Vacuum polarization throughout general subtracted black hole spacetimes

We compute the vacuum polarization of a massless minimally coupled scalar field in a background given by a black hole with subtracted geometry. Extending previous results for the horizon of rotating black holes with no charge, we obtain an analytical expression for the vacuum polarization that is valid throughout the spacetime and for arbitrary rotation and charge parameters. The vacuum polarization diverges at the inner horizon, and the quantum state cannot be extended to the inside of it.

Figure 1

$(64{\pi }^2{m}^2)⟨{\varphi }^2⟩$ at the Kerr-Newman horizon, for $a/r_{+}=\alpha =0.75$. The charge parameter $\delta$ takes the values 0, 0.2, 0.4, 0.7 and 1 for the dotted, small-dashed, medium-dashed, large-dashed and full lines respectively.

Figure 2

$(64{\pi }^2{m}^2)⟨{\varphi }^2⟩$ at the Kerr-Newman horizon, for $\delta =0.4$ (corresponding to $Q/M=0.66$). The rotation parameter $\alpha =a/r_{+}$ takes the values 0.75, 0.4, 0.2, and 0 for the dotted, small-dashed, medium-dashed and full lines respectively.

Figure 3

$(768{\pi }^2{m}^2{\mathrm{\Pi }}_c)⟨{\varphi }^2⟩$ as a function of $x$ for $\alpha =0={\mathrm{\Pi }}_s$. The constant multiplying $R(x)$ in the arbitrary term added to (35) is set as $-{\pi }^2{m}^2{\mathrm{\Pi }}_c$.

Figure 4

$(64{\pi }^2{m}^2)⟨{\varphi }^2⟩$ at the equatorial plane as a function of the dimensionless radial coordinate $x$ for the subtracted Kerr black hole with $\alpha =1/2$. The divergence at $x=-1$ corresponds to the inner horizon.