Vacuum polarization of the quantized massive scalar field in Reissner-Nordström spacetime

The approximation of the renormalized stress-energy tensor of the quantized massive scalar field in Reissner-Nordström spacetime is constructed. It is achieved by functional differentiation of the first two nonvanishing terms of the Schwinger-DeWitt expansion involving the coincidence limit of the Hadamard-Minakshisundaram-DeWitt-Seely coefficients [$a_3$] and [$a_4$] with respect to the metric tensor. It is shown, by comparison with the existing numerical results, that inclusion of the second-order term leads to substantial improvement of the approximation of the exact stress-energy tensor. The approximation to the field fluctuation, $⟨{\varphi }^2⟩$, is constructed and briefly discussed.

Figure 1

This graph shows the rescaled values of the vacuum polarization $⟨{\varphi }^2{⟩}_i$ [$\lambda =16{\pi }^2{M}^2$] at the event horizon for the massive scalar field with $mM=2$ The solid line corresponds to $⟨{\varphi }^2{⟩}_4$, the dashed line corresponds to $⟨{\varphi }^2{⟩}_3$, and the dotted line corresponds to $⟨{\varphi }^2{⟩}_2$.

Figure 2

This graph shows the rescaled values of the vacuum polarization $⟨{\varphi }^2{⟩}_i$ [$\lambda =16{\pi }^2{M}^2$] at the event horizon for the massive scalar field with $mM=2$. The solid line corresponds to $⟨{\varphi }^2{⟩}_4$, the dashed line corresponds to $⟨{\varphi }^2{⟩}_3$, and the dotted line corresponds to $⟨{\varphi }^2{⟩}_2$.

Figure 3

This graph shows the rescaled values of $C_{\theta }^{\theta }$ [$\lambda =90(8M{)}^4{\pi }^2$] as function of $x=(r-r_{+})/M$ for massive scalar field with $mM=2$ and $|Q|/M=0.95$. The solid line corresponds to the improved approximation; whereas, the dashed line to the first-order Schwinger-DeWitt approximation. In the small panel the near horizon behavior of $C_{\theta }^{\theta }$ is displayed.

Figure 4

This graph shows the rescaled values of $D_{\theta }^{\theta }$ [$\lambda =90(8M{)}^4{\pi }^2$] as function of $x=(r-r_{+})/M$ for massive scalar field with $mM=2$ and $|Q|/M=0.95$. The solid line corresponds to the improved approximation; whereas, the dashed line to the first-order Schwinger-DeWitt approximation. In the small panel the near horizon behavior of $D_{\theta }^{\theta }$ is displayed.