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Computation of the renormalized stress-energy tensor is the most seriousobstacle in studying the dynamical, self-consistent, semiclassical evaporationof a black hole in 4D. The difficulty arises from the delicate regularizationprocedure for the stress-energy tensor, combined with the fact that in practicethe modes of the field need be computed numerically. We have developed a newmethod for numerical implementation of the point-splitting regularization in4D, applicable to the renormalized stress-energy tensor as well as to$\left\langle \phi^{2}\right\rangle _{ren}$, namely the renormalized$\left\langle \phi^{2}\right\rangle$. So far we have formulated two variants ofthis method: t-splitting (aimed for stationary backgrounds) and angularsplitting (for spherically-symmetric backgrounds). In this paper we introduceour basic approach, and then focus on the t-splitting variant, which is thesimplest of the two (deferring the angular-splitting variant to a forthcomingpaper). We then use this variant, as a first stage, to calculate $\left\langle\phi^{2}\right\rangle _{ren}$ in Schwarzschild spacetime, for a massless scalarfield in the Boulware state. We compare our results to previous ones, obtainedby a different method, and find full agreement. We discuss how this approachcan be applied (using the angular-splitting variant) to analyze the dynamicalself-consistent evaporation of black holes.

Pragmatic mode-sum regularization method for semiclassical black-hole spacetimes