# A spacetime with pseudo-projective curvature tensor

TL;DR: In this paper, it was shown that a pseudo-projectively flat spacetime with vanishing pseudoprojective curvature tensor obeys Einstein's field equation without cosmological constant is an Euclidean space.

Abstract: The object of the present paper is to study spacetimes admitting pseudo-projective curvature tensor. At first we prove that a pseudo-projectively flat spacetime is Einstein and hence it is of constant curvature and the energy momentum tensor of such a spacetime satisfying Einstein’s field equation with cosmological constant is covariant constant. Next, we prove that if the perfect fluid spacetime with vanishing pseudo-projective curvature tensor obeys Einstein’s field equation without cosmological constant, then the spacetime has constant energy density and isotropic pressure, and the perfect fluid always behaves as a cosmological constant and also such a spacetime is infinitesimally spatially isotropic relative to the unit timelike vector field
U. Moreover, it is shown that a pseudo-projectively flat spacetime satisfying Einstein’s equation without cosmological constant for a purely electromagnetic distribution is an Euclidean space. We also prove that under certain conditions a perfect fluid spacetime with divergence-free pseudo-projective curvature is a Robertson-Walker spacetime and the possible local cosmological structure of such a spacetime is of type I, D or O. We also study dust-like fluid spacetime with vanishing pseudo-projective curvature tensor.

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01 Jan 2015

TL;DR: (2 < p < 4) [200].

Abstract: (2 < p < 4) [200]. (Uq(∫u(1, 1)), oq1/2(2n)) [92]. 1 [273, 79, 304, 119]. 1 + 1 [252]. 2 [352, 318, 226, 40, 233, 157, 299, 60]. 2× 2 [185]. 3 [456, 363, 58, 18, 351]. ∗ [238]. 2 [277]. 3 [350]. p [282]. B−L [427]. α [216, 483]. α− z [322]. N = 2 [507]. D [222]. ẍ+ f(x)ẋ + g(x) = 0 [112, 111, 8, 5, 6]. Eτ,ηgl3 [148]. g [300]. κ [244]. L [205, 117]. L [164]. L∞ [368]. M [539]. P [27]. R [147]. Z2 [565]. Z n 2 [131]. Z2 × Z2 [25]. D(X) [166]. S(N) [110]. ∫l2 [154]. SU(2) [210]. N [196, 242]. O [386]. osp(1|2) [565]. p [113, 468]. p(x) [17]. q [437, 220, 92, 183]. R, d = 1, 2, 3 [279]. SDiff(S) [32]. σ [526]. SLq(2) [185]. SU(N) [490]. τ [440]. U(1) N [507]. Uq(sl 2) [185]. φ 2k [283]. φ [553]. φ4 [365]. ∨ [466]. VOA[M4] [33]. Z [550].

35 citations

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TL;DR: In this paper, a new type of curvature tensor called H-curvature tensors of type (1, 3) was introduced, which is a linear combination of conformal and projective curvatures.

Abstract: In this paper, we introduce a new type of curvature tensor named H-curvature
tensor of type (1, 3) which is a linear combination of conformal and
projective curvature tensors. First we deduce some basic geometric
properties of H-curvature tensor. It is shown that a H-flat Lorentzian
manifold is an almost product manifold. Then we study pseudo H-symmetric
manifolds (PHS)n (n > 3) which recovers some known structures on Lorentzian
manifolds. Also, we provide several interesting results. Among others, we
prove that if an Einstein (PHS)n is a pseudosymmetric (PS)n, then the scalar
curvature of the manifold vanishes and conversely. Moreover, we deal with
pseudo H-symmetric perfect fluid spacetimes and obtain several interesting
results. Also, we present some results of the spacetime satisfying
divergence free H-curvature tensor. Finally, we construct a non-trivial
Lorentzian metric of (PHS)4.

7 citations

### Cites background from "A spacetime with pseudo-projective ..."

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TL;DR: In this article, a necessary and sufficient condition for a spacetime with pseudo symmetric energy-momentum tensor to be a pseudo Ricci symmetric spacetime was given, and several interesting results were obtained.

Abstract: The object of the present paper is to characterize spacetimes with different types of energy–momentum tensor. At first we consider spacetimes with pseudo symmetric energy–momentum tensor T . We obtain a necessary and sufficient condition for a spacetime with pseudo symmetric energy–momentum tensor to be a pseudo Ricci symmetric spacetime. Next we consider the spacetimes with Codazzi type of energy–momentum tensor and several interesting results are pointed out. Moreover, some results related to perfect fluid spacetimes with different forms of energy–momentum tensors have been obtained. We study spacetimes with quadratic Killing energy–momentum tensor T and show that a GRW spacetime with quadratic Killing energy–momentum tensor is an Einstein space. Finally, we have considered general relativistic spacetimes with semisymmetric energy–momentum tensor and obtained some important results.

