Vacuum polarization of the quantized massive scalar field in Reissner-Nordstr\"om spacetime

The approximation of the renormalized stress-energy tensor of the quantized massive scalar field in Reissner-Nordstr"om 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.