We develop a new method in order to classify the Bianchi I spacetimes which admit conformal Killing vectors (CKV). The method is based on two propositions which relate the CKVs of 1+(n-1) decomposable Riemannian spaces with the CKVs of the (n-1) subspace and show that if 1+(n-1) space is conformally flat then the (n-1) spacetime is maximally symmetric. The method is used to study the conformal algebra of the Kasner spacetime and other less known Bianchi type I matter solutions of General Relativity.
The aim of this paper is to explore teleparallel conformal Killing vector fields (CKVFs) of locally rotationally symmetric (LRS) Bianchi type V spacetimes in the context of teleparallel gravity and compare the obtained results with those of general relativity. The general solution of teleparallel conformal Killing's equations is found in terms of some unknown functions of t and x , along with a set of integrability conditions. The integrability conditions are solved in some particular cases to get the final form of teleparallel CKVFs. It is observed that the LRS Bianchi type V spacetimes admit…
An answer to the question: Can, in general, the adoption of a given symmetry induce a further symmetry, which might be hidden at a first level? has been attempted in the context of differential geometry of locally homogeneous spaces. Based on E. Cartan's theory of moving frames, a methodology for finding all symmetries for any n dimensional locally homogeneous space is provided. The analysis is applied to 3 dimensional spaces, whereby the embedding of them into a 4 dimensional Lorentzian manifold is examined and special solutions to Einstein's field equations are recovered. The analysis is mai…
We determine the conformal algebra of Bianchi III and Bianchi V spacetimes or, equivalently, we determine all Bianchi III and Bianchi V spacetimes which admit a proper conformal Killing vector. The algorithm that we use has been developed in Class. Quantum. Grav. 15, 2909 (1998) and concerns the computation of the CKVs of decomposable spacetimes. The main point of this method is that a decomposable space admits a CKV if the reduced space admits a gradient homothetic vector the latter being possible only if the reduced space is flat or a space of constant curvature. We apply this method in a st…
In this paper, we present a complete classification of Locally Rotationally Symmetric (LRS) Bianchi type I spacetimes according to their Conformal Ricci Collineations (CRCs). When the Ricci tensor is non-degenerate, a general form of the vector field generating CRCs is found, subject to some integrability conditions. Solving the integrability conditions in different cases, it is found that the LRS Bianchi type I spacetimes admit 7, 10, 11 or 15- dimensional Lie algebra of CRCs for the choice of non-degenerate Ricci tensor. Moreover, it is found that these spacetimes admit infinite number of CR…
Conformal Killing vectors (CKVs) preserve the spacetime metric up to a factor. Homothetic vectors and Killing vectors are special cases of CKVs. Classification of some classes of spacetimes on the basis of CKVs give interesting results showing how homothetic and Killing vectors which form subsets of the set of CKVs can be recovered as a result of the above classification.
This paper intends to obtain concircular vector fields of Kantowski Sachs and Bianch type III spacetimes. For this purpose, ten conformal Killing equations and their general solution in the form of conformal Killing vector fields are derived along with their conformal factors. The obtained conformal Killing vector fields are then placed in Hessian equations to obtain the final form of concircular vector fields. The existence of concircular symmetry imposes restrictions on the metric functions. The conditions imposing restrictions on these metric functions are obtained as a set of integrability…
Through an exhaustive search, we produce a 5-parameter family of propagation identities for the closed conformal Killing-Yano equation on 2-forms, which hold on an Einstein cosmological vacuum spacetime in any dimension $n>4$. It is well-known that spacetimes admitting a non-degenerate 2-form of this type are exhausted by the Kerr-NUT-(A)dS family of exact higher dimensional black hole solutions. As a consequence, we identify a set of necessary and sufficient conditions ensuring that the cosmological vacuum development of an initial data set for Einstein's field equations admits a closed confo…
The virial theorem is formulated both intrinsically and in local coordinates for a Lagrangian system of mechanical type on a Riemann manifold. An import case studied in this paper is that of an affine virial function associated to a vector field on the configuration manifold. The special cases of a virial function associated to a Killing, a homothetic and a conformal Killing vector field are considered and the corresponding virial theorems are established for this type of functions.
