# A Study On Some Geometric and Physical Properties of Hyper-Generalised Quasi-Einstein Spacetime

Kaushik Chattopadhyay, Arindam Bhattacharyya, Dipankar Debnath

aa r X i v : . [ g r- q c ] J a n A Study On Some Geometric and Physical Properties ofHyper-Generalised Quasi-Einstein Spacetime

Kaushik Chattopadhyay ∗ , Arindam Bhattacharyya and Dipankar Debnath Department of Mathematics,Jadavpur University, Kolkata-700032, IndiaE-mail:[email protected] Department of Mathematics,Jadavpur University, Kolkata-700032, IndiaE-mail:[email protected] Department of Mathematics,Bamanpukur High School(H.S), Nabadwip, IndiaE-mail:[email protected]

Abstract

In the present paper we discuss about a set of geometric and physicalproperties of hyper-generalised quasi-Einstein spacetime. At the beginning wediscuss about pseudosymmetry over a hyper-generalised quasi-Einstein space-time. Here we discuss about W -Ricci pseudosymmetry, Z -Ricci pseudosym-metry, Ricci pseudosymmetry and projective pseudosymmetry over a hyper-generalised quasi-Einstein spacetime. Later on we take over Ricci symmet-ric hyper-generalised quasi-Einstein spacetime and derive a set of importantgeometric and physical theorems over it. Moving further we consider somephysical applications of the hyper-generalised quasi-Einstein spacetime. Lastlywe prove the existence of a hyper-generalised quasi-Einstein spacetime by con-structing a non-trivial example. M.S.C.2010:

Keywords: W -curvature tensor, Z tensor, projective curvature tensor, Rieman-nian curvature tensor, hyper generalised quasi-Einstein spacetime, Einstein equa-tion, heat ﬂux, stress tensor. The General theory of Relativity is unarguably the most beautiful theory theWorld of Physics has ever produced. It is the most powerful result of the humanintellect. This is an extremely important theory to study the nature of this uni-verse, cosmology and gravity. Three most important things that Modern Scientists/Mathematical Physicists can learn from special to general relativity are as follows:(i) The laws of Physics should be the same in every inertial reference frame, i. ∗ This is the corresponding author. e., the abandonment of the privileged states of inertial frame of reference.(ii) The acceptance of the dynamical role of the metric g , i.e., the study of non-linear behaviour of nature.and(iii) The spacetime has to be considered as a class of semi-Riemannian geometry.The semi-Riemannian geometry has become more and more relevant and signiﬁcantin dealing with the nature of this universe with every passing day.The theory of general relativity is mainly studied on a semi-Riemannian manifoldwhich sometimes is not an Einstein spacetime. Thus it was always necessary toexpand the concept of Einstein manifold to quasi-Einstein, then generalised quasi-Einstein, mixed generalised quasi-Einstein and lastly to hyper-generalised quasi-Einstein manifold. We demonstrate the introduction to this procedure as follows:An Einstein manifold is a Riemannian or pseudo-Riemannian manifold whoseRicci tensor S of type (0 ,

2) is non-zero and proportional to the metric tensor. Ein-stein manifolds form a natural subclass of various classes of Riemannian or semi-Riemannian manifolds by a curvature condition imposed on their Ricci tensor [4].Also in Riemannian geometry as well as in general relativity theory, the Einsteinmanifold plays a very important role.M. C. Chaki and R. K. Maity had given the notion of quasi Einstein mani-fold [18] in 2000. A non ﬂat n -dimensional Riemannian manifold ( M n , g ), n ( > S of type (0 , S ( X, Y ) = αg ( X, Y ) + βA ( X ) A ( Y ) , (1.1)where for all vector ﬁelds X , g ( X, ξ ) = A ( X ) , g ( ξ , ξ ) = 1 . (1.2)That is, A being the associated 1-form, ξ is generally known as the generator ofthe manifold. α and β are associated nonzero scalar functions. This manifold is de-noted by ( QE ) n . Clearly, for β = 0, this manifold reduces an Einstein manifold. Wecan note that Robertson-Walker spacetimes are quasi-Einstein spacetimes. In therecent papers [2], [19], the application of quasi-Einstein spacetime and generalisedquasi-Einstein spacetime in general relativity have been studied. Many more workshave been done in the spacetime of general relativity [3], [13], [14], [21], [22], [23],[24], [25], [26].Then M. C. Chaki initiated the notion of generalized quasi-Einstein manifold[17] in 2001. A Riemannian manifold of dimension n ( >

2) is said to be generalizedquasi Einstein manifold if its Ricci tensor S of type (0 ,

2) is not identically zero andsatisﬁes the following condition S ( X, Y ) = αg ( X, Y ) + βA ( X ) A ( Y ) + γ [ A ( X ) B ( Y ) + A ( Y ) B ( X )] , (1.3)where α , β and γ are real valued, nonzero scalar functions on ( M n , g ), A and B arecalled two non zero 1-forms such that g ( X, ξ ) = A ( X ) , g ( X, ξ ) = B ( X ) , g ( ξ , ξ ) = 0 , g ( ξ , ξ ) = 1 , g ( ξ , ξ ) = 1 . (1.4)Here ξ and ξ are two unit vector ﬁelds which are orthogonal to each other. α , β and γ are called associated scalars, A and B are called associated 1-forms. ξ and ξ are two generators of the manifold. This manifold is denoted by ( GQE ) n . Clearly,for γ = 0, then it takes the form of a quasi-Einstein manifold and for β = γ = 0, ittakes the form of an Einstein manifold.The notion of hyper-generalized quasi-Einstein manifold has been introducedby A. A. Shaikh, C. ¨Ozg¨ur and A. Patra [1] in 2011. According to them, a Rie-mannian manifold ( M n , g ), ( n >

2) is said to be a hyper-generalized quasi-Einsteinmanifold if its Ricci tensor S of type (0 ,

