Next-to-leading term of the renormalized stress-energy tensor of the quantized massive scalar field in Schwarzschild spacetime. The back reaction

The next-to-leading term of the renormalized stress-energy tensor of the quantized massive field with an arbitrary curvature coupling in the spacetime of the Schwarzschild black hole is constructed. It is achieved by functional differentiation of the DeWitt-Schwinger effective action involving coincidence limit of the Hadamard-Minakshisundaram-DeWitt-Seely coefficients $a_3$ and $a_4$. 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 even in the closest vicinity of the event horizon. The back reaction of the quantized field upon the Schwarzschild black hole is briefly discussed.

Figure 1

This graph shows the rescaled $T_t^{(2)t}$ [$\lambda =(8M{)}^4{\pi }^2{m}^4$] component of the stress-energy tensor of the massive scalar field as a function of $z=r/M$ plotted for a few exemplary values of the coupling parameter $\xi$. Top to bottom (at the maximum) the curves are plotted for $\xi =0.2i$ $(i=0,\dots ,8)$ and for $\xi =1/6$.

Figure 2

This graph shows the rescaled $T_r^{(2)r}$ [$\lambda =(8M{)}^4{\pi }^2{m}^4$] component of the stress-energy tensor of the massive scalar field as a function of $z=r/M$ plotted for a few exemplary values of the coupling parameter $\xi$. Top to bottom (at the maximum) the curves are plotted for $\xi =0.2i$ $(i=0,\dots ,8)$ and for $\xi =1/6$.

Figure 3

This graph shows the rescaled $T_{\theta }^{(2)\theta }$ [$\lambda =(4M{)}^8{\pi }^2{m}^4$] component of the stress-energy tensor of the massive scalar field as a function of $z=r/M$ plotted for a few exemplary values of the coupling parameter $\xi$. Top to bottom (at the maximum) the curves are plotted for $\xi =0.2i$ $(i=0,\dots ,8)$ and for $\xi =1/6$.

Figure 4

The curves in this figure display the conformal part of the stress-energy tensor, $C_{\theta }^{\theta }$, for the massive scalar field with $mM=2$ in the vicinity of the Schwarzschild black hole. From top to bottom at the event horizon ($r_{+}=2M$) the curves correspond to the first-order approximation and the improved approximation.

Figure 5

The curves in this figure display the nonconformal part of the stress-energy tensor, $D_{\theta }^{\theta }$, for the massive scalar field with $mM=2$ in the vicinity of the Schwarzschild black hole. From top to bottom at the event horizon ($r_{+}=2M$) the curves correspond to the improved approximation and to the first-order approximation.