Hawking temperature for near-equilibrium black holes

We discuss the Hawking temperature of near-equilibrium black holes using a semiclassical analysis. We introduce a useful expansion method for slowly evolving spacetime, and evaluate the Bogoliubov coefficients using the saddle point approximation. For a spacetime whose evolution is sufficiently slow, such as a black hole with slowly changing mass, we find that the temperature is determined by the surface gravity of the past horizon. As an example of applications of these results, we study the Hawking temperature of black holes with null shell accretion in asymptotically flat space and the AdS-Vaidya spacetime. We discuss implications of our results in the context of the AdS/CFT correspondence.

Figure 1

$\kappa (u)$ as a function of time ($u=v$) at the boundary (left) and behaviors of $\kappa (u)$ near the final temperature ${\kappa }_{\mathrm{f}}$ (right). Time scales of injection are taken to be $\Delta v=0.1$, 0.2, 0.5, 1, 2, which are shorter than a typical time scale given by temperature. We set $m_0=1/2$ and $\Delta m=15/2$, which give ${\kappa }_{\mathrm{i}}=2$ and ${\kappa }_{\mathrm{f}}=4$, respectively.

Figure 2

Relaxation rate for final temperatures ${\kappa }_{\mathrm{f}}=4$, 8, 16 for the case of null shell accretion ($\Delta v=0$). The initial temperature is given by ${\kappa }_{\mathrm{i}}=2$. In any case, an exponential decay phase starts after a transient phase, and the decay rate in the exponential decay phase is proportional to inverse of the final temperature.

Figure 3

$\kappa (u)$ and a quasistatic temperature $2\pi T_{\mathrm{qs}}=2(2m(v){)}^{1/4}$ for various injection times $\Delta v$.