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.     

Ricci Collineations of Static Space Times with Maximal Symmetric Transverse Spaces

A complete classification of static space times with maximal symmetric transverse spaces is provided, according to their Ricci collineations. The classification is made when one component of Ricci collineation vector field V is non-zero (cases 1～4), two components of V are non-zero (cases 5～10), and three components of V are non-zero (cases 11～14), respectivily. Both non-degenerate (det R_ab ≠0) as well as the degenerate (det R_ab=0) cases are discussed and some new metrics are found.     

Ricci and Matter Collineations of Locally Rotationally Symmetric Space-Times

A new method is presented for the determination of Ricci Collineations (RC) and Matter Collineations (MC) of a given spacetime, in the cases where the Ricci tensor and the energy momentum tensor are non-degenerate and have a similar form with the metric. This method reduces the problem of finding the RCs and the MCs to that of determining the KVs whereas at the same time uses already known results on the motions of the metric. We employ this method to determine all hypersurface homogeneous locally rotationally symmetric spacetimes, which admit proper RCs and MCs. We also give the corresponding collineation vectors. These results conclude a long due open problem, which has been considered many times partially in the literature.     

Comment: Towards a Complete Classification of Spherically Symmetric Lorentzian Manifolds According to Their Ricci Collineations

General expressions for the components of the Ricci collineation vector are derived and the related constraints are obtained. These constraints are then solved to obtain Ricci collineations and the related constraints on the Ricci tensor components for all spacetime manifolds (degenerate or non-degenerate, diagonal or non-diagonal) admitting symmetries larger than so(3) and already known results are recovered. A complete solution is achieved for the spacetime manifolds admitting so(3) as the maximal symmetry group with non-degenerate and non diagonal Ricci tensor components. It is interesting to point out that there appear cases with finite number of Ricci collineations although the Ricci tensor is degenerate and also the cases with infinitely many Ricci collineations even in the case of non-degenerate Ricci tensor. Interestingly, it is found that the spacetime manifolds with so(3) as maximal symmetry group may admit two extra proper Ricci collineations, although they do not admit a G ₅ as the maximal symmetry group. Examples are provided which show and clarify some comments made by Camci et al. [Camci, U., and Branes, A. (2002). Class. Quantum Grav. 19, 393–404]. Theorems are proved which correct the earlier claims made in [Carot, J., Nunez, L. A., and Percoco, U. (1997). Gen. Relativ. Gravit. 29, 1223–1237; Contreras, G., Núñez, L. A., and Percolo, U. (2000). Gen. Relativ. Gravit. 32, 285–294].     

Einstein vacuum field equations with a single non-null Killing vector

It is shown that, in the case where there is a single non-null Killing vector, the vacuum Einstein field equations imply that there is a Ricci collineation in the quotient 3-space. Using coordinates adapted to the collineation vector, we derive a fourth order partial differential equation involving the metric of the quotient 3-space and we show that if this equation is satisfied, the Ernst potential may be obtained by integrating a total Riccati equation and a straightforward set of total differential equations. We also show that if the collineation vector is null, the metric of the quotient 3-space may be expressed in terms of two real Clebsch potentials. Finally in the special case where the collineation vector is the generator of a timelike homothetic motion we reduce the field equations to a single second order partial differential equation of non-Painlevé type in two independent variables and obtain Petrov type III solution of Robinson-Trautman type.     

Conformal Ricci Collineations of Static Spherically Symmetric Spacetimes

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

Ricci Symmetries of Static Space-Times with Maximal Symmetric Transverse Spaces

In the paper [M. Akbar and R.G. Cai, Commun. Theor. Phys. 45 （2006） 95], a complete classification is provided with at least one component of the vector field V is zero. In this paper, I consider the vector field V with all non-zero components and the static space times with maximal symmetric transverse spaces are classified according to their Ricci collineations. These are investigated for non-degenerate Ricci tensor det R ≠0. It turns out that the only collineations admitted by these spaces can be ten, seven, six or four. It also covers our previous results as a spacial case. Some new metrics admitting proper Ricci collineations are also investigated.     

The preferred null symmetries of a null electromagnetic field interacting with a null free gravitational field

The real null vector 1 ^a of the Newman-Penrose formalism is preferred to correspond to a geometrical symmetry as well as a dynamical symmetry. The 16 types of geometrical symmetries expressed through the vanishing of the Lie derivatives of certain tensor fields with respect to 1 ^a are examined separately. Two types of dynamical symmetries are imposed simultaneously on 1 ^a : A null electromagnetic field and a null gravitational field are both chosen to have the same propagation vector 1 ^a . By adopting freedom conditions on 1 ^a , it is shown that the symmetries of the null electromagnetic field are shared neither by the free gravitational field nor by the gravitational potentials. In fact the following five preferred null symmmetries are found to be proper: motion, affine collineation, special curvature collineation, curvature collineation, and Ricci collineation. The scalars characterizing the coupled fields are found to be constant with respect to 1 ^a .     

