Hawking radiation, local temperatures, and nonequilibrium thermodynamics of the black holes with non-Killing horizon

Recently, a class of stationary black hole solutions with non-Killing horizon in the asymptotic AdS bulk space (i.e., nonequilibrium black funnel) was constructed to describe the far from equilibrium heat transport and particle transport between the boundary black holes via AdS/CFT correspondence. It is generally believed that the temperature of a black hole with non-Killing horizon can not be properly defined by the conventional methods used in the equilibrium black holes with Killing horizon. In this study, we calculate the spectrum of Hawking radiation of the nonequilibrium black funnel using the Damour-Ruffini method. Our results indicate that the spectrum and the temperatures as well as the chemical potentials of the nonequilibrium black funnel do depend on one of the spatial coordinates. This is different from the equilibrium black holes with Killing horizon, where the temperatures are uniform. Therefore, the black hole with non-Killing horizon can be overall in nonequilibrium steady state while the Hawking temperature of the black funnel can be viewed as the local temperature and the corresponding Hawking radiation can be regarded as being in the local equilibrium with the horizon of the black funnel. By AdS/CFT, we discuss some possible implications of our results of local Hawking temperature for the nonequilibrium thermodynamics of dual conformal field theory. We further discuss the nonequilibrium thermodynamics of the black funnel, where the first law can be formulated as the entropy production rate being equal to the sum of the changes of the entropies from the system (black funnel) and environments while the second law is given by the entropy production being larger than or equal to zero. We found the time arrow emerged from the nonequilibrium black hole heat and particle transport dissipation. We also discuss how the nonequilibrium dissipation may influence the evaporation process of the black funnel.