Quantum effects near the Cauchy horizon of a Reissner-Nordström black hole

We consider a massless, minimally coupled quantum scalar field on a Reissner-Nordström black hole background, and we study the leading asymptotic behavior of the expectation value of the stress energy tensor operator $⟨{\widehat{T}}_{\mu \nu }{⟩}_{\mathrm{ren}}$ and of $⟨{\widehat{\mathrm{\Phi }}}^2{⟩}_{\mathrm{ren}}$ near the inner horizon, in both the Unruh and the Hartle-Hawking quantum states. We find that the coefficients of the expected leading-order divergences of these expectation values vanish, indicating that the modifications of the classical geometry due to quantum vacuum effects might be weaker than expected. In addition, we calculate the leading-order divergences of $⟨{\widehat{T}}_{\mu \nu }{⟩}_{\mathrm{ren}}$ and of $⟨{\widehat{\mathrm{\Phi }}}^2{⟩}_{\mathrm{ren}}$ in the Boulware state near the outer (event) horizon, and we obtain analytical expressions that correspond to previous numerical results.

Figure 1

Penrose diagram of Reissner-Nordström spacetime. In the exterior region, region I in the figure, we use the exterior Eddington-Finkelstein coordinates, while in the interior, region II in the figure, we use the interior Eddington-Finkelstein coordinates. In addition, the Kruskal coordinate system is shown and is defined in both regions I and II. The red-framed area denotes the region in the eternal Reissner-Nordström spacetime which concerns this paper, i.e., regions I and II.