Renormalized charged scalar current in the Reissner–Nordström–de Sitter spacetime

We perform a Hadamard point-split renormalization of the current density $j_{\nu }$ of a charged scalar field in a Reissner–Nordström–de Sitter spacetime. We compute numerically the expectation values of the components $j_r$ and $j_t$ in the Unruh state in the exterior region and study their dependence on the field parameters and the position.

Figure 1

Penrose diagram for Reissner–Nordström–de Sitter spacetime. I and III constitute the exterior region, with III being the region beyond the cosmological horizon ${\mathcal{H}}_c$, i.e., out of causal contact with the BH interior. The latter is represented by regions II and IV, which are separated from the exterior by the event horizon ${\mathcal{H}}^R$ and from each other by the Cauchy horizon ${\mathcal{CH}}^R$. The curvature singularity is represented by the wiggled line. The green line indicates a Cauchy surface $\mathrm{\Sigma }$ for the region $\mathrm{I}\cup \mathrm{II}\cup \mathrm{III}$. Here we focus on region $\mathrm{I}$ and the event horizon ${\mathcal{H}}^R$.

Figure 2

The $v$-component of the current of a conformally coupled scalar field on the event horizon as a function of the scalar field charge in the Unruh vacuum for $Q=0.95M$ and $\mathrm{\Lambda }=0.14{\mathrm{M}}^{-2}$. The smaller graph shows the results for $Q=0.95M$ for small $qQ$, the dashed line indicates a $q^2$-fit.

Figure 3

The $v$-component of the current of a charged scalar field of charge $qQ=0.1$ at the event horizon as a function of the scalar field mass in the Unruh vacuum for $Q=0.95M$ and $\mathrm{\Lambda }=0.14{\mathrm{M}}^{-2}$.

Figure 4

$r^2⟨j_{r_{*}}{⟩}_{\mathrm{U}}$ for the conformally coupled scalar at different radii $r$. This quantity must be constant for the current conservation. The vertical lines mark $r_{+}$ and $r_c$, respectively, where the first and last point are computed. $Q=0.95M$, $\mathrm{\Lambda }=0.14{\mathrm{M}}^{-2}$, and $qQ=0.1$.

Figure 5

$⟨j_t{⟩}_{\mathrm{U}}$ for the conformally coupled scalar in region $\mathrm{I}$ as a function of $r$ for $qQ=0.1$, $Q=0.95M$, and $\mathrm{\Lambda }=0.14{\mathrm{M}}^{-2}$. The first and last point are results obtained at $r_{+}$ and $r_c$, respectively, including realistic errors. $r_{+}$ and $r_c$ are also marked by the vertical lines.