Rotating black hole and quintessence
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We discuss spherically symmetric exact solutions of the Einstein equations for quintessential matter surrounding a black hole, which has an additional parameter ($\omega$) due to the quintessential matter, apart from the mass ($M$). In turn, we employ the Newman\(-\)Janis complex transformation to this spherical quintessence black hole solution and present a rotating counterpart that is identified, for $\alpha=-e^2 \neq 0$ and $\omega=1/3$, exactly as the Kerr\(-\)Newman black hole, and as the Kerr black hole when $\alpha=0$. Interestingly, for a given value of parameter $\omega$, there exists a critical rotation parameter ($a=a_{E}$), which corresponds to an extremal black hole with degenerate horizons, while for $a<a_{E}$, it describes a non-extremal black hole with Cauchy and event horizons, and no black hole for $a>a_{E}$. We find that the extremal value $a_E$ is also influenced by the parameter $\omega$ and so is the ergoregion.
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