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.     

Comment: Note on Matter Collineations in Kantowski-Sachs, Bianchi Types I and III Spacetimes

We show that the classification of Kantowski-Sachs, Bianchi Types I and III spacetimes admitting Matter Collineations (MCs) presented in a recent paper by Camci et al. [Camci, U., and Sharif, M. Matter Collineations in Kantowski-Sachs, Bianchi Types I and III Spacetimes, (2003) Gen. Rel. Grav. 35, 97-109.] is incomplete. Furthermore for these spacetimes and when the Einstein tensor is non-degenerate, we give the complete Lie Algebra of MCs and the algebraic constraints on the spatial components of the Einstein tensor.     

Classification of Cylindrically Symmetric Static Spacetimes According to Their Ricci Collineations

A complete classification of cylindrically symmetric static Lorentzian manifolds according to their Ricci collineations (RCs) is provided. The Lie algebras of RCs for the non-degenerate Ricci tensor have dimensions 3 to 10, excluding 8 and 9. For the degenerate tensor the algebra is mostly but not always infinite dimensional; there are cases of 10-, 5-, 4- and 3-dimensional algebras. The RCs are compared with the Killing vectors (KVs) and homothetic motions (HMs). The (non-linear) constraints corresponding to the Lie algebras are solved to construct examples which include some exact solutions admitting proper RCs. Their physical interpretation is given. The classification of plane symmetric static spacetimes emerges as a special case of this classification when the cylinder is unfolded.     

Metric and Ricci tensors for a certain class of space-times ofD type

The class of space-times has been determined at the connection level, assuming the existence of some symmetrical relations between the Ricci rotation coefficients. It has been assumed, for instance, that at least two shear-free congruences of null geodesics exist. We have shown that onlyD type or conformally flat space-times can belong to this class. The theorem has been proved that a system of coordinates exists in which the metric tensor can depend on two coordinates, only. The metric tensor has been determined with an accuracy to two functions, each of which is a function of only one coordinate. Linear, second-order differential expressions have been found for these two functions. They determine the Ricci tensor. Several solutions of the Einstein-Maxwell equations with a cosmological constant are given.On leave from the Institute of Theoretical Physics, Warsaw University, Warsaw, Poland.     

Connecting T-duality invariant theories

We show that the vanishing of the one-loop beta-functional of the doubled formalism (which describes string theory on a torus fibration in which the fibres are doubled) is the same as the equation of motion of the recently proposed generalised metric formulation of double field theory restricted to this background: both are the vanishing of a generalised Ricci tensor. That this tensor arises in both backgrounds indicates the importance of a new doubled differential geometry for understanding both constructions.     

An elementary derivation of the five dimensional Myers–Perry metric

We derive the five dimensional Myers–Perry metric via an elementary method to solve the vacuum Einstein field equations directly. This method firstly proposed by Clotz is very simple since it merely involves four components of Ricci tensor and only requires us to deal with some equations without second derivatives’ terms when the metric ansatz is assumed to take an appropriate form.     

Congruences of Fluids in a Finslerian Anisotropic Space-Time

We derive the generalized Raychauduri equation concepts of expansion, shear and vorticity. We give the Ricci tensor of a constant-curvature Randers–Finsler space metric whose first term is the Robertson–Walker metric.Dedicated to the memory of Professor Nikolaos Danikas.     

Gravitation,Electromagnetism and Cosmological Constant in Purely Affine Gravity

The Ferraris-Kijowski purely affine Lagrangian for the electromagnetic field, that has the form of the Maxwell Lagrangian with the metric tensor replaced by the symmetrized Ricci tensor, is dynamically equivalent to the metric Einstein-Maxwell Lagrangian, except the zero-field limit, for which the metric tensor is not well-defined. This feature indicates that, for the Ferraris-Kijowski model to be physical, there must exist a background field that depends on the Ricci tensor. The simplest possibility, supported by recent astronomical observations, is the cosmological constant, generated in the purely affine formulation of gravity by the Eddington Lagrangian. In this paper we combine the electromagnetic field and the cosmological constant in the purely affine formulation. We show that the sum of the two affine (Eddington and Ferraris-Kijowski) Lagrangians is dynamically inequivalent to the sum of the analogous (ΛCDM and Einstein-Maxwell) Lagrangians in the metric-affine/metric formulation. We also show that such a construction is valid, like the affine Einstein-Born-Infeld formulation, only for weak electromagnetic fields, on the order of the magnetic field in outer space of the Solar System. Therefore the purely affine formulation that combines gravity, electromagnetism and cosmological constant cannot be a simple sum of affine terms corresponding separately to these fields. A quite complicated form of the affine equivalent of the metric Einstein-Maxwell-Λ Lagrangian suggests that Nature can be described by a simpler affine Lagrangian, leading to modifications of the Einstein-Maxwell-ΛCDM theory for electromagnetic fields that contribute to the spacetime curvature on the same order as the cosmological constant.     

