Killing Equations Research Articles

Laplacian Operators and Q-curvature on Conformally Einstein Manifolds

A new definition of canonical conformal differential operators P k (k = 1,2,...), with leading term a kth power of the Laplacian, is given for conformally Einstein manifolds of any signature. These act between density bundles and, more generally, between weighted tractor bundles of any rank. By construction these factor into a power of a fundamental Laplacian associated to Einstein metrics. There are natural conformal Laplacian operators on density bundles due to Graham–Jenne–Mason–Sparling (GJMS). It is shown that on conformally Einstein manifolds these agree with the P k operators and hence on Einstein manifolds the GJMS operators factor into a product of second-order Laplacian type operators. In even dimension n the GJMS operators are defined only for 1 ≤ k ≤ n/2 and so, on conformally Einstein manifolds, the P k give an extension of this family of operators to operators of all even orders. For n even and k > n/2 the operators P k are each given by a natural formula in terms of an Einstein metric but they are not natural conformally invariant operators in the usual sense. They are shown to be nevertheless canonical objects on conformally Einstein structures. There are generalisations of these results to operators between weighted tractor bundles. It is shown that on Einstein manifolds the Branson Q-curvature is constant and an explicit formula for the constant is given in terms of the scalar curvature. As part of development, conformally invariant tractor equations equivalent to the conformal Killing equation are presented.

Geometrically motivated hyperbolic coordinate conditions for numerical relativity: Analysis, issues and implementations

We study the implications of adopting hyperbolic-driver coordinate conditions motivated by geometrical considerations. In particular, conditions that minimize the rate of change of the metric variables. We analyze the properties of the resulting system of equations and their effect when implementing excision techniques. We find that commonly used coordinate conditions lead to a characteristic structure at the excision surface where some modes are not of outflow type with respect to any excision boundary chosen inside the horizon. Thus, boundary conditions are required for these modes. Unfortunately, the specification of these conditions is a delicate issue as the outflow modes involve both gauge and main variables. As an alternative to these driver equations, we examine conditions derived from extremizing a scalar constructed from Killing's equation and present specific numerical examples.

THE LIE DERIVATIVE OF SPINOR FIELDS: THEORY AND APPLICATIONS

Starting from the general concept of a Lie derivative of an arbitrary differentiable map, we develop a systematic theory of Lie differentiation in the framework of reductive G-structures P on a principal bundle Q. It is shown that these structures admit a canonical decomposition of the pull-back vector bundle [Formula: see text] over P. For classical G-structures, i.e. reductive G-subbundles of the linear frame bundle, such a decomposition defines an infinitesimal canonical lift. This lift extends to a prolongation Γ-structure on P. In this general geometric framework the concept of a Lie derivative of spinor fields is reviewed. On specializing to the case of the Kosmann lift, we recover Kosmann's original definition. We also show that in the case of a reductive G-structure one can introduce a "reductive Lie derivative" with respect to a certain class of generalized infinitesimal automorphisms, and, as an interesting by-product, prove a result due to Bourguignon and Gauduchon in a more general manner. Next, we give a new characterization as well as a generalization of the Killing equation, and propose a geometric reinterpretation of Penrose's Lie derivative of "spinor fields". Finally, we present an important application of the theory of the Lie derivative of spinor fields to the calculus of variations.

On Noether symmetries and form invariance of mechanico-electrical systems

This Letter focuses on form invariance and Noether symmetries of mechanico-electrical systems. Based on the invariance of Hamiltonian actions for mechanico-electrical systems under the infinitesimal transformation of the coordinates, the electric quantities and the time, the authors present the Noether symmetry transformation, the Noether quasi-symmetry transformation, the generalized Noether quasi-symmetry transformation and the general Killing equations of Lagrange mechanico-electrical systems and Lagrange–Maxwell mechanico-electrical systems. Using the invariance of the differential equations, satisfied by physical quantities, such as Lagrangian, non-potential general forces, under the infinitesimal transformation, the authors propose the definition and criterions of the form invariance for mechanico-electrical systems. The Letter also demonstrates connection between the Noether symmetries and the form invariance of mechanico-electrical systems. An example is designed to illustrate these results.

Isometries of a D3-brane space

We obtain the Killing equations and the corresponding infinitesimal isometries for ten-dimensional space generated by a large number of coincident D3-branes. In a convenient limit this space becomes AdS5 × S5 which is relevant for the AdS/CFT correspondence. In this case, using Poincaré coordinates, we also write the Killing equations and infinitesimal isometries. Then we obtain a simple realization of the isomorphism between AdS isometries and the boundary conformal group.

