Shear-free perfect fluids with a solenoidal electric curvature

Carminati, John; Karimian, H R; Bergh, N Van den; Vu, Khai T.

doi:10.1088/0264-9381/29/10/105010

Cited by 8 publications

(22 citation statements)

References 24 publications

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“…For the system of equations yielded by the extended tetrad formalism, imposing the existence of a barotropic equation of state p = p(µ) as well as the vanishing of the shear results in new chains of integrability conditions. The procedure of building up the sequence of integrability conditions has been carried out in several papers and for details of their derivation we refer the reader for example to [39]. The final result of this procedure, taking into account all Jacobi equations and Einstein field equations, the 18 Ricci equations, the contracted Bianchi equations, the 'Ė', 'Ḣ' and '∇ · E ' Bianchi equations and all integrability conditions on µ, θ,u α and ω (the [∂ 1 , ∂ 3 ]ω and [∂ 2 , ∂ 3 ]ω relations being equivalent with the '∇ · H' equations) is presented in Appendix 1; see also [40], or [22] for the compact '1+3 covariant form' of some of these equations.…”

Section: Notations and Fundamental Equationsmentioning

confidence: 99%

“…Analogously, repeating the argument with (46) in the place of (45), we get R 2 = Q 1 . By propagating the equations (39,40) we obtain a homogeneous system in θ and λ 5 (p + µ) −1 , whose necessarily vanishing determinant leads us to a third degree polynomial equation inU 3 with basic coefficients. Again (cf.…”

Section: Linear Equation Of Statementioning

confidence: 99%

“…. , h 7 are functions of µ defined by (39,40,41,42) in the case of a linear equation of state correspond respectively to equations ( 26), ( 27), ( 28) and ( 23) of [30].…”

Section: Formulation In Terms Of Basic Variablesmentioning

confidence: 99%

“…Let us assume that U3 is not basic, as otherwise Theorem 2 applies. Since the equation of state is linear, the determinant of the linear system (39,40) in U1 , U2…”

Section: Remark 4 ([30]mentioning

confidence: 99%

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Shear-free perfect fluids with a barotropic equation of state in general relativity: the present status

Bergh

Slobodeanu

2016

Class. Quantum Grav.

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Section: Notations and Fundamental Equationsmentioning

confidence: 99%

Section: Linear Equation Of Statementioning

confidence: 99%

“…. , h 7 are functions of µ defined by (39,40,41,42) in the case of a linear equation of state correspond respectively to equations ( 26), ( 27), ( 28) and ( 23) of [30].…”

Section: Formulation In Terms Of Basic Variablesmentioning

confidence: 99%

“…Let us assume that U3 is not basic, as otherwise Theorem 2 applies. Since the equation of state is linear, the determinant of the linear system (39,40) in U1 , U2…”

Section: Remark 4 ([30]mentioning

confidence: 99%

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Shear-free perfect fluids with a barotropic equation of state in general relativity: the present status

Bergh

Slobodeanu

2016

Class. Quantum Grav.

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“…Also in the case of a divergence-free electric part of the Weyl tensor [29], or the case with a γ-law equation of state where the magnetic part of the Weyl tensor is divergence-free [30], and one where the equation of state is more generic [31]. Nzioki, Goswami, Dunsby and Ellis proved the case in which the Einstein field equations are linearized with respect to the FLRW background [32].…”

Section: Introductionmentioning

confidence: 99%

Reviving the shear-free perfect fluid conjecture in general relativity

Sikhonde¹,

Dunsby²

2017

Class. Quantum Grav.

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Abstract. Employing a Mathematica symbolic computer algebra package called xTensor, we present (1 + 3)-covariant special case proofs of the shear-free perfect fluid conjecture in General Relativity. We first present the case where the pressure is constant, and where the acceleration is parallel to the vorticity vector. These cases were first presented in their covariant form by Senovilla et. al. We then provide a covariant proof for the case where the acceleration and vorticity vectors are orthogonal, which leads to the existence of a Killing vector along the vorticity. This Killing vector satisfies the new constraint equations resulting from the vanishing of the shear. Furthermore, it is shown that in order for the conjecture to be true, this Killing vector must have a vanishing spatially projected directional covariant derivative along the velocity vector field. This in turn implies the existence of another basic vector field along the direction of the vorticity for the conjecture to hold. Finally, we show that in general, there exist a basic vector field parallel to the acceleration for which the conjecture is true. IntroductionThis paper presents a method of investigating shear-free perfect fluid solutions in General Relativity. This method is used to examine a number of special cases. The shear-free perfect fluid conjecture in General Relativity states that, a shear-free velocity vector field of a barotropic perfect fluid with µ + p = 0 and p = p(µ), is either expansion or rotation free [1], where µ and p are the energy density and pressure of the fluid respectively. The first indication that the vanishing of the shear can have restrictive properties between the rotation and the expansion of a cosmological model was given by Gödel [2,3]. Some of the well known examples of these cosmological models include: the Einstein (static)Reviving The Shear-Free Perfect Fluid Conjecture In General Relativity 2 universe, FLRW universes (expanding models) and the Gödel (purely rotating) universe.The requirement that the matter equation of state be of a barotropic perfect fluid produces new additional constraints on the full Einstein field equations, for instance all expanding and non-rotating shear-free perfect fluids with a barotropic equation of state are known to exist [4]. In contrast, not all non-expanding and rotating shear-free perfect fluids with a barotropic equation of state are known [5], but a fair amount of these models are known to exist especially all stiff rotating axisymmetric and stationary perfect fluids belong in this class. A simultaneously expanding and rotating model (Bianchi IX) was found to have an equation of state of the form p = −µ = constant, which is of the cosmological constant type and is excluded in this discussion [6].The validity of the above mentioned conjecture would be a striking feature of the full Einstein field equations, while in Newtonian cosmology there exist shear-free perfect fluids with a barotropic equation of state which are both expanding and rotating simultaneously [1,[7...

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Harmonic morphisms and shear-free perfect fluids coupled with gravity

Slobodeanu

2013

Annali di Matematica

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Shear-free perfect fluids with a solenoidal electric curvature

Abstract: We prove that the vorticity or the expansion vanishes for any shear-free perfect fluid solution of the Einstein field equations where the pressure satisfies a barotropic equation of state and the spatial divergence of the electric part of the Weyl tensor is zero.

Cited by 8 publications

References 24 publications

Shear-free perfect fluids with a barotropic equation of state in general relativity: the present status

Shear-free perfect fluids with a barotropic equation of state in general relativity: the present status

Reviving the shear-free perfect fluid conjecture in general relativity

Harmonic morphisms and shear-free perfect fluids coupled with gravity

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