Conformal Killing Vector Research Articles

Thermodynamical equilibrium and spacetime geometry

In relativistic theory of irreversible thermodynamical processes near equilibrium, generally a series of assumptions is made having, in particular, the consequence that the temperature vector is a Killing vector. We show that, in contrast to usual approaches, in equilibrium (i) the temperature vector can also be a conformal Killing vector, (ii) as an implication of the Killing property of the temperature vector, most assumptions made can be derived, without restricting the matter configuration to a perfect fluid, (iii) for non-vanishing rotation of the fluid, the heat-flow is unequal to zero, (iv) for vanishing acceleration of the fluid the Friedmann radiation cosmos is the only physically significant solution of Einstein's equations and (v) the equilibrium conditions are of the Cattaneo form such that a causal propagation of temperature can be expected.

Generalized Korn's inequality and conformal Killing vectors

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.

The conformal Penrose limit and the resolution of the pp-curvature singularities

We consider the exact solutions of the supergravity theories in various dimensions in which the spacetime has the form Md × SD−d where Md is an Einstein space admitting a conformal Killing vector and SD−d is a sphere of an appropriate dimension. We show that if the cosmological constant of Md is negative and the conformal Killing vector is spacelike, then such solutions will have a conformal Penrose limit: M(0)d × SD−d where M(0)d is a generalized d-dimensional AdS plane wave. We study the properties of the limiting solutions and find that M(0)d has 1/4 supersymmetry as well as a Virasoro symmetry. We also describe how the pp-curvature singularity of M(0)d is resolved in the particular case of the D6-branes of D = 10 type IIA supergravity theory. This distinguished case provides an interesting generalization of the plane waves in D = 11 supergravity theory and suggests a duality between the SU(2) gauged d = 8 supergravity theory of Salam and Sezgin on M(0)8 and the d = 7 ungauged supergravity theory on its pp-wave boundary.

Conformally symmetric vacuum solutions of the gravitational field equations in the brane-world models

A class of exact solutions of the gravitational field equations in the vacuum on the brane are obtained by assuming the existence of a conformal Killing vector field, with non-static and non-central symmetry. In this case, the general solution of the field equations can be obtained in a parametric form in terms of the Bessel functions. The behavior of the basic physical parameters describing the non-local effects generated by the gravitational field of the bulk (dark radiation and dark pressure) is also considered in detail, and the equation of state satisfied at infinity by these quantities is derived. As a physical application of the obtained solutions we consider the behavior of the angular velocity of a test particle moving in a stable circular orbit. The tangential velocity of the particle is a monotonically increasing function of the radial distance and, in the limit of large values of the radial coordinate, tends to a constant value, which is independent on the parameters describing the model. Therefore, a brane geometry admitting a one-parameter group of conformal motions may provide an explanation for the dynamics of the neutral hydrogen clouds at large distances from the galactic center, which is usually explained by postulating the existence of the dark matter.

ON A CONFORMAL KILLING VECTOR FIELD IN A COMPACT ALMOST KAHLERIAN MANIFOLD

In this paper, we will prove that in a compact almost Kahlerian manifold M n , any conformal Killing vector fleld is Killing if n ‚ 4.

Conformal killing vector fields on spacetime solutions of Einstein's equations and initial data

Working within the ADM formalism we have established results characterizing a conformal Killing vector (CKV) field on a spacetime solution of Einstein's field equations and the initial data on a spacelike hypersurface. We have considered the cases (i) the CKV is tangential to spacelike slices, (ii) the CKV is closed and the initial hypersurface is totally umbilical and has constant mean curvature (CMC), and (iii) spacetime is a perfect fluid, has a non-vanishing CKV transverse to a compact CMC hypersurface orthogonal to the 4-velocity. Finally, we show that a non-null CKV that is also a geodesic vector field is either homothetic or locally closed.

