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Killing spinors and separability of Rarita–Schwinger’s equation in type {2,2} backgrounds
Journal of Mathematical Physics
Abstract
It is shown that in a type {2,2} solution of the Einstein vacuum field equations, which admits a two‐index Killing spinor, a differential operator can be constructed that maps a solution of the Rarita–Schwinger equation into a solution of its complex conjugate. Furthermore, by considering the Plebański–Demiański metric, which contains all the vacuum type {2,2} metrics, it is shown that the separable solutions of the Rarita–Schwinger equation are eigenfunctions of a certain differential operator with the Starobinsky constant as the eigenvalue. © 1995 American Institute of Physics.
... The most commonly used equation in the study of the perturbations of Kerr black holes is the Teukolsky master equation (TME). This is a wave equation for the extreme spin-weight Newman-Penrose components (with respect to the Kinnersley tetrad) of the electromagnetic field strength or the linearized Weyl tensor, depending on the value of the spin parameter (Note that certain components of the spin-1/2 and spin-3/2 fields are also solutions of the TME with the corresponding value of the spin parameter [1][2][3][4]). Nevertheless, the other components of the electromagnetic field strength and the linearized Weyl tensor also satisfy analogous wave equations [5][6][7]. ...
... From the transformation properties of the spherical harmonics and of E μ P and J μ P under P it follows that E 0 0 and J 0 0 are real. If the fourth order Runge-Kutta method is applied, as in the present work, then the numerically computed values of E 0 0 and J 0 0 should converge to zero at the rate ∼ ( R) 4 , if the grid spacing R goes to 0. We verified this behaviour in several [14,15], is that the dependence of E 0 0 and J 0 0 on R follows the law ∼ ( R) 4 quite accurately, whereas in the case of the charge balances there are relatively large subleading corrections. ...
... sin θ e iϕ , (C. 4) and the operators J ± are the standard so(3) Lie algebra elements that raise and lower the z-component of the spin. J ± act generally as ...
The late-time behaviour of the solutions of the Fackerell–Ipser equation (which is a wave equation for the spin-zero component of the electromagnetic field strength tensor) on the closure of the domain of outer communication of sub-extremal Kerr spacetime is studied numerically. Within the Kerr family, the case of Schwarzschild background is also considered. Horizon-penetrating compactified hyperboloidal coordinates are used, which allow the behaviour of the solutions to be observed at the event horizon and at future null infinity as well. For the initial data, pure multipole configurations that have compact support and are either stationary or non-stationary are taken. It is found that with such initial data the solutions of the Fackerell–Ipser equation converge at late times either to a known static solution (up to a constant factor) or to zero. As the limit is approached, the solutions exhibit a quasinormal ringdown and finally a power-law decay. The exponents characterizing the power-law decay of the spherical harmonic components of the field variable are extracted from the numerical data for various values of the parameters of the initial data, and based on the results a proposal for a Price’s law relevant to the Fackerell–Ipser equation is made. Certain conserved energy and angular momentum currents are used to verify the numerical implementation of the underlying mathematical model. In the construction of these currents a discrete symmetry of the Fackerell–Ipser equation, which is the product of an equatorial reflection and a complex conjugation, is also taken into account.
... It is well known that on the Kerr spacetime in 4 dimensions, the field equations for spin-s fields for s = 0, 1/2, 1, 3/2, 2, namely, the Klein-Gordon equation, the Dirac equation, the Maxwell equation, the Rarita-Schwinger equation, and the linearized Einstein equation, are separable [4][5][6][7][8][9][10][11][12]. In contrast, on the Myers-Perry spacetime, which is a higher-dimensional generalization of the Kerr spacetime, the only spin-s field equations for s = 0, 1/2, 1 are known to be separable; the separability of the equations for s = 3/2 and 2 have not been revealed (see the magnificent wrap-up [13] of progress until 2017, references therein, and Refs. ...
... We introduce the notations y a = (t, r) and z i = (θ, ϕ, ψ) suitable for warped product spaces, and rewrite Eqs. (11) and (12) as ...
