruence, with a singular ring uniformly contracting to a point and expanding then to infinity. Electromagnetic and complex eikonal field distributions are naturally associated with the obtained congruence, with electric charge being necesssarily unit (``elementary''). We conjecture that the corresponding solution to the Einstein-Maxwell equations could describe the process of continious transition of the naked ringlike singularitiy into a rotating black hole and vice versa, under a particular current radius of the singular ring.

Complex Kerr geometry, twistors and the Dirac electron

The Kerr spinning particle displays some remarkable relations to the Dirac electron, and has a reach spinor structure which is based on a twistorial description of the Kerr congruence determined by the Kerr theorem. We consider the relation between this spinor-twistorial structure and spinors of the Dirac equation, and show that the Dirac equation may naturally be incorporated into Kerr–Schild formalism as a master equation controlling the twistorial structure of Kerr geometry. As a result, the Dirac electron acquires an extended spacetime structure having a clear coordinate description with natural incorporation of a gravitational field. The relation between the Dirac wave function and Kerr geometry is realized via a chain of links: Dirac wave function ⇒ Complex Kerr Source ⇒ Kerr Theorem ⇒ Real Kerr geometry. As a result, the wave function acquires the role of an ‘order parameter’ which controls spin, dynamics and twistorial polarization of Kerr–Newman spacetime.

THE KERR THEOREM AND MULTI-PARTICLE KERR–SCHILD SOLUTIONS

We discuss an extended version of the Kerr theorem which allows one to construct the exact solutions of the Einstein–Maxwell field equations from a holomorphic generating function F of twistor variables. The exact multi-particle Kerr–Schild solutions are obtained from generating function of the form [Formula: see text], where Fi are partial generating functions for the spinning particles i = 1 ⋯ k. Solutions have an unusual multi-sheeted structure. Twistorial structures of the ith and jth particles do not feel each other, forming a type of its internal space. Gravitational and electromagnetic interaction of the particles occurs via the light-like singular twistor lines. As a result, each particle turns out to be "dressed" by singular pp-strings connecting it to other particles. We argue that this solution may have a relation to quantum theory.

Asymptotic twistor theory and the Kerr theorem

We first review asymptotic twistor theory with its real subspace of null asymptotic twistors: a five-dimensional CR manifold. This is followed by a description of the Kerr theorem (the identification of shear-free null congruences, in Minkowski space, with the zeros of holomorphic functions of three variables) and an asymptotic version of the Kerr theorem that produces regular asymptotically shear-free null geodesic congruences in arbitrary asymptotically flat Einstein or Einstein–Maxwell spacetimes. A surprising aspect of this work is the role played by analytic curves in H-space, each curve generating an asymptotically flat null geodesic congruence. Also there is a discussion of the physical space realizations of the two associated five- and three-dimensional CR manifolds.

Kerr-Schild approach to the boosted Kerr solution

Using a complex representation of the Debney-Kerr-Schild solutions and the Kerr theorem we analyze the boosted Kerr geometries and give the exact and explicit expressions for the metrics, the principal null congruences, the coordinate systems and the location of the singularities for an arbitrary value and orientation of the boost with respect to the angular momentum. In the limiting, ultrarelativistic case we obtain lightlike solutions possessing diverging and twisting principal null congruences and having, contrary to the known $\mathrm{pp}$-wave limiting solutions, a nonzero value of the total angular momentum. The implications of the above results in various related fields are discussed.

Twistor geometry of light rays

The geometry of real and complex light rays (mainly in Minkowski space) is studied using twistor methods. The properties of weak and strong incidence between rays are examined, some apparent anomalies arising for neighbouring rays being explained. Various apparently different interpretations of a complex surface in projective twistor space are given (one coming from the Kerr theorem describing shear-free congruences in Minkowski space). The relations between them are analysed in terms of the caustics of shear-free congruences, null hypersurfaces and a twistor description of spacelike or timelike 2-surfaces. The situation for curved spacetime is also considered. The relation between ray geometry in Minkowski space and a 6-quadric of signature (++++----), with its triality properties, is explored.

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.

Riemannian twistors and Hermitian structures on low-dimensional space forms

A Riemannian Kerr Theorem is described, which associates to an integrable Hermitian structure (with singularities) on S4 a complex analytic surface in CP3. To such a Hermitian structure can be associated a harmonic morphism defined on each of the four-dimensional space forms with values in a Riemann surface. This unifies the classifications for the three-dimensional space forms of Baird and Wood and the recent classifications for R4 and S4 by Wood, into one simple equation. As a consequence we derive the classification for harmonic morphisms defined on four-dimensional hyperbolic space with values in a Riemann surface and a new implicit form for harmonic morphisms defined on three-dimensional hyperbolic space. Unification of the Kerr Theorems for S4 and for compactified Minkowski space M is achieved at the complex level after suitably embedding the spaces in compactified, complexified Minkowski space.