Conformal Killing Vector Research Articles

Computation of the conformal algebra of 1 + 3 decomposable spacetimes

The conformal algebra of a 1 + 3 decomposable spacetime can be computed from the conformal Killing vectors (CKV) of the 3-space. It is shown that the general form of such a 3-CKV is the sum of a gradient CKV and a Killing or homothetic 3-vector. It is proved that spaces of constant curvature always admit such conformal Killing vectors. As an example, the complete conformal algebra of a Gödel-type spacetime is computed. Finally it is shown that this method can be extended to compute the conformal algebra of more general non-decomposable spacetimes.

Inflation in a self-interacting gas universe

We show that a de Sitter spacetime is a solution of Einstein's field equations with the energy momentum tensor of a self-interacting, classical Maxwell-Boltzmann gas in collisional equilibrium. The self-interaction is described by a four-force which is quadratic in the (spatially projected) particle four-momenta. This force does not preserve the particle number and gives rise to an exponential increase in the comoving entropy of the universe while the temperature of the latter remains constant. These properties of a gas universe are related to the existence of a ``projector-conformal'' timelike Killing vector representing a symmetry which is ``in between'' the symmetries characterized by a Killing vector and those characterized by a conformal Killing vector.

Null Killing vectors and reductions of the self-duality equations

We find all null conformal Killing vectors in with the metric . We classify one and two-dimensional totally null algebras generated by such vectors. Reductions of the self-dual Yang-Mills equations are obtained for gauge fields invariant with respect to these algebras. PACS Numbers: 0210, 1130

On `Diagonal spacetimes admitting inheriting conformal Killing vector fields'

In a recent paper it is stated that if a perfect-fluid diagonal spacetime with barotropic equation of state satisfying the energy positive condition admits a proper inheriting conformal Killing vector field, then the solution is either FLRW, a very special plane-symmetric spacetime, or a solution which is conformal to a Bianchi type I spacetime. Nevertheless, this result is more general than the authors claim because in the process of obtaining the solutions one of the Einstein equations was incidentally forgotten, and therefore, their statement applies not only to perfect fluids but to more general anisotropic sources with two equal pressures. Thanks to the remaining equation, it is possible to establish a more restrictive statement that characterizes the Petrov type-D Allnutt solution.

Group theoretical derivation of Liouville action for Regge surfaces

We show that the structure of the Liouville action on a two-dimensional Regge surface of the topology of the sphere and of the torus is determined by the invariance under the transformations induced by the conformal Killing vector fields and under modular transformations.

A Generalisation of Brinkmann's Theorem

Brinkmann has shown that conformally related Einstein spaces are either Ricci flat pp-waves or conformally flat spaces with one as de Sitter space and the other being flat. We herewith show that if gikand \(\overline g _{ik} \left( { = \left( {1/w^2 } \right)g_{ik} } \right)\)are distinct metrics for which \(G_{ik} = \overline G _{ik}\), then both represent (generalised) pp-waves; and w,iis a homothetic conformal Killing vector of gik. It is further observed that in the situation \(G_{ik} = \overline G _{ik} + \Lambda \overline g _{ik}\)we can have many (and also non-conformally flat) conformally related solutions, unlike in Einstein spaces. In particular for perfect fluid coupling, we show that the conformally related solutions have to be Friedman-Robertson-Walker metrics with equation of state as \(\mu + 3p = 0,\mu = p\), or pp-waves depending on whether w,iis timelike, spacelike or null.

Irreducible Decomposition of Einstein's Equations in Spacetimes with Symmetries

In this paper we develop a method to write Einstein's field equations for general matter in an irreducible form that fully incorporates the character of a symmetry introduced in spacetime. We express our method using the framework of double congruences and thus the formalism is covariant. The basic notion on which it is based is that of the geometrisation of a general symmetry. We apply our general scheme to the case of a spacelike conformal Killing vector field on the spacetime manifold regarding the situations of a perfect fluid and anisotropic matter.