It has been revealed that the first order symmetry operator for the linearized Einstein equation on a vacuum spacetime can be constructed from a Killing-Yano 3-form. This might be used to construct all or part of solutions to the field equation. In this paper, we perform a mode decomposition of a metric perturbation on the Schwarzschild spacetime and the Myers-Perry spacetime with equal angular momenta in 5 dimensions, and investigate the action of the symmetry operator on specific modes concretely. We show that on such spacetimes, there is no transition between the modes of a metric perturbation by the action of the symmetry operator, and it ends up being the linear combination of the infinitesimal transformations of isometry.
... It is well known that on the Kerr spacetime in 4 dimensions, the field equations for spin-s fields for s = 0, 1/2, 1, 3/2, 2, namely, the Klein-Gordon equation, the Dirac equation, the Maxwell equation, the Rarita-Schwinger equation, and the linearized Einstein equation, are separable [4][5][6][7][8][9][10][11][12]. In contrast, on the Myers-Perry spacetime, which is a higher-dimensional generalization of the Kerr spacetime, the only spin-s field equations for s = 0, 1/2, 1 are known to be separable; the separability of the equations for s = 3/2 and 2 have not been revealed (see the magnificent wrap-up [13] of progress until 2017, references therein, and Refs. ...
... We introduce the notations y a = (t, r) and z i = (θ, φ, ψ) suitable for warped product spaces, and rewrite Eqs. (11) and (12) as ...
It has been revealed that the first order symmetry operator for the linearized Einstein equation on a vacuum spacetime can be constructed from a Killing-Yano 3-form. This might be used to construct all or part of solutions to the field equation. In this paper, we perform a mode decomposition of a metric perturbation on the Schwarzschild spacetime and the Myers-Perry spacetime with equal angular momenta in 5 dimensions, and investigate the action of the symmetry operator on specific modes concretely. We show that on such spacetimes, there is no transition between the modes of a metric perturbation by the action of the symmetry operator, and it ends up being the linear combination of the infinitesimal transformations of isometry.
... We cannot, in general, assume any relation between the gauge terms in the two different shear-free equations. We let ω be defined by (34) such that {φ 1 , φ 2 , ω} is an eigenbasis for C. Since each φ i is an eigenvector of C (29) becomes ...
... The separation constants obtained in this procedure can be given an intrinsic characterisation in terms of eigenvalues of symmetry operators. Torres del Castillo [33] has shown how, for Maxwell fields, the Starobinsky constant is given by the symmetry operator obtained via Debye potential methods in the Plebański-Demiański background and Silva-Ortigoza [34] has presented a similar analysis for the Rarita-Schwinger (spin- 3 2 ) equation. Recently Kalnins, Williams and Miller [35] have given a detailed account of the separation of variables for electromagnetic and gravitational perturbations in the Kerr space-time using Hertz potentials. ...
By using conformal Killing-Yano tensors, and their generalisations, we obtain scalar potentials for both the source-free Maxwell and massless Dirac equations. For each of these equations we construct, from conformal Killing-Yano tensors, symmetry operators that map any solution to another.
... For spins s = 1/2, 1, 3/2, 2 the field equations are Dirac-Weyl, Maxwell, Rarita-Schwinger, and linearized gravity, respectively. For spins 0, 1/2, 1, 2, see Teukolsky (1973), for the spin-3/2 case, Torres del Castillo and Silva-Ortigoza (1992), see also Silva-Ortigoza (1995). Working in the Kinnersley principal tetrad, let φ 0 , φ 2 be the Newman-Penrose scalars of spin weights 1, −1 for a Maxwell test field on the Kerr background, and letΨ 0 ,Ψ 4 denote the linearized Weyl scalars of spin weights 2, −2 for a solution of the linearized vacuum Einstein equations on the Kerr background, see Aksteiner and Andersson (2011) for details. ...
A generalization of the mode stability result of Whiting (1989) for the Teukolsky equation is proved for the case of real frequencies. The main result of the paper states that a separated solution of the Teukolsky equation governing massless test fields on the Kerr spacetime, which is purely outgoing at infinity, and purely ingoing at the horizon, must vanish. This has the consequence, that for real frequencies, there are linearly independent fundamental solutions of the radial Teukolsky equation which are purely ingoing at the horizon, and purely outgoing at infinity, respectively. This fact yields a representation formula for solutions of the inhomogenous Teukolsky equation.
