Conformally recurrent space-times admitting a proper conformal vector field
30 Apr 2014-Communications of The Korean Mathematical Society (The Korean Mathematical Society)-Vol. 29, Iss: 2, pp 319-329
TL;DR: In this paper, the authors studied the properties of conformally recur- rent pseudo Riemannian manifolds admitting a proper conformal vector field with respect to the scalar field, focusing particularly on the 4-dimensional Lorentzian case.
Abstract: In this paper we study the properties of conformally recur- rent pseudo Riemannian manifolds admitting a proper conformal vector field with respect to the scalar field �, focusing particularly on the 4- dimensional Lorentzian case. Some general properties already proven by one of the present authors for pseudo conformally symmetric manifolds endowed with a conformal vector field are proven also in the case, and some new others are stated. Moreover interesting results are pointed out; for example, it is proven that the Ricci tensor under certain conditions is Weyl compatible: this notion was recently introduced and investigated by one of the present authors. Further we study conformally recurrent 4-dimensional Lorentzian manifolds (space-times) admitting a conformal vector field: it is proven that the covectorj is null and unique up to scaling; moreover it is shown that the same vector is an eigenvector of the Ricci tensor. Finally, it is stated that such space-time is of Petrov type N with respect toj.
Citations
More filters
Dissertation•
01 Jan 2017
TL;DR: In this paper, a tensor by combining Riemann-Christoffel curvature tensor, Ricci tensor and the metric tensor is presented, and with the help of algebraic classification, the equivalence of different geometric structures is proved.
Abstract: In the literature we see that after introducing a geometric structure by imposing some restrictions on Riemann–Christoffel curvature tensor, the same type structures given by imposing same restriction on other curvature tensors being studied. The main object of the present paper is to study the equivalency of various geometric structures obtained by same restriction imposing on different curvature tensors. In this purpose we present a tensor by combining Riemann–Christoffel curvature tensor, Ricci tensor, the metric tensor and scalar curvature which describe various curvature tensors as its particular cases. Then with the help of this generalized tensor and using algebraic classification we prove the equivalency of different geometric structures (see Theorems 6.3, 6.4, 6.5, 6.6 and 6.7; Tables 1 and 2). Mathematics Subject Classification (2010). 53C15, 53C21, 53C25, 53C35.
11 citations
Additional excerpts
..., conformally recurent ([6], [20], [40], [62], [176], [223], [294], [304]), projectively recurrent [7], concircularly recurrent ([123], [131]), conharmonically recurrent [132] and quasi-conformally recurrent [65]) manifold....
[...]
6 citations
References
More filters
Book•
19 May 2003TL;DR: A survey of the known solutions of Einstein's field equations for vacuum, Einstein-Maxwell, pure radiation and perfect fluid sources can be found in this paper, where the solutions are ordered by their symmetry group, their algebraic structure (Petrov type) or other invariant properties such as special subspaces or tensor fields and embedding properties.
Abstract: A paperback edition of a classic text, this book gives a unique survey of the known solutions of Einstein's field equations for vacuum, Einstein-Maxwell, pure radiation and perfect fluid sources. It introduces the foundations of differential geometry and Riemannian geometry and the methods used to characterize, find or construct solutions. The solutions are then considered, ordered by their symmetry group, their algebraic structure (Petrov type) or other invariant properties such as special subspaces or tensor fields and embedding properties. Includes all the developments in the field since the first edition and contains six completely new chapters, covering topics including generation methods and their application, colliding waves, classification of metrics by invariants and treatments of homothetic motions. This book is an important resource for graduates and researchers in relativity, theoretical physics, astrophysics and mathematics. It can also be used as an introductory text on some mathematical aspects of general relativity.
3,502 citations
Book•
01 Jan 1990TL;DR: Geometry, Topology and physics, Second Edition as mentioned in this paper is an excellent introduction to differential geometry and topology for postgraduate students and researchers in theoretical and mathematical physics, with a considerably expanded first chapter, reviewing aspects of path integral quantization and gauge theories.
