Tunnelling from Gödel black holes

We consider the spacetime structure of Kerr-Gödel black holes, analyzing their parameter space in detail. We apply the tunnelling method to compute their temperature and compare the results to previous calculations obtained via other methods. We claim that it is not possible to have the closed timelike curve (CTC) horizon in between the two black hole horizons and include a discussion of issues that occur when the radius of the CTC horizon is smaller than the radius of both black hole horizons.

Figure 1

3d plot of $\sqrt{x_{\mathrm{ctc}}}-\sqrt{x_{+}}$ in terms of $L$ and $J$ (i.e. compares location of the CTC horzion to largest black hole horizon) Note: Regions when $J^2>1$ are negative which means the CTC horizon is inside the black hole horizon. The peak at $J=0$ corresponds to the infinite CTC horizon and indicates regular 5d Kerr.

Figure 2

Plots of $\sqrt{x_{\mathrm{ctc}}}$, $\sqrt{x_{-}}$, $\sqrt{x_{+}}$ it terms of $L$ and when $J=-L$, Note: Dashed line corresponds to $\sqrt{x_{\mathrm{ctc}}}$ and solid lines are $\sqrt{x_{-}}$, $\sqrt{x_{+}}$. Notice the CTC’s horizon is either outside both horizons or inside both horizons but never in between.

Figure 3

Plots of $\sqrt{x_{\mathrm{ctc}}}$, $\sqrt{x_{-}}$, $\sqrt{x_{+}}$ in terms of (a) $L$ when $J=1$, (b) $L$ when $J=-2L-1$ Note: The dashed line corresponds to $\sqrt{x_{\mathrm{CTC}}}$ and solid lines are $\sqrt{x_{-}}$, $\sqrt{x_{+}}$ (these are extremal cases so $\sqrt{x_{-}}=\sqrt{x_{+}}$). Notice, black hole horizons do not exist for (a) when $L>0$ and (b) when $L>0$. Also in (b) $\sqrt{x_{\mathrm{CTC}}}$ is infinite at $L=-\frac{1}{2}$ because $J=0$ which is 5d Kerr.

Figure 4

Plot of the horizon behavior in terms of $J$ and $L$. The white region corresponds to naked singularities (no black hole horizons). In the gray region the CTC horizon is outside both black hole horizons. In the black region the CTC horizon is inside both black hole horizons. The white line corresponds to the special case when the CTC horizon is in coincidence with a black hole horizon (outer horizon in gray region, inner horizon in black region, and both at the special points $J=-L=\pm 1$).

Figure 5

Plot of horizon behavior in terms of $J$ and $L$. The curve corresponds to $r_{\mathrm{CTC}}=\sqrt{2m}$. In the black region $r_{\mathrm{CTC}}>\sqrt{2m}$ and in the gray region $r_{\mathrm{CTC}}<\sqrt{2m}$.