... The aim of the present paper is to construct conserved currents associated with time translation, axial rotation and scaling symmetry for the Teukolsky Master Equation (TME) [1,2,3,4], which is a wave equation that governs the evolution of the extreme spin weight Newman-Penrose components [5,6] in Kinnersley tetrad of the Maxwell, the linearized gravitational, spin-1/2 (neutrino) or spin-3/2 fields in Kerr spacetime, and plays an important role in the analysis of these fields. ...
Conserved currents associated with the time translation and axial symmetries of the Kerr spacetime and with scaling symmetry are constructed for the Teukolsky Master Equation (TME). Three partly different approaches are taken, of which the third one applies only to the spacetime symmetries. The results yielded by the three approaches, which correspond to three variants of Noether's theorem, are essentially the same, nevertheless. The construction includes the embedding of the TME into a larger system of equations, which admits a Lagrangian and turns out to consist of two TMEs with opposite spin weight. The currents thus involve two independent solutions of the TME with opposite spin weights. The first approach provides an example of the application of an extension of Noether's theorem to nonvariational differential equations. This extension is also reviewed in general form. The variant of Noether's theorem applied in the third approach is a generalization of the standard construction of conserved currents associated with spacetime symmetries in general relativity, in which the currents are obtained by the contraction of the symmetric energy-momentum tensor with the relevant Killing vector fields. Symmetries and conserved currents related to boundary conditions are introduced as well, and Noether's theorem and its variant for nonvariational differential equations are extended to them. The extension of the latter variant is used to construct conserved currents related to the Sommerfeld boundary condition.
... This is a wave equation for the extreme spin-weight Newman-Penrose components (with respect to the Kinnersley tetrad) of the electromagnetic field strength or the linearized Weyl tensor, depending on the value of the spin parameter. (Note that certain components of the spin-1/2 and spin-3/2 fields are also solutions of the TME with the corresponding value of the spin parameter [2,3,4,5]). Nevertheless, the other components of the electromagnetic field strength and the linearized Weyl tensor also satisfy analogous wave equations [1,112,113]. ...
The late-time behaviour of the solutions of the Fackerell-Ipser equation (which is a wave equation for the spin-zero component of the electromagnetic field strength tensor) on the closure of the domain of outer communication of sub-extremal Kerr spacetime is studied numerically. Within the Kerr family, the case of Schwarzschild background is also considered. Horizon-penetrating compactified hyperboloidal coordinates are used, which allow the behaviour of the solutions to be observed at the event horizon and at future null infinity as well. For the initial data, pure multipole configurations that have compact support and are either stationary or non-stationary are taken. It is found that with such initial data the solutions of the Fackerell-Ipser equation converge at late times either to a known static solution (up to a constant factor) or to zero. As the limit is approached, the solutions exhibit a quasinormal ringdown and finally a power-law decay. The exponents characterizing the power-law decay of the spherical harmonic components of the field variable are extracted from the numerical data for various values of the parameters of the initial data, and based on the results a proposal for a Price's law relevant to the Fackerell-Ipser equation is made. Certain conserved energy and angular momentum currents are used to verify the numerical implementation of the underlying mathematical model. In the construction of these currents a discrete symmetry of the Fackerell-Ipser equation, which is the product of an equatorial reflection and a complex conjugation, is also taken into account.
... In this paper we consider this issue for Rarita-Schwinger field ψ µ . (For some related discussions see [7][8][9].) We only consider the case where ψ µ is a Dirac vector-spinor, and in sections 2 and 3 we only consider the case where the spacetime dimension D is even. In section 4 D is set 4, and in section 5 we do not impose any restriction on D. We put no assumption on the signature of the background metric. ...
We consider first order linear operators commuting with the operator appearing in the linearized equation of motion of Rarita-Schwinger fields which comes directly from the action. First we consider a simplified operator giving an equation equivalent to the original equation, and classify first order operators commuting with it in four dimensions. In general such operators are symmetry operators of the original operator, but we find that some of them commute with it. We extend this result in four dimensions to arbitrary dimensions and give first order commuting operators constructed of odd rank Killing-Yano and even rank closed conformal Killing-Yano tensors with additional conditions.
... In this paper we consider this issue for Rarita-Schwinger field ψ µ . (For some related discussions see [7,8,9].) We only consider the case where ψ µ is a Dirac vector-spinor, and in section 2 and 3 we only consider the case where the spacetime dimension D is even. ...