A new type of conserved quantities for Birkhoffian systems*

This paper gives a new type of conserved quantities for Birkhoffian systems. First, the equations of motion of the systems are established, and the definitions and criterions of Mei symmetries are given. Next, an existence theorem for the n ew type of conserved quantities of the systems is given, and the generalized Killing equations used to determine the infinitesimal generators are deduced. Finally, the inverse theorem of the conservation law is established.

An algorithm for finding the Lie symmetries of tensors

The vector fields along which the metric, the Ricci and the energy-momentum tensors remain invariant under the Lie transport are called the vectors (KVs), the Ricci collineations (RCs) and the matter collineations(MCs), respectively. In this paper we show that the solution of the Killing equations for a second-rank, diagonal, symmetric tensor for which two components are equal provides an algorithm from which we can find the exact nature of KVs, RCs and MCs, and the dimension of the Lie algebras generated by them for all plane-symmetric static space-times.

Killing spinors in supergravity with 4-fluxes

We study the spinorial Killing equation of supergravity involving a torsion 3-form T as well as a flux 4-form F. In dimension seven, we construct explicit families of compact solutions out of 3-Sasakian geometries, nearly parallel G2-geometries and on the homogeneous Aloff–Wallach space. The constraint F · Ψ = 0 defines a non-empty subfamily of solutions. We investigate the constraint T · Ψ = 0, too, and show that it singles out a very special choice of numerical parameters in the Killing equation, which can also be justified geometrically.

New SUSY and motions in hot NUT–Reissner–Nordström spinning space

We study geodesic motion of pseudo-classical spinning particles in the hot NUT–Reissner–Nordström space. We investigate the generalized Killing equations for spinning spaces and derive the constants of motion in terms of the solutions of these equations. We give an analysis of the motion on a cone and on a plane.

Magnetic black holes with string-loop corrections

String-loop corrections to magnetic black holes are studied. 4D effective action is obtained by compactification of the heterotic string theory on the manifold K3×T2 or on a suitable orbifold yielding N=1 supersymmetry in 6D. In the resulting 4D theory with N=2 local supersymmetry, the prepotential receives only one-string-loop perturbative correction. The loop-corrected black hole is obtained in two approaches: (i) by solving the system of the Einstein-Maxwell equations of motion derived from the loop-corrected effective action and (ii) by solving the system of spinor Killing equations (conditions for the supersymmetry variations of the fermions to vanish) and Maxwell equations. We consider a particular tree-level solution with the magnetic charges adjusted so that the moduli connected with the metric of the internal two-torus are constant. In this case, the loop correction to the prepotential is independent of coordinates, and it is possible to solve the system of the Einstein-Maxwell and spinor Killing equations in the first order in string coupling analytically. The set of supersymmetric solutions of the loop-corrected spinor Killing equations is contained in a larger set of solutions of the equations of motion derived from the string-loop-corrected effective action. Loop corrections to the metric and dilaton are large at small distances from the center of the black hole.

Spinning Particles in Reissner-Nordström-de Sitter Spacetime

We study the geodesic motion of pseudo-classical spinning particles in the Reissner-Nordstrom-de Sitter spacetime. We investigate the generalized Killing equations for spinning space and derive the constants of motion in terms of the solutions of these equations. We discuss bound state orbits in a plane.

A local existence theorem for the Einstein–Dirac equation

We studied the Einstein–Dirac equation as well as the weak Killing equation on Riemannian spin manifolds with codimension one foliation. We prove that, for any manifold M n admitting real Killing spinors (resp. parallel spinors), there exist warped product metrics η ̄ on M n× R such that (M n× R, η ̄ ) admit Einstein spinors (resp. weak Killing spinors). To prove the result we split the Einstein–Dirac equation into evolution equations and constraints, by means of Cartan’s frame formalism, and apply the local preservation property of constraints.

(Conformal) Killing Vectors in the Newman-Penrose Formalism

Finding (conformal) Killing vectors of a given metric can be a difficult task. This paper presents an efficient technique for finding Killing, homothetic, or even proper conformal Killing vectors in the Newman-Penrose (NP) formalism. Leaning on, and extending, results previously derived in the GHP formalism we show that the (conformal) Killing equations can be replaced by a set of equations involving the commutators of the Lie derivative with the four NP differential operators, applied to the four coordinates. It is crucial that these operators refer to a preferred tetrad relative to the (conformal) Killing vectors, a notion to be defined. The equations can then be readily solved for the Lie derivative of the coordinates, i.e. for the components of the (conformal) Killing vectors. Some of these equations become trivial if some coordinates are chosen intrinsically (where possible), i.e. if they are somehow tied to the Riemann tensor and its covariant derivatives. If part of the tetrad, i.e. part of null directions and gauge, can be defined intrinsically then that part is generally preferred relative to any Killing vector. This is also true relative to a homothetic vector or a proper conformal Killing vector provided we make a further restriction on that intrinsic part of the tetrad. If because of null isotropy or gauge isotropy, where part of the tetrad cannot even in principle be defined intrinsically, the tetrad is defined only up to (usually) one null rotation parameter and/or a gauge factor, then the NP-Lie equations become slightly more involved and must be solved for the Lie derivative of the null rotation parameter and/or of the gauge factor as well. However, the general method remains the same and is still much more efficient than conventional methods. Several explicit examples are given to illustrate the method.

