On a type of spacetime
01 Jan 2017-International Journal of Geometric Methods in Modern Physics (World Scientific Publishing Company)-Vol. 14, Iss: 01, pp 1750003
TL;DR: In this article, the authors studied a conharmonically flat spacetime with cyclic parallel Ricci tensor and proved that the energy-momentum tensor is cyclic-parallel and conversely, the integral curves of the vector field U are geodesics.
Abstract: The object of the present paper is to study a spacetime admitting conharmonic curvature tensor and some geometric properties related to this spacetime. It is shown that in a conharmonically flat spacetime with cyclic parallel Ricci tensor, the energy–momentum tensor is cyclic parallel and conversely. Finally, we prove that for a radiative perfect fluid spacetime if the energy–momentum tensor satisfying the Einstein’s equations without cosmological constant is generalized recurrent, then the fluid has vanishing vorticity and the integral curves of the vector field U are geodesics.
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TL;DR: In this paper, the authors consider almost geodesic lines of manifolds with non-symmetric linear connection and prove some existence theorems for special almost-geodesic mappings of the second type between generalized Riemannian spaces as well as between generalized classical and hyperbolic Kahler spaces.
Abstract: We deal with almost geodesic lines of manifolds with non-symmetric linear connection. Also, we consider special almost geodesic mappings of the second type between Eisenhart’s generalized Riemannian spaces as well as between generalized classical (elliptic) and hyperbolic Kahler spaces. These mappings are generalizations of holomorphically projective mappings between generalized classical and hyperbolic Kahler spaces. We prove some existence theorems for special almost geodesic mappings of the second type between generalized Riemannian spaces as well as between generalized classical and hyperbolic Kahler spaces. Finally, we find some invariant geometric objects with respect to these mappings.
11 citations
03 Oct 2019
TL;DR: In this article, the authors define a new type of quarter-symmetric non-metric connection on an $LP$-Sasakian manifold and prove its existence.
Abstract: We define a new type of quarter-symmetric non-metric $\xi$-connection on an $LP$-Sasakian manifold and prove its existence. We provide its application in the general theory of relativity. To validate the existence of the quarter-symmetric non-metric $\xi$-connection on an $LP$-Sasakian manifold, we give a non-trivial example in dimension $4$ and verify our results.
10 citations
Cites background from "On a type of spacetime"
...We cite ([8], [14], [16], [30]) and their references....
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...[8] also studied the properties of the isotropic and homogeneous space-times....
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TL;DR: In this article, the effects of conharmonic flatness are studied and applied to Friedmann-Lemaitre-Robertson-Walker spacetime, and it is shown that imposing too much extra symmetry can cause the problem to become somewhat trivial.
Abstract: To get exact solutions to Einstein’s field equations in general relativity, one has to impose some symmetry requirements. Otherwise, the equations are too difficult to solve. However, sometimes, the imposition of too much extra symmetry can cause the problem to become somewhat trivial. As a typical example to illustrate this, the effects of conharmonic flatness are studied and applied to Friedmann–Lemaitre–Robertson–Walker spacetime. Hence, we need to impose some symmetry to make the problem tractable, but not too much so as to make it too simple.
6 citations
Cites background from "On a type of spacetime"
...[27] showed that, in a conharmonically flat spacetime with cyclic parallel Ricci tensor, the energy–momentum tensor is cyclic parallel and conversely....
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TL;DR: In this paper, it was shown that every type of vacuum solution of Einstein's equations admits a quadratic first integral of the null geodesic equations (conformal Killing tensor of valence 2), which is independent of the metric and of any Killing vectors arising from symmetries.
Abstract: It is shown that every type {22} vacuum solution of Einstein's equations admits a quadratic first integral of the null geodesic equations (conformal Killing tensor of valence 2), which is independent of the metric and of any Killing vectors arising from symmetries. In particular, the charged Kerr solution (with or without cosmological constant) is shown to admit a Killing tensor of valence 2. The Killing tensor, together with the metric and the two Killing vectors, provides a method of explicitly integrating the geodesics of the (charged) Kerr solution, thus shedding some light on a result due to Carter.
