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

The full computation of the renormalized expectation values $\langle\Phi^{2}\rangle_{ren}$ and $\langle\hat{T}_{\mu\nu}\rangle_{ren}$ in 4D black hole interiors has been a long standing challenge, which has impeded the investigation of quantum effects on the internal structure of black holes for decades. Employing a recently developed mode sum renormalization scheme to numerically implement the point-splitting method, we report here the first computation of $\langle\Phi^{2}\rangle_{ren}$ in Unruh state in the region inside the event horizon of a 4D Schwarzschild black hole. We further present its Hartle-Hawking counterpart, which we calculated using the same method, and obtain a fairly good agreement with previous results attained using an entirely different method by Candelas and Jensen in 1986. Our results further agree upon approaching the event horizon when compared with previous results calculated outside the black hole. Finally, the results we obtained for Hartle-Hawking state at the event horizon agree with previous analytical results published by Candelas 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 4D black hole interiors.