Complete Classification of Conformal Killing Symmetries of Plane-Symmetric Static Spacetimes in Teleparallel Theory

We have studied the conformal, homothetic and Killing vectors in the context of teleparallel theory of gravitation for plane-symmetric static spacetimes. We have solved completely the non-linear coupled teleparallel conformal Killing equations. This yields the general form of teleparallel conformal vectors along with the conformal factor for all possible cases of metric functions. We have found four solutions which are divided into one Killing symmetries and three conformal Killing symmetries. One of these teleparalel conformal vectors depends on x only and other is a function of all spacetime coordinates. The three conformal Killing symmetries contain three proper homothetic symmetries where the conformal factor is an arbitrary non-zero constant.     

Noether versus Killing symmetry of conformally flat Friedmann metric

In a recent study Noether symmetries of some static spacetime metrics in comparison with Killing vectors of corresponding spacetimes were studied. It was shown that Noether symmetries provide additional conservation laws that are not given by Killing vectors. In an attempt to understand how Noether symmetries compare with conformal Killing vectors, we find the Noether symmetries of the flat Friedmann cosmological model. We show that the conformally transformed flat Friedman model admits additional conservation laws not given by the Killing or conformal Killing vectors. Inter alia, these additional conserved quantities provide a mechanism to twice reduce the geodesic equations via the associated Noether symmetries.     

Nonlinear electrodynamics coupled to teleparallel theory of gravity

Using nonlinear electrodynamics coupled to teleparallel theory of gravity, regular charged spherically symmetric solutions are obtained. The nonlinear theory is reduced to the Maxwell one in the weak limit and the solutions correspond to charged spacetimes. One of the obtained solutions contains an arbitrary function which we call general solution since we can generate from it the other solutions. The metric associated with these spacetimes is the same, i.e., regular charged static spherically symmetric black hole. In calculating the energy content of the general solution using the gravitational energy--momentum within the framework of the teleparallel geometry, we find that the resulting form depends on the arbitrary function. Using the regularized expression of the gravitational energy--momentum we obtain the value of energy.     

Conformal Killing Vectors in LRS Bianchi Type Ⅴ Spacetimes

In this note,we investigate conformal Killing vectors(CKVs)of locally rotationally symmetric(LRS)Bianchi type V spacetimes.Subject to some integrability conditions,CKVs up to implicit functions of(t,x)are obtained.Solving these integrability conditions in some particular cases,the CKVs are completely determined,obtaining a classification of LRS Bianchi type V spacetimes.The inheriting conformal Killing vectors of LRS Bianchi type V spacetimes are also discussed.     

Conformal Killing Vectors in LRS Bianchi Type V Spacetimes

In this note, we investigate conformal Killing vectors (CKVs) of locally rotationally symmetric (LRS) Bianchi type V spacetimes. Subject to some integrability conditions, CKVs up to implicit functions of (t,x) are obtained. Solving these integrability conditions in some particular cases, the CKVs are completely determined, obtaining a classification of LRS Bianchi type V spacetimes. The inheriting conformal Killing vectors of LRS Bianchi type V spacetimes are also discussed.     

Teleparallel versions of Friedmann and Lewis–Papapetrou spacetimes

This paper is devoted to investigating the teleparallel versions of the Friedmann models as well as the Lewis–Papapetrou solution. We obtain the tetrad and the torsion fields for both spacetimes. It is shown that the axial-vector vanishes for the Friedmann models. We discuss the different possibilities for the axial-vector, depending on the arbitrary functions ω and ψ in the Lewis–Papapetrou metric. The vector related to spin has also been evaluated.     