Symmetry algebras of Killing vector fields and conformal Killing vectors fields can be extended to Killing-Yano and conformal Killing-Yano superalgebras in constant curvature manifolds. By defining $\mathbb{Z}$-gradations and filtrations of these superalgebras, we show that the second cohomology groups of them are trivial and they cannot be deformed to other Lie superalgebras. This shows the rigidity of Killing-Yano and conformal Killing-Yano superalgebras and reveals the fact that they correspond to geometric invariants of constant curvature manifolds. We discuss the structures and dimensions…
The "fully charged" spin-2 field solution is presented. This is an analog of the Coulomb solution in electrodynamics and represents the "non-waving" part of the spin-2 field theory. Basic facts and definitions of the spin--2 field and conformal Yano-Killing tensors are introduced. Application of those two objects provides a precise definition of quasi-local gravitational charge. Next, the 3+1 decomposition leads to the construction of the momentary gravitational charges on initial surface which is applicable for Schwarzschild-like spacetimes.
It is natural to expect and simple to prove that every conformally flat space possess the maximal number of conformal Killing vector fields (CKVs). On the other hand, it is interesting to ask whether the converse is true. Is conformal flatness a necessary condition for the existence of the maximal number of CKVs? In this review article it is proven that the answer is yes, a space admits the maximal number of CKVs if, and only if, it is conformally flat.
Direct integration technique is used to study the proper conformal vector fields in non conformally flat Bianchi type-1 space-times. Using the above mentioned technique we have shown that a very special class of the above space-time admits proper conformal vector fields.
Koutras has proposed some methods to construct reducible proper conformal Killing tensors and Killing tensors (which are, in general, irreducible) when a pair of orthogonal conformal Killing vectors exist in a given space. We give the completely general result demonstrating that this severe restriction of orthogonality is unnecessary. In addition we correct and extend some results concerning Killing tensors constructed from a single conformal Killing vector. A number of examples demonstrate how it is possible to construct a much larger class of reducible proper conformal Killing tensors and Ki…
We investigate proper conformal vector fields in non conformally flat Kantowski-Sachs and Bianchi type III space-times using direct integration technique. Using the above mentioned technique we show that very special classes of the above space-times admit proper conformal vector fields.
We find necessary and sufficient conditions ensuring that the vacuum development of an initial data set of the Einstein's field equations admits a conformal Killing vector. We refer to these conditions as conformal Killing initial data (CKID) and they extend the well-known Killing initial data (KID) that have been known for a long time. The procedure used to find the CKID is a classical argument, which is reviewed and presented in a form that may have an independent interest, based on identifying a suitable propagation identity and checking the well-posedness of the corresponding initial value…
In this paper, conformal motions are studied in plane symmetric static spacetimes. The general solution of conformal Killing equations and the general form of the conformal Killing vector for these spacetimes are presented. All possibilities for the existence of conformal motions in these spacetimes are exhausted.
Some years ago Koutras presented a method of constructing a conformal Killing tensor from a pair of orthogonal conformal Killing vectors. When the vector associated with the conformal Killing tensor is a gradient, a Killing tensor (in general irreducible) can then be constructed. In this paper it is shown that the severe restriction of orthogonality is unnecessary and thus it is possible that many more Killing tensors can be constructed in this way. We also extend, and in one case correct, some results on Killing tensors constructed from a single conformal Killing vector. Weir's result that, f…
The formal solution of the second order Killing tensor equations for the general pp-wave spacetime is given. The Killing tensor equations are integrated fully for some specific pp-wave spacetimes. In particular, the complete solution is given for the conformally flat plane wave spacetimes and we find that irreducible Killing tensors arise for specific classes. The maximum number of independent irreducible Killing tensors admitted by a conformally flat plane wave spacetime is shown to be six. It is shown that every pp-wave spacetime that admits an homothety will admit a Killing tensor of Koutra…
A global extension theorem is established for isotropic singularities in polytropic perfect fluid Bianchi space-times. When an extension is possible, the limiting behaviour of the physical space-time near the singularity is analysed.
Symmetric conformal Killing tensors and (skew-symmetric) conformal Yano-Killing tensors for Euclidean Taub-NUT metric are given in explicit form. Relations between Yano and CYK tensors in terms of conformal rescaling are discussed.
