Conformal Ricci Collineations Research Articles

Non-static spherically symmetric spacetimes and their conformal Ricci collineations

For a perfect fluid matter, we present a study of conformal Ricci collineations (CRCs) of non-static spherically symmetric spacetimes. For non-degenerate Ricci tenor, a vector field generating CRCs is found subject to certain integrability conditions. These conditions are then solved in various cases by imposing certain restrictions on the Ricci tensor components. It is found that non-static spherically symmetric spacetimes admit 5, 6 or 15 CRCs. In the degenerate case, it is concluded that such spacetimes always admit infinite number of CRCs.

Conformal Ricci Collineations of Static Space–Times with Maximal Symmetric Transverse Spaces

We explore conformal Ricci collineations (CRCs) for static space–times with maximal symmetric transverse spaces. Solving the CRC equations in the degenerate and nondegenerate cases, we show that the dimension of the Lie algebra of CRCs for these space–times can be 6, 7, or 15 for a nondegenerate Ricci tensor, while a degenerate Ricci tensor produces an infinite number of CRCs.

Concircular vector fields for Kantowski–Sachs and Bianchi type-III spacetimes

This paper intends to obtain concircular vector fields (CVFs) 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 (CKVFs) 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 conditions. It is shown that Kantowski–Sachs and Bianchi type-III spacetimes admit four-, six-, or fifteen-dimensional concircular vector fields. It is established that for Einstein spaces, every conformal Killing vector field is a concircular vector field. Moreover, it is explored that every concircular vector field obtained here is also a conformal Ricci collineation.

Conformal Collineations of the Ricci and Energy–Momentum Tensors in Static Plane Symmetric Space–Times

We provide a complete classification of static plane symmetric space–times according to conformal Ricci collineations (CRCs) and conformal matter collineations (CMCs) in both the degenerate and nondegenerate cases. In the case of a nondegenerate Ricci tensor, we find a general form of the vector field generating CRCs in terms of unknown functions of t and x subject to some integrability conditions. We then solve the integrability conditions in different cases depending upon the nature of the Ricci tensor and conclude that the static plane symmetric space–times have a 7-, 10- or 15-dimensional Lie algebra of CRCs. Moreover, we find that these space–times admit an infinite number of CRCs if the Ricci tensor is degenerate. We use a similar procedure to study CMCs in the case of a degenerate or nondegenerate matter tensor. We obtain the exact form of some static plane symmetric space–time metrics that admit nontrivial CRCs and CMCs. Finally, we present some physical applications of our obtained results by considering a perfect fluid as a source of the energy–momentum tensor.

Classification of Kantowski–Sachs metric via conformal Ricci collineations

In this paper, we present a classification of the Kantowski–Sachs spacetime metric according to its conformal Ricci collineations (CRCs). Solving the CRC equations, it is shown that the Kantowski–Sachs metric admits 15-dimensional Lie algebra of CRCs when its Ricci tensor is non-degenerate and an infinite dimensional group of CRCs when the Ricci tensor is degenerate. Some examples of Kantowski–Sachs metric admitting nontrivial CRCs are presented and their physical interpretation is provided.

Classification of locally rotationally symmetric Bianchi-I space–times using conformal Ricci collineations

We present a complete classification of locally rotationally symmetric (LRS) Bianchi-I space–times in accordance with their conformal Ricci collineations (CRCs). In the case where the Ricci tensor is nondegenerate, we find a general form of the vector field generating CRCs subject to some integrability conditions. Solving the integrability conditions in different cases, we find that the LRS Bianchi-I space–times admit 7-, 10-, 11-, or 15-dimensional Lie algebras of CRCs in the case where the Ricci tensor is nondegenerate. Moreover, we find that these space–times admit an infinite number of CRCs if the Ricci tensor is degenerate. We give some examples of LRS Bianchi-I space–times that admit nontrivial CRCs and are models of a perfect fluid.

Conformal Ricci collineations in LRS Bianchi type V spacetimes with perfect fluid matter

Considering the perfect fluid as a source of energy–momentum tensor, we have classified locally rotationally symmetric (LRS) Bianchi type V spacetimes according to their conformal Ricci collineations (CRCs). It is shown that the LRS Bianchi type V spacetimes with perfect fluid matter admit 9- or 15-dimensional Lie algebra of CRCs when the Ricci tensor is non-degenerate, while the group of CRCs is infinite for degenerate Ricci tensor.

On symmetries of generalized Robertson-Walker spacetimes and applications

ABSTRACTThe purpose of the present article is to study and characterize several types of symmetries of generalized Robertson-Walker spacetimes. Conformal vector fields, curvature and Ricci collineations are studied. Many implications for existence of these symmetries on generalized Robertson-Walker spacetimes are obtained. Finally, Ricci solitons on generalized Robertson-Walker spacetimes admitting conformal vector fields are investigated.

Concircular vector fields for plane-symmetric static spacetimes

In this paper, we investigate concircular vector fields (CVFs) of static plane symmetric four-dimensional Lorentzian manifold. Ten conformal Killing equations and their general form of conformal Killing vector fields (CKVFs) are derived along with their conformal factor. These CKVFs are then placed into the conformal Ricci collineation equations to obtain the final form of CVFs. 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 conditions. It is shown that plane-symmetric static spacetimes admit four, six, seven or fifteen-dimensional concircular vector fields. Analysis of our results are also given in the light of some established results in the literature.

Conformal Ricci Collineations of Static Spherically Symmetric Spacetimes

Conformal Ricci collineations of static spherically symmetric spacetimes are studied. The general form of the vector fields generating conformal Ricci collineations is found when the Ricci tensor is non-degenerate, in which case the number of independent conformal Ricci collineations is 15, the maximum number for four-dimensional manifolds. In the degenerate case it is found that the static spherically symmetric spacetimes always have an infinite number of conformal Ricci collineations. Some examples are provided which admit non-trivial conformal Ricci collineations, and perfect fluid source of the matter.

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.

LETTER: Conformal Ricci Collineations of Space-Times

We study conformal vector fields on space-times which in addition are compatible with the Ricci tensor (so-called conformal Ricci collineations). In the case of Einstein metrics any conformal vector field is automatically a Ricci collineation as well. For Riemannian manifolds, conformal Ricci collineation were called concircular vector fields and studied in the relationship with the geometry of geodesic circles. Here we obtain a partial classification of space-times carrying proper conformal Ricci collineations. There are examples which are not Einstein metrics.

Einstein–Maxwell equations and the conformal Ricci collineations

The Einstein–Maxwell field equations for non-null electromagnetic fields are studied under the assumption of admitting a conformal Ricci collineation. It is shown that a non-null electromagnetic field does not admit any conformal Ricci collineation, unless the generators of the symmetry groups are Killing vector fields. Furthermore, it is shown that the energy-momentum tensor of a non-null electromagnetic field can admit a conformal Ricci collineation, if and only if the collineation is homothetic. The restrictions on non-null Maxwell field, its sources, and its invariants implied by the symmetry condition are calculated. An example of a space-time satisfying the Einstein–Maxwell equations, and admitting a homothetic conformal vector field is also given.