Killing Yano Research Articles

Extended superalgebras from twistor and Killing spinors

The basic first-order differential operators of spin geometry that are Dirac operator and twistor operator are considered. Special types of spinors defined from these operators such as twistor spinors and Killing spinors are discussed. Symmetry operators of massless and massive Dirac equations are introduced and relevant symmetry operators of twistor spinors and Killing spinors are constructed from Killing–Yano (KY) and conformal Killing–Yano (CKY) forms in constant curvature and Einstein manifolds. The squaring map of spinors gives KY and CKY forms for Killing and twistor spinors, respectively. They constitute a graded Lie algebra structure in some special cases. By using the graded Lie algebra structure of KY and CKY forms, extended Killing and conformal superalgebras are constructed in constant curvature and Einstein manifolds.

Killing spinors and related symmetries in six dimensions

Benefiting from the index spinorial formalism, the Killing spinor equation is integrated in six-dimensional spacetimes. The integrability conditions for the existence of a Killing spinor are worked out and the Killing spinors are classified in two algebraic types, in the first type the scalar curvature of the spacetime must be negative, while in the second type the spacetime must be an Einstein manifold. In addition, the equations that define Killing-Yano (KY) and closed conformal Killing-Yano (CCKY) tensors are expressed in the index notation and, as consequence, all non-vanishing KY and CCKY tensors that can be generated from a Killing spinor are made explicit.

A simple test for spacetime symmetry

This paper presents a simple method for investigating spacetime symmetry for a given metric. The method makes use of the curvature conditions that are obtained from the Killing equations. We use the solutions of the curvature conditions to compute an upper bound on the number of Killing vector fields, as well as Killing–Yano (KY) tensors and closed conformal KY tensors. We also use them in the integration of the Killing equations. By means of the method, we thoroughly investigate KY symmetry of type D vacuum solutions such as the Kerr metric in four dimensions. The method is also applied to a large variety of physical metrics in four and five dimensions.

Basic gravitational currents and Killing–Yano forms

It has been shown that for each Killing-Yano (KY)-form accepted by an $n$-dimensional (pseudo)Riemannian manifold of arbitrary signature, two basic gravitational currents can be defined. Conservation of the currents are explicitly proved by showing co-exactness of the one and co-closedness of the other. Some general geometrical facts implied by these conservation laws are also elucidated. In particular, the conservation of the one-form currents implies that the scalar curvature of the manifold is a flow invariant for all of its Killing vector fields. It also directly follows that, while all KY-forms and their Hodge duals on a constant curvature manifold are the eigenforms of the Laplace-Beltrami operator, for an Einstein manifold this is certain only for KY 1-forms, $(n-1)$-forms and their Hodge duals.

Killing–Yano forms of a class of spherically symmetric space-times: A unified generation of higher forms

Killing–Yano (KY) 2- and 3-forms of a class of spherically symmetric space-times that includes the well-known Minkowski, Schwarzschild, Reissner–Nordstrøm, Robertson–Walker, and six different forms of de Sitter space-times as special cases are derived in a unified and exhaustive manner. It is directly proved that while the Schwarzschild and Reissner–Nordstrøm space-times do not accept any KY 3-form and they accept only one 2-form, the Robertson–Walker space-time admits four KY 2-forms and only one KY 3-form. Maximal number of KY forms is obtained for Minkowski and all known forms of de Sitter space-times. Complete lists comprising explicit expressions of KY forms are given.

First-order symmetries of the Dirac equation in a curved background: a unified dynamical symmetry condition**Dedicated to Professor T Dereli on the occasion of his 60th birthday.

It has been shown that, for all dimensions and signatures, the most general first-order linear symmetry operators for the Dirac equation including interaction with Maxwell field in a curved background are given in terms of Killing–Yano (KY) forms. As a general gauge invariant condition it is found that among all KY forms of the underlying (pseudo) Riemannian manifold, only those which Clifford commute with the Maxwell field take part in the symmetry operator. It is also proved that associated with each KY form taking part in the symmetry operator, one can define a quadratic function of velocities which is a geodesic invariant as well as a constant of motion for the classical trajectory. Some geometrical and physical implications of the existence of KY forms are also elucidated.

Killing-Yano Tensors and Superintegrable Systems

Killing-Yano (KY) and Killing tensors of the four types of metrics, that represent the two-dimensional spaces given in Darboux’s classification, are obtained. It is proved that all spaces admit dual manifolds and their KY tensors are calculated.

ON THE RELATIONSHIP BETWEEN KILLING-YANO TENSORS AND ELECTROMAGNETIC FIELDS ON CURVED SPACES

The existence of skew symmetric Killing-Yano (KY) tensors of order 2 has been investigated on curved spaces. The integrability conditions of KY tensor including Carter's algebraic relation of symmetric tensor with electromagnetic field have been transcribed in Newman-Penrose formalism. The KY bivectors are classified according to their nullity in electrovac space-times. It is shown that the non-null (or null) electromagnetic field implies to the existence of non-null (or null) KY tensor. Thus Collinson's theorems on the existence of KY tensors on vacuum space-times have been generalized on electrovac space-times. Chandrasekhar's theorem on vacuum type D space-times has also been generalized on the existence of non-null KY tensor on electrovac type D or non-vacuum type D space-time filled with dust. All theorems presented here have been strengthened by giving examples for known space-times. It is also shown that most of KY tensors discussed here are eigen-KY-bivectors of the respective curvature tensors.