We consider first order linear operators commuting with the operator appearing in the linearized equation of motion of Rarita-Schwinger fields which comes directly from the action. First we consider a simplified operator giving an equation equivalent to the original equation, and classify first order operators commuting with it in four dimensions. In general such operators are symmetry operators of the original operator, but we find that some of them commute with it. We extend this result in four dimensions to arbitrary dimensions and give first order commuting operators constructed of odd rank Killing-Yano and even rank closed conformal Killing-Yano tensors with additional conditions.
... For spins s = 1/2, 1, 3/2, 2 the field equations are Dirac-Weyl, Maxwell, Rarita-Schwinger, and linearized gravity, respectively. For spins 0, 1/2, 1, 2, see Teukolsky (1973), for the spin-3/2 case, Torres del Castillo and Silva-Ortigoza (1992), see also Silva-Ortigoza (1995). Working in the Kinnersley principal tetrad, let φ 0 , φ 2 be the Newman-Penrose scalars of spin weights 1, −1 for a Maxwell test field on the Kerr background, and letΨ 0 ,Ψ 4 denote the linearized Weyl scalars of spin weights 2, −2 for a solution of the linearized vacuum Einstein equations on the Kerr background, see Aksteiner and Andersson (2011) for details. ...
In the here presented work we discuss a series of results that are all in one way or another connected to the phenomenon of trapping in black hole spacetimes. First we present a comprehensive review of the Kerr-Newman-Taub-NUT-de-Sitter family of black hole spacetimes and their most important properties. From there we go into a detailed analysis of the bahaviour of null geodesics in the exterior region of a sub-extremal Kerr spacetime. We show that most well known fundamental properties of null geodesics can be represented in one plot. In particular, one can see immediately that the ergoregion and trapping are separated in phase space. We then consider the sets of future/past trapped null geodesics in the exterior region of a sub-extremal Kerr-Newman-Taub-NUT spacetime. We show that from the point of view of any timelike observer outside of such a black hole, trapping can be understood as two smooth sets of spacelike directions on the celestial sphere of the observer. Therefore the topological structure of the trapped set on the celestial sphere of any observer is identical to that in Schwarzschild. We discuss how this is relevant to the black hole stability problem. In a further development of these observations we introduce the notion of what it means for the shadow of two observers to be degenerate. We show that, away from the axis of symmetry, no continuous degeneration exists between the shadows of observers at any point in the exterior region of any Kerr-Newman black hole spacetime of unit mass. Therefore, except possibly for discrete changes, an observer can, by measuring the black holes shadow, determine the angular momentum and the charge of the black hole under observation, as well as the observer's radial position and angle of elevation above the equatorial plane. Furthermore, his/her relative velocity compared to a standard observer can also be measured. On the other hand, the black hole shadow does not allow for a full parameter resolution in the case of a Kerr-Newman-Taub-NUT black hole, as a continuous degeneration relating specific angular momentum, electric charge, NUT charge and elevation angle exists in this case. We then use the celestial sphere to show that trapping is a generic feature of any black hole spacetime. In the last chapter we then prove a generalization of the mode stability result of Whiting (1989) for the Teukolsky equation for the case of real frequencies. The main result of the last chapter states that a separated solution of the Teukolsky equation governing massless test fields on the Kerr spacetime, which is purely outgoing at infinity, and purely ingoing at the horizon, must vanish. This has the consequence, that for real frequencies, there are linearly independent fundamental solutions of the radial Teukolsky equation which are purely ingoing at the horizon, and purely outgoing at infinity, respectively. This fact yields a representation formula for solutions of the inhomogenous Teukolsky equation, and was recently used by Shlapentokh-Rothman (2015) for the scalar wave equation.
... We do not review here the corresponding calculations. The first two are easy to perform and we refer to the original papers [204,205] for the separability of massless Dirac equation; see also [356,343] for a discussion of electromagnetic and Rarita-Swinger perturbations. ...
