{'fraction_non_alphanumeric': 0.04769054137596043, 'fraction_numerical': 0.023403691601165767, 'mean_word_length': 4.084867534800179, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Shearfree perfect fluids with solenoidal magnetic curvature and a gamma-law equation of state

We show that shearfree perfect fluids obeying an equation of state p=(gamma -1) mu are non-rotating or non-expanding under the assumption that the spatial divergence of the magnetic part of the Weyl tensor is zero.


Introduction
The shear-free fluid conjecture claims that for any general relativistic perfect fluid, in which the energy density µ and the pressure p satisfy a barotropic equation of state p = p(µ) with p + µ = 0 ‡, necessarily the expansion θ or the vorticity ω vanishes. The conjecture has been demonstrated in a number of particular cases: constant p ('dust' with a cosmological constant) [10,9,16], spatial homogeneity [1,11], dp/dµ = 1/3 [20], dp/dµ = −1/3 or 1/9 [8,21], aligned vorticity and acceleration [22], vanishing magnetic part of the Weyl tensor H = 0 [7], θ = θ(µ) [12], θ = θ(ω) [18], Petrov types N [3] and III [4], models in which there is a conformal Killing vector parallel to the fluid flow [6], purely magnetic perfect fluids [8]. It should be noted that this conjecture is not true in Newtonian theory and hence if true in general relativity, then it would highlight essential differences, like that of a well-defined universal time, between the two theories [17]. Although in most of the above mentioned results a tetrad formalism was used, recently it has been shown that a covariant approach can also be successfully employed with considerable potential for future analysis [17,18,5].
Motivated by the continued absence of a general proof of the conjecture (or lack of counter-example), we have decided, as an important step forward, to investigate a direct generalisation of Collins' 1984 result that ωθ = 0 holds when the magnetic part H of the Weyl curvature (with respect to the fluid congruence) vanishes. In our analysis, we shall make the weaker assumption that H is solenoidal, that is, that its spatial divergence (see below) vanishes. Interestingly, it has been recently shown [19,2] that the assumption of third order restrictions, such as divH = 0 and/or divE = 0, leads to physically interesting families of perfect fluid solutions. Also, in a classification attempt of these fluids the shearfree sub-family would appear to be a natural first candidate for further investigation. Although a large part of our analysis holds for a general barotropic equation of state p = p(µ) (such as the positive definiteness of the matrix and the ensuing existence of a Killing vector parallel to the vorticity), at a certain point, we shall need to simplify matters by assuming a γ-law equation of state (with a possible non-vanishing cosmological term "effectively" present by allowing p = (γ − 1)µ + constant). In the general barotropic case with D b H ab = 0, technical difficulties of a very different nature are present and a number of extra subcases remain to be investigated (work in progress).