KIDs are Non-Generic

We prove that the space-time developments of generic solutions of the vacuum constraint Einstein equations do not possess any global or local Killing vectors, when Cauchy data are prescribed on an asymptotically flat Cauchy surface, or on a compact Cauchy surface with mean curvature close to a constant, or for CMC asymptotically hyperbolic initial data sets. More generally, we show that nonexistence of global symmetries implies, generically, non-existence of local ones. As part of the argument, we prove that generic metrics do not possess any local or global conformal Killing vectors.

Bi-conformal vector fields and their applications

We introduce a concept of bi-conformal transformation, as a generalization of conformal ones, by allowing two orthogonal parts of a manifold with metric g to be scaled by different conformal factors. In particular, we study their infinitesimal version, called bi-conformal vector fields. We show that these are characterized by the differential conditions and , where P and Π are orthogonal projectors (P + Π = g). Keeping P and Π fixed, the set of bi-conformal vector fields is a Lie algebra which can be finite or infinite dimensional according to the dimensionality of the projectors. We determine (i) when an infinite-dimensional case is feasible and its properties, and (ii) a normal system for the generators in the finite-dimensional case. Its integrability conditions are also analysed, which in particular provides the maximum number of linearly independent solutions. We identify the corresponding maximal spaces, and show a necessary geometric condition for a metric tensor to be a double-twisted product. More general ‘breakable’ spaces are briefly considered. Many known symmetries are included, such as conformal Killing vectors, Kerr–Schild vector fields, kinematic self-similarity, causal symmetries and rigid motions.

Conformal symmetry classes for pp-wave spacetimes

We determine conformal symmetry classes for the pp-wave spacetimes. This refines the isometry classification scheme given by Sippel and Goenner (1986 Gen. Rel. Grav. 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 each type N isometry class of Sippel and Goenner and present new isometry classes.

Asymptotically anti-de Sitter space-times: symmetries and conservation laws revisited

In this short note, we verify explicitly in static coordinates that the non trivial asymptotic Killing vectors at spatial infinity for anti-de Sitter space-times correspond one to one to the conformal Killing vectors of the conformally flat metric induced on the boundary. The fall-off conditions for the metric perturbations that guarantee finiteness of the associated conserved charges are derived.

Bowen–York tensors

For a conformally flat 3-space, we derive a family of linear second-order partial differential operators which sends vectors into trace-free, symmetric 2-tensors. These maps, which are parametrized by conformal Killing vectors on the 3-space, are such that the divergence of the resulting tensor field depends only on the divergence of the original vector field. In particular, these maps send source-free electric fields into TT tensors. Moreover, if the original vector field is the Coulomb field on , the resulting tensor fields on are nothing but the family of TT tensors originally written by Bowen and York.

Gauge theories on sphere and Killing vectors

We provide a general method for studying manifestly O( n+1) covariant formulation of p-form gauge theories by stereographically projecting these theories, defined in flat Euclidean space, onto the surface of a hypersphere. The gauge fields in the two descriptions are mapped by conformal Killing vectors while conformal Killing spinors are necessary for the matter fields, allowing for a very transparent analysis and compact presentation of results. General expressions for these Killing vectors and spinors are given. The familiar results for a vector gauge theory are reproduced.

A charged spherically symmetric solution

We find a solution of the Einstein-Maxwell system of field equations for a class of accelerating, expanding and shearing spherically symmetric metrics. This solution depends on a particularansatz for the line element. The radial behaviour of the solution is fully specified while the temporal behaviour is given in terms of a quadrature. By setting the charge contribution to zero we regain an (uncharged) perfect fluid solution found previously with the equation of statep = μ + constant, which is a generalisation of a stiff equation of state. Our class of charged shearing solutions is characterised geometrically by a conformal Killing vector.