Abstract: Differential geometry and topology have become essential tools for many theoretical physicists. In particular, they are indispensable in theoretical studies of condensed matter physics, gravity, and particle physics. Geometry, Topology and Physics, Second Edition introduces the ideas and techniques of differential geometry and topology at a level suitable for postgraduate students and researchers in these fields.The second edition of this popular and established text incorporates a number of changes designed to meet the needs of the reader and reflect the development of the subject. The book features a considerably expanded first chapter, reviewing aspects of path integral quantization and gauge theories. Chapter 2 introduces the mathematical concepts of maps, vector spaces, and topology. The following chapters focus on more elaborate concepts in geometry and topology and discuss the application of these concepts to liquid crystals, superfluid helium, general relativity, and bosonic string theory. Later chapters unify geometry and topology, exploring fiber bundles, characteristic classes, and index theorems. New to this second edition is the proof of the index theorem in terms of supersymmetric quantum mechanics. The final two chapters are devoted to the most fascinating applications of geometry and topology in contemporary physics, namely the study of anomalies in gauge field theories and the analysis of Polakov's bosonic string theory from the geometrical point of view.Geometry, Topology and Physics, Second Edition is an ideal introduction to differential geometry and topology for postgraduate students and researchers in theoretical and mathematical physics.
1,968 citations
Posted Content•
TL;DR: In this paper, it is shown that the exterior Schwarzschild solution itself provides necessary conditions for the types of the density distributions to be considered inside the mass, in order to obtain exact solutions or equations of state compatible with the structure of general relativity.
Abstract: We examine various well known exact solutions available in the literature to investigate the recent criterion obtained in ref. [20] which should be fulfilled by any static and spherically symmetric solution in the state of hydrostatic equilibrium. It is seen that this criterion is fulfilled only by (i) the regular solutions having a vanishing surface density together with the pressure, and (ii) the singular solutions corresponding to a non-vanishing density at the surface of the configuration . On the other hand, the regular solutions corresponding to a non-vanishing surface density do not fulfill this criterion. Based upon this investigation, we point out that the exterior Schwarzschild solution itself provides necessary conditions for the types of the density distributions to be considered inside the mass, in order to obtain exact solutions or equations of state compatible with the structure of general relativity. The regular solutions with finite centre and non-zero surface densities which do not fulfill the criterion [20], in fact, can not meet the requirement of the `actual mass' set up by exterior Schwarzschild solution. The only regular solution which could be possible in this regard is represented by uniform (homogeneous) density distribution.
The criterion [20] provides a necessary and sufficient condition for any static and spherical configuration (including core-envelope models) to be compatible with the structure of general relativity. Thus, it may find application to construct the appropriate core-envelope models of stellar objects like neutron stars and may be used to test various equations of state for dense nuclear matter and the models of relativistic stellar structures like star clusters.
791 citations
"Conformally recurrent space-times a..." refers background in this paper
...If σj ≡ ∇jσ 6= 0 the motion is called proper, if σ is constant the vector is called homothetic [31]....
[...]
...It is always assumed that ∇jgkl = 0 (Levi Civita connection) and that the space-matter content is described by the stress-energy tensor and related to the Ricci tensor by Einstein’s equations Rkl − R 2 gkl = κTkl, being κ = 8πG c(4) the Einstein gravitational constant (see [9], [30], [31])....
[...]
...An n-dimensional pseudo Riemannian manifold is said to admit an infinitesimal conformal vector field (or a proper conformal motion) ξ if the Lie derivative of the metric gij along ξ satisfies the following condition (see [31] page 564 and [12]): (3) £ξgij ≡ ∇iξj +∇jξi = 2σgij , where σ is a scalar function....
[...]
...Finally, from σCjklm = 0 and σ Rim = λσm a direct calculations brings (see also Hall’s theorem in [20] and [31]) σσRjklm = (λ− R 6 )σkσl from which it is σ[pRk]jlmσ σ = 0 and the Riemannian tensor is algebraically special....
[...]
...When k satisfies condition b) the Weyl tensor is named algebraically special (see [12], [27], [30] and [31])....
[...]
TL;DR: In this paper, the emission of gravitational waves from a finite isolated axially symmetrical material system in otherwise empty space has been investigated by consideration of the metric : Einstein vacuum field equations have been solved by an expansion in negative powers of r which represents radial distance in a well-defined sense.
Abstract: THE emission of gravitational waves from a finite isolated axially symmetrical material system in otherwise empty space has been investigated by consideration of the metric : Einstein vacuum field equations have been solved by an expansion in negative powers of r which represents radial distance in a well-defined sense. In this expansion it has been assumed that only outgoing waves are present. The expression : represents mass in the static case, and forms a suitable generalization of this static concept to the dynamical case.
738 citations