The study of higher-dimensional black holes is a subject which has recently attracted a vast interest. Perhaps one of the most surprising discoveries is a realization that the properties of higher-dimensional black holes with the spherical horizon topology are very similar to the properties of the well known four-dimensional Kerr metric. This remarkable result stems from the existence of a single object called the principal tensor. In our review we discuss explicit and hidden symmetries of higher-dimensional black holes. We start with the overview of the Liouville theory of completely integrable systems and introduce Killing and Killing-Yano objects representing explicit and hidden symmetries. We demonstrate that the principal tensor can be used as a 'seed object' which generates all these symmetries. It determines the form of the black hole geometry, as well as guarantees its remarkable properties, such as special algebraic type of the spacetime, complete integrability of geodesic motion, and separability of the Hamilton-Jacobi, Klein-Gordon, and Dirac equations. The review also contains a discussion of different applications of the developed formalism and its possible generalizations.
... For spins s = 1/2, 1, 3/2, 2 the field equations are Dirac-Weyl, Maxwell, Rarita-Schwinger, and linearized gravity, respectively. For spins 0, 1/2, 1, 2, see [29], for the spin-3/2 case, see [31], see also [25]. Working in the Kinnersley principal tetrad, let φ 0 , φ 2 be the Newman-Penrose scalars of spin weights 1, −1 for a Maxwell test field on the Kerr background, and letΨ 0 ,Ψ 4 denote the linearized Weyl scalars of spin weights 2, −2 for a solution of the linearized vacuum Einstein equations on the Kerr background, see [1] for details. ...
A generalization of the mode stability result of Whiting (1989) for the Teukolsky equation is proved for the case of real frequencies. The main result of the paper states that a separated solution of the Teukolsky equation governing massless test fields on the Kerr spacetime, which is purely outgoing at infinity, and purely ingoing at the horizon, must vanish. This has the consequence, that for real frequencies, there are linearly independent fundamental solutions of the radial Teukolsky equation which are purely ingoing at the horizon, and purely outgoing at infinity, respectively. This fact yields a representation formula for solutions of the inhomogenous Teukolsky equation.
... Returning to the framework of linearized supergravity and expressing the complete solution of the Rarita-Schwinger equation in terms of Debye potentials, the spin 3/2 perturbations of certain background solutions of the Einstein and Einstein-Maxwell equations have been analyzed in [30,31,32,40,41]. ...
We present an account of the development of supergravity in México.
Conserved currents associated with the time translation and axial symmetries of the Kerr spacetime are constructed for the Teukolsky Master Equation (TME). Three partly different approaches are taken, in which three variants of Noether's theorem are applied. The results yielded by the three approaches are essentially the same, nevertheless. The construction includes the embedding of the TME into a larger system of equations, which admits a Lagrangian and turns out to consist of two TMEs with opposite spin weight. The currents thus involve two independent solutions of the TME with opposite spin weights. The first approach provides an example of the application of an extension of Noether's theorem to nonvariational differential equations. This extension is also reviewed in general form. The variant of Noether's theorem applied in the third approach is a generalization of the standard construction of conserved currents associated with spacetime symmetries in general relativity, in which the currents are obtained as the contraction of the symmetric energy-momentum tensor with the relevant Killing vector fields.
It is shown that the separation constant not related to isometries, which appears in the solution of the Klein-Gordon equation in the Carter A background, can be defined in an invariant way as the eigenvalue of a second-order differential operator made out of a two-index Killing spinor, with the eigenfunctions being the separable solutions.
It is shown that the Dirac equation is separable by variables in a five-dimensional rotating Kerr-(anti-)de Sitter black hole with two independent angular momenta. A first-order symmetry operator that commutes with the Dirac operator is constructed in terms of a rank-3 Killing-Yano tensor whose square is a second-order symmetric Stäckel-Killing tensor admitted by the five-dimensional Kerr-(anti-)de Sitter spacetime. We highlight the construction procedure of such a symmetry operator. In addition, the first law of black hole thermodynamics has been extended to the case that the cosmological constant can be viewed as a thermodynamical variable.
Using the fact that in any type-{2 2} (i.e. type D) vacuum spacetime each component of the Maxwell spinor field satisfies a decoupled equation and there exists a two-index Killing spinor, it is shown, by a simple argument, that one can construct a symmetry operator for the Maxwell equations. An analogous result is also obtained for the Weyl neutrino field.
It is shown that the separation constant not related to isometries that appears in the solution of the spin- perturbations of the Carter A solution, can be defined in an invariant way as the eigenvalue of a certain differential operator made out of a two-index Killing spinor. Furthermore, when the electromagnetic field of the background spacetime is absent, we obtain analogous results for gravitational perturbations.
