Canonical quantization of the electromagnetic field on the Kerr background

We investigate the canonical quantization of the electromagnetic field on the Kerr background. We give new expressions for the expectation value of the electromagnetic stress-energy tensor in various quantum states and give a physical interpretation of the separate terms appearing in them. We numerically calculate the luminosity for some of these states. We investigate the form of the renormalized stress-energy tensor close to the horizon when the electromagnetic field is in the past Boulware state.

Figure 1

Positive polarization contribution to: (top left) $\frac{1}{4\pi }⟨{\widehat{T}}_{tr}{⟩}^{U^{-}-B^{-}}$, (top right) $\frac{1}{4\pi }⟨{\widehat{T}}_{t\varphi }{⟩}^{U^{-}-B^{-}}$, (bottom left) $\frac{1}{4\pi }\Delta ⟨{\widehat{T}}_{tr}{⟩}^{CC{H}^{-}-U^{-}}$ and (bottom right)$\frac{1}{4\pi }⟨{\widehat{T}}_{t\varphi }{⟩}^{CC{H}^{-}-U^{-}}$.

Figure 2

(top left) $\frac{1}{4\pi }\Delta ⟨{\widehat{T}}_{tr}{⟩}^{U^{-}-B^{-}}$, (top right) $\frac{1}{4\pi }\Delta ⟨{\widehat{T}}_{r\varphi }{⟩}^{U^{-}-B^{-}}$, (bottom left) $\frac{1}{4\pi }{\Delta }^2⟨{\widehat{T}}_{t\theta }{⟩}^{U^{-}-B^{-}}$ and (bottom right) $\frac{1}{4\pi }{\Delta }^2⟨{\widehat{T}}_{\theta \varphi }{⟩}^{U^{-}-B^{-}}$. The viewing angle of the bottom two graphs is different from that of the top two in order to make more visible the region far from the horizon.

Figure 3

(top left) $\frac{1}{4\pi }\Delta ⟨{\widehat{T}}_{tr}{⟩}^{CC{H}^{-}-U^{-}}$, (top right) $\frac{1}{4\pi }\Delta ⟨{\widehat{T}}_{r\varphi }{⟩}^{CC{H}^{-}-U^{-}}$,( bottom left) $\frac{1}{4\pi }{\Delta }^2⟨{\widehat{T}}_{t\theta }{⟩}^{CC{H}^{-}-U^{-}}$ and (bottom right) $\frac{1}{4\pi }{\Delta }^2⟨{\widehat{T}}_{\theta \varphi }{⟩}^{CC{H}^{-}-U^{-}}$.

Figure 4

${10}^5\times \Delta ⟨{\widehat{T}}_{tr}{⟩}^{U^{-}-B^{-}}$ at $r\sim 190M$. This quantity is plotted as a function of $\theta$ in the graph on the left. The graph on the right uses polar coordinates: the absolute value of this quantity is the radial coordinate and the vertical axis is the axis of symmetry of the Kerr black hole. This graph is for a $t=\mathrm{const}$, $\varphi =\mathrm{const},\mathrm{const}+\pi$ slice of the space-time.

Figure 5

(top left) $\frac{1}{4\pi }{\Delta }^2⟨{\widehat{T}}_{tt}{⟩}^{CC{H}^{-}-B^{-}}$, (top right) $\frac{1}{4\pi }{\Delta }^2⟨{\widehat{T}}_{t\varphi }{⟩}^{CC{H}^{-}-B^{-}}$, (bottom left) $\frac{1}{4\pi }{\Delta }^2⟨{\widehat{T}}_{\theta \theta }{⟩}^{CC{H}^{-}-B^{-}}$ and (bottom right) $\frac{1}{4\pi }{\Delta }^2⟨{\widehat{T}}_{\varphi \varphi }{⟩}^{CC{H}^{-}-B^{-}}$.

Figure 6

$\frac{1}{4\pi }{\Delta }^2⟨{\widehat{T}}_{\theta \theta }{⟩}^{CC{H}^{-}-B^{-}}$ and $\frac{1}{4\pi }{\Delta }^2⟨{\widehat{T}}_{\theta \theta }{⟩}^{U^{-}-B^{-}}$

Figure 7

Graphs on the left and on the right: $\frac{1}{4\pi }\Delta ⟨{\widehat{T}}_{t_{+}{\varphi }_{+}}{⟩}^{CC{H}^{-}-B^{-}}$ (3-d surface), $\frac{1}{4\pi }\Delta T_{t_{+}{\varphi }_{+}}^{(\mathrm{th},\mathrm{RR})}$, $\frac{1}{4\pi }\Delta T_{t_{+}{\varphi }_{+}}^{(\mathrm{th},\mathrm{ZAMO})}$ and $\frac{1}{4\pi }\Delta T_{t_{+}{\varphi }_{+}}^{(\mathrm{th},\mathrm{Carter})}$. The graph on the right reaches further in $r$, has a different viewing angle and it also includes the speed-of-light (SOL) and the stationary limit (SL) surfaces.

Figure 8

Plots of $({\Omega }_{\mathrm{ZEFO}}^{CC{H}^{-}-B^{-}}-{\Omega }_{+})$ (graph on the left) and $({\Omega }_{\mathrm{ZEFO}}^{CC{H}^{-}-B^{-}}-{\Omega }_{+})/\Delta$ (graph on the right) together with the corresponding plots with the angular velocities of a RRO, ZAMO and Carter observer.