SYMMETRIES OF THE ELECTROMAGNETIC FIELDS IN GENERAL RELATIVITY

Using the null tetrad approach of Newman and Penrose, the symmetries of the electro-magnetic fields are investigated. It is found that null electromagnetic fields admit Maxwell collineation, and the existence of Ricci collineation and motion is possible only under certain conditions on the spin-coefficients.     

Ricci recurrent space-times with torsion

A Ricci recurrent space-time with covariantly constant stress tensor is an Einstein space-time. We extend this result to Ricci recurrent space-times with torsion. The result is applied to the case of Riemann-Cartan space-times with spin density.     

Correspondence Between Ricci and Other Dark Energies

Purpose of the present paper is to view the correspondence between Ricci and other dark energies. We have considered the Ricci dark energy in presence of dark matter in non-interacting situation. Subsequently, we have derived the pressure and energy density for Ricci dark energy. The equation of state parameter has been generated from these pressure and energy density. Next, we have considered the correspondence between Ricci and other dark energy models, namely tachyonic field, DBI-essence and new agegraphic dark energy without any interaction and investigated possible cosmological consequences.     

Ricci Collineations in Perfect Fluid Bianchi V Spacetime

The Bianchi V spacetimes with perfect-fluid matter are classified according to their Ricci collineations. We have found that in the degenerate case there are infinitely many Ricci collineations whereas a subcase gives a finite number of Ricci collineations which are five. In the non-degenerate case the group of Ricci collineations is finite, i.e. four or five or six or seven. Also, all results obtained satisfy the energy conditions.     

Mass Under the Ricci Flow

In this paper, we study the change of the ADM mass of an ALE space along the Ricci flow. Thus we first show that the ALE property is preserved under the Ricci flow. Then, we show that the mass is invariant under the flow in dimension three (similar results hold in higher dimension with more assumptions). A consequence of this result is the following. Let (M, g) be an ALE manifold of dimension n = 3. If m(g) ≠ 0, then the Ricci flow starting at g can not have Euclidean space as its (uniform) limit. Partially supported by NSF and NSFC. The research is partially supported by the National Natural Science Foundation of China 10631020 and SRFDP 20060003002.     

Differential Harnack Inequalities Under a Coupled Ricci Flow

In this paper, we prove several differential Harnack inequalities under a coupled Ricci flow. As applications, we get Harnack inequalities for positive solutions of backward heat-type equations with potentials under the coupled Ricci flow. We also derive Perelman??s differential Harnack inequality for fundamental solution of the conjugate heat equation under the Ricci flow.     

Ricci recurrent space times

The algebraic restrictions on the Ricci tensor in a Ricci-recurrent space-time are determined. The restrictions imposed on the Petrov type of the Weyl tensor are also given.     

Letter: Comment on Ricci Collineations for Spherically Symmetric Space-Times

It is shown that the results of the paper Contreras, G., Nunez, L. A., Percoco, U. Ricci Collineations for Non-degenerate, Diagonal and Spherically Symmetric Ricci Tensors (2000). Gen. Rel. Grav. 32, 285-294 concerning the Ricci Collineations in spherically symmetric space-times with non-degenerate and diagonal Ricci tensor do not cover all possible cases. Furthermore the complete algebra of Ricci Collineations of certain Robertson-Walker metrics of vanishing spatial curvature are given.     

On Ricci tensors of Randers metrics

In this paper, we study Randers metrics and find a condition on the Ricci tensors of these metrics for being Berwaldian. This generalizes Shen’s Theorem which says that every R-flat complete Randers metric is locally Minkowskian. Then we find a necessary and sufficient condition on the Ricci tensors under which a Randers metric of scalar flag curvature is of zero flag curvature.     

Ricci flow for homogeneous compact models of the universe

Using quaternions, we give a concise derivation of the Ricci tensor for homogeneous spaces with topology of the 3-dimensional sphere. We derive explicit and numerical solutions for the Ricci flow PDE and discuss their properties. In the collapse (or expansion) of these models, the interplay of the various components of the Ricci tensor are studied.     

Ricci nilsoliton black holes

We follow a constructive approach and find higher-dimensional black holes with Ricci nilsoliton horizons. The spacetimes are solutions to the Einstein equation with a negative cosmological constant and generalise therefore, Anti-de Sitter black hole spacetimes. The approach combines a work by Lauret–which relates the so-called Ricci nilsolitons and Einstein solvmanifolds–and an earlier work by the author. The resulting black hole spacetimes are asymptotically Einstein solvmanifolds and thus, are examples of solutions which are not asymptotically Anti-de Sitter. We show that any nilpotent group in dimension n≤6$n\le 6$ has a corresponding Ricci nilsoliton black hole solution in dimension (n+2)$(n+2)$. Furthermore, we show that in dimensions (n+2)>8$(n+2)>8$, there exist an infinite number of locally distinct Ricci nilsoliton black hole metrics.