Curvature and Weyl collineations of Bianchi type V spacetimes

The objective of this paper is twofold: (a) First the curvature collineations of the Bianchi type V spacetimes are studied using rank argument of curvature matrix. It is found that the rank of the 6×6 curvature matrix is 3, 4, 5 or 6 for these spacetimes. In one of the rank 3 cases the Bianchi type V spacetime admits proper curvature collineations which form infinite dimensional Lie algebra. (b) Then the Weyl collineations of the Bianchi type V spacetimes are investigated using rank argument of the Weyl matrix. It is obtained that the rank of the 6×6 Weyl matrix for Bianchi type V spacetimes is 0, 4 or 6. It is further shown that these spacetimes do not admit proper Weyl collineations, except in the trivial rank 0 case, which obviously form infinite dimensional Lie algebra. In some special cases it is found that these spacetimes admit Weyl collineations in addition to the Killing vectors, which are in fact proper conformal Killing vectors. The obtained conformal Killing vectors form four-dimensional Lie algebra.     

A Note on Classification of Spatially Homogeneous Rotating Space-Times According to Their Teleparallel Killing Vector Fields in Teleparallel Theory of Gravitation

In this paper we classify spatially homogeneous rotating space-timesaccording to their teleparallel Killing vector fields using direct integration technique. It turns out that the dimension of the teleparallel Killing vector fields is 5 or 10. In the case of 10 teleparallel Killing vector fields the space-time becomes Minkowski and all the torsion components are zero. Teleparallel Killing vector fields in this case are exactly the same as in general relativity. In the cases of 5 teleparallel Killing vector fields we get two more conservation laws in the teleparallel theory of gravitation. Here we also discuss some well-known examples of spatially homogeneous rotating space-times according to their teleparallel Killing vector fields.     

A note on teleparallel Killing vector fields in Bianchi type VIII and IX spaceben times in teleparallel theory of gravitation

In this paper we classify Bianchi type VIII and IX space—times according to their teleparallel Killing vector fields in the teleparallel theory of gravitation by using a direct integration technique. It turns out that the dimensions of the teleparallel Killing vector fields are either 4 or 5. From the above study we have shown that the Killing vector fields for Bianchi type VIII and IX space—times in the context of teleparallel theory are different from that in general relativity.     

Spherically symmetric, self-similar spacetimes

In this paper, we show that self-similarity with respect to the existence of a (purely radial) homothetic Killing vector field for spherically symmetric spacetimes implies the separability of the spacetime metric in terms of the co-moving coordinates (and vice versa) and that the metric is, uniquely, the one recently reported in (Class. Quantam Grav. 18: 2147–2162; 2001). This spacetime, in general, has non-vanishing energy flux and shear. An interesting feature of this spacetime, in contrast to other self-similar spherically symmetric spacetimes (not reducible to our form) is that it has an arbitrary radial distribution of matter.     

Conformal Symmetries of Spherical Spacetimes

We investigate the conformal geometry of spherically symmetric spacetimes in general without specifying the form of the matter distribution. The general conformal Killing symmetry is obtained subject to a number of integrability conditions. Previous results relating to static spacetimes are shown to be a special case of our solution. The general inheriting conformal symmetry vector, which maps fluid flow lines conformally onto fluid flow lines, is generated and the integrability conditions are shown to be satisfied. We show that there exists a hypersurface orthogonal conformal Killing vector in an exact solution of Einstein’s equations for a relativistic fluid which is expanding, accelerating and shearing.     

Obtaining a class of type O pure radiation metrics with a negative cosmological constant,using invariant operators

Using the generalised invariant formalism we derive a special subclass of conformally flat spacetimes whose Ricci tensor has a pure radiation and a Ricci scalar component. The method used is a development of the methods used earlier for pure radiation spacetimes of Petrov types O and N, respectively. In this paper we demonstrate how to handle, in the generalised invariant formalism, spacetimes with isotropy freedom and rich Killing vector structure. Once the spacetimes have been constructed, it is straightforward to deduce their Karlhede classification: the Karlhede algorithm terminates at the fourth derivative order, and the spacetimes all have one degree of null isotropy and three, four or five Killing vectors.     

