Infinite Quantum Twisting at the Cauchy Horizon of Rotating Black Holes

We present a numerical calculation of the expectation value of the quantum angular-momentum current flux density for a scalar field in the Unruh state near the inner horizon of a Kerr–de Sitter black hole. Our results indicate that this flux diverges as $V_{-}^{-1}$ in a suitable Kruskal coordinate such that $V_{-}=0$ at the inner horizon. Depending on the parameter values of the scalar field and black hole that we consider, and depending on the polar angle (latitude), this flux can have different signs. In the near extremal cases considered, the angle average of the expectation value of the quantum angular momentum current flux is of the opposite sign as the angular momentum of the background itself, suggesting that, in the cases considered, quantum effects tend to decrease the total angular momentum of the spheres away from the extremal value. We also numerically calculate the energy flux component, which provides the leading order divergence of the quantum stress energy tensor, dominant over the classical stress energy tensor, at the inner horizon. Taking our expectation value of the quantum stress tensor as the source in the semiclassical Einstein equation, our analysis suggests that the spheres approaching the inner horizon can undergo an infinite twisting due to quantum effects along latitudes separating regions of infinite expansion and contraction.

Figure 1

Infinite twisting between regions of diverging expansion and contraction on a cross section of the IH.

Figure 2

Carter-Penrose diagram of KdS. The orange line $\mathrm{\Sigma }$ is a Cauchy surface for regions I, II, and III. The red lines indicate the ranges of different Kruskal coordinates, while the blue lines illustrate $u$ and $v$ in the corresponding block. The wavy line represents the timelike ring singularity of KdS (at $r=0$ and $\theta =\pi /2$).

Figure 3

The $\cos \theta$ dependence of ${⟨T_{v{\phi }_{-}}⟩}_{\mathrm{U}}^{\mathrm{IH}}$ for $\mathrm{\Lambda }=1/30$ and $a=0.95$ (blue), 0.975 (orange), 1 (green), and 1.007 (red). Note that $a_{\max }\approx 1.012$.

Figure 4

${⟨{\widehat{T}}_{vv}⟩}_{\mathrm{U}}^{\mathrm{IH}}$ for $\theta =0$, $\mathrm{\Lambda }=1/270$ as a function of $a$. The vertical line indicates $a_{\max }$. The inset is a closeup near extremality.