2) is non-zero and satisﬁes the followingcondition S ( X, Y ) = αg ( X, Y ) + βA ( X ) A ( Y ) + γ [ A ( X ) B ( Y ) + A ( Y ) B ( X )]+ δ [ A ( X ) D ( Y ) + A ( Y ) D ( X )] , (1.5)for all X, Y, Z ∈ χ ( M ). Here α , β , γ , δ are non-zero scalar functions on ( M n , g ). A , B , D are non-zero 1-forms such that g ( X, ξ ) = A ( X ) , g ( X, ξ ) = B ( X ) , g ( X, ξ ) = D ( X ) , (1.6)where ξ , ξ , ξ are mutually orthogonal unit vector ﬁelds. i. e., g ( ξ , ξ ) = g ( ξ , ξ ) = g ( ξ , ξ ) = 0; g ( ξ , ξ ) = g ( ξ , ξ ) = g ( ξ , ξ ) = 1 . (1.7)An n -dimensional hyper-generalized quasi-Einstein manifold is generally denotedas ( HGQE ) n . Shaikh, ¨Ozg¨ur and Patra in [1] studied on hyper-generalized quasi-Einstein manifolds with some geometric properties of it. G¨uler and Demirbaˇg [20]dealt with some Ricci conditions on hyper-generalized quasi-Einstein manifolds, D.Debnath [9] proved few theorems about the properties of the hyper-generalized quasi-Einstein manifolds.The concept of perfect ﬂuid spacetime arose while discussing the structure ofthis universe. Perfect ﬂuids are often used in the general relativity to model theidealised distribution of matter, such as the interior of a star or isotropic pressure.In general relativity the matter content of the spacetime is described by the energy-momentum tensor. The matter content is assumed to be a ﬂuid having density andpressure and possessing dynamical and kinematical quantities like velocity, acceler-ation, vorticity, shear and expansion. The energy-momentum tensor T of a perfectﬂuid spacetime is given by the following equation [11] , [16] T ( X, Y ) = ( σ + p ) A ( X ) A ( Y ) + pg ( X, Y ) . (1.8)Here g ( X, ξ ) = A ( X ) , A ( ξ ) = −

1, for any

X, Y . p and σ are called the isotropicpressure and the energy density respectively. ξ being the unit timelike velocityvector ﬁeld.The Einstein ﬁeld equation [6] is given by S ( X, Y ) − r g ( X, Y ) + λg ( X, Y ) = kT ( X, Y ); ∀ X, Y ∈ T M, (1.9)here r being the scalar curvature, S being the Ricci tensor of type (0 , k and λ arethe gravitational constant and cosmological constant respectively. From Einstein’sﬁeld equation it follows that energy momentum tensor is a symmetric (0 ,

2) typetensor of divergence zero.In the year of 2012, Mantica and Molinari[7] deﬁned a new generalized symmet-ric (0 ,

2) tensor called Z tensor. According to them it is given as, Z ( X, Y ) = S ( X, Y ) + φg ( X, Y ) , (1.10)where φ is an arbitrary scalar function. A set of properties of Z tensor have beenstudied in the papers [7] and [8].Like the Z curvature tensor projective curvature tensor also plays a very signiﬁcantrole in studying diﬀerent properties of semi-Riemannian geometry. Let M n ( n ≥ P ( X, Y ) Z = R ( X, Y ) Z − n − { S ( Y, Z ) X − S ( X, Z ) Y } . (1.11)Hyper-generalized quasi-Einstein manifolds is considered as the base space ofgeneral relativistic viscous ﬂuid spacetime, which inspired us to take a look on somegeometric properties of the ( HGQE ) spacetime under certain conditions which westudy on sections 2, 3, 4 and 5. Then we discuss about Ricci symmetric hyper-generalized quasi-Einstein spacetimes in section 6. In section 7 we derive a resultabout the energy-momentum tensor on hyper-generalized quasi-Einstein spacetimeof constant curvature with cyclic parallel Ricci tensor. Also the spacetime haswide applications in general relativistic viscous ﬂuid spacetime admitting heat ﬂuxand stress, which motivated us to discuss about some physical applications of an( HGQE ) spacetime in section 8. Finally in section 9 we construct a non-trivialexample of an ( HGQE ) spacetime to prove the existence of such spacetime. W -Ricci pseudosymmetric ( H GQE ) spacetime The W -curvature tensor was introduced by G. P. Pokhariyal and R. S. Mishra[12] in 1970 and they studied some properties of it. A W -curvature tensor on amanifold ( M n , g ), n ( >

3) is deﬁned by W ( X, Y ) Z = R ( X, Y ) Z − n − g ( Y, Z ) QX − g ( X, Z ) QY ] . (2.1)Here r being the curvature tensor and Q is the Ricci operator deﬁned by g ( QX, Y ) = S ( X, Y ) , ∀ X, Y .For an (

HGQE ) quasi-Einstein spacetime the (0 , W curvature tensor takesthe following form, W ( X, Y, Z, W ) = R ( X, Y, Z, W ) −

13 [ g ( Y, Z ) S ( X, W ) − g ( X, Z ) S ( Y, W )] . (2.2)Firstly, we take a hyper generalized quasi-Einstein spacetime satisfying the condition W .S = F S Q ( g, S ). Here F S being a certain function on the set U S = { x ∈ M : S = rn g at x } and Q ( g, S ) being the Tachibana tensor working on the metric tensor andthe Ricci tensor. This spacetime is called W -Ricci pseudosymmetric ( HGQE ) .Now for all X, Y, Z ∈ χ ( M ); S ( W ( X, Y ) Z, W ) + S ( Z, W ( X, Y ) W )= F S [ g ( Y, Z ) S ( X, W ) − g ( X, Z ) S ( Y, W )+ g ( Y, W ) S ( Z, X ) − g ( X, W ) S ( Y, Z )] . (2.3)From the equation (1.5) from the equation (2.3) we get, αg ( W ( X, Y ) Z, W ) + βA ( W ( X, Y ) Z ) A ( W )+ γ [ A ( W ( X, Y ) Z ) B ( W ) + B ( W ( X, Y ) Z ) A ( W )]+ δ [ A ( W ( X, Y ) Z ) D ( W ) + D ( W ( X, Y ) Z ) A ( W )]+ αg ( W ( X, Y ) W, Z ) + βA ( W ( X, Y ) W ) A ( Z )+ γ [ A ( W ( X, Y ) W ) B ( Z ) + B ( W ( X, Y ) W ) A ( Z )]+ δ [ A ( W ( X, Y ) W ) D ( Z ) + D ( W ( X, Y ) W ) A ( Z )]= F S [ g ( Y, Z ) S ( X, W ) − g ( X, Z ) S ( Y, W )+ g ( Y, W ) S ( Z, X ) − g ( X, W ) S ( Y, Z )] . (2.4)Contracting the equation over X and W and putting Z = ξ we get, α [ 43 { ( α − β ) A ( Y ) − γB ( Y ) − δD ( Y ) } − r A ( Y )] −

