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The full computation of the renormalized expectation values$\langle\Phi^{2}\rangle_{ren}$ and $\langle\hat{T}_{\mu\nu}\rangle_{ren}$ in 4Dblack hole interiors has been a long standing challenge, which has impeded theinvestigation of quantum effects on the internal structure of black holes fordecades. Employing a recently developed mode sum renormalization scheme tonumerically implement the point-splitting method, we report here the firstcomputation of $\langle\Phi^{2}\rangle_{ren}$ in Unruh state in the regioninside the event horizon of a 4D Schwarzschild black hole. We further presentits Hartle-Hawking counterpart, which we calculated using the same method, andobtain a fairly good agreement with previous results attained using an entirelydifferent method by Candelas and Jensen in 1986. Our results further agree uponapproaching the event horizon when compared with previous results calculatedoutside the black hole. Finally, the results we obtained for Hartle-Hawkingstate at the event horizon agree with previous analytical results published byCandelas in 1980. This work sets the stage for further explorations of$\langle\Phi^{2}\rangle_{ren}$ and $\langle\hat{T}_{\mu\nu}\rangle_{ren}$ in 4Dblack hole interiors.

Mode-sum renormalization of $\langle\Phi^{2}\rangle$ for a quantum scalar field inside a Schwarzschild black hole