Paper detail

The Kiselev black hole is neither perfect fluid, nor is it quintessence

The Kiselev black hole spacetime, \[ ds^2 = - \left(1-{2m\over r} - {K\over r^{1+3w}} \right) dt^2 + {dr^2\over1-{2m\over r} - {K\over r^{1+3w}}} + r^2 \,dΩ_2^2, \] is an extremely popular toy model, with over 200 direct and indirect citations as of 2019. Unfortunately, despite repeated assertions to the contrary, this is not a perfect fluid spacetime. The relative pressure anisotropy and average pressure are easily calculated to satisfy \[ Δ= {Δp\over \bar p} = {p_r - p_t \over {1\over3} (p_r+2p_t)} =- {3(1+w)\over 2 w}; \qquad\qquad {\bar p\over ρ} = {{1\over3} (p_r + 2p_t)\over ρ} = w. \] The relative pressure anisotropy $Δ$ is generally a non-zero constant, (unless $w=-1$, corresponding to Schwarzschild-(anti)-de Sitter spacetime). Kiselev's original paper was very careful to point this out in the calculation, but then in the discussion made a somewhat unfortunate choice of terminology which has (with very limited exceptions) been copied into the subsequent literature. Perhaps worse, Kiselev's use of the word "quintessence" does not match the standard usage in the cosmology community, leading to another level of unfortunate and unnecessary confusion. Very few of the subsequent follow-up papers get these points right, so a brief explicit comment is warranted.