13 [ − γB ( Y ) − δD ( Y )]+ γ A ( Y ) + γR ( ξ , Y, ξ , ξ ) − αγ B ( Y ) + δ A ( Y )+ δR ( ξ , Y, ξ , ξ ) − δα D ( Y )= F S [ rA ( Y ) − { ( α − β ) A ( Y ) − γB ( Y ) − δD ( Y ) } ] . (2.5)Putting X = ξ , Z = ξ , W = ξ in the equation (2.4) we get, δR ( ξ , Y, ξ , ξ ) + γR ( ξ , Y, ξ , ξ ) − α B ( Y )( − δ ) − α D ( Y )( − γ ) + δ B ( Y )( α − β ) + γ D ( Y )( α − β )= F S [ B ( Y )( − δ ) + D ( Y )( − γ )] . (2.6)combining the equations (2.5) and (2.6) we get, α [ 43 { ( α − β ) A ( Y ) − γB ( Y ) − δD ( Y ) } − r A ( Y )] −

13 [ − γB ( Y ) − δD ( Y )]+ γ A ( Y ) − αγ B ( Y ) + δ A ( Y ) − δα D ( Y ) − F S [ B ( Y )( − δ ) + D ( Y )( − γ )] − α B ( Y )( − δ ) − α D ( Y )( − γ ) + δ B ( Y )( α − β ) + γ D ( Y )( α − β )= F S [ rA ( Y ) − { ( α − β ) A ( Y ) − γB ( Y ) − δD ( Y ) } ] . (2.7)setting γ = δ yields, α [ 43 { ( α − β ) A ( Y ) − γB ( Y ) − γD ( Y ) } − r A ( Y )] −

13 [ − γB ( Y ) − γD ( Y )]+ γ A ( Y ) − αγ B ( Y ) + γ A ( Y ) − γα D ( Y ) − F S [ B ( Y )( − γ ) + D ( Y )( − γ )] − α B ( Y )( − γ ) − α D ( Y )( − γ ) + γ B ( Y )( α − β ) + γ D ( Y )( α − β )= F S [ rA ( Y ) − { ( α − β ) A ( Y ) − γB ( Y ) − γD ( Y ) } ] . (2.8)Putting Y = ξ in the above equation we get, − βF S = αβ − γ . (2.9)Again putting Y = ξ in the equation (2.8) we get, γ ( α + 3 F S ) = 0 , (2.10)which gives either γ = 0 or F S = − α . Now, if γ = 0 then since γ = δ thus γ = δ = 0. Hence the spacetime becomes a quasi-Einstein spacetime. On the otherhand if F S = − α then from the equation (2.9) we get γ = 0. Again since γ = δ thus γ = δ = 0. Thus in both the cases the manifold is reduced to a quasi-Einsteinspacetime. Hence we conclude the following theorem as: Theorem 2.1: A W -Ricci pseudosymmetric ( HGQE ) spacetime is a ( QE ) space-time if γ = δ . ( H GQE ) spacetime A semi-Riemannian manifold ( M n , g ) , n ≥

3, is called Z-Ricci pseudosymmetric iﬀthe following relation holds: Z · K = F K P ( g, K ) , (3.1)on the set U K = { x ∈ M : P ( g, K ) = 0 at x } , where K is the Ricci operator deﬁnedby S ( X, Y ) = g ( KX, Y ) and F K is a smooth function on U K . The operator P isdeﬁned by the following way: P ( g, K )( W ; X, Y ) = K (( X ∧ g Y ) W ) (3.2)for all vector ﬁelds X, Y, W .Now if the spacetime is a Z-Ricci pseudosymmetric then from the equation (3.1)we get, ( Z ( X, Y ) · K ) W = F K P ( g, K )( W ; X, Y ) , (3.3)which takes the following form, Z ( Y, KW ) X − Z ( X, KW ) Y − Z ( Y, W ) KX − Z ( X, W ) KY = F K { g ( Y, W ) KX − g ( X, W ) KY } . (3.4)With the help of the equation (1 .