6 citations

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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.

5 citations

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TL;DR: In this paper, the authors studied weak Ricci symmetric spacetimes and showed that the local cosmological structure of a weakly Ricci-symmetric perfect fluid spacetime can be identified as Petrov type I, D or O. They also showed the nonexistence of radiation era in such a spacetime.

Abstract: The objective of the present paper is to study weakly Ricci symmetric spacetimes. Among others, we prove that a weakly Ricci symmetric spacetime obeying Einstein’s field equation without cosmological constant represents stiff matter. Moreover, it is shown that the local cosmological structure of a weakly Ricci symmetric perfect fluid spacetime can be identified as Petrov type I, D or O. Next, we prove that a dust and dark fluid weakly Ricci symmetric spacetime satisfying Einstein’s field equation without cosmological constant is vacuum. Finally, we show the non-existence of radiation era in such a spacetime.

2 citations

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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.

242 citations

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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.

85 citations

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TL;DR: WeakWeakly Z-symmetric (WZS) as mentioned in this paper is a Riemannian manifold that includes weakly-, pseudo-and pseudo projective Ricci symmetric manifolds.

Abstract: We introduce a new kind of Riemannian manifold that includes weakly-, pseudo- and pseudo projective Ricci symmetric manifolds. The manifold is defined through a generalization of the so called Z tensor; it is named weakly Z-symmetric and is denoted by (WZS) n .I f theZ tensor is singular we give condi- tions for the existence of a proper concircular vector. For non singular Z tensors, we study the closedness property of the associated covectors and give sufficient conditions for the existence of a proper concircular vector in the conformally har- monic case, and the general form of the Ricci tensor. For conformally flat (WZS) n manifolds, we derive the local form of the metric tensor.

57 citations

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TL;DR: In this article, the authors consider time reversal transformations to obtain twofold orthogonal splittings of any tensor on a Lorentzian space of arbitrary dimension n. They study the cases where one of these parts vanishes in detail, i.e., purely electric or magnetic spacetimes.

Abstract: We consider time reversal transformations to obtain twofold orthogonal splittings of any tensor on a Lorentzian space of arbitrary dimension n. Applied to the Weyl tensor of a spacetime, this leads to a definition of its electric and magnetic parts relative to an observer (defined by a unit timelike vector field u), in any dimension. We study the cases where one of these parts vanishes in detail, i.e., purely electric (PE) or magnetic (PM) spacetimes. We generalize several results from four to higher dimensions and discuss new features of higher dimensions. For instance, we prove that the only permitted Weyl types are G, Ii and D, and discuss the possible relation of u with the Weyl aligned null directions (WANDs); we provide invariant conditions that characterize PE/PM spacetimes, such as Bel–Debever-like criteria, or constraints on scalar invariants, and connect the PE/PM parts to the kinematic quantities of u; we present conditions under which direct product spacetimes (and certain warps) are PE/PM, which enables us to construct explicit examples. In particular, it is also shown that all static spacetimes are necessarily PE, while stationary spacetimes (such as spinning black holes) are in general neither PE nor PM. Whereas ample classes of PE spacetimes exist, PM solutions are elusive; specifically, we prove that PM Einstein spacetimes of type D do not exist, in any dimension. Finally, we derive corresponding results for the electric/magnetic parts of the Riemann tensor, which is useful when considering spacetimes with matter fields, and moreover leads to first examples of PM spacetimes in higher dimensions. We also note in passing that PE/PM Weyl (or Riemann) tensors provide examples of minimal tensors, and we make the connection hereof with the recently proved alignment theorem (Hervik 2011 Class. Quantum Grav. 28 215009). This in turn sheds new light on the classification of the Weyl tensors based on null alignment, providing a further invariant characterization that distinguishes the (minimal) types G/I/D from the (non-minimal) types II/III/N.

55 citations

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TL;DR: In this article, the Bianchi identity of the new "Codazzi deviation tensor" is shown to be equivalent to a Bianchi tensor on the Riemann tensor.

Abstract: Derdzinski and Shen’s theorem on the restrictions posed by a Codazzi tensor on the Riemann tensor holds more generally when a Riemann-compatible tensor exists. Several properties are shown to remain valid in this broader setting. Riemann compatibility is equivalent to the Bianchi identity of the new “Codazzi deviation tensor”, with a geometric significance. The general properties are studied, with their implications on Pontryagin forms. Examples are given of manifolds with Riemann-compatible tensors, in particular those generated by geodesic mappings. Compatibility is extended to generalized curvature tensors, with an application to Weyl’s tensor and general relativity.

48 citations