Shear-free perfect fluids with linear equation of state

Abstract

Abstract. We prove that shear-free perfect fluid solutions of Einstein's field equations must be either expansion-free or non-rotating (as conjectured by Ellis and Treciokas) for all linear equations of state p = wρ except for w ∈ − IntroductionFor the Einstein field equations with perfect fluid sources, an important insight into the structure of solutions can be obtained under two fairly general and observationally defensible assumptions: the equation of state for the pressure p and energy density ρ is barotropic (i.e. p = p(ρ)) and the flow is shear-free (i.e. fluid's velocity is a transversally conformal vector). While the former restriction is specific to isoentropic fluids, the latter is essentially a kinematic condition of isotropy. In cosmology, the shear-free condition expresses the isotropy of the relative recessional motion of the galaxies (but allows the red shift and the cosmic microwave background radiation to be anisotropic) and is a common feature of standard spacetimes, such as the FRW and Gödel models. In kinetic theory this characterizes [35] the velocity of collision-dominated gases with isotropic distribution function, under the Einstein-Boltzmann equations.The first result indicating the nature of solutions in this context was stated by Gödel ([12]): a shear-free dust fluid (i.e. p = 0, in particular a congruence of timelike geodesics) in a spatially homogeneous spacetime of type IX cannot both expand and rotate. This contrasts Newtonian cosmology where expanding and rotating (shear-free) solutions exist and can avoid the singularity formation [24]. Gödel's result was later proved [15] to remain true without symmetry assumptions and it can be seen as a timelike analogue of the Goldberg-Sachs theorem [18]. After proving that the same conclusion holds for radiation fluids (i.e. p = If the velocity vector field of a self-gravitating barotropic perfect fluid (p + ρ = 0 and p = p(ρ)) is shear-free, then either the expansion or the rotation of the fluid vanishes.2010 Mathematics Subject Classification. 53B30, 53C43, 53Z05, 83C55.

Reviving the shear-free perfect fluid conjecture in general relativity

“…Complementary, propagating the constraint equation (C 5 ) a along u and making use of the following relations: (A.21),(27d),(27a) and (69), and employing the identities (B.7) to (B.11) this results in equation (33) which shows that ǫ a bc∇…”

“…Definition: A spatially projected tensorial object ζ on a manifold (M, g, u) is called basic if its spatially projected Lie derivative along u is zero [33]. Proof: In the following discussion will shall assume that Θω and p ′ are non-zero, then we will show that this leads to an inconsistency unless there exist a Killing vector along the vorticity.…”

“…Recently, Slobodeanu [33] provided a proof of the conjecture for a γ-law equation of state with a zero cosmological constant which excludes the following cases (γ − 1 = − 1 5 , − 1 6 The above mentioned proof shows that whenever the acceleration is orthogonal to the vorticity vector, there is a Killing vector along the vorticity under which the constraints resulting from the vanishing of the shear are satisfied and remain so under time-propagation. With further analysis, we show that this Killing vector must have the property that its spatially projected covariant directional derivative along u a must vanish in order for the conjecture to be true, which essentially means that there exist another vector field along the direction of the vorticity with a vanishing spatially projected Lie derivative along the velocity, implying that it is basic.…”

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

“…Some immediate properties of basic functions are provided by the following lemma, the proof of which is easily checked: In the case of a γ-law equation of state a length scale λ −1 was introduced in [30], enabling one to write pressure and energy density as p = r−3 3 λ r , µ = λ r with r = 3γ. This not only leads to a simplification of the equations, but also plays a key role in some of the arguments, such as in Proposition 5 of [30]. In order to generalise this proposition to the case of a general barotropic equation of state and to formulate similar useful criteria, we will introduce the function λ = λ(µ) as follows,…”

“…Since then the conjecture has been proved also in a large number of special cases, such as dp/dµ = − 1 3 [10,21,29,36]; θ = θ(ω) [32]; Petrov types N [2] and III [3,4]; the existence of a conformal Killing vector parallel to the fluid flow [9]; the Weyl tensor having either a divergence-free electric part [39], or a divergence-free magnetic part, in combination with an equation of state which is of the γ-law type [38] or which is sufficiently generic [5], and in the case where the Einstein field equations are linearised about a FLRW background [25] . A major step has been achieved recently by the second author [30] proving the conjecture for an arbitrary γ-law equation of state (except for the cases γ − 1 = − 1 5 , − 1 6 , − 1 11 , − 1 21 , 1 15 , 1 4 ) and a vanishing cosmological constant. In this approach, reminiscent of Pantilie's classification result on Einstein manifolds [27], the Einstein field equations were seen as a second order differential system in the length scale function, with the integrability conditions for this system allowing one to prove the conjecture via some sufficient conditions in terms of basic functions, i.e.…”