Thermodynamics phase transition and Hawking radiation of the Schwarzschild black hole with quintessence-like matter and a deficit solid angle

In this paper, we investigate the thermodynamics and Hawking radiation of Schwarzschild black hole with quintessence-like matter and deficit solid angle. From the metric of the black hole, we derive the expressions of temperature and specific heat using the laws of black hole thermodynamics. Using the null geodesics method and Parikh–Wilczeck tunneling method, we derive the expressions of Boltzmann factor and the change of Bekenstein–Hawking entropy for the black hole. The behaviors of the temperature, specific heat, Boltzmann factor and the change of Bekenstein entropy versus the deficit solid angle (\(\epsilon ^{2}\)) and the density of static spherically symmetric quintessence-like matter (\(\rho _{0}\)) were explicitly plotted. The results show that, when the deficit solid angle (\(\epsilon ^{2}\)) and the density of static spherically symmetric quintessence-like matter at \(r=1\) (\(\rho _{0}\)) vanish \((\rho _{0}=\epsilon =0)\), these four thermodynamics quantities are reduced to those obtained for the simple case of Schwarzschild black hole. For low entropies, the presence of quintessence-like matter induces a first order phase transition of the black hole and for the higher values of the entropies, we observe the second order phase transition. When increasing \(\rho _{0}\), the transition points are shifted to lower entropies. The same thing is observed when increasing \(\epsilon ^{2}\). In the absence of quintessence-like matter (\(\rho _{0}=0\)), these transition phenomena disappear. Moreover the rate of radiation decreases when increasing \(\rho _{0}\) or \((\epsilon ^2)\).