Tetrad choice and relevant equations
We consider shear-free perfect fluid solutions of the Einstein field equations where u is the future-pointing (timelike) unit tangent vector to the flow, µ and p are the energy density and pressure of the fluid and h ab = g ab + u a u b is the projection tensor into the rest space of the observers with 4-velocity u. The vanishing of the shear can be expressed by where θ is the (rate of) volume expansion,u the acceleration and ω is the vorticity. Throughout, we will assume familiarity with the notations and conventions of the orthonormal tetrad formalism of [14]. Crucial in the successful investigation of any problem with tetrad formalisms, is the choice of the tetrad alignment as it can dramatically alter the appearance and complexity of the resulting complete set of equations. We have chosen our tetrad as follows. We begin by aligning e 0 and e 3 with u and ω, respectively, such that ω = ωe 3 = 0. The relevant variables then become µ, p, ω, θ,u α and the quantities Ω α , (which determine the rotation of the triad e α with respect to a set of Fermi propagated axes), together with the quantities n αβ and a α . Latin indices will be spacetime indices while Greek indices will take the values 1, 2 and 3. Further, it is advantageous to replace n αβ and a α with the new variables q α , r α and n α , defined by n α+1 α−1 = (r α + q α )/2, a α = (r α − q α )/2, n αα = n α+1 + n α−1 , expressions which have to be read modulo 3 (for example α = 3 gives n 12 = (q 3 + r 3 )/2.) The relation between these quantities and the Ricci rotation coefficients can be deduced from the commutators (65). We will also use, as extra (extension) variables, the components of the spatial gradient of the expansion and the (3+1) covariant divergence of the acceleration, j ≡u a ;a = ∂ αu α +u αu α −u α (r α − q α ).
As neither the Jacobi identities nor the field equations contain expressions for the evolution of the Ω α , it is good practice to choose the triad e α such that Ω + ω = 0. The fact that this is always possible follows from the results of the action of the commutators [∂ 3 , ∂ 1 ] and [∂ 3 , ∂ 2 ] on p. One finds that Ω 1 = Ω 2 = 0 after which a rotation in the (12)-plane can be chosen such that Ω 3 + ω = 0. Herewith, the tetrad is determined up to rotations in the (12)-plane over an angle α satisfying ∂ 0 α = 0. Noticing that the evolution equations for the quantities n 11 − n 22 and n 12 are identical (see [14]), allows one to further fix the tetrad by making either n 11 − n 22 = 0 or n 12 = 0. Henceforth our choice will be We express the vanishing of the spatial divergence of the magnetic part of the Weyl tensor, H ab = C acbd u c u d , by means of the Bianchi identity [13], Here the spatial derivative operator is defined by This shows that when D b H ab = 0, the vorticity is an eigenvector of the electric part of the Weyl tensor, E ab = C acbd u c u d , with eigenvalue −(µ + p)/3. With our choice of tetrad this implies In order to relate the components of E ab with the spatial gradient of the acceleration, we will make use of the Ricci identity [13] where S ab stands for the spatially projected and trace-free part of S ab . The basic equations of the formalism are now the Einstein field equations and the Jacobi equations, which we present, using the simplifications above, in the appendix. Henceforth we will assume γ = 1 [10,16,9], γ = 4 3 [20], γ = 2 3 [21], γ = 10 9 [21] and, of course γ = 0 (cf. introduction).
We shall argue by contradiction in order to establish that a shearfree fluid under the given conditions satisfies ωθ = 0. The important step is contained in section 3, where we prove first that, if ωθ = 0, a Killing vector exists parallel to the vorticity. This results in a great simplification of the governing equations and, although this subcase still contains all the intricacies of the 'general' problem, it allows us to complete the proof in section 4.
3. Proof of the existence of a Killing vector parallel to ω for ωθ = 0 First notice that the equations (73,76) immediately lead to evolution equations for the variables r α and q α , while (66) and the (0α) field equations (78-80) give us the spatial derivatives of ω together with the algebraic restriction The evolution equation for n follows from (74): Acting with the commutators [∂ 0 , ∂ α ] and [∂ 1 , ∂ 2 ] on the pressure and using (72) together with the conservation laws leads to a first set of evolution equations for the acceleration and vorticity: The spatial derivatives of the acceleration can be obtained from (10), using (5): Next we act with the [∂ 0 , ∂ α ] commutators on ω and θ and use the propagation of (16) along u in order to obtain expressions for the evolution of z α and the spatial gradient of j: and Now we may evaluate α [∂ 0 , ∂ α ]u α by using (5) and (21)(22)(23) and this leads to the evolution of j as Next a long calculation, involving the propagation of the field equations (81-86) along u and the use of the [∂ 0 , ∂ α ] commutators on r β , q β ,u β together with the [∂ α , ∂ β ] commutators on θ and ω, allows one to obtain algebraic expressions for the directional derivatives ∂ α z β , as well as evolution equations for j and E αβ (the latter can equally well be obtained by writing out the 'dot E' Bianchi identities [13]). The resulting expressions can be simplified by the introduction of the following non-linear combinations of kinematic quantities, each having a clear geometric meaning, We obtain: a) the evolution of j: b) the evolution of E αβ : c) the spatial derivatives of z α : The evolution of the new variables U, V , W and Z is given by The equations given so far describe the behaviour of a general shearfree perfect fluid with a γ-law equation of state (and hence can be useful in further investigations of the ωθ = 0 conjecture). It is only at this stage that we introduce the condition that the magnetic part of the Weyl tensor is divergence-free. Imposing (8,9) we deduce from (40, 41) where the factor γ − 4 3 can be divided out (γ = 4 3 corresponds to a pure radiation perfect fluid, in which case the conjecture ωθ = 0 is known to hold [20]).

z 3 + 2θu 3 = 0
If z 3 + 2θu 3 = 0 then (52) implies thatu 3 z 1 =u 3 z 2 = 0. Ifu 3 = 0 propagation of the conditions z 1 = z 2 = 0 along u gives, using (30), which again would imply that the vorticity and acceleration are parallel. We therefore conclude thatu 3 = 0 and hence also z 3 = 0. By (80) one also has then n 33 = 0. Furthermore, with (26,27) and (49,50) the conditionsu 3 = z 3 = 0 yield such that, unless acceleration and vorticity are parallel, at least one of q 3 or r 3 = 0. Without loss of generality we can suppose q 3 = 0. If r 3 = 0 then z 1 =u 1 = 0, after which (30) would imply (9γ − 10)ω(z 2 + θu 2 ) = 0. The spatial gradients of θ and p would then be parallel (with the coefficient of proportionality depending on θ only), which again would imply that the flow is irrotational. We conclude that both q 3 = r 3 = 0. A straightforward calculation shows then that there is a Killing vector along e 3 , although in the sequel this property will not be explicitly used.

Discussion
We have demonstrated that a shearfree perfect fluid, obeying an equation of state p = (γ−1)µ and satisfying the 'solenoidal' condition divH = 0, is either expansion free or vorticity free. Our proof relies heavily on the subcases discussed earlier in the literature for various values of γ. In addition, we have presented an interesting framework which shows promise for successfully mounting an assault on the shearfree conjecture with the assumption of just a gamma law. Presently we are trying to generalise our result to include an arbitrary barotropic equation of state p = p(µ) with the solenoidal condition. Preliminary work shows that almost all relations in the present work can be generalised (in particular the fact that ωθ = 0 would imply the existence of a Killing vector parallel to the vorticity). Some technical difficulties still remain to be dealt with in the final stage of the proof. This is primarily due to the resulting more complicated nature of equation (63) and its subsequent derivatives. A proof of the conjecture would be very desirable, as the classification of the shearfree case would form a necessary and natural first step in the study of purely 'solenoidal' perfect fluids.