Hawking radiation from rotating AdS black holes in conformal gravity

Abstract

We extend to study Hawking radiation via tunneling in conformal gravity. We adopt Parikh-Wilczek’s semi-classical tunneling method and the method of complex-path integral to investigate Hawking radiation from new rotating AdS black holes in conformal gravity. In this paper, the research on Hawking radiation from the rotating black holes is done in a general system, not limited in dragging coordinate systems any longer. Moreover, there existed some shortcomings in the previous derivation of geodesic equations. Different from the massless case, they used a different approach to derive the geodesic equation of the massive particles. Even the treatment was inconsistent with the variation principle of action. To remedy the shortcoming, we improve treatment to deduce the geodesic equations of massive and massless particles in a unified and self-consistent way. In addition, we also recover the Hawking temperature resorting to the complex-path integral method.

Appendix

In this appendix, we will show that there is another kind of coordinate system that can do the tunneling calculation.

If we introduce the following coordinate transformations

$$\begin{aligned} &d\tilde{t} = dt -\frac{\sqrt{r^2 +a^2}\sqrt{(r^2 +a^2)\Delta_\theta-\Delta_r}}{ \Delta_r\sqrt{\Delta_\theta}}dr , \end{aligned}$$

(69)

$$\begin{aligned} &d\tilde{\phi} = d\phi-\frac{a\sqrt{1 +g^2r^2}\sqrt{(r^2 +a^2)\Delta_\theta-\Delta_r}}{ \Delta_r\sqrt{(r^2 +a^2)\Delta_\theta}}dr , \end{aligned}$$

(70)

the metric (1) turns into a new form

$$\begin{aligned} ds^2 &= -\Delta_\theta dt^2 +\frac {\varSigma d\theta^2}{\Delta_\theta} \\ &\quad {} +\frac{\Delta_\theta(r^2 +a^2) \sin^2\theta}{\varXi^2}\bigl(d\phi-a g^2dt\bigr)^2 \\ &\quad {}+ \biggl[\frac{\sqrt{\Delta_\theta(r^2 +a^2) -\Delta_r}}{\sqrt{\varSigma}\varXi}\bigl(\Delta_\theta dt -a \sin^2\theta d\phi\bigr) \\ & \quad {}+\sqrt{\frac{\varSigma}{\Delta_\theta(r^2 +a^2)}}dr \biggr]^2. \end{aligned}$$

(71)

One can repeat the same procedure in the main text to derive the geodesic equations and the tunneling rate of massive particles, which is omitted here. Now the dragging angular velocity is

$$ \varOmega= -\frac{g_{{t}{\phi}}}{g_{{\phi}{\phi}}} = \frac{a\Delta_\theta [(r^2 +a^2)(1 +g^2r^2) -\Delta_r ]}{\Delta_\theta(r^2 +a^2)^2 -\Delta_r a^2\sin^2\theta} . $$

(72)

After performing a dragging coordinate transformation dϕ=Ωdt, the metric (71) is then changed into the following form

$$\begin{aligned} ds^2 &= -\frac{\Delta_r\Delta _\theta\varSigma}{\Delta_\theta(r^2 +a^2)^2 -\Delta_r a^2\sin ^2\theta}dt^2 \\ &\quad {}+\frac{2\varSigma\sqrt{\Delta_\theta(r^2 +a^2) [(r^2 +a^2)(1 +g^2r^2) - \Delta_r ]}}{\Delta_\theta(r^2 +a^2)^2 -\Delta_r a^2\sin^2\theta}dt dr \\ & \quad {}+\frac{\varSigma}{(r^2 +a^2)\Delta_\theta}dr^2 +\frac{\varSigma}{\Delta_\theta}d \theta^2. \end{aligned}$$

(73)

The metric (73) components in the dragging coordinate system can’t satisfy Eq. (28) due to

$$ \frac{\partial }{\partial \theta} \biggl({-}\frac{g_{{t}r}}{g_{{t}{t}}} \biggr) \neq\frac{\partial }{\partial r} \biggl({-}\frac{g_{{t}\theta}}{g_{{t}{t}}} \biggr) = 0. $$

(74)

Different from the metric (27), the metric (73) can’t satisfy Landau’s condition of the coordinate clock synchronization (Landau and Lifshitz 1987), however, one can still finish the tunneling analysis without any difficulty in this coordinate system too.