Vacuum polarization of the quantized massive scalar field in the global monopole spacetime: The renormalized quantum stress energy tensor

This paper is devoted to the construction of the renormalized quantum stress energy tensor ${⟨T_{\mu }^{\nu }⟩}_{\mathrm{ren}}$ for a massive scalar field with arbitrary coupling to the gravitational field of a pointlike global monopole, using the Schwinger-DeWitt approximation, up to second order in the inverse mass $\mu$ of the field. The given stress energy tensor is constructed by functional differentiation with respect to the metric tensor of the one-loop effective action of a sufficiently massive scalar field, such that the Compton length of the quantum field is much less than the characteristic radius of the curvature of the background geometry. The results are obtained for a general curvature coupling parameter $\xi$ and specified to the more physical cases of minimal and conformal coupling, showing that, in this specific case, the quantum massive scalar field in the global monopole spacetime violates all the pointwise energy conditions.

Figure 1

Dependence on the coupling constant $\xi$ of the rescaled time component of the renormalized stress energy tensor $⟨T_0⟩=96{\pi }^2{\mu }^2{⟨T_t^t⟩}_{\mathrm{ren}}$ for a massive scalar field in the pointlike global monopole spacetime. The values of the parameters used in the calculations are $r=\frac{1}{3}$ and $1-{\alpha }^2={10}^{-5}$.

Figure 2

Dependence on the coupling constant $\xi$ and distance from the monopole’s core $r$ of the rescaled time component of the renormalized stress energy tensor $⟨T_0⟩=96{\pi }^2{\mu }^2{⟨T_t^t⟩}_{\mathrm{ren}}$ for a massive scalar field in the pointlike global monopole spacetime. The value of the parameter used in the calculations is $1-{\alpha }^2={10}^{-5}$.

Figure 3

Dependence on the distance from the monopole’s core $r$ of the rescaled time component of the renormalized stress energy tensor $⟨T_0⟩=96{\pi }^2{\mu }^2{⟨T_t^t⟩}_{\mathrm{ren}}$ for a massive scalar field in the pointlike global monopole spacetime. The values of the parameters used in the calculations are $1-{\alpha }^2={10}^{-5}$ and $\xi =0$ (top plot) and $\xi =\frac{1}{6}$ (bottom plot).

Figure 4

Dependence on the coupling constant $\xi$ of the rescaled radial component of the renormalized stress energy tensor $⟨T_1⟩=96{\pi }^2{\mu }^2{⟨T_r^r⟩}_{\mathrm{ren}}$ for a massive scalar field in the pointlike global monopole spacetime. The values of the parameters used in the calculations are $r=\frac{1}{3}$ and $1-{\alpha }^2={10}^{-5}$.

Figure 5

Dependence on the coupling constant $\xi$ and the distance $r$ from the monopole’s core of the rescaled radial component of the renormalized stress energy tensor $⟨T_1⟩=96{\pi }^2{\mu }^2{⟨T_r^r⟩}_{\mathrm{ren}}$ for a massive scalar field in the pointlike global monopole spacetime. The value of the parameter used in the calculations is $1-{\alpha }^2={10}^{-5}$.

Figure 6

Dependence on the distance from the monopole’s core $r$ of the rescaled radial component of the renormalized stress energy tensor $⟨T_1⟩=96{\pi }^2{\mu }^2{⟨T_r^r⟩}_{\mathrm{ren}}$ for a massive scalar field in the pointlike global monopole spacetime. The values of the parameters used in the calculations are $1-{\alpha }^2={10}^{-5}$ and $\xi =0$ (top plot) and $\xi =\frac{1}{6}$ (bottom plot).