It is shown that if the background spacetime is a type-D solution of the Einstein vacuum-field equations with cosmological constant or of the Einstein - Maxwell field equations without cosmological constant, an operator can be constructed that maps a solution of the Rarita - Schwinger equation into another solution. In the case of the spin- fields in a type-D electrovac spacetime, we show that the maximal spin weight components, which obey separable decoupled equations, satisfy certain differential relations that are equivalent to the Teukolsky - Starobinsky identities fulfilled by the separated functions. It is also shown that the real and imaginary parts of the Starobinsky constants can be obtained from these relations.
It is shown that each component of the Dirac field satisfies a decoupled equation, which admits separable solutions, when the background spacetime is the Bertotti–Robinson metric, which is a solution of the Einstein vacuum field equations with a cosmological constant. Furthermore, the seperated functions appearing in the solutions are shown to obey identities of the Teukolsky–Starobinsky type and the separable solutions are shown to be eigenfunctions of a certain differential operator.
It is shown that the Dirac equation is separable by variables in a five-dimensional rotating Kerr-(anti-)de Sitter black hole with two independent angular momenta. A first order symmetry operator that commutes with the Dirac operator is constructed in terms of a rank-three Killing-Yano tensor whose square is a second order symmetric Stackel-Killing tensor admitted by the five-dimensional Kerr-(anti-)de Sitter spacetime. We highlight the construction procedure of such a symmetry operator. In addition, the first law of black hole thermodynamics has been extended to the case that the cosmological constant can be viewed as a thermodynamical variable. Comment: 24 pages, iopart.cls. Published version with references updated
The Dirac equation for the electron around a five-dimensional rotating black hole with two different angular momenta is separated into purely radial and purely angular equations. The general solution is expressed as a superposition of solutions derived from these two decoupled ordinary differential equations. By separating variables for the massive Klein-Gordon equation in the same space-time background, I derive a simple and elegant form for the Stackel-Killing tensor, which can be easily written as the square of a rank-three Killing-Yano tensor. I have also explicitly constructed a symmetry operator that commutes with the scalar Laplacian by using the Stackel-Killing tensor, and the one with the Dirac operator by the Killing-Yano tensor admitted by the five-dimensional Myers-Perry metric, respectively.
For Einstein-Maxwell fields for which the Weyl spinor is of type {2, 2}, and the electromagnetic field spinor is of type {1, 1} with its principal null directions coaligned with those of the Weyl spinor, the integrability conditions for the existence of a certain valence two Killing tensor are shown to reduce to a simple criterion involving the ratio of the amplitude of the Weyl spinor to the amplitude of a certain test solution of the spin two zero restmass field equations. The charged Kerr solution provides an example of a spacetime for which the criterion is satisfied; the chargedC-metric provides an example for which it is not.
The Kerr solution describes, in Einstein's theory, the gravitational field of a rotating black hole. The axial symmetry and stationarity of the solution are shown here to arise in a simple way from properties of the curvature tensor.
Associated with the charged Kerr solution of the Einstein gravitational field equation there is a Killing tensor of valence two. The Killing tensor, which is related to the angular momentum of the field source, is shown to yield a quadratic first integral of the equation of the motion for charged test particles.
Separable wave equations with source terms are presented for electromagnetic and gravitational perturbations of an uncharged, rotating black hole. These equations describe the radiative field completely, and also part of the nonradiative field. Nontrivial, source-free, stationary perturbations are shown not to exist. The barrier integral governing synchrotron radiation from particles in circular orbits is shown to be the same as for scalar radiation. Future applications (stability of rotating black holes, "spin-down," superradiant scattering, floating orbits) are outlined.
Expressions for the complete solution of the Rarita–Schwinger equation in terms of complex scalar potentials are obtained by means of Wald’s method of adjoint operators. The background space‐time is required to be an algebraically special solution of the Einstein vacuum field equations with cosmological constant or a solution of the Einstein–Maxwell equations such that one principal null direction of the electromagnetic field is geodetic and shear‐free.
We derive necessary and sufficient conditions that a second-order co-variant differential operator be a symmetry operator of Maxwell's equations in a general curved space-time background. It is found that such operators are naturally formulated in terms of conformal Killing vectors, tensors and spinors. Operators of this type play a role in the solution of Maxwell's equations via separation of variables in the Kerr background space-time.
