Thermodynamic properties and matter accretion properties of Kiselev black hole
Authors: Wei Yi-Huan ,   Department of Physics, School of Mathematics and Physics, Bohai University, Jinzhou 121000, China
Published: December 2020

Kiselev black hole possesses the two horizons, i.e. the inner horizon and outer horizon. In some cases, the so-called outer horizon of black hole is actually a cosmic horizon. In this paper, Kiselev space-time with black hole horizon and cosmic horizon is considered. The radius of black hole horizon and the radius of cosmic horizon are approximately obtained to be <inline-formula><tex-math id="M1">\begin{document}$r_{\rm B} \approx 2M \left[ 1 + \left(2M/{\lambda}\right)^{-(3w_{\rm {\rm q}}+1)} \right]$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182055_M1.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182055_M1.png"/></alternatives></inline-formula> and <inline-formula><tex-math id="M2">\begin{document}$r_{\rm C} \approx \lambda + \dfrac{2M} {3w_{\rm {\rm q}}+1}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182055_M2.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182055_M2.png"/></alternatives></inline-formula> with <inline-formula><tex-math id="M3">\begin{document}$M \ll \lambda$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182055_M3.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182055_M3.png"/></alternatives></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$w_{\rm q}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182055_M4.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182055_M4.png"/></alternatives></inline-formula> a parameter. The energy density of the Kiselev spacetime near the cosmic horizon is approximately proportional to <inline-formula><tex-math id="M5">\begin{document}$w_{\rm q}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182055_M5.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182055_M5.png"/></alternatives></inline-formula>, so the energy densities with some different <inline-formula><tex-math id="Z-20190315022608-2">\begin{document}$ w_{\rm q}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182055_Z-20190315022608-2.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182055_Z-20190315022608-2.png"/></alternatives></inline-formula> have the same order of magnitude in the range <inline-formula><tex-math id="M6">\begin{document}$-1<w_{\rm q} < - 1/3$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182055_M6.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182055_M6.png"/></alternatives></inline-formula>. Near the black hole horizon, it increases rapidly with the increase of <inline-formula><tex-math id="M7">\begin{document}$w_{\rm q}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182055_M7.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182055_M7.png"/></alternatives></inline-formula>.　　The thermodynamic properties of the systems with black hole horizon and cosmic horizon as boundary are studied. The first law of thermodynamics for the two systems is given in a unified way. Similarly, Smarr relation for the mass of Kiselev black hole is also obtained. For <inline-formula><tex-math id="M8">\begin{document}$M \ll \lambda $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182055_M8.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182055_M8.png"/></alternatives></inline-formula>, the work done by the fluid on the cosmic horizon and the thermal energy flux flowing into the cosmic horizon of Kiselev spacetime are calculated approximately. In the range of <inline-formula><tex-math id="M9">\begin{document}$-1 < w_{\rm q} < - 1/3 $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182055_M9.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182055_M9.png"/></alternatives></inline-formula>, the thermal energy always flows out of the cosmic horizon. The work done by the fluid on the black hole horizon is much smaller than the change in the energy of black hole, <inline-formula><tex-math id="M10">\begin{document}$\Delta w_{\rm B} \ll \Delta r_{\rm B}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182055_M10.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182055_M10.png"/></alternatives></inline-formula>. This indicates that the energy increase of black hole comes mainly from the thermal energy flowing into the black hole through its outer horizon. The problem of accreting the pressureless fluid into Kiselev black hole is discussed. One can find that there are the zero gravity surfaces between the black hole horizon and cosmic horizon of Kiselev spacetime, the radii of which increase with the decrease of <inline-formula><tex-math id="M11">\begin{document}$w_{\rm q}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182055_M11.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182055_M11.png"/></alternatives></inline-formula>. For <inline-formula><tex-math id="M12">\begin{document}$w_{\rm q}=-\dfrac{2}{3}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182055_M12.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182055_M12.png"/></alternatives></inline-formula> and <inline-formula><tex-math id="M13">\begin{document}$w_{\rm q}=-1$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182055_M13.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182055_M13.png"/></alternatives></inline-formula>, the accretion radii of Kiselev black hole are respectively determined to be <inline-formula><tex-math id="M14">\begin{document}$r_0 \approx 1.6 \times 10^{4}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182055_M14.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182055_M14.png"/></alternatives></inline-formula> (l.y.) and <inline-formula><tex-math id="M15">\begin{document}$r_0 \approx 1.2 \times 10^{6}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182055_M15.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182055_M15.png"/></alternatives></inline-formula> (l.y.). On condition that the accretion energy density is proportional to the background energy density, <inline-formula><tex-math id="M16">\begin{document}$\rho_{\rm {mB}} = \eta_{\rm B} \rho_{\rm B}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182055_M16.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182055_M16.png"/></alternatives></inline-formula> with <inline-formula><tex-math id="M17">\begin{document}$\eta_{\rm B}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182055_M17.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182055_M17.png"/></alternatives></inline-formula> being a proportionality coefficient, the accretion rate of Kiselev black hole is given as <inline-formula><tex-math id="M18">\begin{document}$\chi_{\rm B} = - \dfrac{3 \eta_{\rm B} w_{\rm q}} {2} \left(\dfrac{2M}{\lambda}\right)^{-(3w_{\rm q} + 1)}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182055_M18.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182055_M18.png"/></alternatives></inline-formula>. For <inline-formula><tex-math id="M19">\begin{document}$w_{\rm q}= - 2/3 $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182055_M19.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182055_M19.png"/></alternatives></inline-formula>, the accretion rate of the black hole takes its maximum <inline-formula><tex-math id="M20">\begin{document}$\chi_{\rm max} \approx 1.2 \times 10^{- 6} \eta_{\rm B} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182055_M20.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182055_M20.png"/></alternatives></inline-formula>; for <inline-formula><tex-math id="M21">\begin{document}$w_{\rm q}= - 1$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182055_M21.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182055_M21.png"/></alternatives></inline-formula>, the accretion rate takes its minimum <inline-formula><tex-math id="M22">\begin{document}$\chi_ {\rm {min}} \approx 1.2 \times 10 ^ {-8} \eta_{\rm B} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182055_M22.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182055_M22.png"/></alternatives></inline-formula>. On the assumption that <inline-formula><tex-math id="M23">\begin{document}$\eta_{\rm B}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182055_M23.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182055_M23.png"/></alternatives></inline-formula> changes slowly enough with <inline-formula><tex-math id="M24">\begin{document}$w_{\rm q}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182055_M24.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182055_M24.png"/></alternatives></inline-formula>, the accretion rate of Kiselev black hole increases with the increase of <inline-formula><tex-math id="M25">\begin{document}$w_{\rm q}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182055_M25.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182055_M25.png"/></alternatives></inline-formula>.