5) we demonstrate the Ricci operator by the fol-lowing equation: KX = αX + βA ( X ) ξ + γ [ A ( X ) ξ + B ( X ) ξ ] + δ [ A ( X ) ξ + D ( X ) ξ ] . (3.5)Applying the equation (3.5) in equation (3.4) we get, Z ( Y, αW ) X + βA ( W ) Z ( Y, ξ ) X + γA ( W ) Z ( Y, ξ ) X + γB ( W ) Z ( Y, ξ ) X + δA ( W ) Z ( Y, ξ ) X + δD ( W ) Z ( Y, ξ ) X − [ Z ( X, αW ) Y + βA ( W ) Z ( X, ξ ) Y + γA ( W ) Z ( X, ξ ) Y + γB ( W ) Z ( X, ξ ) Y + δA ( W ) Z ( X, ξ ) Y + δD ( W ) Z ( X, ξ ) Y ]= { F K g ( Y, W ) + Z ( Y, W ) } KX − { F K g ( X, W ) − Z ( X, W ) } KY. (3.6)With the help of the equation (1.10) this further implies, αZ ( Y, W ) X + βA ( W ) { ( α − β + φ ) A ( Y ) − γB ( Y ) − δD ( Y ) } X + γA ( W ) { ( α + φ ) B ( Y ) + γA ( Y ) } X + γB ( W ) { ( α − β + φ ) A ( Y ) − γB ( Y ) − δD ( Y ) } X + δA ( W ) { ( α + φ ) D ( Y ) + γA ( Y ) } X + δD ( W ) { ( α − β + φ ) A ( Y ) − γB ( Y ) − δD ( Y ) } X − αZ ( X, W ) Y − βA ( W ) { ( α − β + φ ) A ( X ) − γB ( X ) − δD ( X ) } Y − γA ( W ) { ( α + φ ) B ( X ) + γA ( X ) } Y − γB ( W ) { ( α − β + φ ) A ( X ) − γB ( X ) − δD ( X ) } Y − δA ( W ) { ( α + φ ) D ( X ) + γA ( X ) } Y − δD ( W ) { ( α − β + φ ) A ( X ) − γB ( X ) − δD ( X ) } Y = { F K g ( Y, W ) + S ( Y, W ) + φg ( Y, W ) } KX −{ F K g ( X, W ) − S ( X, W ) − φg ( X, W ) } KY. (3.7)Putting X = ξ , Y = ξ equation (3.7) yields, α { ( α + φ ) B ( W ) + γA ( W ) } ξ + βA ( W ) {− γ } ξ + γA ( W ) { α + φ } ξ + γB ( W ) {− γ } ξ + δD ( W ) {− γ } ξ − α { ( α − β + φ ) A ( W ) − γB ( W ) − δD ( W ) } ξ + βA ( W ) { α − β + φ ) } ξ + γ A ( W ) ξ + γB ( W ) { α − β + φ ) } ξ + δ A ( W ) ξ δD ( W ) { α − β + φ ) } ξ = { F K B ( W ) + φB ( W ) + αB ( W ) + γA ( W ) } Kξ −{ F K A ( W ) − φA ( W ) − ( α − β ) A ( W ) + γB ( W ) + δD ( W ) } Kξ . (3.8)Taking inner product with ξ to both the sides of the equation (3.8) we get, A ( W ) { αγ + φγ } + B ( W ) {− γ − F K ( α − β ) + φβ + αβ } − D ( W ) { δγ } = 0 . (3.9)Putting W = ξ in equation (3.9) we get, − γδ = 0 . (3.10)That means at least one of γ or δ must be zero. Which means the manifold isreduced to a generalized quasi-Einstein spacetime. This allows us to derive the nexttheorem as: Theorem 3.1:

A Z-Ricci pseudosymmetric ( HGQE ) spacetime is a ( GQE ) spacetime. ( H GQE ) spacetime A semi-Riemannian manifold M n ( n ≥

3) is called Ricci-pseudosymmetric if thefollowing relation ( R ( X, Y ) · S )( Z, W ) = F S Q ( g, S ) (4.1)holds on U S = { x ∈ M : S = rn at x } and L S is a function on U S . From the equation(4.1) we get, S ( R ( X, Y ) Z, W ) + S ( Z, R ( X, Y ) W )= F S [ g ( Y, Z ) S ( X, W ) − g ( X, Z ) S ( Y, W )+ g ( Y, W ) S ( Z, X ) − g ( X, W ) S ( Y, Z )] . (4.2)Using the equation (1.5) we demonstrate the equation (4.2) as follows: β [ A ( R ( X, Y ) Z ) A ( W ) + A ( Z ) A ( R ( X, Y ) W )]+ γ [ A ( R ( X, Y ) Z ) B ( W ) + A ( W ) B ( R ( X, Y ) Z ) + A ( Z ) B ( R ( X, Y ) W )+ A ( R ( X, Y ) W ) B ( Z )] + δ [ A ( R ( X, Y ) Z ) D ( W ) + A ( W ) D ( R ( X, Y ) Z )+ A ( Z ) D ( R ( X, Y ) W ) + A ( R ( X, Y ) W ) D ( Z )]= F S [ β { g ( Y, Z ) A ( X ) A ( W ) − g ( X, Z ) A ( Y ) A ( W ) + g ( Y, Z ) A ( Z ) A ( X ) − g ( X, W ) A ( Y ) A ( Z ) } + γ { g ( Y, Z )[ A ( X ) B ( W ) + A ( W ) B ( X )] − g ( X, Z )[ A ( Y ) B ( W ) + A ( W ) B ( Y )] + g ( Y, W )[ A ( X ) B ( Z ) + A ( Z ) B ( X )] − g ( X, W )[ A ( Y ) B ( Z ) + A ( Z ) B ( Y )] } + δ { g ( Y, Z )[ A ( X ) D ( W ) + A ( W ) D ( X )] − g ( X, Z )[ A ( Y ) D ( W ) + A ( W ) D ( Y )] + g ( Y, Z )[ A ( X ) D ( Z ) + A ( Z ) D ( X )] − g ( X, W )[ A ( Y ) D ( Z ) + A ( Z ) D ( Y )] } ] . (4.3)Contracting equation (4.3) over X and W we get, β [ A ( R ( ξ , Y ) Z ) − A ( Z ) S ( Y, ξ )] + γ [ A ( R ( ξ , Y ) Z )+ B ( R ( ξ , Y ) Z ) − A ( Z ) S ( Y, ξ ) − B ( Z ) S ( Y, ξ )]+ δ [ A ( R ( X, Y ) Z ) D ( W ) + D ( R ( ξ , Y ) Z ) − A ( Z ) S ( Y, ξ ) − S ( Y, ξ ) D ( Z )]= F S { β [ − g ( Y, Z ) − A ( Y ) A ( Z )] − γ [ A ( Y ) B ( Z ) A ( Z ) B ( Y )] − δ [ A ( Y ) D ( Z ) + A ( Z ) D ( Y )] } . (4.4)Setting Z = ξ in equation (4 .