The results of the theory of complexified V4’s which admit a null string are reexamined by using systematically the Bianchi identities. The formal reasons which permit the integration of the Gμν=0 equations to one differential constraint in the case of the HH structures are exhibited. Some new results concerning complexified Einstein–Weyl equations in the presence of a null string are given as well.
We show that there exist a coordinate system and null tetrad for the space-times admitting a two-parameter abelian orthogonally transitive isometry group and a pair of shearfree geodesic null congruences in which the neutrino equation is solvable by separation of variables if and only if the Weyl tensor is Petrov type D. The massive Dirac equation is separable if in addition the conformal factor satisfies a certain functional equation. As a corollary, we deduce that the neutrino equation is separable in a canonical system of coordinates and tetrad for the solution of Einstein’s type D vacuum or electrovac field equations with cosmological constant admitting a nonsingular aligned Maxwell field and that the Dirac equation is separable only in the subclass of Carter’s [A˜] solutions and the Debever–McLenaghan null orbit solution A0. We also compute the symmetry operators which arise from the above separability properties.
A simple proof is given of a general result which shows how to construct solutions of a coupled system of linear self-adjoint partial differential equations once one has succeeded in deriving a decoupled equation in the manner described below. In the case of electromagnetic and gravitational perturbations of algebraically special vacuum space-times, this procedure yields the formulas of Cohen and Kegeles and of Chrzanowski.
The Kerr family of solutions of the Einstein and Einstein-Maxwell equations is the most general class of solutions known at present which could represent the field of a rotating neutral or electrically charged body in asymptotically flat space. When the charge and specific angular momentum are small compared with the mass, the part of the manifold which is stationary in the strict sense is incomplete at a Killing horizon. Analytically extended manifolds are constructed in order to remove this incompleteness. Some general methods for the analysis of causal behavior are described and applied. It is shown that in all except the spherically symmetric cases there is nontrivial causality violation, i.e., there are closed timelike lines which are not removable by taking a covering space; moreover, when the charge or angular momentum is so large that there are no Killing horizons, this causality violation is of the most flagrant possible kind in that it is possible to connect any event to any other by a future-directed timelike line. Although the symmetries provide only three constants of the motion, a fourth one turns out to be obtainable from the unexpected separability of the Hamilton-Jacobi equation, with the result that the equations, not only of geodesics but also of charged-particle orbits, can be integrated completely in terms of explicit quadratures. This makes it possible to prove that in the extended manifolds all geodesics which do not reach the central ring singularities are complete, and also that those timelike or null geodesics which do reach the singularities are entirely confined to the equator, with the further restriction, in the charged case, that they be null with a certain uniquely determined direction. The physical significance of these results is briefly discussed.
The no-hair conjecture is proved for the uncharged black holes of supergravity theory. The equations governing the spin-32 field in the Kerr geometry are decoupled and separated. All solutions that cannot be brought to zero by supersymmetry transformations are shown to satisfy the Teukolsky equations for s=+/-32. Such solutions cannot furnish a new hair for the black hole.
It is shown that the variables of Dirac's equation for the electron can be separated in the Kerr geometry and the solution may be expressed in terms of certain radial and angular functions satisfying decoupled equations. Dirac's equation is written in accordance with the Newman-Penrose (1962) formalism, in the Kerr geometry, and in the Boyer-Lindquist (1967) coordinates. The variables are separated, and the solution is reduced to the solution of a pair of decoupled equations. It is concluded that the general solution can be expressed as a linear superposition of the various solutions belonging to the different characteristic values of the second constant of separation.
It is shown that previous results concerning test massless fields on algebraically special vacuum backgrounds can be extended to the case of massless spin- (3)/(2) Rarita--Schwinger fields. A decoupled equation is derived from the Rarita--Schwinger equation on an algebraically special vacuum space-time and it is shown that all the components of the field can be obtained from a scalar potential that obeys a wavelike equation. In the case of type D metrics, identities of the Teukolsky--Starobinsky-type are obtained. Some relations induced by Killing spinors are also included.