Existence of Constant Mean Curvature Foliations in Spacetimes with Two-Dimensional Local Symmetry

It is shown that in a class of maximal globally hyperbolic spacetimes admitting two local Killing vectors, the past (defined with respect to an appropriate time orientation) of any compact constant mean curvature hypersurface can be covered by a foliation of compact constant mean curvature hypersurfaces. Moreover, the mean curvature of the leaves of this foliation takes on arbitrarily negative values and so the initial singularity in these spacetimes is a crushing singularity. The simplest examples occur when the spatial topology is that of a torus, with the standard global Killing vectors, but more exotic topologies are also covered. In the course of the proof it is shown that in this class of spacetimes a kind of positive mass theorem holds. The symmetry singles out a compact surface passing through any given point of spacetime and the Hawking mass of any such surface is non-negative. If the Hawking mass of any one of these surfaces is zero then the entire spacetime is flat. Received: 15 July 1996 / Accepted: 12 March 1997     

Timelike Killing vectors and ergo surfaces in non-asymptotically flat spacetimes

Ergo surfaces are investigated in spacetimes with a cosmological constant. We find the existence of multiple timelike Killing vectors, each corresponding to a distinct ergo surface, with no one being preferred. Using a kinematic invariant, which provides a measure of hypersurface orthogonality, we explore its potential role in selecting a preferred timelike Killing vector and consequently a unique ergo surface.     

Concircular Curvature Tensor and Fluid Spacetimes

In the differential geometry of certain F-structures, the importance of concircular curvature tensor is very well known. The relativistic significance of this tensor has been explored here. The spacetimes satisfying Einstein field equations and with vanishing concircular curvature tensor are considered and the existence of Killing and conformal Killing vectors have been established for such spacetimes. Perfect fluid spacetimes with vanishing concircular curvature tensor have also been considered. The divergence of concircular curvature tensor is studied in detail and it is seen, among other results, that if the divergence of the concircular tensor is zero and the Ricci tensor is of Codazzi type then the resulting spacetime is of constant curvature. For a perfect fluid spacetime to possess divergence-free concircular curvature tensor, a necessary and sufficient condition has been obtained in terms of Friedmann-Robertson-Walker model.     

Quasi-local energy for spherically symmetric spacetimes

We present two complementary approaches for determining the reference for the covariant Hamiltonian boundary term quasi-local energy and test them on spherically symmetric spacetimes. On the one hand, we isometrically match the 2-surface and extremize the energy. This can be done in two ways, which we call programs I (without constraint) and II (with additional constraints). On the other hand, we match the orthonormal 4-frames of the dynamic and the reference spacetimes. Then, if we further specify the observer by requiring the reference displacement to be the timelike Killing vector of the reference, the result is the same as program I, and the energy can be positive, zero, or even negative. If, instead, we require that the Lie derivatives of the two-area along the displacement vector in both the dynamic and reference spacetimes to be the same, the result is the same as program II, and it satisfies the usual criteria: the energies are non-negative and vanish only for Minkowski (or anti-de Sitter) spacetime.     

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.     

No-Go theorem in spacetimes with two commuting spacelike killing vectors

Four-dimensional Riemannian spacetimes with two commuting spacelike Killing vectors are studied in Einstein's theory of gravity, and found that no outer apparent horizons exist, provided that the dominant energy condition holds.     

Noether Symmetries Versus Killing Vectors and Isometries of Spacetimes

Symmetries of spacetime manifolds which are given by Killing vectors are compared with the symmetries of the Lagrangians of the respective spacetimes. We find the point generators of the one parameter Lie groups of transformations that leave invariant the action integral corresponding to the Lagrangian (Noether symmetries). In the examples considered, it is shown that the Noether symmetries obtained by considering the Larangians provide additional symmetries which are not provided by the Killing vectors. It is conjectured that these symmetries would always provide a larger Lie algebra of which the KV symmetres will form a subalgebra. PACS: 04.25.-g, 02.20.Sv, 11.30.-j