4) we get, β ( Y, ξ ) + γ [ R ( ξ , Y, ξ , ξ ) + S ( Y, ξ )]+ δ [ R ( ξ , Y, ξ , ξ ) + S ( Y, ξ )]= F S [3 βA ( Y ) + 4 γB ( Y ) + 4 δD ( Y )] . (4.5)Now, putting Z = ξ , W = ξ in the equation (4.3) we have, − γR ( ξ , Y, ξ , ξ ) − δR ( ξ , Y, ξ , ξ )= F S { γ [ D ( Y ) A ( X ) − D ( X ) A ( Y )] + δ [ A ( X ) B ( Y ) − A ( Y ) B ( X )] } . (4.6)If γ = δ then from the equations (4.5) and (4.6) we conclude,[ βS ( Y, ξ ) + γS ( Y, ξ ) + γS ( Y, ξ )] − F S [3 βA ( Y ) + 4 γB ( Y ) + 4 γD ( Y )]= F S { γ [ D ( Y ) A ( X ) − D ( X ) A ( Y )] + γ [ A ( X ) B ( Y ) − A ( Y ) B ( X )] } . (4.7)0Putting X = ξ we get, A ( Y )[ αβ − β + 2 γ ] + B ( Y )[ − βγ + γα ] + D ( Y )[ − βγ + γα ] − F S [3 βA ( Y ) + 4 γB ( Y ) + 4 γD ( Y )] = − F S γ [ D ( Y ) + B ( Y )] . (4.8)Putting Y = ξ in equation (4.8) we get, γ [( α − β ) − F S ] = 0 . (4.9)Again putting Y = ξ in equation (4.8) we get, F S = αβ − β + 2 γ β . (4.10)From the equation (4.9) we have either γ = 0 or F S = α − β . Now, if γ = 0 then since γ = δ , thus γ = δ = 0, implying the manifold reduces to a quasi-Einstein spacetime.Again if F S = α − β then from the equation (4.10) we get γ = 0 and hence γ = δ = 0,which again implies the manifold is reduced to a quasi-Einstein spacetime. Thisallows us to deduce the following theorem: Theorem 4.1:

A Ricci pseudosymmetric ( HGQE ) spacetime is a ( QE ) spacetimeif γ = δ . ( H GQE ) space-time From the equation (1 .

11) we see that for an (

HGQE ) quasi-Einstein spacetime the(0 ,

4) projective curvature tensor takes the following form, P ( X, Y, Z, W ) = R ( X, Y, Z, W ) −

13 [ S ( Y, Z ) g ( X, W ) − S ( X, Z ) g ( Y, W )] . (5.1)A semi-Riemannian manifold M n ( n ≥

3) is called projectively pseudosymmetric ifthe following relation ( P ( X, Y ) · S )( Z, W ) = F S Q ( g, S ) (5.2)holds on U S = { x ∈ M : S = rn at x } and L S is a function on U S . From the equation(5.2) we get, S ( P ( X, Y ) Z, W ) + S ( Z, P ( X, Y ) W )= F S [ g ( Y, Z ) S ( X, W ) − g ( X, Z ) S ( Y, W )+ g ( Y, W ) S ( Z, X ) − g ( X, W ) S ( Y, Z )] . (5.3)1From the equation (1.5) from the equation (5.3) we get, αg ( P ( X, Y ) Z, W ) + βA ( P ( X, Y ) Z ) A ( W )+ γ [ A ( P ( X, Y ) Z ) B ( W ) + B ( P ( X, Y ) Z ) A ( W )]+ δ [ A ( P ( X, Y ) Z ) D ( W ) + D ( P ( X, Y ) Z ) A ( W )]+ αg ( P ( X, Y ) W, Z ) + βA ( P ( X, Y ) W ) A ( Z )+ γ [ A ( P ( X, Y ) W ) B ( Z ) + B ( P ( X, Y ) W ) A ( Z )]+ δ [ A ( P ( X, Y ) W ) D ( Z ) + D ( P ( X, Y ) W ) A ( Z )]= F S [ g ( Y, Z ) S ( X, W ) − g ( X, Z ) S ( Y, W )+ g ( Y, W ) S ( Z, X ) − g ( X, W ) S ( Y, Z )] . (5.4)Putting X = ξ , Z = ξ , W = ξ in the equation (5.4) we get, − δR ( ξ , Y, ξ , ξ ) − γR ( ξ , Y, ξ , ξ )= − αγ D ( Y ) − αδ B ( Y ) + γ αD ( Y ) + δA ( Y ))+ δ αB ( Y ) + γA ( Y )) − γ A ( Y ) − δ A ( Y )+ F S { γD ( Y ) + δB ( Y ) } . (5.5)Contracting equation (5.4) over X, W and putting Z = ξ we get, − βγ B ( Y ) − βδ D ( Y ) − γ A ( Y )+ 4 δ αD ( Y ) + δA ( Y )] + γ ( β − α ) B ( Y ) − δ A ( Y ) − δr D ( Y ) + δ ( β − α ) D ( Y ) − α α − β ) A ( Y ) − γB ( Y ) − δD ( Y )] + αr A ( Y )+ 4 β α − β ) A ( Y ) − γB ( Y ) − δD ( Y )] − βr A ( Y )+ 4 γ αB ( Y ) + γA ( Y )] − γr B ( Y ) − F S { rA ( Y ) − α − β ) A ( Y ) − γB ( Y ) − δD ( Y )] } = − γR ( ξ , Y, ξ , ξ ) − δR ( ξ , Y, ξ , ξ ) . (5.6)2If γ = δ then rom the equations (5 . , (5 . R we get, − βγ B ( Y ) − βγ D ( Y ) − γ A ( Y )+ 4 γ αD ( Y ) + γA ( Y )] + γ ( β − α ) B ( Y ) − γ A ( Y ) − γr D ( Y ) + γ ( β − α ) D ( Y ) − α α − β ) A ( Y ) − γB ( Y ) − γD ( Y )] + αr A ( Y )+ 4 β α − β ) A ( Y ) − γB ( Y ) − γD ( Y )] − βr A ( Y )+ 4 γ αB ( Y ) + γA ( Y )] − γr B ( Y ) − F S { rA ( Y ) − α − β ) A ( Y ) − γB ( Y ) − γD ( Y )] } = −{− αγ D ( Y ) − αγ B ( Y ) + γ αD ( Y ) + γA ( Y ))+ γ αB ( Y ) + γA ( Y )) − γ A ( Y ) − γ A ( Y )+ F S { γD ( Y ) + γB ( Y ) }} . (5.7)Putting Y = ξ in equation (5.7) we get, γ − β + α + 3 F S ) = 0 . (5.8)Again putting Y = ξ in equation (5.7) we get,3 βF S = 2 γ αβ − β . (5.9)From the equation (5.8) we have either γ = 0 or F S = α − β . Now, if γ = 0 then since γ = δ , thus γ = δ = 0, implying the manifold reduces to a quasi-Einstein spacetime.Again if F S = α − β then from the equation (5.9) we get γ = 0 and hence γ = δ = 0,which again implies the manifold is reduced to a quasi-Einstein spacetime. Thisallows us to deduce the following theorem: Theorem 5.1:

