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

Bergh, N Van den; Slobodeanu, Radu

doi:10.1088/0264-9381/33/8/085008

Cited by 8 publications

(21 citation statements)

References 48 publications

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“…we have managed to reproduce two covariant proofs provided in reference [1], these are the constant pressure case and the case where the vorticity is parallel to the acceleration vector field, illuminating some of the steps involved in these proofs. We then provide covariant proofs of the cases where the vorticity is orthogonal to the acceleration and where there exist a basic vector along the acceleration, which were first given in reference [35] using a tetrad formalism.…”

Section: Discussionmentioning

confidence: 99%

“…In this section we present a covariant proof of the theorem that was given in [35], which was demonstrated using the orthonormal tetrad formalism.…”

Section: The Case Where the Acceleration Vector Field Is Orthogonal Tmentioning

confidence: 97%

“…The above equation (35) has been simplified by defining p ′ = ∂p ∂µ and using the following scalar variables, E is the sum of the pressure and the energy density of the fluid, J is the divergence of the acceleration vector, φ is related to the second partial derivative of the pressure with respect to the energy density and ψ is related to the third partial derivative of the pressure with respect to the energy density as follows,…”

Section: Propagation Of the New Constraints Along Umentioning

confidence: 99%

“…which is a 3rd order differential equation for p, this is equivalent to equation (51) in reference [35]. Henceforth, we shall define two force(this is just a name tag for Θ∇ a ξ)…”

Section: The Case Where the Acceleration Vector Field Is Orthogonal Tmentioning

confidence: 99%

“…Proof: A purely spatial vector field is said to be basic if its spatially projected Lie derivative along u vanishes as defined in [35]. Therefore, the spatially projected Lie derivative along u of Ω a is given by;…”

Section: The Existence Of a Basic Vector Field Along The Direction Ofmentioning

confidence: 99%

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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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Section: Discussionmentioning

confidence: 99%

“…In this section we present a covariant proof of the theorem that was given in [35], which was demonstrated using the orthonormal tetrad formalism.…”

Section: The Case Where the Acceleration Vector Field Is Orthogonal Tmentioning

confidence: 97%

Section: Propagation Of the New Constraints Along Umentioning

confidence: 99%

“…which is a 3rd order differential equation for p, this is equivalent to equation (51) in reference [35]. Henceforth, we shall define two force(this is just a name tag for Θ∇ a ξ)…”

Section: The Case Where the Acceleration Vector Field Is Orthogonal Tmentioning

confidence: 99%

Section: The Existence Of a Basic Vector Field Along The Direction Ofmentioning

confidence: 99%

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Reviving the shear-free perfect fluid conjecture in general relativity

Sikhonde¹,

Dunsby²

2017

Class. Quantum Grav.

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Reference frames in general relativity and the galactic rotation curves

Costa,

Natário,

Frutos-Alfaro

et al. 2023

Phys. Rev. D

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Stable and self-consistent charged gravastar model within the framework of $$f(R,\,T)$$ gravity

Bhar¹,

Rej²

2021

Eur. Phys. J. C

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In this work, we discuss the configuration of a gravastar (gravitational vacuum stars) in the context of $$f(R, \,T )$$ f ( R , T ) gravity by employing the Mazur–Mottola conjecture (Mazur and Mottola in Report No. LA-UR-01-5067, 2001; Mazur and Mottola, Proc Natl Acad Sci USA 101:9545, 2004). The gravastar is conceptually a substitute for a black hole theory as available in the literature and it has three regions with different equation of states. By assuming that the gravastar geometry admits a conformal Killing vector, the Einstein–Maxwell field equations have been solved in different regions of the gravastar by taking a specific equation of state as proposed by Mazur and Mottola. We match our interior spacetime to the exterior spherical region which is completely vacuum and described by the Reissner–Nordström geometry. For the particular choice of $$f(R,\,T)$$ f ( R , T ) of $$f(R, \,T )=R+2\gamma T$$ f ( R , T ) = R + 2 γ T , here we analyze various physical properties of the thin shell and also present our results graphically for these properties. The stability analysis of our present model is also studied by introducing a new parameter $$\eta $$ η and we explore the stability regions. Our proposed gravastar model in the presence of charge might be treated as a successful stable alternative of the charged black hole in the context of this version of gravity.

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

Cited by 8 publications

References 48 publications

Reviving the shear-free perfect fluid conjecture in general relativity

Reviving the shear-free perfect fluid conjecture in general relativity

Reference frames in general relativity and the galactic rotation curves

Stable and self-consistent charged gravastar model within the framework of $$f(R,\,T)$$ gravity

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