Anholonomic frames, generalized Killing equations, and anisotropic Taub-NUT spinning spaces

By using anholonomic frames in (pseudo)-Riemannian spaces we define anisotropic extensions of Euclidean Taub-NUT spaces. With respect to coordinate frames such spaces are described by off-diagonal metrics which could be diagonalized by corresponding anholonomic transforms. We define the conditions when the 5D vacuum Einstein equations have as solutions anisotropic Taub-NUT spaces. The generalized Killing equations for the configuration space of anisotropically spinning particles (anisotropic spinning space) are analyzed. Simple solutions of the homogeneous part of these equations are expressed in terms of some anisotropically modified Killing–Yano tensors. The general results are applied to the case of the four-dimensional locally anisotropic Taub-NUT manifold with Euclidean signature. We emphasize that all constructions are for (pseudo)-Riemannian spaces defined by vacuum solutions, with generic anisotropy, of 5D Einstein equations, the solutions being generated by applying the moving frame method.

Spinning Particles in Curved Spacetime

We briefly discuss the relevant equations for the motion of spinning particles in curved spacetime. We describe the generalized Killing equations for spinning spaces and derive the constants of motion. We apply the formalism to solve for the motion of a pseudoclassical spinning particle in Schwarzschild–de Sitter spacetime.

On the symmetries of the Edgar–Ludwig metric

The conformal Killing equations for the most general (non-plane wave) conformally flat pure radiation field are solved to find the conformal Killing vectors. As expected 15 independent conformal Killing vectors exist, but in general the metric admits no Killing or homothetic vectors. However for certain special cases a one-dimensional group of homotheties or motions may exist and in one very special case, overlooked by previous investigators, a two-dimensional homethety group exists. No higher dimensional groups of motions or homotheties are admitted by these metrics.

On the symmetry approach to polynomial conservation laws of one-dimensional Lagrangian systems

In this paper we analyze polynomial conservation laws of one-dimensional non-autonomous Lagrangian dynamical systems x ̈ =−∂ Π(t,x)/∂x . The analysis is based upon application of Noether's theorem which relates the existence of conservation laws to the symmetries of Hamilton's action integral. It is shown that the existence of first integrals depends on the solution of the system of first-order partial differential equations — generalized Killing's equations. General solution of the problem is formally determined. It is demonstrated that the final form of dynamical system and corresponding conservation law depends on the solution of the so-called potential equation. However, the structure of symmetry transformations, which generate particular class of conservation laws, could be prescribed independent of the solution of potential equation. This fact is used to underline phenomenological aspect of symmetry approach. Its pragmatic value is confirmed through several concrete examples.

The classification of plane symmetric spacetimes by isometries

A complete classification of plane symmetric Lorentzian manifolds according to their additional isometries and metrics (or classes of metrics) is obtained by solving the Killing equations. We obtain all metrics (or classes of metrics), that admit the group of motions Gr (where r=3,4,5,6,7 and 10) containing SO(2)⨷R2, the minimal symmetry inherited by the plane symmetric manifolds.

Spinning Particles in NUT–Reissner–Nordstrom Space–Time

We study the geodesic motion of pseudo-classical spinning particles in the NUT–Reissner–Nordstrom space–time. We investigate the generalized Killing equations for spinning space and derive the constants of the motion in terms of the solutions of these equations. We give an analysis of the motion on a cone and on a plane.

Some remarks on the Hijazi inequality and generalizations of the Killing equation for spinors

We generalize the well-known lower estimates for the first eigenvalue of the Dirac operator on a compact Riemannian spin manifold proved by Friedrich [Math. Nachr. 97 (1980) 117–146] and Hijazi [Math. Phys. 104 (1986) 151–162; J. Geom. Phys. 16 (1995) 27–38]. The special solutions of the Einstein–Dirac equation constructed recently by Friedrich/Kim are examples for the limiting case of these inequalities. The discussion of the limiting case of these estimates yields two new field equations generalizing the Killing equation as well as the weak Killing equation for spinor fields. Finally, we discuss the two-and three-dimensional case in more detail.