Shear-free perfect fluids in general relativity: IV. Petrov type III spacetimes

Abstract

It is shown that for any Petrov type III, shear-free, perfect-fluid solution of Einstein's field equations, in which the perfect fluid satisfies a barotropic equation of state such that , the fluid volume expansion is zero. In addition, the fluid must necessarily be rotating.

Shear-free perfect fluids with linear equation of state

“…In this last sub-case Z itself is shear-free and divergence-free 4 . Moreover its 3-dimensional orthogonal distribution is integrable (so Z is aligned with a gradient vector) and minimal.…”

“…Now, taking the derivative along V of Equation ( 44), then replacing β(X)X(ln λ)+β(Y )Y (ln λ) from (44), β(Y )X(ln λ)−β(X)Y (ln λ) from (43) and also X(ln λ) 2 + Y (ln λ) 2 from the trace constraint (17), gives us an equation of the following form: α(r)V (V (ln λ)) 2 + β(r)V (V (ln λ))V (ln λ) 2 + γ(r)V (ln λ) 4 + V (ln λ) 2 P 1 (λ 8−2r , λ r−2 ) + V (V (ln λ))P 2 (λ 8−2r , λ r−2 ) + P 3 (λ 8−2r , λ r−2 ) = 0,…”

“…Since then, many partial results have been obtained by specializing (i) the curvature (divergence-free electric or magnetic part of Weyl curvature [5,37], Petrov types N [3] and III [4]), (ii) kinematical quantities (parallel vorticity and acceleration [38], functionally dependent expansion and energy density [20] or expansion and rotation scalar [33], fluid flow parallel with a conformal vector field [8]), or (iii) the equation of state (p = wρ, with w = 0, ± 1 3 , 1 9 [15,19,32,35,36]). Further aspects supporting the conjecture can be found in [10]; for a recent discussion, see [16].…”

The shear-free perfect fluid conjecture

“…• the case where the expansion scalar and the energy density are functionally related: θ = θ(µ) [10], • the case where the expansion scalar and the vorticity scalar are functionally related: θ = θ(ω) [16], • Petrov types N [3] and III [4], but a general proof is still lacking.…”

“…Jacobi identities and Einstein equations with a choice of tetrad as described in section 1.2. 4 3 θω = 0 (A13)…”