A projectively pseudosymmetric ( HGQE ) spacetime is a ( QE ) spacetime if γ = δ . ( H GQE ) spacetime A semi-Riemannian manifold M n ( n ≥

3) is called Ricci-symmetric if ∇ S = 0, where S is the Ricci tensor of the manifold. Considering the spacetime as a ( HGQE ) .

5) that the manifold becomes Ricci symmetricif it satisﬁes the following relation: ∇ Z S ( X, Y ) = dα ( Z ) g ( X, Y ) + dβ ( Z ) A ( X ) A ( Y )+ β [( ∇ Z A )( X ) A ( Y ) + A ( X )( ∇ Z A )( Y )]+ dγ ( Z )[ A ( X ) B ( Y ) + A ( Y ) B ( X )]+ γ [( ∇ Z A )( X ) B ( Y ) + A ( X )( ∇ Z B )( Y )]+( ∇ Z A )( Y ) B ( X ) + A ( Y )( ∇ Z B )( X )]+ dδ ( Z )[ A ( X ) D ( Y ) + A ( Y ) D ( X )]+ δ [( ∇ Z A )( X ) D ( Y ) + A ( X )( ∇ Z D )( Y )]+( ∇ Z A )( Y ) D ( X ) + A ( Y )( ∇ Z D )( X )] = 0 . (6.1)Putting X = Y = ξ in (6 .

1) we get, − dα ( Z ) + dβ ( Z ) − γ ( ∇ Z B )( ξ ) − δ ( ∇ Z D )( ξ ) = 0 . (6.2)Putting X = Y = ξ in (6 .

1) we get, dα ( Z ) + 2 γ ( ∇ Z A )( ξ ) = 0 . (6.3)Again putting X = Y = ξ in (6 .

1) we get, dα ( Z ) + 2 δ ( ∇ Z A )( ξ ) = 0 . (6.4)Since the vector ﬁelds ξ , ξ , ξ are mutually orthogonal then g ( ξ , ξ ) = g ( ξ , ξ ) = 0,this implies that Z ( g ( ξ , ξ )) = Z ( g ( ξ , ξ )) = 0. Which further implies,( ∇ Z B )( ξ ) = − ( ∇ Z A )( ξ ) (6.5)and ( ∇ Z D )( ξ ) = − ( ∇ Z A )( ξ ) . (6.6)Subtracting equations (6.3),(6.4) from (6.2) and using the relations (6.5) and (6.6)we get, d ( β − α )( Z ) = 0 , ∀ Z ∈ χ ( M ) . (6.7)That implies β − α is a constant.Now contracting the equation (6 .

1) over

X, W and using (6 . , (6 .

6) we get, d ( β − α )( Z ) = 0 , ∀ Z ∈ χ ( M ) . (6.8)From the equations (6.7) and (6.8) it is clear that α, β are constants. So, from theequations (6.3) and (6.4) we get γ ( ∇ Z )( ξ ) = 0 (6.9)4and δ ( ∇ Z )( ξ ) = 0 . (6.10)The equation (6.9) shows γ = 0 or γ ( ∇ Z )( ξ ) = 0. If γ = δ then from (6.1) we get, β [( ∇ Z A )( X ) A ( Y ) + A ( X )( ∇ Z A )( Y ) = 0 . (6.11)Putting x = ξ in (6.11) we get, β ( ∇ Z A )( Y ) = 0 . (6.12)If β = 0 then since γ = δ = 0 the manifold reduces to an Einstein manifold whichis a contradiction. So, β = 0. This implies that,( ∇ Z A )( Y ) = 0 . (6.13)Again if γ = 0 then from (6.9) we get, β ( ∇ Z A )( ξ ) = 0 . (6.14)Using (6.14) and putting X = ξ , Y = ξ in the equation (6.1) we get, dγ ( Z ) = 0 . (6.15)Which imply γ is also a constant. Then, putting X = ξ in (6.1) and using (6.15)we get, γ ( ∇ Z A )( Y ) = 0 . (6.16)Since γ = 0 thus we get the equation (6.13) once again. Hence, we always obtain( ∇ Z A )( Y ) = 0 ∀ Z, Y ∈ χ ( M ) . Which can be written as g ( ∇ Z ξ , Y ) = 0 ∀ Z, Y ∈ χ ( M ) . (6.17)Thus we obtain ∇ Z ξ = 0 . (6.18)Which implies that the generator vector ﬁeld ξ is always parallel. Hence we obtainthe following theorem as: Theorem 6.1:

In a Ricci symmetric ( HGQE ) spacetime with γ = δ the gen-erator vector ﬁeld ξ is always parallel .Again putting Z = ξ in (6.18) we get, ∇ ξ ξ = 0 . (6.19)Which implies that the integral curves of ξ Theorem 6.2:

In a Ricci symmetric ( HGQE ) spacetime with γ = δ the inte-gral curves of the generator vector ﬁeld ξ are geodesics .With the help of the theorems (6.1) and (6.2) we arrive at the following condition R ( X, Y ) ξ = ∇ X ∇ Y ξ − ∇ Y ∇ X ξ − ∇ [ X,Y ] ξ = 0 . (6.20)Hence we obtain the following theorem as: Theorem 6.3:

