Quantum effects in near-extremal charged black hole spacetimes

We compute the semiclassical current and stress-energy fluxes both at the event and Cauchy horizon of a near-extremal Reissner-Nordström black hole. We consider a minimally coupled, massless, weakly charged scalar field in the Unruh state, describing an evaporating black hole. The near-extremal domain allows for an analytical treatment of the scattering problem of the Boulware modes both in the interior and exterior regions. We present this and explicit analytical expressions for <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mo stretchy="false">⟨</a:mo><a:msub><a:mi>j</a:mi><a:mi>v</a:mi></a:msub><a:mo stretchy="false">⟩</a:mo></a:math> and <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mo stretchy="false">⟨</e:mo><e:msub><e:mi>T</e:mi><e:mrow><e:mi>v</e:mi><e:mi>v</e:mi></e:mrow></e:msub><e:mo stretchy="false">⟩</e:mo></e:math> at the horizons, as well as estimates for <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mo stretchy="false">⟨</i:mo><i:msub><i:mi>j</i:mi><i:mi>u</i:mi></i:msub><i:mo stretchy="false">⟩</i:mo></i:math> and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:mo stretchy="false">⟨</m:mo><m:msub><m:mi>T</m:mi><m:mrow><m:mi>u</m:mi><m:mi>u</m:mi></m:mrow></m:msub><m:mo stretchy="false">⟩</m:mo></m:math>. We cross-check the analytical results numerically by bringing the radial Klein-Gordon equation into the form of the confluent Heun equation. Inserting these expectation values as sources to the Einstein-Maxwell equations, we find that at least in the near-extremal regime of small field charge, quantum effects drive the black hole interior away from extremality. Our work generalizes the known results for the real scalar field [Noa Zilberman and Amos Ori, Quantum fluxes at the inner horizon of a near-extremal spherical charged black hole, .] and is in agreement with recent work on charged fields in expanding Reissner–Nordström–de Sitter universes [Christiane Klein , Quantum (dis)charge of black hole interiors, ., Christiane Klein and Jochen Zahn, Renormalized charged scalar current in the Reissner–Nordström–de Sitter spacetime, .].