Algebraic integrability conditions for Killing tensors on constant sectional curvature manifolds

We use an isomorphism between the space of valence two Killing tensors on an n$n$-dimensional constant sectional curvature manifold and the irreducible GL(n+1)$GL(n+1)$-representation space of algebraic curvature tensors in order to translate the Nijenhuis integrability conditions for a Killing tensor into purely algebraic integrability conditions for the corresponding algebraic curvature tensor, resulting in two simple algebraic equations of degree two and three. As a first application of this we construct a new family of integrable Killing tensors.     

On the space-times admitting two shear-free geodesic null congruences

We analyze the space-times admitting two shear-free geodesic null congruences. The integrability conditions are presented in a plain tensorial way as equations on the volume element U of the time-like 2-plane that these directions define. From these we easily deduce significant consequences. We obtain explicit expressions for the Ricci and Weyl tensors in terms of U and its first and second order covariant derivatives. We study the different compatible Petrov-Bel types and give the necessary and sufficient conditions that characterize every type in terms of U. The type D case is analyzed in detail and we show that every type D space-time admitting a 2 + 2 conformal Killing tensor also admits a conformal Killing-Yano tensor.     

LETTER: Note on Killing Yano Tensors Admitted by Spherically Symmetric Static Space-Times

The Killing Yano tensors of order two admitted by a general class of spherically symmetric static space-times are found. All such space-times admit at least one Killing Yano tensor and four special cases exist, one admitting four Killing Yano tensors the others admitting ten Killing Yano tensors. The Killing Yano tensors are used to construct second order non-stationary Killing tensors and the nature of the redundancy of these Killing tensors is discussed with reference to the time dependence of the constituent tensors/vectors.     

A note on Killing tensors

The connection between symmetric and skew-symmetric Killing tensors is studied. Some theorems on skew-symmetric Killing tensors are generalized, and it is shown that in all type-D vacuum metrics admitting a symmetric Killing tensor, this Killing tensor can be given in terms of a skew-symmetric Killing tensor.     

Generalized Killing Tensors

Generalized Killing tensors are defined and the integrability conditions discussed to show that the familiar result that a space of constant curvature admits the maximum number of Killing vectors and second order Killing tensors does not necessarily generalize. The existence of second order Generalized Killing Yano tensors in spherically symmetric static space-times is investigated and a non-redundant example is given. It is proved that the natural vector analogue of the Lenz-Runge vector does not exist.     

A Rainich-Like Approach to the Killing-Yano Tensors

The Rainich problem for Killing-Yano tensors posed by Collinson [1] is solved. In intermediate steps, we first obtain the necessary and sufficient conditions for a 2+2 almost-product structure to determine the principal 2–planes of a Killing-Yano tensor. Then we give the additional conditions on a symmetric Killing tensor for it to be the square of a Killing-Yano tensor. We also analyze a similar problem for the conformal Killing-Yano tensors. Our results show that, in both cases, the principal 2–planes define a Maxwellian structure. The associated Maxwell fields are obtained and we outline how this approach is of interest in studying the spacetimes that admit these kind of first integrals of the geodesic equation.     

Finding Collineations of Kimura Metrics

Kimura investigated static spherically symmetric metrics and found several to have quadratic first integrals. We use REDUCE and the package Dimsym to seek collineations for these metrics. For one metric we find that three proper projective collineations exist, two of which are associated with the two irreducible quadratic first integrals found by Kimura. The third projective collineation is found to have a reducible quadratic first integral. We also find that this metric admits two conformal motions and that the resulting reducible conformal Killing tensors also lead to Kimura's quadratic integrals. We demonstrate that when a Killing tensor is known for a metric we can seek an associated collineation by solving first order equations that give the Killing tensor in terms of the collineation rather than the second order determining equations for collineations. We report less interesting results for other Kimura metrics.     

Three-dimensional space-times

A real version of the Newman-Penrose formalism is developed for (2+1)-dimensional space-times. The complete algebraic classification of the (Ricci) curvature is given. The field equations of Deser, Jackiw, and Templeton, expressing balance between the Einstein and Bach tensors, are reformulated in triad terms. Two exact solutions are obtained, one characterized by a null geodesic eigencongruence of the Ricci tensor, and a second for which all the polynomial curvature invariants are constant.     

The verification of Killing tensor components for metrics in general relativity using the computer algebra system SHEEP

We report on a program, written in the computer algebra system SHEEP, for verifying the components of Killing tensors and conformal Killing tensors. We give some examples, including the components of the Killing tensor admitted by the Kerr metric. We also note that the explicit form of all conformal Killing tensors for a subclass of the Petrov typeD solutions is known.     

Lax Pair Tensors and Integrable Spacetimes

The use of Lax pair tensors as a unifying framework for Killing tensors of arbitrary rank is discussed. Some properties of the tensorial Lax pair formulation are stated. A mechanical system with a well-known Lax representation—the three-particle open Toda lattice—is geometrized by a suitable canonical transformation. In this way the Toda lattice is realized as the geodesic system of a certain Riemannian geometry. By using different canonical transformations we obtain two inequivalent geometries which both represent the original system. Adding a timelike dimension gives four-dimensional spacetimes which admit two Killing vector fields and are completely integrable.     