In a Ricci symmetric ( HGQE ) spacetime with γ = δ the Rie-mannian curvature tensor vanishes at the generator vector ﬁeld .Now contracting (6.20) we get S ( X, ξ ) = 0 . (6.21)Thus from (1.5) we get,( α − β ) A ( X ) − γ [ B ( X ) − D ( X )] = 0 . (6.22)Since γ = δ thus putting X = ξ in (6.22) we get, α = β. (6.23)Again putting X = ξ in (6.22) we get, γ = 0 . (6.24)Thus γ = δ = 0, hence from (1.5) we get, S ( X, Y ) = α [ g ( X, Y ) + A ( X ) A ( Y )] . (6.25)This allows us to arrive at the next theorem as: Theorem 6.4:

Every Ricci symmetric ( HGQE ) spacetime with γ = δ is a ( QE ) spacetime with the scalar functions are constants and equal .Using the equations (6.25) from the equation (1.9) we get, T ( X, Y ) = 2 λ − α k g ( X, Y ) + αk A ( X ) A ( Y ) . (6.26)Since α, λ, k all are constants thus taking derivative to both the sides the equation(6.26) we get, ( ∇ Z T )( X, Y ) = 0 . (6.27)Therefore we see in this case the energy-momentum tensor is covariantly constant.This leads us to the following theorem:6 Theorem 6.5:

In a Ricci-symmetric ( HGQE ) spacetime with γ = δ satisfyingEinstein ﬁeld equation with cosmological constant the energy-momentum tensor iscovariantly constant. Again from (6.26) we observe that since the velocity vector ﬁeld ξ is parallel and α is a constant thus the energy-momentum tensor is of Codazzi type. Hence we derivethe next theorem as: Theorem 6.6:

In a Ricci-symmetric ( HGQE ) spacetime with γ = δ satisfyingEinstein ﬁeld equations with cosmological constant the energy-momentum tensor isof Codazzi type .Now from the equations (1.8), (1.9) and theorem (6.4) we conclude the values of σ and p as, σ = 3 α − λ k , p = 2 λ − α k . (6.28)Since, α, λ, k all are constants thus we get σ, p are also constants. This leads us tothe following theorem: Theorem 6.7:

In a Ricci-symmetric ( HGQE ) spacetime with γ = δ satisfy-ing Einstein ﬁeld equation with cosmological constant the energy density and theisotropic pressure are constants .It is proved [2] that in a perfect ﬂuid spacetime if the energy-momentum tensoris of Codazzi type then the vorticity and shear of the spacetime vanish. Hence wederive the next theorem: Theorem 6.8:

In a Ricci-symmetric ( HGQE ) spacetime with γ = δ satisfyingEinstein ﬁeld equations with cosmological constant the vorticity and the shear tensorvanish. Here we see that the velocity vector ξ is constant over the spacelike hypersurfaceorthogonal to ξ . But it is described in [5] that perfect ﬂuid spacetime that is vortic-ity free and shear free is of petrov type I, D or O . Thus we state the next theorem as: Theorem 6.9:

The local cosmological structure of a Ricci-symmetric ( HGQE ) spacetime with γ = δ satisfying Einstein ﬁeld equation with cosmological constantcan be identiﬁed as petrov type I, D or O .7 ( H GQE ) spacetime with cyclic parallel Riccitensor Consider an (

HGQE ) with cyclic parallel Ricci tensor. Then we get the followingequation, ( ∇ X S )( Y, Z ) + ( ∇ Y S )( X, Z ) + ( ∇ Z S )( X, Y ) = 0 . (7.1)From the equation (1 .

9) we have( ∇ X S )( Y, Z ) = 12 dr ( Z ) g ( X, Y ) + k ( ∇ Z T )( X, Y ) . (7.2)Now, if in an ( HGQE ) with cyclic parallel Ricci tensor the scalar curvature of thespacetime is constant then, dr ( X ) = 0 , (7.3)for all X ∈ χ ( M ). Using the equation (7 .

3) in the equation (8 .

2) we get,( ∇ X S )( Y, Z ) = k ( ∇ Z T )( X, Y ) . (7.4)Now, if the ( HGQE ) spacetime is cyclic parallel then, k { ( ∇ X T )( Y, Z ) + ( ∇ Y T )( X, Z ) + ( ∇ Z T )( X, Y ) } = ( ∇ X S )( Y, Z ) + ( ∇ Y S )( X, Z ) + ( ∇ Z S )( X, Y ) = 0 . (7.5)Since k , being the gravitational constant is always nonzero, from the equation ( ?? )we have ( ∇ X T )( Y, Z ) + ( ∇ Y T )( X, Z ) + ( ∇ Z T )( X, Y ) = 0 . (7.6)This allows us to obtain the following theorem as: Theorem 7.1:

In a hyper-generalised quasi-Einstein spacetime with cyclic parallelRicci tensor if the scalar curvature is constant then the energy-momentum tensor isalso cyclic parallel. ( H GQE ) space-time Here we study some physical applications of the (

HGQE ) spacetime. In [10], [11]G. F. R. Ellis has given the energy momentum tensor of a ﬂuid matter distributionas follows: T ( X, Y ) = ( σ + p ) A ( X ) A ( Y ) + pg ( X, Y ) + A ( X ) B ( Y ) + A ( Y ) B ( X )+ A ( X ) D ( Y ) + A ( Y ) D ( X ) , (8.1)where, g ( X, ξ ) = A ( X ) , g ( X, ξ ) = B ( X ) , g ( X, ξ ) = D ( X ) , A ( ξ ) = − , B ( ξ ) = 1 , D ( ξ ) = 1 ,g ( ξ , ξ ) = 0 , g ( ξ , ξ ) = 0 , g ( ξ , ξ ) = 0 , and σ is the matter density, p is the isotropic pressure, ξ is the timelike velocityvector ﬁeld, ξ is the heat conduction vector ﬁeld and ξ is the stress vector ﬁeld.Combining equation (8.1) with equation (1.9) we get, S ( X, Y ) = ( kp + r − λ ) g ( X, Y ) + k ( σ + p ) A ( X ) A ( Y )+ k [ A ( X ) B ( Y ) + A ( Y ) B ( X )]+ k [ A ( X ) D ( Y ) + A ( Y ) D ( X )] . (8.2)Comparing equation (8.2) with equation (1.9) we get that this spacetime is clearlyan ( HGQE ) spacetime with the constants γ = δ = k. Hence all the results that wederived in the earlier sections are absolutely eﬀective in this spacetime, and thus wederive the following set of theorems:From theorem (2.1) we get,