Uniqueness of the Functional Determinant

The functional determinant of the conformal laplacian and the square of the Dirac operator are known to be extremized at the standard round metric of the four-sphere among all conformal metrics (up to gauge equivalence). In this article we show that this is the unique critical point, thus extending the work of Onofri and Osgood, Phillips and Sarnak for the functional determinant on S 2 which characterized the constant curvature metric as the unique critical point of the determinant. In addition, we introduce a new symmetric two-tensor field which is defined on any conformally flat four-manifold and can be viewed as a fourth order generalization of the Einstein gravitational tensor. As a consequence we prove a Pohozaev identity for manifolds with boundary which admit conformal Killing vector fields.

Incorporation of space–time symmetries in Einstein’s field equations

In the search for exact solutions to Einstein’s field equations the main simplification tool is the introduction of space–time symmetries. Motivated by this fact, we develop a method to write the field equations for general matter in a form that fully incorporates the character of the symmetry. The method is being expressed in a covariant formalism using the framework of a double congruence. The basic notion on which it is based is that of the geometrization of a general symmetry. As a special application of our general method we consider the case of a space-like conformal Killing vector field on the space–time manifold regarding special types of matter fields. New perspectives in General Relativity are discussed.

Inheriting Conformal and Special Conformal Killing Vectors in String Cosmology

In this paper we study the consequences of the existence of conformal and special conformal Killing vectors (CKV, SCKV) for string cloud and string fluid in the context of general relativity. The inheritance symmetries of the string cloud and string fluid are discussed. Einstein's field equations have been solved for static spherically symmetric space-time with cloud and fluid of strings source via CKV.

Lie Groups of Conformal Motions Acting on Null Orbits

Space-times admitting a 3-dimensional Lie group of conformal motions $C_3$ acting on null orbits are studied. Coordinate expressions for the metric and the conformal Killing vectors (CKV) are provided (irrespectively of the matter content) and then all possible perfect fluid solutions are found, although none of these verify the weak and dominant energy conditions over the whole space-time manifold.

Perfect-fluid cosmologies with a proper conformal Killing vector

We study the Einstein field equations for spacetimes admitting a maximal two-dimensional Abelian group of isometries acting orthogonally transitively on spacelike surfaces and, in addition, with at least one conformal Killing vector. The three-dimensional conformal group is restricted to the case when the two-dimensional Abelian isometry subalgebra is an ideal and it is also assumed to act on non-null hypersurfaces (both spacelike and timelike cases are studied). We consider both diagonal and non-diagonal metrics, and find all the perfect-fluid solutions under these assumptions (except those already known). We find four families of solutions, each one containing arbitrary parameters for which no differential equations remain to be integrated. We write the line-elements in a simplified form and perform a detailed study for each of these solutions, giving the kinematical quantities of the fluid velocity vector, the energy density and pressure, values of the parameters for which the energy conditions are fulfilled everywhere, the Petrov type, the singularities in the spacetimes and the Friedmann - Lemaître - Robertson - Walker metrics contained in each family.

LETTER TO THE EDITOR: General Non-Rotating Perfect-Fluid Solution with an Abelian Spacelike C3 Including Only One Isometry

The general solution for non-rotating perfect-fluid spacetimes admitting one Killing vector and two conformal (non-isometric) Killing vectors spanning an abelian three-dimensional conformal algebra (C3) acting on spacelike hypersurfaces is presented. It is of Petrov type D; some properties of the family such as matter contents are given. This family turns out to be an extension of a solution recently given in [9] using completely different methods. The family contains Friedman-Lemaître-Robertson-Walker particular cases and could be useful as a test for the different FLRW perturbation schemes. There are two very interesting limiting cases, one with a non-abelian G2 and another with an abelian G2 acting non-orthogonally transitively on spacelike surfaces and with the fluid velocity non-orthogonal to the group orbits. No examples are known to the authors in these classes.