The purpose of this paper is to revisit the Bianchi identities existing for the Riemann and Weyl tensors in the combined framework of the formal theory of systems of partial differential equations (Spencer cohomology, differential systems, formal integrability) and Algebraic Analysis (homological algebra, differential modules, duality). In particular, we prove that the n 2 (n 2 -- 1)(n -- 2)/24 generating Bianchi identities for the Riemann tensor are first order and can be easily described by means of the Spencer cohomology of the first order Killing symbol in arbitrary dimension n $\ge$ 2. Si…
We provide a generalization of the Lie algebra of conformal Killing vector fields to conformal Killing-Yano forms. A new Lie bracket for conformal Killing-Yano forms that corresponds to slightly modified Schouten-Nijenhuis bracket of differential forms is proposed. We show that conformal Killing-Yano forms satisfy a graded Lie algebra in constant curvature manifolds. It is also proven that normal conformal Killing-Yano forms in Einstein manifolds also satisfy a graded Lie algebra. The constructed graded Lie algebras reduce to the graded Lie algebra of Killing-Yano forms and the Lie algebras of…
We analyze Killing Initial Data on Cauchy surfaces in conformally rescaled vacuum space-times satisfying Friedrich's conformal field equations. As an application, we derive the KID equations on a spacelike $\mathcal{J}^-$.
The purpose of this paper is to find conformal vector fields of some perfect fluid Kantowski-Sachs and Bianchi type III space-times in the f(R) theory of gravity using direct integration technique. In this study there exists only eight cases. Studying each case in detail, we found that in two cases proper conformal vector fields exist while in the rest of six cases conformal vector fields become Killing vector fields. The dimension of conformal vector fields is either 4 or 6.
We show how, for all dimensions and signatures, a symmetry operator for the massless Dirac equation can be constructed from a conformal Killing-Yano tensor of arbitrary degree.
Local scalar QFT (in Weyl algebraic approach) is constructed on degenerate semi-Riemannian manifolds corresponding to Killing horizons in spacetime. Covariance properties of the $C^*$-algebra of observables with respect to the conformal group $PSL(2,\bR)$ are studied.It is shown that, in addition to the state studied by Guido, Longo, Roberts and Verch for bifurcated Killing horizons, which is conformally invariant and KMS at Hawking temperature with respect to the Killing flow and defines a conformal net of von Neumann algebras, there is a further wide class of algebraic (coherent) states repr…
The integrability conditions for the existence of a conformal Killing-Yano tensor of arbitrary order are worked out in all dimensions and expressed in terms of the Weyl tensor. As a consequence, the integrability conditions for the existence of a Killing-Yano tensor are also obtained. By means of such conditions, it is shown that in certain Einstein spaces one can use a conformal Killing-Yano tensor of order p to generate a Killing-Yano tensor of order (p-1). Finally, it is proved that in maximally symmetric spaces the covariant derivative of a Killing-Yano tensor is a closed conformal Killing…
The solutions of generalized Killing equation have been obtained for line element with initial $t^2 \oplus so(3)$ symmetry. The coefficients of the metric $g$ corresponding to these vector fields are written down.
We study the space-times admitting a null conformal Killing-Yano tensor whose divergence defines a Killing vector. We analyze the similitudes and differences with the recently studied non null case (Gen. Relativ. Grav. (2015) {\bf 47} 1911). The results by Barnes concerning the Birkhoff theorem for the case of null orbits are analyzed and generalized.
Assuming the existence of a single rank-2 closed conformal Killing-Yano tensor with a certain symmetry we show that there exist mutually commuting rank-2 Killing tensors and Killing vectors. We also discuss the condition of separation of variables for the geodesic Hamilton-Jacobi equations.
Axisymmetric spacetimes with a conformal symmetry are studied and it is shown that, if there is no further conformal symmetry, the axial Killing vector and the conformal Killing vector must commute. As a direct consequence, in conformally stationary and axisymmetric spacetimes, no restriction is made by assuming that the axial symmetry and the conformal timelike symmetry commute. Furthermore, we prove that in axisymmetric spacetimes with another symmetry (such as stationary and axisymmetric or cylindrically symmetric spacetimes) and a conformal symmetry, the commutator of the axial Killing vec…
Certain dynamical black hole solutions can be mapped to static spacetimes by conformal metric transformations. This mapping provides a physical link between the conformal Killing horizon of the dynamical black hole and a Killing horizon of the static spacetime. We show how this conformal relation can be used to derive thermodynamic properties for the dynamical black holes. Although these horizons are defined quasi-locally and can be located by local experiments, they are distinct from other popular notions of quasi-local horizons such as apparent horizons. Thus in the dynamical Vaidya spacetim…
Using twistor methods, we explicitly construct all local forms of four--dimensional real analytic neutral signature anti--self--dual conformal structures $(M,[g])$ with a null conformal Killing vector. We show that $M$ is foliated by anti-self-dual null surfaces, and the two-dimensional leaf space inherits a natural projective structure. The twistor space of this projective structure is the quotient of the twistor space of $(M,[g])$ by the group action induced by the conformal Killing vector. We obtain a local classification which branches according to whether or not the conformal Killing vect…
We analyze the main geometric conditions imposed by the hypothesis of the Jebsen-Birkhoff theorem. We show that the result (existence of an additional Killing vector) does not necessarily require a three-dimensional isometry group on two-dimensional orbits but only the existence of a conformal Killing-Yano tensor. In this approach the (additional) isometry appears as the known invariant Killing vector that the ${\cal D}$-metrics admit.