564 citations
TL;DR: In this article, it was shown that the existence of a curvature collineation (CC) is a necessary condition for a covariant generator of field conservation laws in the theory of general relativity.
Abstract: A Riemannian space Vn is said to admit a particular symmetry which we call a ``curvature collineation'' (CC) if there exists a vector ξi for which £ξRjkmi=0, where Rjkmi is the Riemann curvature tensor and £ξ denotes the Lie derivative. The investigation of this symmetry property of space‐time is strongly motivated by the all‐important role of the Riemannian curvature tensor in the theory of general relativity. For space‐times with zero Ricci tensor, it follows that the more familiar symmetries such as projective and conformal collineations (including affine collineations, motions, conformal and homothetic motions) are subcases of CC. In a V4 with vanishing scalar curvature R, a covariant conservation law generator is obtained as a consequence of the existence of a CC. This generator is shown to be directly related to a generator obtained by means of a direct construction by Sachs for null electromagnetic radiation fields. For pure null‐gravitational space‐times (implying vanishing Ricci tensor) which admit CC, a similar covariant conservation law generator is shown to exist. In addition it is found that such space‐times admit the more general generator (recently mentioned by Komar for the case of Killing vectors) of the form (−g Tijkmξiξjξk);m=0, involving the Bel‐Robinson tensor Tijkm. Also it is found that the identity of Komar, [−g(ξi;j−ξj;i)];i;j=0, which serves as a covariant generator of field conservation laws in the theory of general relativity appears in a natural manner as an essentially trivial necessary condition for the existence of a CC in space‐time. In addition it is shown that for a particular class of CC,£ξK is proportional to K, where K is the Riemannian curvature defined at a point in terms of two vectors, one of which is the CC vector. It is also shown that a space‐time which admits certain types of CC also admits cubic first integrals for mass particles with geodesic trajectories. Finally, a class of null electromagnetic space‐times is analyzed in detail to obtain the explicit CC vectors which they admit.
259 citations
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209 citations
"On a type of spacetime" refers background in this paper
...But this means that the vector field ρ corresponding to the 1-form E defined by g(X, ρ) = E(X) is a proper concircular vector field [19, 23]....
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01 Jan 1988
126 citations
"On a type of spacetime" refers background in this paper
...In this connection we mention the works of Defever et al. [8], Shaikh et al. [20], Kaigorodov [12], De and Ghosh [5], Chaki and Roy [4], De et al. [7], Guha and Chakraborty [11], Prvanovic [16, 17], Tamássy and Binh [21] and many others....
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...Also if in (1) the 1-form A is replaced by 2A, then the manifold is called a generalized pseudo Ricci symmetric manifold introduced by Chaki and Koley [3]....
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...If in (1) the 1-form A is replaced by 2A, B and D are replaced by A, then the manifold is called a pseudo Ricci symmetric manifold introduced by Chaki [2]....
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TL;DR: In this article, the authors consider the case where the flow lines of a perfect fluid form a time-like shear-free normal congruence and show that all the degenerate fields admit at least a one-parameter group of local isometries with space-like trajectories.
Abstract: Flows of a perfect fluid in which the flow-lines form a time-like shear-free normal congruence are investigated. The space-time is quite severely restricted by this condition on the flow: it must be of Petrov Type I and is either static or degenerate. All the degenerate fields are classified and the field equations solved completely, except in one class where one ordinary differential equation remains to be solved. This class contains the spherically symmetric non-uniform density fields and their analogues with planar or hyperbolic symmetry. The type D fields admit at least a one-parameter group of local isometries with space-like trajectories. All vacuum fields which admit a time-like shear-free normal congruence are shown to be static. Finally, shear-free perfect fluid flows which possess spherical or a related symmetry are considered, and all uniform density solutions and a few non-uniform density solutions are found. The exact solutions are tabulated in section 7.
87 citations
"On a type of spacetime" refers background in this paper
...Again Barnes [1] has proved that if a perfect fluid spacetime is shear free, vorticity free and the velocity vector field is hypersurface orthogonal and the energy density is constant over a hypersurface orthogonal to the velocity vector field, then the possible local cosmological structure of the spacetime are of Petrov type I, D or O....
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