Some remarks on the geometry of superspace supergravity

In this note, we first give a quick presentation of the supergeometry underlying supergravity theories, using an intrinsic differential geometric language. For this, we adopt the point of view of Cartan geometries, and rely as well on the work of John Lott, who has found a unified geometrical interpretation of the torsion constraints for many supergravity theories, based on the use of H-structures. In this framework, the constraints amount to requiring first-order integrability of H-structures, for a specific supergroup H.The supergroup H used by Lott is not the usual diagonal representation of the Lorentz group on superspace, but an extension of the latter. This extension appears to be natural and it can be related to the super-Poincaré group. We also observe that the constraints arising from the requirement of first-order integrability have basically the same form, in any spacetime dimension.Looking at supergravity from an affine viewpoint (i.e. as a gauge theory for the super-Poincaré group), we show that requiring first-order integrability amounts to requiring the equivalence, up to gauge transformations, between infinitesimal gauge supertranslations acting on the supervielbein and infinitesimal superdiffeomorphisms acting on the supervielbein.The latter action is performed through a covariant Lie derivative, whose expression involves naturally the supertorsion tensor. We use this expression to show that the term added to the spin connection, in the supercovariant derivative of d=11 supergravity, has a natural superspace origin. In particular, the 4-form field strength is related to a specific component of the supertorsion tensor.We conclude by some general remarks concerning Killing spinors in geometry and supergravity, discussing their possible interpretations, as Killing vector fields on a specific supermanifold on one hand, and as parallel spinors for an appropriate connection on the other hand. We show that this last interpretation is very natural from the point of view of Klein and Cartan geometries.     

Homothetic killing tensors

A vectorfield approach to Killing tensors (KTs) is used to analyse the conventional definition of conformal Killing tensors (CKTs). It is shown that the CKT associated with a KT is removable and that so far as constants of the motion are concerned the homothetic Killing tensor defined here is more useful. A geometric interpretation of CKTs is also given.     

New integrable (2+1)- and (3+1)-dimensional sinh-Gordon equations with constant and time-dependent coefficients

In this work, we mainly address two new integrable (2+1)- and (3+1)-dimensional sinh-Gordon equations, which naturally appear in surface theory and fluid dynamics. The first equation includes constant coefficients, while the other is characterized with time-dependent coefficients. It is of further value to investigate the integrability of each model. This study puts forward a Painlevé test to reveal the Painlevé integrability. We show that the first equation passes the Painlevé test to confirm its integrability. However, the compatibility conditions of the second model with time-dependent coefficients provides the relation between these coefficients to ensure its integrability. We show that the dispersion relations of the two equations are distinct, whereas the phase shifts are identical. We apply the simplified Hirota's method where four sets of multiple soliton are derived for these equations.     

Proper Conformal Killing Vectors in Kantowski-Sachs Metric

This paper deals with the existence of proper conformal Killing vectors(CKVs) in Kantowski-Sachs metric.Subject to some integrability conditions, the general form of vector filed generating CKVs and the conformal factor is presented. The integrability conditions are solved generally as well as in some particular cases to show that the nonconformally flat Kantowski-Sachs metric admits two proper CKVs, while it admits a 15-dimensional Lie algebra of CKVs in the case when it becomes conformally flat. The inheriting conformal Killing vectors(ICKVs), which map fluid lines conformally, are also investigated.     

Conservation laws and stress–energy–momentum tensors for systems with background fields

This article attempts to delineate the roles played by non-dynamical background structures and Killing symmetries in the construction of stress–energy–momentum tensors generated from a diffeomorphism invariant action density. An intrinsic coordinate independent approach puts into perspective a number of spurious arguments that have historically lead to the main contenders, viz the Belinfante–Rosenfeld stress–energy–momentum tensor derived from a Noether current and the Einstein–Hilbert stress–energy–momentum tensor derived in the context of Einstein’s theory of general relativity. Emphasis is placed on the role played by non-dynamical background (phenomenological) structures that discriminate between properties of these tensors particularly in the context of electrodynamics in media. These tensors are used to construct conservation laws in the presence of Killing Lie-symmetric background fields.     

An (n+1)-Dimensional Integrable Model: Painlevé Test

A (3+1)-dimensional model is proposed at first. The integrability of the model is proved by using the standard Painlevk analysis. Then the model is extended to (n + 1)-dimensional case. The model is space-isotropic and is a type of the modified KdV extension in higher dimensions.     

On the relationship between Killing tensors and Killing-Yano tensors

The conditions that a Killing tensor of order 2 be the contracted product of a Killing-Yano tensor of order 2 with itself are found. All Killing-Yano tensors of order 2 admitted in empty space-times are obtained     

The Casimir effect of Reissner-NordstrSm black hole

The stress tensor of a massless scalar field satisfying a mixed boundary condition in a （1 ＋ 1）-dimensional Reissner- Nordstrom black hole background is calculated by using Wald＇s axiom. We find that Dirichlet stress tensor and Neumann stress tensor can be deduced by changing the coefficients of the stress tensor calculated under a mixed boundary condition. The stress tensors satisfying Dirichlet and Neumann boundary conditions are discussed. In addition, we also find that the stress tensor in conformal flat spacetime background differs from that in flat spacetime only by a constant.     

Third-order Killing tensors in the Schwarzschild space-time

The general solution for the third-order Killing tensor equation in the Schwarzschild space-time is written down. It follows that the Schwarzschild metric admits only redundant Killing tensors of order 3.This work was carried out under the auspices of the National Group for Mathematical Physics of C. N. R.     

On the evolution equations for a self-gravitating charged scalar field

We consider a complex scalar field minimally coupled to gravity and to a U(1) gauge symmetry and we construct of a first order symmetric hyperbolic evolution system for the Einstein-Maxwell-Klein-Gordon system. Our analysis is based on a $1+3$ tetrad formalism which makes use of the components of the Weyl tensor as one of the unknowns. In order to ensure the symmetric hyperbolicity of the evolution equations, implied by the Bianchi identity, we introduce a tensor of rank 3 corresponding to the covariant derivative of the Faraday tensor, and two tensors of rank 2 for the covariant derivative of the vector potential and the scalar field.