Properties of (skew-symmetric) conformal Yano--Killing tensors are reviewed. Explicit forms of three symmetric conformal Killing tensors in Kerr spacetime are obtained from the Yano--Killing tensor. The relation between spin-2 fields and solutions to the Maxwell equations is used in the construction of a new conserved quantity which is quadratic in terms of the Weyl tensor. The formula obtained is similar to the functional obtained from the Bel--Robinson tensor and is examined in Kerr spacetime. A new interpretation of the conserved quantity obtained is proposed.
We determine conformal symmetry classes for the pp-wave spacetimes. This refines the isometry classification scheme given by Sippel and Goenner (1986 {\it Gen. Rel. Grav.} {\bf 18} 1229). It is shown that every conformal Killing vector for the null fluid type $N$ pp-wave spacetimes is a conformal Ricci collineation. The maximum number of proper non-special conformal Killing vectors in a type $N$ pp-wave spacetime is shown to be three, and we determine the form of a particular set of type $N$ pp-wave spacetimes admitting such conformal Killing vectors. We determine the conformal symmetries of e…
We obtain necessary and sufficient conditions for an initial data set for the vacuum conformal Einstein field equations to give rise to a spacetime development in possession of a Killing spinor. The fact that the conformal Einstein field equations are used in our derivation allows for the possibility of the initial hypersurface $\mathcal{S}$ intersecting non-trivially with (or even being a subset of) null infinity $\mathscr{I}$. For conciseness, these conditions are derived assuming that the initial hypersurface is spacelike. Hence, in particular, these conformal Killing spinor initial data eq…
In this paper, we find all the Conformal Killing Vectors (CKVs) and their Lie Algebra for the recently reported [cqg1] spherically symmetric, shear-free separable metric spacetimes with non-vanishing energy or heat flux. We also solve the geodesic equatios of motion for the spacetime under consideration.
We present a new method for constructing conformal Yano-Killing tensors in five-di\-men\-sio\-nal Anti-de Sitter space-time. The found tensors are represented in two different coordinate systems. We also discuss, in terms of CYK tensors, global charges which are well defined for asymptotically (five-dimensional) Anti-de Sitter space-time. Additionally in Appendix we present our own derivation of conformal Killing one-forms in four-dimensional Anti-de Sitter space-time as an application of the Theorem presented in the paper.
Given an initial-boundary value problem for an anti-de Sitter-like spacetime, we analyse conditions on the conformal boundary ensuring the existence of Killing vectors in the spacetime arising from this problem. This analysis makes use of a system of conformal wave equations describing the propagation of the Killing equation first considered by Paetz. We identify an obstruction tensor constructed from Killing vector candidate and the Cotton tensor of the conformal boundary whose vanishing is a necessary condition for the existence of Killing vectors in the spacetime. This obstruction tensor va…
We generalize our recent method for constructing Killing tensors of the second rank to conformal Killing tensors. The method is intended for foliated spacetimes of arbitrary dimension $m$, which have a set of conformal Killing vectors. It applies to foliations of a more general structure than in previous literature. The basic idea is to start with reducible Killing tensors in slices constructed from a set of conformal Killing vectors and the induced metric, and then lift them to the whole manifold. Integrability conditions are derived that ensure this, and a constructive lifting procedure is p…
We study first BGG operators and their solutions on homogeneous conformal geometries. We focus on conformal Killing tensors, conformal Killing--Yano forms and twistor spinors in particular. We develop an invariant calculus that allows us to find solutions explicitly using only algebraic computations. We also discuss applications to holonomy reductions and conserved quantities of conformal circles. We demonstrate our result on examples of homogeneous conformal geometries coming mostly from general relativity.