Symmetry mappings in Einstein-Maxwell space-times

General properties of Einstein-Maxwell spaces, with both null and nonnull source-free Maxwell fields, are examined when these space-times admit various kinds of symmetry mappings. These include Killing, homothetic and conformal vector fields, curvature and Ricci collineations, and mappings belonging to the family of contracted Ricci collineations. In particular, the behavior of the electromagnetic field tensor is examined under these symmetry mappings. Examples are given of such space-times which admit proper curvature and proper Ricci collineations. Examples are also given of such space-times in which the metric tensor admits homothetic and other motions, but in which the corresponding Lie derivatives of the electromagnetic Maxwell tensor are not just proportional to the Maxwell tensor.On leave from Mathematics Department, Monash University, Clayton, Victoria, 3168, Australia.     

On asymptotically flat space-times

The allowed asymptotic behavior of the Ricci tensor is determined for asymptotically flat space-times. With the aid of Penrose's conformai technique the asymptotic behavior of the components of the metric tensor, Weyl tensor, and spin coefficients in a suitable frame is calculated for such a space-time. For Einstein-Maxwell space-times these results reduce to those of Exton, Newman, Penrose, Unti, and Kozarzewski.     

Dynamical Conformal Transformation and Classical Euclidean Wormholes

We investigate the necessary condition for the existence of classical Euclidean wormholes in a conformally non-invariant gravitational model minimally coupled to an scalar field. It is shown that while the original Ricci tensor with positive eigenvalues does not allow the Euclidean wormholes to occur, under dynamical conformal transformations the Ricci tensor, with respect to the original metric, is dynamically coupled with the conformal field and its eigenvalues may become negative allowing the Euclidean wormholes to occur. Therefore, it is conjectured that dynamical conformal transformations may provide us with effective forms of matter sources leading to Euclidean wormholes in conformally non-invariant systems.     

A Complete Classification of Curvature Collineations of Cylindrically Symmetric Static Metrics

Curvature collineations are symmetry directions for the Riemann tensor, as isometries are for the metric tensor and Ricci collineations are for the Ricci tensor. Complete listings of many metrics possessing some minimal symmetry have been given for a number of symmetry groups for the latter two symmetries. It is shown that a claimed complete listing of cylindrically symmetric static metrics by their curvature collineations [1] was actually incomplete and is completed here. It turns out that in this complete list, unlike the previous claim, there are curvature collineations that are distinct from the set of isometries and of Ricci collineations. The physical interpretation of some of the metrics obtained is given.     

Quadratic metric‐affine gravity

We consider spacetime to be a connected real 4‐manifold equipped with a Lorentzian metric and an affine connection. The 10 independent components of the (symmetric) metric tensor and the 64 connection coefficients are the unknowns of our theory. We introduce an action which is (purely) quadratic in curvature and study the resulting system of Euler–Lagrange equations. In the first part of the paper we look for Riemannian solutions, i.e. solutions whose connection is Levi‐Civita. We find two classes of Riemannian solutions: 1) Einstein spaces, and 2) spacetimes with pp‐wave metric of parallel Ricci curvature. We prove that for a generic quadratic action these are the only Riemannian solutions. In the second part of the paper we look for non‐Riemannian solutions. We define the notion of a “Weyl pseudoinstanton” (metric compatible spacetime whose curvature is purely of Weyl type) and prove that a Weyl pseudoinstanton is a solution of our field equations. Using the pseudoinstanton approach we construct explicitly a non‐Riemannian solution which is a wave of torsion in a spacetime with Minkowski metric. We discuss the possibility of using this non‐Riemannian solution as a mathematical model for the neutrino.     

On the Rainich geometrization of scalar meson fields

Similarly as in the Rainich geometrization of an electromagnetic field, the author finds a system of differential equations for the metric tensor, equivalent to the equations of the gravitational and scalar meson field, and shows how to find the wave function of the meson field if the Ricci tensor is known.     

Weyl,curvature, ricci,and metric tensor symmetries

Weyl symmetries for some specific spherically symmetric static spacetimes are derived and compared with metric, Ricci, and curvature tensor symmetries.     

Higher-dimensional homogeneous cosmological models as autonomous dynamical systems

The field equations for homogeneous models in an arbitrary number of dimensions form a Hamiltonian system with constraint forces. Due to the monotonic behavior of the determinant of the induced metric, the evolution of the system can be interpreted as the motion of a particle in an explicitly time-dependent potential. Considering vacuum models, we show that this explicit time-dependence can be eliminated. Using the scaling properties of the Ricci tensor we obtain an autonomous system, for which we can also find a Liapunov function in terms of the n-dimensional Ricci 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.     

Einstein equation at singularities

Einstein’s equation is rewritten in an equivalent form, which remains valid at the singularities in some major cases. These cases include the Schwarzschild singularity, the Friedmann-Lemaître-Robertson-Walker Big Bang singularity, isotropic singularities, and a class of warped product singularities. This equation is constructed in terms of the Ricci part of the Riemann curvature (as the Kulkarni-Nomizu product between Einstein’s equation and the metric tensor).