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Analytical approximation of the stress-energy tensor of a quantized scalar field in static spherically symmetric spacetimes

Analytical approximations for ${< ϕ^2 >}$ and ${< T^μ_ν >}$ of a quantized scalar field in static spherically symmetric spacetimes are obtained. The field is assumed to be both massive and massless, with an arbitrary coupling $ξ$ to the scalar curvature, and in a zero temperature vacuum state. The expressions for ${< ϕ^2 >}$ and ${< T^μ_ν >}$ are divided into low- and high-frequency parts. The contributions of the high-frequency modes to these quantities are calculated for an arbitrary quantum state. As an example, the low-frequency contributions to ${< ϕ^2 >}$ and ${< T^μ_ν >}$ are calculated in asymptotically flat spacetimes in a quantum state corresponding to the Minkowski vacuum (Boulware quantum state). The limits of the applicability of these approximations are discussed.