Fayos and Sopuerta have recently set up a formalism for studying vacuum spacetimes with an isometry, a formalism that is centred around the bivector corresponding to the Killing vector and that adapts the tetrad to the bivector. Steele has generalized their approach to include the homothetic case. Here, we generalize this formalism to arbitrary spacetimes and to homothetic and conformal Killing vectors but do not insist on aligning the tetrad with the bivector. The most efficient way to use the formalism to find conformal Killing vectors (proper or not) of a given spacetime is to combine it wi…
Conformal rescaling of conformal Yano--Killing tensors and relations between Yano and CYK tensors are discussed. Pullback of these objects to a submanifold is used to construct all solutions of a CYK equation in anti-de Sitter and de Sitter spacetimes. Properties of asymptotic conformal Yano--Killing tensors are examined for asymptotic anti-de Sitter spacetimes. Explicit asymptotic forms of them are derived. The results are used to construct asymptotic charges in asymptotic AdS spacetime. Well known examples like Schwarzschild-AdS, Kerr-AdS and NUT-AdS are examined carefully in the constructio…
In this paper, we investigate conformal Killing's vectors (CKVs) admitted by some plane symmetric spacetimes. Ten conformal Killing's equations and their general forms of CKVs are derived along with their conformal factor. The existence of conformal Killing's symmetry imposes restrictions on the metric functions. The conditions imposing restrictions on these metric functions are obtained as a set of integrability conditions. Considering the cases of time-like and inheriting CKVs, we obtain spacetimes admitting plane conformal symmetry. Integrability conditions are solved completely for some kn…
We present the explicit expressions for the conformal Killing-Yano tensors for the Plebanski-Demianski family of type D solutions in four dimensions. Some physically important special cases are discussed in more detail. In particular, it is demonstrated how the conformal Killing-Yano tensor becomes the Killing-Yano tensor for the solutions without acceleration. A possible generalization into higher dimensions is studied. Whereas the transition from the nonaccelerating to accelerating solutions in four dimensions is achieved by the conformal rescaling of the metric, we show that such a procedur…
We study the Lie point symmetries of a general class of partial differential equations (PDE) of second order. An equation from this class naturally defines a second-order symmetric tensor (metric). In the case the PDE is linear on the first derivatives we show that the Lie point symmetries are given by the conformal algebra of the metric modulo a constraint involving the linear part of the PDE. Important elements in this class are the Klein--Gordon equation and the Laplace equation. We apply the general results and determine the Lie point symmetries of these equations in various general classe…
Korn's inequality plays an important role in linear elasticity theory. This inequality bounds the norm of the derivatives of the displacement vector by the norm of the linearized strain tensor. The kernel of the linearized strain tensor are the infinitesimal rigid-body translations and rotations (Killing vectors). We generalize this inequality by replacing the linearized strain tensor by its trace free part. That is, we obtain a stronger inequality in which the kernel of the relevant operator are the conformal Killing vectors. The new inequality has applications in General Relativity.
In four spacetime dimensions, all ${\cal N} =1$ supergravity-matter systems can be formulated in the so-called $\mathsf{U}(1)$ superspace proposed by Howe in 1981. This paper is devoted to the study of those geometric structures which characterise a background $\mathsf{U}(1)$ superspace and are important in the context of supersymmetric field theory in curved space. We introduce (conformal) Killing tensor superfields $\ell_{(α_1 \dots α_m) ({\dot α}_1 \dots {\dot α}_n)}$, with $m$ and $n$ non-negative integers, $m+n>0$, and elaborate on their significance in the following cases: (i) $m=n=1$; (…
We consider the problem of picking a physically motivated vacuum state on a spherically symmetric spacetime with an extra conformal Killing vector, as opposed to an extra Killing vector as in the Schwarzschild case. Considering a conformal symmetry instead of a symmetry allows us to consider spacetimes that are dynamical and not static (like Schwarzschild). The extra conformal symmetry allows us to calculate the response of particle detectors however. We look at the specific example of a self-similar LTB spacetime that represents a spherically symmetric but inhomogeneous cosmology. We remark t…
A generalisation of a known theorem concerning the computation of the conformal algebra in 1+(n-1) decomposable spaces is presented. It is shown that the general form of Conformal Vector Fields (CVF) is the sum of a gradient CVF and a Killing or Homothetic (n-1)-vector. A simple criterion is established which enables one to check if a 1+(n-1) decomposable spacetime admits proper CVF. As an example, the complete conformal algebra of a Gödel-type spacetime is computed.