It is found that an operator of the form iγ5γμ[fμν∇ν-(1/6)γνγρfμν;ρ] commutes with the Dirac operator γμ∇ν whenever fμν is an antisymmetric tensor satisfying the Penrose-Floyd equation fμ(ν;ρ)=0. Such a tensor exists notably in the Kerr solutions and in the flat-space limit wherein the operator can be interpreted as the square root of the ordinary total squared angular momentum Casimir operator of the rotation group.
It is shown that in a type‐D space‐time that admits a two‐index Killing spinor a differential operator can be constructed that maps a solution of the Maxwell equations into another solution. By considering as a background the Plebański–Demiański metric, which includes all the vacuum type‐D metrics, this operator is used to obtain all the components of the electromagnetic field and the vector potential. The separated functions appearing in the solutions are shown to obey identities of the Teukolsky–Starobinsky type and the separable solutions are shown to be eigensolutions of a certain differential operator with the ‘‘Starobinsky constant’’ as the eigenvalue.
It is shown that in all the type D solutions to the Einstein vacuum field equations with cosmological constant, the one‐variable functions appearing in the separable maximal spin‐weight components of the neutrino, electromagnetic, and gravitational perturbations are related by certain differential operators and the corresponding proportionality constants are obtained. It is also shown that analogous relations hold in the case of perturbations by a Rarita–Schwinger field if the cosmological constant vanishes.
The equations for the spin‐ (3)/(2) perturbations of the solutions of the Einstein–Maxwell equations given by the linearized O(2) extended supergravity are considered. It is shown that for each geodetic and shear‐free principal null direction of the background electromagnetic field there exists a gauge‐invariant quantity made out of the spin‐ (3)/(2) field that satisfies a decoupled equation. In the case of type‐D solutions with a nonsingular aligned electromagnetic field it is explicitly shown that the decoupled equations associated with the two principal null directions admit separable solutions and the separated functions obey certain differential relations.
The most general first-order differential operator that commutes with
the Dirac operator and hence permits the construction of quantum numbers
is given. Necessary and sufficient conditions for its existence are
expressed in terms of the generalized Killing tensors of Yano. As a
special case we obtain an extension to curved space-time of a covariant
description of spin.
We determine the conditions necessary for a solution of the supergravity field equations with infinitesimal spin-3/2 field to be a pure gauge transformation of an Einstein vacuum field. The analysis depends on the Petrov classification of the curvature tensor and uses two-component spinor calculus. For general type I, the type II, and typeD, the necessary conditions found are also shown to be sufficient, and the explicit form of the gauge transformation can be given.
A new general class of solutions of the Einstein-Maxwell equations is presented. It depends on seven arbitrary parameters that group in a natural way into three complex parameters m + in, a + ib, e + ig, and the cosmological constant λ. They correspond to mass, NUT parameter, angular momentum per unit mass, acceleration, and electric and magnetic charge. The metric is in general stationary and axially symmetric. These solutions are of type D and contain as special cases all known solutions of type D belonging to this class. The known solutions are recovered by performing limiting transitions. An appropriate limit of our solutions describes an electromagnetic field in flat spacetime. We investigate the properties of that field. Its singular region corresponds in general to two circles moving with uniform acceleration in the positive and negative directions along the axis of symmetry. One can easily extend our solutions to the complex domain. Then it turns out that the metric can be written in a double Kerr-Schild form.
This paper investigates the relationship between Killing Tensors and separable systems for the geodesic Hamilton-Jacobi equation in Riemannian and Lorentzian manifolds: locally, a separable system consists of the vector and covector associated with a separable coordinate. It is shown that there are only two types of separable system, those associated with local symmetry groups and those which can be obtained by a simple transformation from orthogonal systems. Some sufficient conditions for existence are given and some global problems are enumerated. The results are illustrated with a demonstration that the existence of separable systems in a certain class of {2, 2} space-times is a consequence of the algebraic properties of the Weyl tensor.
It is shown that every type {22} vacuum solution of Einstein's equations admits a quadratic first integral of the null geodesic equations (conformal Killing tensor of valence 2), which is independent of the metric and of any Killing vectors arising from symmetries. In particular, the charged Kerr solution (with or without cosmological constant) is shown to admit a Killing tensor of valence 2. The Killing tensor, together with the metric and the two Killing vectors, provides a method of explicitly integrating the geodesics of the (charged) Kerr solution, thus shedding some light on a result due to Carter.