Stationary and axisymmetric perfect fluids with one conformal Killing vector

We study the stationary and axisymmetric non-convective differentially rotating perfect-fluid solutions of Einstein's field equations admitting one conformal symmetry. We analyse the two inequivalent Lie algebras not exhaustively considered in our previous work and show that the general solution for each Lie algebra depends on one arbitrary function of one of the coordinates while a set of three ordinary differential equations for four unknowns remains to be solved. The conformal Killing vector of these solutions is necessarily homothetic. We summarize in a table all the possible solutions for all the allowed Lie algebras and also add a corrigendum to an erroneous statement in our previous paper concerning the differentially rotating character of one of the solutions presented.

Anisotropic universes with conformal motion

By imposing natural geometrical and kinematical conditions on a conformal Killing vector in Bianchi I spacetime, we show that a class of axisymmetric metrics admits a conformal motion. This class contains new exact solutions of Einstein's equations, including anisotropic radiation universes whose expansion isotropizes at late times.

Conformal symmetries of conformal-reducible spacetimes with non-zero Weyl tensor

By using the fact that 2 + 2 reducible spacetimes which are not conformally flat can admit no proper conformal Killing vectors, it is shown how all conformal symmetries of 2 + 2 conformal reducible spacetimes, including spherically symmetric and plane symmetric spacetimes, with non-zero Weyl tensor may be found. The method applies to any spacetime which is conformal to a 2 + 2 reducible spacetime with non-zero Weyl tensor.

Space-times admitting a three-dimensional conformal group

Perfect fluid space-times admitting a three-dimensional Lie group of conformal motions containing a two-dimensional Abelian Lie subgroup of isometries are studied. Demanding that the conformal Killing vector be proper (i.e., not homothetic nor Killing), all such space-times are classified according to the structure of their corresponding three-dimensional conformal Lie group and the nature of their corresponding orbits (that are assumed to be non-null). Each metric is then explicitly displayed in coordinates adapted to the symmetry vectors. Attention is then restricted to the diagonal case, and exact perfect fluid solutions are obtained in both the cases in which the fluid four-velocity is tangential or orthogonal to the conformal orbits, as well as in the more general “tilting” case.

SPACE–TIMES WITH A TWO-DIMENSIONAL GROUP OF CONFORMAL MOTIONS

The necessary and sufficient conditions for a space–time to admit a two-dimensional group of conformal motions (and, in particular, of homothetic motions) acting on nonnull orbits are found in the compacted spin-coefficient formalism. Although the discussion is restricted to the case of spacelike orbits, similar results are readily obtained for timelike orbits via the (modified) Sachs star operation. A number of theorems are obtained dealing with such topics as the Gaussian curvature of the group orbits, orthogonal transitivity, and hypersurface orthogonality of the conformal Killing vectors. A simple proof is presented of a generalization of a theorem due to Papapetrou.

General solution and classification of conformal motions in static spherical spacetimes

We find the complete and explicit general solution of the conformal Killing equation in static spherically symmetric spacetimes, thus unifying and generalizing previous special cases. For non-conformally-flat spacetimes, there are at most two proper conformal motions. There are three classes of such spacetimes, and one or both of the conformal Killing vectors is non-inheriting. One of the classes includes self-similar spacetimes (i.e. with a homothetic motion). The conformally flat spacetimes (including the Schwarzschild interior metric) fall into three classes, and their eleven proper conformal Killing vectors are given in full. The only spacetimes with conformal motion that are regular at the centre are conformally flat. An addendum for this article has been published in 1996 Class. Quantum Grav. 13 317

REDUCED PHASE SPACE QUANTIZATION OF ASHTEKAR’S GRAVITY ON A DE SITTER BACKGROUND

We solve perturbative constraints and eliminate gauge freedom for Ashtekar’s gravity on a de Sitter background. We show that the reduced phase space consists of transverse, traceless, symmetric fluctuations of the triad and transverse, traceless, symmetric fluctuations of the connection. A part of the gauge freedom corresponding to the conformal Killing vectors of the three-manifold can be fixed only by imposing conditions on the Lagrange multiplier. The reduced phase space is equivalent to that of ADM gravity on the same background.