Renormalized stress tensor in Kerr space-time: Numerical results for the Hartle-Hawking vacuum

We show that the pathology which afflicts the Hartle-Hawking vacuum on the Kerr black hole space-time can be regarded as due to rigid rotation of the state with the horizon in the sense that, when the region outside the speed-of-light surface is removed by introducing a mirror, there is a state with the defining features of the Hartle-Hawking vacuum. In addition, we show that, when the field is in this state, the expectation value of the energy-momentum stress tensor measured by an observer close to the horizon and rigidly rotating with it corresponds to that of a thermal distribution at the Hawking temperature rigidly rotating with the horizon.

Figure 1

A graph of $⟨{\widehat{\varphi }}^2{⟩}_{\mathrm{RRO}}^{H_{\mathcal{M}}}$, the expectation value of ${\widehat{\varphi }}^2$ measured by an RRO when the field is in the state $|H_{\mathcal{M}}⟩$. The dark line gives the value on the horizon for a thermal distribution at the Hawking temperature rigidly rotating with the horizon. The units are such that $M=1$.

Figure 2

Graphs of the components of $⟨{\widehat{T}}_{\mu \nu }{⟩}_{\mathrm{RRO}}^{H_{\mathcal{M}}}$, the expectation value of the energy-momentum stress tensor measured by an RRO when the field is in the state $|H_{\mathcal{M}}⟩$. The dark line gives the value on the horizon for a thermal distribution at the Hawking temperature rigidly rotating with the horizon. The units are such that $M=1$.

Figure 3

A graph of $(\overline{\Omega }-\Omega )/\Delta$, where $\overline{\Omega }$ is the rate of rotation of the frame in which there is no flux of energy. The black line gives the value on the horizon for $\overline{\Omega }={\Omega }_{+}$, the blue line gives the value for $\overline{\Omega }=\Omega$ and the green line gives the value for $\overline{\Omega }={\Omega }_{\mathrm{Carter}}$.

Figure 4

A space-time diagram of the region bounded by the surfaces $S_t$, $S_{t^{\prime }}$, and the mirror.

Figure 5

A space-time diagram of the region bounded by the surfaces $S_0$, $S_v$, a hypersurface of constant Boyer-Lindquist radius, and the mirror.

Figure 6

A graph of the complex eigenfrequencies corresponding to solutions of the Klein-Gordon equation which are null in the field theory inside a mirror of constant Boyer-Lindquist radius $r_0$ for a black hole with $a=0.95M$. As $r_0$ decreases, the frequencies pass into the lower complex plane through the point $\tilde{\omega }=0$ before the minimum radius of the speed-of-light surface, $r_{\mathrm{SOL}}(\pi /2)=1.675855M$, is reached. The eigenfrequency is given in units in which $M=1$.

Figure 7

Schematic diagrams of the horizon (blue line), static limit surface (green line), and speed-of-light surface (red line) for $a=0.5M$, $a=\sqrt{2(\sqrt{2}-1)}M$, and $a=0.95M$. The diagrams show a $t=\mathrm{\text{constant}}$, $\phi =\mathrm{\text{constant}}$ slice with the Boyer-Lindquist coordinates $r$ and $\theta$ used as plane polar coordinate.