Conserved currents of massless fields of spin s ≥ ½

A complete and explicit classification of all locally constructed conserved currents and underlying conserved tensors is obtained for massless linear symmetric spinor fields of any spin s>0 in four dimensional flat spacetime. These results generalize the recent classification in the spin s=1 case of all conserved currents locally constructed from the electromagnetic spinor field. The present classification yields spin s>0 analogs of the well-known electromagnetic stress-energy tensor and Lipkin's zilch tensor, as well as a spin s>0 analog of a novel chiral tensor found in the spin s=1 case. The chiral tensor possesses odd parity under a duality symmetry (i.e., a phase rotation) on the spin s field, in contrast to the even parity of the stress-energy and zilch tensors. As a main result, it is shown that every locally constructed conserved current for each s>0 is equivalent to a sum of elementary linear conserved currents, quadratic conserved currents associated to the stress-energy, zilch, and chiral tensors, and higher derivative extensions of these currents in which the spin s field is replaced by its repeated conformally-weighted Lie derivatives with respect to conformal Killing vectors of flat spacetime. Moreover, all of the currents have a direct, unified characterization in terms of Killing spinors. The cases s=2, s=1/2 and s=3/2 provide a complete set of conserved quantities for propagation of gravitons (i.e., linearized gravity waves), neutrinos and gravitinos, respectively, on flat spacetime. The physical meaning of the zilch and chiral quantities is discussed.

Killing tensors and conformal Killing tensors from conformal Killing vectors

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 that it is possible to construct a much larger class of reducible proper conformal Killing tensors and Killing tensors than permitted by the Koutras algorithms. In particular, by showing that all conformal Killing tensors are reducible in conformally flat spaces, we have a method of constructing all conformal Killing tensors, and hence all the Killing tensors (which will in general be irreducible) of conformally flat spaces using their conformal Killing vectors.

INHERITING SPACELIKE CONFORMAL KILLING VECTORS IN STRING COSMOLOGY

We study the consequences of the existence of spacelike conformal Killing vectors (SpCKV) parallel to xa for cosmic strings and string fluid in the context of general relativity. The inheritance symmetries of the cosmic strings and string fluid are discussed in the case of SpCKV. Furthermore we examine proper homothetic spacelike Killing vectors for the cosmic strings and string fluid.

Conformal symmetry inheritance in null fluid spacetimes

We define inheriting conformal Killing vectors for null fluid spacetimes and find the maximum dimension of the associated inheriting Lie algebra. We show that for non-conformally flat null fluid spacetimes, the maximum dimension of the inheriting algebra is seven and for conformally flat null fluid spacetimes the maximum dimension is eight. In addition, it is shown that there are two distinct classes of non-conformally flat generalized plane wave spacetimes which possess the maximum dimension, and one class in the conformally flat case.

Spacelike Ricci inheritance vectors in a model of string cloud and string fluid stress tensor

We study the consequences of the existence of spacelike Ricci inheritance vectors (SpRIVs) parallel to xa for a model of string cloud and string fluid stress tensor in the context of general relativity. Necessary and sufficient conditions are derived for a spacetime with a model of string cloud and string fluid stress tensor to admit a SpRIV, and a SpRIV which is also a spacelike conformal Killing vector. Also, some results are obtained.

The maximum dimension of the inheriting algebra in perfect fluid space–times

We determine the maximum dimension of the Lie algebra of inheriting conformal Killing vectors in perfect fluid space–times. For the case of conformally flat space–times the maximum dimension is eight and for the case of nonconformally flat space–times the maximum dimension is found to be five. We illustrate each case with examples.

FINDING CONFORMAL KILLING VECTORS FROM THE INVARIANT CLASSIFICATION SCHEME

The invariant classification (Karlhede classification)1 of a particular spacetime fixes (up to isotropy) an intrinsic tetrad for the spacetime, in terms of the Riemann tensor and its derivatives. Exploiting this tetrad, it has been shown how to determine the number of Killing vectors which exist 2, and also whether a proper homothetic Killing vector exists 3. The first part of the talk outlined a procedure for finding explicitly homothetic and Killing vectors by exploiting the symmetry properties of the intrinsic tetrad 4,5. In general, it is not always possible to use this same tetrad to investigate proper conformal Killing vectors by the same methods. In this talk it was suggested to modify the invariant classification procedure so that only conformally invariant conditions are used for fixing the tetrad in terms of the Riemann tensor and its derivatives. This would mean that the existence of proper conformal Killing vectors could be deduced directly from the modified invariant classification scheme, and then found explicitly 6.