Theorem 8.1: A W -Ricci pseudosymmetric viscous ﬂuid ( HGQE ) spacetimeis a ( QE ) spacetime. From theorem (3.1) we get,

Theorem 8.2: A Z -Ricci pseudosymmetric viscous ﬂuid ( HGQE ) spacetime isa ( GQE ) spacetime. From theorem (4.1) we get,

Theorem 8.3:

A Ricci pseudosymmetric viscous ﬂuid ( HGQE ) spacetime is a ( QE ) spacetime. From theorem (5.1) we get,

Theorem 8.4:

A projectively pseudosymmetric viscous ﬂuid ( HGQE ) spacetimeis a ( QE ) spacetime. From theorem (6.1) we get,

Theorem 8.5:

In a Ricci symmetric ( HGQE ) viscous ﬂuid spacetime the gen-erator vector ﬁeld ξ is always parallel .From theorem (6.2) we get, Theorem 8.6:

In a Ricci symmetric ( HGQE ) viscous ﬂuid spacetime the in- tegral curves of the generator vector ﬁeld ξ are geodesics .From theorem (6.3) we get, Theorem 8.7:

In a Ricci symmetric ( HGQE ) viscous ﬂuid spacetime the Rie-mannian curvature tensor vanishes at the generator vector ﬁeld .From theorem (6.4) we get, Theorem 8.8:

Every Ricci symmetric ( HGQE ) viscous ﬂuid spacetime is a ( QE ) spacetime with the scalar functions are constants and equal .From theorem (6.5) we get, Theorem 8.9:

In a Ricci symmetric ( HGQE ) viscous ﬂuid spacetime satisfy-ing Einstein ﬁeld equation with cosmological constant the energy-momentum tensoris covariantly constant .From theorem (6.6) we get, Theorem 8.10:

In a Ricci symmetric ( HGQE ) viscous ﬂuid spacetime satisfy-ing Einstein ﬁeld equation with cosmological constant the energy-momentum tensoris of Codazzi type .From theorem (6.7) we get, Theorem 8.11:

In a Ricci symmetric ( HGQE ) viscous ﬂuid spacetime satisfy-ing Einstein ﬁeld equation with cosmological constant the energy density and theisotropic pressure are constants .From theorem (6.8) we get, Theorem 8.12:

In a Ricci symmetric ( HGQE ) viscous ﬂuid spacetime satisfy-ing Einstein ﬁeld equation with cosmological constant the vorticity and the sheartensor vanish .From theorem (6.9) we get, Theorem 8.13:

The local cosmological structure of a Ricci-symmetric ( HGQE ) viscous ﬂuid spacetime satisfying Einstein ﬁeld equation with cosmological constantcan be identiﬁed as petrov type I, D or O .0 ( H GQE ) spacetime Finally we give a non-trivial example to establish the existence of (

HGQE ) space-time non-trivially. For this we consider a metric known as Lorentzian metric g on M given by ds = g ij dx i dx j = − kr ( dt ) + cr − ( dr ) + r ( dθ ) + ( r sin θ ) ( dφ ) , where i, j = 1 , , , k, c are constants. Thus we obtain the nonzero com-ponents of Christofell symbols, curvature tensors and Ricci tensors as follows:Γ = 4 r − c, Γ = − r , Γ = c r ( c − r ) , Γ = Γ = 1 r , Γ = cot θ, Γ = (4 r − c )(sin θ ) , Γ = − sin(2 θ )2 (9.1) R = − k ( c − r ) r ( c − r ) , R = k ( c − r )2 r , R = k ( c − r )(sin θ ) r R = c r − c ) , R = c (sin θ ) r − c ) , R = r ( c − r )(sin θ ) R = − kr , R = − r ( c − r ) , R = − , R = − θ ) (9.2)From (9 .

1) and (9 .

2) it follows that M is a Lorentzian manifold of nonzero scalarcurvature (= − r ). Now we will prove that this is an ( HGQE ) manifold.We consider α, β, γ and δ as the associated scalars and we consider them as fol-lows: α = − r , β = − r , γ = 3 r , δ = − r (9.3)and the associated 1-forms are as follows : A i ( x ) = q k r for i = 1 q r c − r ) for i = 20 for i = 3 , B i ( x ) = ( q kr for i = 10 for i = 2 , , D i ( x ) = ( q kr for i = 10 for i = 2 , , i ) R = αg + βA A + γ [ A B + B A ] + δ [ A D + D A ]( ii ) R = αg + βA A + γ [ A B + B A ] + δ [ A D + D A ]( iii ) R = αg + βA A + γ [ A B + B A ] + δ [ A D + D A ]( iv ) R = αg + βA A + γ [ A B + B A ] + δ [ A D + D A ]As every Ricci tensor other than R , R , R and R are zero, so we obtain R ij = αg ij + βA i A j + γ [ A i B j + B i A j ] + δ [ A i D j + D i A j ] , i, j = 1 , , , . Consequently, scalar curvature = 4 α − β = − r . Hence, ( M , g ) is a hyper-generalized quasi Einstein manifold. Conclusion:

The general theory of relativity is the most prominent ﬂagship ofmodern physics. It deals with the curvature of spacetime. As hyper-generalizedquasi-Einstein spacetime is considered as the base space of the ﬂuid matter distri-bution. Thus it has been very necessary to study about the geometric and physicalapplications of hyper-generalized quasi-Einstein spacetime. It deals with the rela-tivistic viscous ﬂuid spacetime admitting heat ﬂux and stress. The general theory ofrelativity describes gravity as a geometric property of spacetime. The curvature ofspacetime is directly related to the energy-momentum tensor. Also we know the cos-mological constant to be of homogeneous energy density which causes the expansionof the universe to accelerate. So, here we obtain a set of geometric and physical prop-erties of hyper-generalized quasi-Einstein spacetimes and give a non-trivial exampleto establish the existence of hyper-generalized quasi-Einstein spacetime non-trivially.

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